Math Without Tears

Abstract Learning and Fractions

2008-08-04T05:42:00.005-05:00

Shifting gears; thinking about getting ready for another school year (yes, school starts early in Texas); taking math seminars (yes, using a new math book always requires seminars to learn how to teach this new math) -- a lot is starting to happen.

This brings me to a book which I've referenced before: Math Coach: A Parent's Guide to Helping Children Succeed in Math, written by Wayne A Wickelgren, Ph.D. Dr. Wickelgren is a former MIT cognitive psychologist specializing in learning. As I'm going to be sitting through a lot of "new" stuff for teaching this "new" math book, I'm finding that I am thinking about some of what Dr. Wickelgren has written. He knows how children learn. I've written other posts about Wickelgren's book, but today I'd like to quote a few things he has to say about "Abstract Learning and Fractions".

Wickelgren says:

"While abstract explanations of fractions are often helpful and interesting to children, they are not necessary for multiplying and dividing fractions. For almost all children, following the simple abstract rules for such operations is the best course. The best way to justify the rules for multiplying and dividing fractions is to say mathematicians devised these rules because they work to make everything come out correctly. There is no good way of explaining the meaning of multiplying and dividing fractions using objects or other concrete representations. Indeed, I believe attempting such explanations will only confuse your child, so I do not discuss them."

Wow, doesn't this just fly in the face of everything the "new math" people say about teaching math. The "new math" folks tell us we must use objects or other concrete representations before we introduce anything and before we explain. Dr. Wickelgren says this (using objects or other concrete representastions to explain the meaning of multipllying and dividing fractions) will confuse your child.

"Abstract, mathematical principles like the rules for multiplying fractions are not necessarily harder for people to understand than concrete, real world examples. The capacity for abstract thinking is what sets human intelligence above that of other animal species. To make numerical abstractions apply to the world, people first relate numbers to objects and operations to actions. But having done that, people should immerse themselves in the abstractions themselves without continually translating them into the "real"world."

Wow, "people first relate numbers to objects . . . . and operations to actiions." And then, they should just immerse themselves in the abstractions themselves . . . that sounds like practice, practice, practice to me.

This is the best part:

"It's like learning a second language. Once a person becomes fluent, he or she need not translate every word of the new language into the native one, since each word and phrase in the new language has taken on a meaning of its own. Indeed, it would be slow and cumbersome to do so. Similarly, after learning the language of math, it's most efficient -- and easier -- just to use the language, mastering it through continued use."

I remember learning a language. It was just so exciting when I realized that I was starting to think in complete phrases, rather than translating each word. I could understand! I was thinking whole phrases. I got it!!!

The same is true for math. Once a student "understands" how to use the fractions and manipulate the fractions, they don't need to use all kinds of tricks of manipulating objects each time to do "real life" stories every time. That is so slow and cumbersome (especially for the really sharp kids that get it quickly). The students just need to practice using the procedures. That's what Wickelgren calls "mastering it through continued use."

And I have found this to be true. Students pay attention to what they are doing, as they do the steps, and suddenly, in doing the steps often, they begin to understand. They gain understanding by doing, by repetition, by practice. Would that be drill? Sounds like it to me.

Dancing in South Carolina

2008-07-30T18:27:00.014-05:00

This story and video are probably going to surprise many, many readers from around the country. But if you are aware of the addition of the curriculum Everyday Mathematics into schools around the country, you need to take this seriously. AND IF YOU ARE FROM SOUTH CAROLINA, WATCH CAREFULLY. THIS IS HAPPENING IN YOUR STATE.

Is this really the best method they can think of for teaching "sequence" to children?

Go here to read the story.

In reading the comments at the bottom of this article, you get the idea of the embarrassment of the citizens of S.C. Considering that 50% of the students in S.C. drop out of school before graduating, you can understand their frustration that their tax money is paying for this kind of teacher training.

And you also see an example of a "brainwashed" teacher who actually believes that Everyday Mathematics raises the bar over other curricula. Any other curricula, according to her, is sub par. She obviously has no idea that college professors have been speaking out strongly for years against using these and other "new math" books.

(The teacher makes it clear that the "dance" number and song aren't actually a part of Everyday Mathematics curriculum -- it's something teachers from all across the country use to teach "sequence".)

Here is another report of this same story.

"New Math" comes to your town?

2008-07-29T18:29:00.010-05:00

Oh, my goodness!!! How did I miss this!

There is a wonderful post and thread over on Kitchen Table Math -the Sequel. The post, which is about Everyday Mathematics, is by Barry Garelick, one of the best.

Go here to read the entire thing.

Oh, the teacher's reference manual has a good section on how to use the calculator! That eases my mind greatly!

Remember: Words and promises are not important. Textbooks make claims and promises all the time. The promises of change are just to lull us all to sleep. Then they come back with the same stuff. It's redefined, so that makes it 'not fuzzy". How much of the traditional algorithms are they really including and having children learn and practice?

N.Y. Algebra Test still being reviewed

2008-07-28T13:22:00.011-05:00

Evidently the recent results of the Regents new integrated algebra test is causing some worry and it seems people are unsure what to do about the results.

The test was administered in June of this year and was apparently so difficult that a student could get a raw score of only 30 and pass. That's 30 out of 87 points! (I wonder if Medical Schools do that!)

We have to understand the difference in the "scaled score" and the "raw score". Questions on the test are weighted. The N.Y. state Education Department uses a "scale score", which gives students more credit for answering certain harder questions.

When a raw scale is used, there is a low cut off level for passing which indicates the test is more difficult than if there was a high cut off level, which would require more and easier questions to be answered correctly.

According to a report in "The Buffalo News: World and Nation", Jonathan Burman of the state Education Department said the following:

"It takes 30 raw score points out of 87 to get a passing score of 65. Some have said this is too low. But you will find that it was a challenging test and the questions that must be answered are appropriate. Many students still did not pass at that level."

The test is mostly taken by ninth graders.

Of interest is that Westminster Charter School in Buffalo teaches integrated algebra to eighth graders. According to their teacher, Patricia Frey, all 14 of her average to slightly above average students passed the test. She stated that she "did not think the test was all that daunting for pupils who are going to be proceeding to geometry the following semester."

GOOD FOR THEM!!

The statewide failure rate has not been calculated. The test is still being reviewed.

Is anyone out there knowledgeable about this and able to shed further light on what is happening?

Read the whole article in "The Buffalo News: World and Nation" here.

___________________________________________

Think about this -- 30 out of 87 questions is passing, but that's OK because they were hard questions. OK, 30 out of 87 is 34%-35% and that is a score of 65 (because the questions were hard). You see, they can make the questions easier and then raise the number of problems a student must answer. Would that make us feel better? Hmmmm.

I have an idea -- How about teaching the students Algebra and how about starting by spending time in the lower grades teaching them math algorithms and requiring some practice, practice, practice.

Good for you, you 8th graders at Westminster Charter School.

As I have said before, you can make a test say anything you want: Just change the way it is scored by changing the weight of certain questions.

The Purpose is NOT to Get the Right Answer . . .

2008-07-28T01:02:00.013-05:00

"The teacher opened her book and read to me that the purpose of the exercise was not to get the right answer..."

I hope that got everyone's attention. Read on to get more information about this quote from a parent.

_____________________________

I was only slightly familiar with the American Policy Center. I had heard the name but that was about all. However, I had never heard of "The DeWeese Reports", nor had I heard of Tom DeWeese, the author. Wow! What a little gold mine I have stumbled upon! I recommend this site to all readers. There are many articles worthy of mention here, but I will reference one that discusses the failures of public education.

Tom DeWeese has an article on "ABSOLUTES" entitled "Why is Public Education Failing?" dated December 17, 2007. In this article, DeWeese addresses the source of the problem of that failure. He discusses restructuring of the classroom, block scheduling, cooperative learning, and individual subjects of Math, English, Reading, and Literature. And his direct quote from a book used in classrooms entitled "The Book of Questions" - - well, that quote will just curl your ear!!

His introductory paragraph follows:

"It's a fact. Most of today's school children can barely read or write. They can't perform math problems without a calculator. They barely know who the Founding Fathers were and know even less of their achievements. Most can't tell you the name of the President of the United States. It's pure and simple; today's children aren't coming out of school with an academics education."

He further states about math:

"Perhaps the most bizarre of all of the school restructuring programs is mathematics. Math is an exact science, loaded with absolutes. There can be no way to question that certain numbers add up to specific totals. Geometric statements and reasons must lead to absolute conclusions. . . .

"Instead, Fuzzy Math teaches students to "appreciate" math, but they can't solve the problems. . "

DeWeese goes on to discuss the social and political issues that are pervasive throughout the texts, leaving little if any room or time for important study of the math concepts and procedures. He states:

"In many of these textbooks, there is literally no math. Instead there are lessons asking children to list "threats to animals," including destruction of habitat, poisons, and hunting. . .

Read the entire article here. It includes the entire opening quote which began this post.

And as you pull up the article, be sure to look for the little "More Articles" button at the top right, where you will find many, additional writings on related subjects by Tom DeWeese and other authors, all of whom are brave to take on the education establishment. I will try to pull some up to my sidebar later.

Making Wise Decisions

Take some time to be educated on some of these underlying issues and sources of problems in public education today. The knowledge we glean from these articles will help us make wise judgments about our children's educations. Our students are not able to make these kinds of judgment themselves, and they depend on and trust us as parents to know what's best and to make wise and sensible decisions for them.

Too Bad: Idaho jumps on the bandwagon for fuzzy math!

2008-07-27T18:10:00.007-05:00

The Idaho Statesman has a letter (dated July 12, 2008) in it's "letter to the editor" page about the state's math. It seems that the state of Idaho has adopted "fuzzy math" curriculum, according to this writer. Read the letter here under the heading Math Initiative (You will have to scroll down a little to get to the letter.)

The writer asks the question, "Why do educators ignore the obvious success of other nations to teach math to the masses?" She is also puzzled, as am I, why Idaho is jumping on the bandwagon for fuzzy math when other states are jumping off!

I think I can answer that question. Educators breed more educators in the universities. Eventually, you have enough people being (mis)educated and being told that the "new math" way is the only way students can really ever understand math. It's hard for a person coming through the system to buck the system, if they ever even see the light at all. For a person to make a career of education, especially Math education, he/she is taught so much "fuzzy" stuff, that they might not even recognize "good common sense" when they see it. And when people show them the statistics of the failures across the country, they just shake their heads and put their fingers in their ears! Even when college professors speak out in huge numbers, the "new math" folks turn a deaf ear and continue to peddle their way as the only way. I know because it happened to me. I didn't even get an audience. There is no reasoning in education departments.

I tried to find information about what Math Curricula are being used in Idaho. Anybody out there in Idaho know for sure? From what I've located on the web, it seems that Everyday Math and Investigations are two. Can anyone verify these? Are there others? I want to be fair and list them all!

Division of Fractions, Part 2; Real Life Problems for Division of Fraction Study

2008-07-24T13:31:00.008-05:00

The real-world-problem-solving folks think that to study real world problems, the students have to be deeply involved in an hour-long problem solving exercise! Or an all week problem solving exercise!!

The examples of short, real life problems, in this post, are proof that real life examples can be used and are used all the time in a traditional classroom. They are a much more efficient use of the students' time and cover examples that make sense to a student. In other words, students can see and understand how they would use a math procedure (division of fractions) that you are teaching.

Students can not only learn division of fractions in 5th grade, but they can also learn to identify when and how to use that procedure in real life. Here are some examples that make sense to children.

Remember, from my last post, that students are learning how to reword a division of fraction problem, without rewriting it. We want students to learn that for division, we are really trying to find "How many ___'s are in ___?" So we use 3 circle manipulatives for our first example. "There are 3 huge cookies. Each child will get 1/2 of a cookie. How many 1/2's will there be in 3 cookies?" I suggest that you also have fractional pieces for 1/2, 1/4, 1/3, etc., for the students to use. Have the students use the pieces to determine how many 1/2'a are in 3 cookies.

I suggest using whole numbers when introducing division of fractions. Repeat the same idea for a few more "whole cookie" problems. Or use stories for pizzas.

You can then progress to yards of rope or ribbon. "The boy scouts had 5 yards of rope. Each scout needs 1/2 yard of rope. How many 1/2 yards can the scouts get from their 5 yards of rope?" Or for the girls: Each girl needs 1/3 of a yard of ribbon for the project. If there are 6 yards of ribbon, how many 1/3 yards are there in the 6 yards of ribbon.

Using rope and ribbon examples are a little different in a child's mind, and a number line and small yardages are recommended for the first examples.

Back to the circular manipulatives, tape thirds and fourths together to form 2/3's and 3/4's. This can be done ahead of time by the teacher, or students can tape their own thirds and fourths together. Here is a real life problem: If you have 2/3 of a pizza, how many 1/6's are in that 2/3's? Write the problem on the board or overhead using the division sign and have the students reword the problem "How many 1/6's are in 2/3?" Have the students use their fraction manipulatives to solve.

As students work over several days, they will become very quick to reword the problem, and will be able to use fractional manipulatives to demonstrate.

Notice that you are teaching children what division of fractions looks like in real life. This is before we even present the procedure of using the reciprocal to solve the problem.

Fraction rods can be used as manipulatives for teaching division of fractions. Also useful is a product called "Fraction Tiles in a Tray"

Be creative with your real-life stories, and there are so many things in real life that students can relate to. Stories of dividing fruit (apples, oranges) are every day events in my family. There are ample opportunities for homeschooling parents or for parents who are trying to help their child understand and catch up with the concept of division of fractions. If you have 1/2 of a cantaloupe, it's so easy to just discuss "how many1/4's are in this half of a cantaloupe?" If your student catch on quickly, go on to use 1/8's and 1/10's when using 1/2 of a cantaloupe.

Then as students progress to use the reciprocal, to invert the fraction and to multiply to solve the problem, there are different real-life problems which students are now ready for. Here are some examples.

Mother needs 3/4's of a yard of ribbon to make a bow. If she has 16 yards of ribbon, how many bows can she make. Discuss this problem with students. "What are we trying to find?" We are trying to find "how many _ _ _ are in _ _ _?"

It may be helpful at first to start them with the incomplete question, but most students will be able to ask the question because they have learned to recognize what it is that we are trying to find and they will be able to reword the problem. If it were my classroom, I'd have each student write the problem (16 divided by 3/4) and have students read the question out loud, maybe even write the question "How many 3/4's are in 16?" Yes, I know this is writing, but we are not using volumes of paper here, nor are we using an hour per problem. We are practicing and training the brain to think "How many _ _ _ are in _ _ _?"

Yes, 5th graders can learn to divide fractions and they can "see" it and they can understand examples in real life when they need to use division of fractions to solve the problem. Your student is smarter than the "fuzzy math" people think. If your student is struggling to understand, I promise you that your student wants to understand. So, take some time, invest some time, in helping him/her. Practice at home with them -- set aside 15-30 minutes a day to practice on this. They can get it. It's like finally understanding long division! It's liberating!

Teaching division of fractions

2008-07-23T13:24:00.004-05:00

In a previous post, I mentioned that, according to the "new math" folks, 5th graders are not able to learn division of fractions. These 5th graders are just not supposed to be able to grasp that concept!!

Now, this is almost funny to me, yet it's sad. Talk about "dumbing-down". That's exactly what this is. Why, I've taught division of fractions for 17 years in 5th grade and students are able to learn division of fractions. They understand it too. And since I rather suspect that "understanding" division of fractions is what the "new math" people really mean, I want to share with you how to be sure that your student understands. It is a procedure that I have used for the last 8 years very successfully. And it is important to "hang" division of fractions to something the student already understands.

If your student is entering 5th grade, he/she may or may not be expected to learn division of fractions. Some curricula (fuzzy, new-math types) will wait until 6th grade because that's when your child can learn it!!

Here's how to guarantee that your child learns and understands division of fractions, whether they are learning it in 5th or 6th grade. Remember it must be "tied" to a previous concept, and that concept is division of whole numbers. If you are helping your child review or relearn "division of fractions" here is what you must do.

Teach your child to reread or (more properly) reword every division problem -- I'm speaking of division of whole numbers.

Example: 12 divided by 4 (Write the problem sideways using the division sign. Or teach your child to rewrite the division problem using the division sign if it is written another way.)

Your child needs to reread/reword that problem as follows: "How many 4's are in 12?" because that is what we are really trying to discover. I show the students that we are almost reading the problem backwards (and we are indeed mentioning the numbers in reverse order). But do not rewrite the problem. It is important that your student see the problem written as a proper division problem as he/she rewords it.

Have your student repeat this activity early on as he/she is learning division of whole numbers. Write a problem. Ask the question, "What are we trying to discover?" Then ask your student to reword the problem. ("How many 6's are in 24?" or "How many 9's are in 63?") Use craft sticks, pennies, or other small manipulative items to work on this if your student needs to "see" it. This rewording of the problem needs to become second nature to your child before division of fractions is introduced.

Using manipulatives can be helpful indeed, but as soon as the student learns multiplication facts, the manipulatives should be used less and less. And the facts need to be learned before division is taught. Remember, Division is the process of searching for the missing factor. So students must know the two factors for 63 (9 and 7), hence learning the multiplication facts from memory is so important.

Do not assume because you have your student reword a division problem a few times that he/she will do it automatically. Practice, practice, practice it. This needs to become a part of your child's thought process every time he/she works a division problem. It will be invaluable later.

Now, when it's time to learn, or review/relearn division of fractions, the student will be used to seeing the division sign in the problem 3 divided by 1/2 and the rewording will come easily. Have your student read the problem "How many one halves are in 3?" Remind your child: "That's what we are really trying to find out, how many 1/2's are in 3." Starting with whole numbers is really smart. Ask your child, "What are we trying to find out?" (how many 1/2's are in 3, or how many 1/3's are in 2)

[Later on you can try 1/2 divided by 1/4 or 2/3 divided by 1/6. Have the student read "how many 1/4's are in 1/2?" or "how many 1/6's are in 2/3's?" ]

Now, hopefully you have some fractional pieces, so that your child can use manipulatives to determine how many 1/2's are in 3 or how many 1/3s are in 2 [or later, how many 1/4's are in 1/2 or how many 1/6's are in 2/3].

Remember the important thing is that students learn to think "how many ----'s are in ----?"

There are so many (real life) problems that we could generate which will show students how we are going to use division of fractions, but I will save that for another post. And it will come shortly. (And by the way, you will notice that we can use real life situations in a traditional classroom without getting bogged down for hours or days.)

Do You Live in a New-New Math City? State?

2008-07-17T14:44:00.003-05:00

I've added a new link on the right to help you identify if you live in a "new-new math" city or state. There are even a couple of links with England and Canada.

You also are invited to contact them with information on your city if yours is not on the list. Send them your story, or your child's story.

Head First, Calculator Second

2008-07-17T12:15:00.004-05:00

Wow, did I locate an excellent page on "fuzzy math" procedures and thoughts, entitled "Everyday (Fuzzy) Math is Dumbing Down our Children", written by Ian Shapira. You people in Virginia may have already read Ian Shapira's writings.

Some of his observations about Everyday Mathematics follow:

"There is a 23-page chapter that teaches nothing but how to use a calculator." Shapira goes on to explain his own take on why so much space is used for teaching how to use a calculator: ". . . the odd algorithmic methods taught in the book for solving math problems are so confusing and unworkable that the students must resort to using a calculator in order to solve math problems."

"The most surprising thing is that the total number of pages in the book devoted to teaching algorithms using whole numbers is 11 pages! That's correct! There are only 11 pages in a 400 page book devoted to explaining algorithms using whole numbers. Only 3 of those pages offer instruction on standard algorithms. . . . On page 50 the book states: 'Finding a percent of a number is the same as multiplying the number by the percent. Usually, it's easiest to change the percent to a decimal and use a calculator.'"

"The preferred Everyday Math are crutches. The crutches are needed because the students are not taught the standard algorithms. The lack of skill in standard algorithms ends up crippling their ability to solve math problems without their crutches. The EDM crutches become cumbersome and hold children back when they are later exposed to to more advanced math problems. Their crippled minds are unable to sprint ahead in math, because they trip all over the crutches imposed upon them by EDM."

"One of the alternative algorithms that is a standard method taught in Everyday Math was authored by a first grader!"

". . . students are expected to invent their own algorithms. Adding to the silliness, the authors of Everyday Math expect the children to invent their own algorithms before they are taught any standard algorithms."

". . . The Everyday Mathematics advocates admit that the standard algorithms used for the past 100 years are 'highly efficient'. One might ask: If the standard algorithms are 'highly efficient,' why replace them with invented and other non-traditional algorithms? The reason is that the Everyday Math advocates are not satisfied with a 'highly efficient' method. They want the 'most efficient' method. In their view the most efficient method is 'mental arithmetic or a calculator.'"

There is so, so much here to read. This article helps all of us identify "fuzzy math" programs, whatever they may be named. Everyday Math is only one, and they all do their damage. The sad thing is, once the damage is done, it's very difficult to go back and redo 4,5,8 years of damage.

Fuzzy Math -- trying to make math more "interesting"

2008-07-17T07:54:00.015-05:00

Here is another "public comment to the national math panel" from yet another college professor and author, J. Martin Rochester, Ph. D.

In his letter, Professor Rochester states the following:

"Fuzzy math . . . has been driven by the same constructivist paradigm and same dumbing-down, populist impulses that gave us the now discredited "whole-language" pedagogy in English. That is, in place of the old maxim 'no pain, no gain,' we now have the new maxim in K-12, 'if it ain't fun, it can't be done.' Under the guise of 'critical thinking' and 'problem solving,' which are ubiquitous buzzwords in every discipline in today's schools, fuzzy math is trying to make math more 'interesting'. . . . The new math deemphasizes and devalues direct instruction, drill and practice, basic computation skills, and getting it right -- getting precise, correct answers. Forget rigor -- the key concern here is to alleviate bordom and drudgery for mathphobes and those who suffer from math anxiety."

Dr. Rochester's letter, short but dead on, is must reading!!!

New Battles Every Year

2008-07-17T05:31:00.005-05:00

A few years back, when I first got word that there were math wars going on, it was because a parent told me what she was learning by searching the internet. She had discovered that our middle school curriculum was rated quite low by Mathematically Correct. And why was she searching? Her daughter was struggling through the 6th grade curriculum and she was initially looking for something to help her daughter.

As I searched, I came across "math wars" and "fuzzy math" and "new math" and "new, new math" and were my eyes opened to what had been going on for years in California, in New York, in Illinois, in Plano, in Penfield, in Utah . . . it just kept going on and on. And I realized that I had been in the dark about all of this. That was back in 2004, 2005.

And as I searched to learn what was causing all of the wars, I read about the college professors who had spoken out years earlier, and who were continuing to speak out and who were taking an active part in making known their concerns of what was happening at the universities because of the math taught at lower levels.

I saw what I thought was the beginning of some success in stamping out the math curricula which were causing all of the confusion and problems, and I breathed with a sigh of relief that perhaps things were getting better for those folks who had fought so relentlessly to get rid of their bad math programs, (after, of course, so much damage had been done to their own children).

* * * * * * *

Now I see that "math wars" are springing up again and sadly now another group of students is suffering and struggling in places such as Florida and Missouri.

And now another group of college professors is speaking out against what they see happening.

The "fuzzy math authors" do not give up so easily. They have their fingers in their ears and masks over their eyes. They refuse to believe that their precious, new math curricula are the cause because after all, "research shows" that students need to learn by discovering and investigating, that students remember best what they figure out on their own without any interference from the teacher, that students need to know they are valued.

In a Missourian article entitled "Math Professors Seek Change in State's K-12 Math Curriculum", college professors are quoted expressing their concerns with the state's math standards and the curriculum. Below is a quote from Missouri Univerisity math professor Adam Helfer:

"One of the most painful things for me as a math professor at Missou is to work with students who have native ability in math but are not going to be able to capitalize on it because their K-12 preparation is inadequate. There is just nothing that can be done at the college level to make up for this -- it's far too late."

Another MU math professor, Alex Koldobsky, is also quoted in the article:

"I have been teaching Calculus I for the last few years and I clearly see the deterioration of computational and algebraic skills of incoming freshmen. Instead of working on the concepts of calculus, the majority of the students have to think for a long time about every elementary arithmetic and algebraic step, which at this point have to be automatic for them."

More than 50 math professors signed the letter, critical of the "student centered focus" which dominates the Missouri K-12 standards -- which repeatedly prescribes that students 'explore', 'investigate', 'develop models', and 'conduct experiments'.

Go here to read the entire article, and while reading it, take the time to go to the side link to the 9-page letter (5 pages of which are signatures of college professors) dated May 5, 2008, sent to the Missouri Department of Elementary and Secondary Education.

If you live in the state of Missouri your children may be affected by the weak standards and "fuzzy math" curriculum. If you are in other states, you need to be vigilant to what is being taught in your state. This is not going away.

"Fuzzy Math" Faces Revolt in Texas

2008-07-16T22:55:00.005-05:00

Because I teach in a private school, this all slipped right past me! I've just discovered that my own state had at one time adopted a "fuzzy math" curriculum, Everyday Mathematics.

I am so proud of the Texas state school board. They have responded to the complaints of many people in a number of school districts, and they have dropped that curriculum. Thank you, Terri Leo and others on the board for listening to parents, teachers, and district board members.

Go here to read the story. Be sure to scroll down and read the letters from parents at the bottom. Thank you Elizabeth Carson for letting us all know who Jesse Arnett is and why he writes with such strong support of Everyday Math.

Here is the story at Edwatch. Scroll through this page because there is so much linked here.

Interactive Concept Development

2008-07-15T14:56:00.004-05:00

I'm reading through the Overview and Implementation Guide for enVisionMath, by Scott Foresman-Addison Wesley (Texas). Because I am used to a traditional, instructivist curriculum, where the teacher guides the student step by step through the lesson, teaching algorithms to help students with knowledge and understanding, I just naturally have some concerns. So, how "new math", how "fuzzy math" is this math textbook?

(My principal stated, as she was giving me the materials, that there were many group activities for developing new concepts and I might want to look over them. Hmmm.)

Several things JUMP out at me.

On about every page so far: "Research says . . ." followed by an explanation of what enVisionMath provides. (There are lots of "Research says . . ." explanations.)

I notice that this seems to be TAKS (our Texas test) driven, and I see that all of the 20 topics are designed to be covered prior to the spring date for students' taking that test.

[One of my prior posts (Expectations Need to be Measurable and Concepts Need Time) referred to an article by William H. Schmidt, where he explains that "top-achieving countries" focus on fewer concepts so that teachers can cover them in depth, rather than the many (up to 20) that our country's math curriculum force teachers to cover.]

And I see that these 20 topics are covered in about 127 lessons (so they can be covered prior to the TAKS), some topics being covered in as few as 4 lessons. So some of the topics are small, bite-sized that hopefully are reinforced throughout the year. I hope there is built-in reinforcement. I'm hoping . . .

Uh-ohhh! Here it is! Interactive Learning

Research Says that students learn best when they have opportunities to interact with teachers and with other students. . . [Research says] Problem-based instruction (before making math concepts explicit) enhances learning because it gets students actively engaged in thinking about a problem and shows students that their thinking is valued.

Teachers are instructed to pose the problem, asking students to work in groups on a problem and share their thinking "before receiving teacher guidance that makes the math explicit".

Before making math concepts explicit ???

And I see lots of writing, writing to list and explain the steps you used, writing to explain what you and your partner decided to do to get the answer and why you chose that method.

Oh my!!!

And while these interactive groups are working, teachers are to be making sure that students are discussing what they are doing and that they are using the proper language and vocabulary in their discussions. How on earth can a teacher be listening to 10 pairs of students at one time to be sure they are using proper math language????

This is why teacher directed instruction is so important. When I teach (prior to the group practice) and when I make concepts explicit (prior to group practice), I can also make sure students practice explicit oral vocabulary (prior to group practice).

More later . . .

Why memorize facts?

2008-07-01T10:09:00.005-05:00

One day, few years back, my principal called me in to her office. She knew I had been researching "new math" and "fuzzy math" and she had a related question.

It seems that a parent of a younger-aged student had questioned why their daughter needed to learn math facts. The teacher of the girl's class had been working on speed drills for math facts and the father was disputing the need for knowing the facts. My principal wanted to get my take on the subject.

The battle is on-going indeed.

So, is memorization of math facts important? You or your student will be confronted with this dilemma, sooner or later, and we need to be prepared. "New math" tells you your child doesn't need to be forced to do the pencil and paper work of the "drill" of traditional math. Your child needs to learn what "8 X 9" actually means, and he/she can always calculate by drawing groups. But your child will understand how to do it, and that's what is important!

According to an article in the March 14, 2008 issue of The New York Times, a recent report, presented to Education Secretary Margaret Spellings by the National Mathematics Advisory Panel (March 2008), addressed the importance of mastering the basic facts.

The report stated that it is important for students to master their basic math facts well enough that their recall becomes automatic, stored in their long-term memory, leaving room in their working memory to take in new math processes.

"For all content areas, practice allows students to achieve automaticity of basic skills -- the fast, accurate and effortless processing of content information -- which frees up working memory for more complex aspects of problem solving," the report said.

According to the NYT, the report found that "to prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills."

The report also said that prekindergarten-to-eighth-grade math curriculum should be streamlined and should focus attention on skills such as the handling of whole numbers and fractions.

Read the entire article here.

By even the 5th grade, the number of steps in problems necessitates that students know those facts quickly so they don't get bogged down in recalculating the facts several times while solving the problem.

The above report stated that by 3rd grade students need to have addition and subtraction facts mastered. By 5th grade, students need to have multiplication and subtraction facts mastered so that they are quickly put to use.

So work with your student on math facts. This is one thing students can do over the summer. Practice, practice, practice.

Girls Learn Differently?

2008-06-30T14:34:00.007-05:00

Well, evidence does show a difference in the way girls learn and the way boys learn.

Many years ago, I read a very interesting article (It's been so long I can't even remember where.) revealing that studies suggested that girls learn math best when they can talk and discuss it aloud.

As an experiment, I have often allowed girls opportunities to work with a friend. I've watched the girls as they progress from problem to problem and they were off task very little. Rather, they helped keep each other focused. But the interesting thing to me was they manner in which they used reasoning skills to solve problems.

I've just read an article that is promoting "all girls' schools and classes." I'm not supporting this type of school, although I have nothing against it. But the information on how girls' brains and how boys' brains work is of interest. I'm passing along the link here.

So if you have an opportunity to try this, it's pretty simple to do. There just needs to be a couple of rules to make it work. Talking is required, but the discussions are about the assignment, not about "what we are going to do this weekend," or "who we saw at the mall".

I've had girls in my class, ones who were quick to pickup new concepts, ask me, "Mrs. . . ., may I help her with problem 2?" I allow it and find that rarely do girls dislike working with a buddy.

Gabby's Story, Part 2

2008-06-30T05:46:00.004-05:00

(If you're reading this blog for the first time in a few days, refer to the previous entry before reading this second part.)

It was easy to figure out how Gabby felt about something. And you knew, everybody knew, when Gabby didn't get something in Math. She spoke right up and told you, for any and all to hear, in no uncertain terms, that she didn't get it.

"I don't get it!" she would say.

The good thing about her was that she really wanted to understand. And I've found this to be true for all students, not only those who speak up. They want to understand. They want to get what everyone else gets. They hope it will make sense.

So when Gabby blurted out, right in the middle of class, it was NOT to annoy me, or to disrupt the class, or to get attention and be the class clown. She truly wanted help.

And so it was, in one of the last weeks of school, as she was at my desk, getting my help, and talking herself through what she knew to do, and solving the problem, she told me this story:

During a previous year, she had been unable to understand long division, as it was being introduced. She had spoken up (boldly, and loudly, I'm sure, because that was Gabby's style) to say that she didn't understand, at which time the teacher had sent her to the board. And apparently she struggled and struggled as she worked at the board, trying to remember what to do next. And it took her a long time to understand with her teacher going through the steps and telling Gabby what to do from her desk. And when at last she did it properly, the teacher had said, "Well, FINALLY she gets it!"

(I need to say right now that I know the teacher, and I'm not going to make judgments on what might have happened before or during the episode. I don't know what she did to teach the students or how she presented that lesson. And I do not get into discussions with students about what a previous teacher has said or done. That conversation is "off limits".)

So I told Gabby that I was sorry that it had happened to her and said something like," I'm can tell that it made you feel bad."

Gabby continued, "I went home and told my mom and she was mad about what happened. And it really made me feel bad and hurt me a lot to be embarrassed in front of the class."

Notice.

It had hurt her.
It had made her feel bad.
It had embarrassed her.
BUT, it had not stopped her from asking questions, thank goodness.

Putting this last piece of the puzzle with all of the other parts of Gabby's math experience in 5th grade, I marveled at her boldness and determination, all the more. She had been willing to take the risk. She had been willing to be "first" to ask for help. She had been willing to be the "only" one, if necessary, in order to be able to understand.

She needed extra time with new concepts and with word problems. She needed to be able to move objects around and handle them. She needed to talk about Math as she worked it. And when she was allowed all of that extra time, she could "get it". And it gave her confidence.

Gabby was one of those students I've written about before. She learned it by "doing" it, by repeatedly "going through the steps", by the actual effort of moving the pencil and doing it. And in her case, by talking out loud about it. And because she "got it," she felt successful.

And one last time, let me say that when she got it, everyone knew she got it, because she announced it with great elation.

She had to work hard for what she got, but she she didn't mind working. And she never made less than a B all year.

If kids think you are willing to help them, they don't mind asking for help. If they think you will keep explaining it until they understand, they don't mind asking questions. If they think there is hope, they don't mind working. And when they know that you, the teacher, or the parent, or a helper, care, they will risk being the only one because they really do want to know and they want to succeed.

"I'm Not Good at Math"

2008-06-28T09:11:00.009-05:00

Gabby, I'll call her, entered fifth grade like a ball of fire. She was very outspoken about what she liked, what she disliked, and what she thought about everything in general. She announced early on, "I'm not good at math."

I'm used to these comments, but I've learned that only a few students are brave enough to speak them aloud. And I'm so thankful for those few kids who are, because they afford me the chance to explain some rules for my math classes:

It's OK not to understand something in math. I will never fuss at you if you don't understand.
If you don't understand, you must tell me so I can help you. When I know you don't understand, I'll think of another way to teach you, and even another, until you get it.
No one is "dumb". Even if you feel "dumb", you are not "dumb". It just means that no one has ever explained it so you could understand.

My first response to Miss Gabby, or to any student who makes that bold announcement, was, "Well, it's going to be different this year. I'm going to make sure that you understand so you can get it."

I usually have two or three students per class who feel they are not good in math. Some students will go so far as to say, "I hate math." For others, it's just the defeated, "I've never been good at math," or the rip-roaring "I stink at math."

Now, I have to say that I truly love having the bold students in my classes because they break the ice. They get the ball rolling. They make it "OK" for anyone else to have trouble. They make it "permissible" for other students to speak up. These "go-get-em" kids really are an asset because even though I have told the students that I want them to ask questions, I've found that nobody wants to be the "first" or to appear to be the "only" one that needs help. And for the extremely shy student who also happens to be weak in math, it is grueling to expose himself/herself. He/she would prefer to just blend in with the furniture as to be found not "getting it."

So, I take advantage of the first opportunity and reply to the "Miss Gabbies", "Good for you for telling me. Look at all of these other kids in the room. Some of them are probably wanting to ask me something also. So now we're going to show them how it helps you to ask me."

Now, I know these "Gabby" types. Not only are they very vocal about what they don't get, but they are equally vocal about what they do get. So, I know what's going to happen next. And sure enough after a little bit of extra help at her desk, Gabby suddenly exclaims, "Oh, I get it!" or "Oh, that's easy!"

Then, I might quicklly say, "Now, try this one." And I mentally construct a similar problem, one that Gabby can solve, so she can have some immediate success and confidence-building.

Students soon learn that if they ask for help, I'm not going to come right out and tell them the answer or what to do. I'm going to get them talking about the problem. If it is a word problem, I'm going to have them read it to me, in parts. (Students who struggle with word problems usually make the mistake of reading the whole problem, without seeing the different parts.) After the student reads the first part, I'll stop her and ask her to tell me what she knows from that part. I sometimes ask students to draw a picture (maybe a map) of it. Then I ask her to read the next part. And I find out if she can properly explain what that tells her. So often a student at this point will suddenly exclaim, "Oh, so that means . . .", and they are able to tell me exactly what they need to do to solve. This was especially true for Gabby. She would inevitably talk her way right through the problem. Her face showed that the "light bulb" had come on. At this point, I always then tell her, or any other student, "See, you didn't need me at all. You figured it out by yourself!"

That response from me is important -- it makes the students see that they can think through the steps if they are on their own, and it gives them confidence.

Gabby was always quite vocal about her "light bulb" moments. The whole class would know that she had gotten it; in fact, they would know the exact moment the light came on for her!!!. Toward the end of the year, she would bring her book to me or if I was near, call me to her desk, and the question would begin with something like this: "OK, here's what it says. There are 3 girls . . ." and she would immediately start talking and explaining to herself and to me . . . and then she would stop dead in the middle of a sentence . . .

"It's so easy when you help me!" she would say. And then I would answer, "But I didn't even say a thing. You did it all by yourself!"

During these one-on-one times with Gabby, I might pull out yarn (to be used to help figure the perimeter), or some counting sticks (to be used for perpendicular or parallel lines or for sets), or pie pieces to help with fractions or percent, whatever I could find, to give her extended hands-on time. And this became her "discovery" time, but it wasn't in a large group, and it wasn't a huge, hour-long time. It was one of the many such "discovery" moments that occur during any good math lesson, moments that many children need and take advantage of, to nail down a concept, to "discover"; and that discovery time needs a reservoir of knowledge from which to draw. One bit of knowledge (the new stuff) hangs onto another bit of knowledge (the old stuff) and some students need just a little more time, connecting those bits, at which time -- "DISCOVERY!!"

So Gabby continued throughout the year, readily acknowledging her deficiencies and her need for help, and talking herself through most all of the things she needed help with.

My goals for her, as they are for any student who enters my room afraid of math, were that she start talking (not particularly hard for Gabby) about what she is doing, or about what the problem is telling her, and that she learn to think, really think, through what she was reading and how she could arrive at the solution/answer. She had to learn that I probably would not tell her what to do, but would help her learn to analyze what she needed to know.

She wavered throughout the year, back and forth from being discouraged to being positive, but her confidence grew. And she continually thanked me for helping her, and I continued to tell her that she hadn't really needed me, that she had figured it out by herself.

And one day near the end of the year, she told me something, an event from a previous school year, that made my heart break for her, but I'll have to wait until next post to pick up that part of Gabby's story. It showed me that even though a person may seem tough on the outside, he or she may be crying on the inside, and those "heart tears" are just as real as the ones all the rest of us can see on the outside.

Knowing Leads to Understanding

2008-06-27T09:25:00.011-05:00

You have to "know" math before you can "understand" math; you have to "know" math before you can "do" math; you have to "know" math before you can "solve math problems".

The above is a paraphrase from a long article by William G. Quirk, Ph. D. in Mathematics.

In this article, Quirk explains that even though the National Council of Teachers of Mathematics has toned down their push for "new math", "constructivist" math, fuzzy math, (probably in an attempt to calm the uproar caused by their original 1989 Standards), they still push student-centered "discovery learning"

[On April 12,2000, The National Council of Teachers of Mathematics (NCTM) released Principles and Standards for School Mathematics (PSSM), a 402 page revision of the NCTM Standards.]

Translation: NCTM sees the error of their ways in the 1989 Standards and so here comes the 2000 Standards in which they pretend to drop all of this "fuzzy" stuff, such as emphasizing calculator skills or student-invented procedures and now appear to emphasize mastery of basic facts.

I think the NCTM must be more patient that the opponents are persistent. They know that if they just give the appearance of change, the "math wars" will subside, we will drop our guard, and then they will be able to come back with terms redefined and no one will notice. They they will be free to unleash, once again, their confusing, dumbed-down ideas on a new and unsuspecting group of children, families, and school districts.

Oh, yes, they say, we are all for "putting arithmetic back into mathematics." We are all for "teachers emphasizing the fundamentals of computation." And the public, with a sigh of relief, smiles and thinks "Oh, isn't that wonderful! The "new math" is gone! Traditional math will return at last! Our children are now safe!"

And we fail to read the fine print:

"When calculators can do multidigit long division in a microsecond, graph complicated functions at the push of a button, and instantaneously calculate derivatives and integrals, serious questions arise about what is important in the mathematics curriculum and what it means to learn mathematics. More than ever, mathematics must include the mastery of concepts instead of mere memorization and the following of procedures. More than ever, school mathematics must include an understanding of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium." --NCTM

Quirk helps us understand that although NCTM says they want to emphasize "understanding", they fail to understand how the brain works. Says Quirk,

". . . they still fail to recognize that specific math content must first be stored in the brain as a necessary precondition for understanding to occur. Although rarely the preferred method, intentional memorization is sometimes the most efficient approach. The first objective is to get it into the brain! Then newly remembered math knowledge can be connected to previously remembered math knowledge and understanding becomes possible. You have to "know math" before you can "understand math", "do math", or "solve math problems.

"Similar to the orignial NCTM Standards, PSSM fails to clearly acknowledge that the abililty to instantly recall basic number facts is an essential preskill, necessary to free up the mind, first for mastery of the standard algorithms of multidigit computation, and next for mastery of fractions. Then, once this knowledge is also instantly available in memory, the mind is again free to focus on the next task level, algebra."

. . . . . .

OK, it boils down to what you want your student to be able to do:

Master the basic facts or Derive basic facts when needed

If basic facts are mastered, students can proceed quickly through multidigit computations, particularly when learning a new concept. Their minds are free to focus on what is new, rather than having to also repeatedly derive methods and facts to help them get through the steps.

. . . . .

The entire article really upset me because it makes statements about students' inability to learn how to properly use division of fractions. I've taught division of fractions for 15 years very successfully using Saxon Math. I strongly refute the statement by PSSM that the "process can seem very remote and mysterious to many students."

When presented properly and taught properly, students are indeed able to grasp the meaning of "invert and multiply". Students are indeed able to learn and understand the concept of dividing 1/2 by 1/4, or 1/3 by 1/6 and "the reasoning" of "How many 1/4's are in 1/2?" And students are indeed able to identify the types of story problems where division of fractions is the preferred method to find the solution.

Read the entire article here.

. . . . . . . .

The article concludes with some excerpts from Roger Howe, Professor of Mathematics at Yale University, several of which will be quoted here:

"An important feature of algorithms is that they are automatic and do not require thought once mastered. Thus learning algorithms frees up the brain to struggle with higher level tasks."

". . . we suspect it is impractical to ask all children personally to devise an accurate, efficient, and general method for dealing with addition of any numbers -- even more so with the other operations. Therefore, we hope that experimental periods during which private algorithms may be developed would be brought to closure with the presentation of and practice with standard algorithms."

"We do not think it wise for students to be left with untested private algorithms for arithmetic operations -- such algorithms may only be valid for some subclass of problems. The virtue of standard algorithms -- that they are guaranteed to work for all problems of the types they deal with -- deserves emphasis."

The "Real" Contest

2008-06-26T06:51:00.007-05:00

Have you ever gotten into a "battle" only to find the opponent has changed and redefined all of the terms?

Have you noticed that the publishers often try to change the focus of the disagreements brought by parents. Or they change definitions and throw out new "standards" so that you and I may not even know what the real issues are.

As a lowly intermediate, elementary school math teacher, I often feel that I do not have the qualifications to step up and confront more "learned" people. I think that no one would even give me any time. However, if you won't listen to me, will you at least hear the words of math professors, who know far more than I, and who strongly oppose the teaching of the "fuzzy math"??

A few years ago when I first became aware of the "math wars" that had occurred, and were still occurring, all across the country, I was most impressed by the pleas of professors to rectify and rewrite curricula, professors such as Dr. Wayne Bishop, Dr. James Milgram, Dr. Wu, Dr. Bas Braams, and many more.

Below are a few quotes from Ralph A. Raimi, professor emeritus of mathematics of the University of Rochester, describing the "real contest" in this "math wars" saga and the folly of expecting children to "discover" formulas and procedures on their own.

"But there is a contest, a serious one, and not the one suggested by catch phrases handed out by the publishers of the reform programs. It is not a contest between rote-memorization of meaningless symbols and deep understanding of problem-solving strategies. . . "

"The real contest in Penfield - and hundreds of other school districts across the country - is between mathematics and non-mathematics, between academic content and childish time-wasting, between what children can learn and what the present Penfield curriculum is pretending to have them "develop." A good mathematics program takes advantage of the mathematical discoveries of thousands of years of civilized effort, while Penfield has them counting with sticks, starting history all over again."

And the following from Raimi is scripted so purely and simply, so precisely and beautifully and is certainly worthy of notice. I'll even go so far as to say it is worthy of putting to memory.

"The systems of decimal and fraction notation are marvels of compressed information, intellectual advances that Euclid did not have available. Arithmetic is not trivial mathematics, and it certainly will not be "discovered" by school children. It must be taught and practiced."

The entire article, entitled "Why American Kids Aren't Learning Math," may be found here.

The Ant and the Grasshopper (a modern day story)

2008-06-26T05:20:00.014-05:00

Alongside a trail in the open Field of Mathe, a Grasshopper was lazing about, discovering and coloring patterns to his heart's content. A lone Ant passed by, bearing along, with great toil and rigor, a bag of corn he was planning to store, yet singing as he labored. Because he knew the facts, he had accurately analyzed his problem, quickly calculated his needs, while carefully considering the 'rithms' of the seasons.

"Why not stop and discover patterns with me," said the Grasshopper, "instead of toiling and moiling (grasshopper speak for "drilling and killing") your life away? You'll be much happier sitting and chitting about your work than making your repetitious rounds." The Grasshopper continued, "That Trail of Boredom you are on becomes the Path of Monotony just over the hill, you know." (Surely, he thought, this was a factor the Ant had not considered.)

But the Ant was undeterred. And he wasted not a step, still singing as he 'plotted' along, for practice had made his search for the missing factors all the easier.

"By the way, my feelers are 'Connected' to my feelings, you see?" said the Grasshopper, everyday still lazing on the trail. "I'm feeling soooo good about my discoveries. I need to be writing about the work I'm going to be doing after I finish discovering what I'm supposed to learn to do. Care to join me?"

"No, thank you. I'm preparing for what's ahead," said the Ant, as he 'Sing(ingly)Poured' his full bag into his storage bin. And off he went to glean more from even higher p(m)aths. "Quit your Trail lazin' days, Mr. Grasshopper. Get up and get to work."

"Oh," said the Grasshopper, "there's still plenty of time. But for now, this writing about my discoveries is just about filling up my hours. But I'll be 'Blazing this Trail' later."

Sadly, the Grasshopper never got past his Trail lazing days. You see, though over 'Andover' he tried, he found that he had to continually 'spiral back' to the place of his discoveries in the '(o)Penfield'. He never made it off the Lazing Trail nor did he ever reach the top of the Field of Mathe.

However, the Ant had all he needed. Not only did he successfully reach the top of the Field, he became the King Ant of the Field of Mathe, all due to his rigorous work and efficient use of time. And he ruled over the Grasshopper all his remaining days.

Hilarious Math

2008-06-25T14:14:00.010-05:00

This is just a light-hearted attempt to give us all a few good laughs today.

So, I share with you some quotes from an old book entitled Kids Sure Rite Funny, written by Art Linkletter of Kid's Say the Darndest Things TV fame. Growing up, I always enjoyed watching him interview those little children, knowing that the parents were in the audience and that Art Linkletter was going to do his best to get those children to say something funny or embarrassing.

In his book, Kids Sure Rite Funny, Linkletter shares from the writings of kids across the country. The book was published in 1962. If you should come across one at a garage sale, grab it. You will be entertained for years to come. I don't know how many times I've read these quotes, but they are funny every time.

(All spellings are the children's, as so spelled in the book.)

Hope you have a good laugh.

________________________________________

"A hypotenoose is a humane device for hanging criminals from a 90 degree angle."

________________________________________

"If your triangles get four sides, you have wrectangles."

________________________________________

"In area, a circle is a pie or square."

________________________________________

"Why I am taking algebra is because I hear that some thoughts cannot be thought without thinking in algebra. Although I have never had such thoughts, I am expecting."

________________________________________

"The minuend is the number from which the minuet is subtracted."

________________________________________

"Square objects are rectangular while round ones are tubercular."

________________________________________

"When rulers are not human, they have twelve inched feet."

________________________________________

"A tangerine is a line going past a circle."

________________________________________

"If you are looking for an X, algebra will help you find it."

________________________________________

"Axioms and postulates are the same. We have both of them in case we forget the word for one of them."

________________________________________

NCTM Focal Points (Standards)

2008-06-25T06:53:00.010-05:00

According to this bulletin from the National Council of Teachers of Mathematics, there are apparently three (3) focal points identified for each grade level, PreK-8.

Here are the focal points, followed by excerpts from NCTM's explanation.

the use of the mathematics to solve problems;
an application of logical reasoning to justify procedures and solutions; and
an involvement in the design and analysis of multiple representations to learn, make connections among, and communicate about the ideas within and outside of mathematics.

"These curriculum focal points should be considered as major instructional goals and desirable learning expectations, not as a list of objectives for students to master."

". . .this set of curriculum focal points has been designed with the intention of providing a three-year middle school program that includes a full year of general mathematics in each of grades 6, 7, and 8."

Go here to read entire bulletin.

________________________________________

My observations:

No matter how good these focal points (expectations) sound, don't kid yourself that they intend for your child to master them. These focal points are not to be considered "as a list of objectives for students to master." That isn't considered to be important to NCTM. Notice, however, that it is considered important that they learn to communicate about the ideas they are learning. And don't be surprised if that communication is written.

And . . .

If states and districts follow these suggested standards, your students will likely fall short of the goal of most families -- completion of pre-algebra and algebra by the end of 8th grade.

Group Discovery ???

2008-06-24T19:44:00.006-05:00

One of the marks of "new math" is discovery learning. But it's more than just discovery learning; it's "group discovery learning". Students are expected to come to knowledge of mathematical facts and procedures through "communal discovery".

You've heard it before. Only what your child discovers himself will he really grasp and remember. He must "own it" to be able to "use it and truly know it".

Now, let me see. If we want everyone to make the discovery himself, why would we put them in a group? That is exactly the opposite way you have every person discovery on his own. The whole group discusses and comes to an action, tried and agreed upon by the group. (Of course, someone in the group had to make the suggestion, to begin with, but never mind that.)

Now, remember through all of this, the purpose is for each student to make the discovery for himself -- so he can "own" that knowledge.

If you're in a group, how can all students come to "discovery" -- of a math procedure -- at the exact same moment? Someone in the group will come to the discovery first. Must he or she remain absolutely quiet? Must he hide his procedure? Remember this is a group discovery.

So, if someone else in the group makes the discovery, I guess no one else ever gets to "own" anything! Be First or Lose Your Shot at It.

By the way, the teacher is very quietly staying to the side, offering no guidance. And once the answer is agreed upon, the teacher is not allowed to check to see that the procedures were used correctly. Later, no drill will be given for students to use for practice. And there is no memorization of facts -- the facts can be 'rediscovered" on the spot, whenever they are needed.

In a traditional classroom, a teacher gives students an opportunity for small discoveries throughout the lesson, but he or she carefully directs the lesson and teaches precise methods, so that all students learn the most efficient procedures. Then the students practice and practice the steps under the supervision of the teacher. Following the guided practice is the independent practice, during which kids do more practice. Teachers are available to guide students and reteach if necessary. It is during this practice that some students "discover" and learn. And as the procedures are practiced over the next days and weeks, still other students may finally make the discovery on their own. But that "discovery" is just as real as the "discovery" made earlier by other students. In the meantime, the student has been able to successfully get the proper answer on the work because the teacher had taught just that.

I disagree that there is no discovery in a traditional classroom. There is. I see it every day.

And I don't think that it is possible for all students in a new math room to "discover" every single fact and procedure without input from the teacher.

Pay Attention to a Child's Individual Bent

2008-06-24T16:57:00.008-05:00

This is an especially fun story for me to recall because it involves my precious nephew. He was my first nephew and I was in college. We were all amazed to see his early interest in numbers and marveled at his insatiable appetite to learn about them. No one forced him to sit down and listen. No one made him memorize anything. His parents just sort of scratched their heads and watched it all unfold.

He had an early desire to know more and more. We soon figured out that as he was mulling over the answers to his questions, he was properly putting it all together in a way none of us imagined. On his own he figured out that the symbols for the numbers meant something.

He didn't fit any of the models I was studying in my college childhood education and psychology courses. None of us knew how he did it. He was just too little to explain to us what was happening in his mind.

Here's how it played out. It started with his asking us "What is 3 and 5?" "What is 4 and 7?" We never knew where that idea to ask those questions came from, but as long as we were willing to answer, he kept firing those questions. It was nonstop and it never seemed to be enough. What 3 1/2 - 4 year old normally does this?

I think we all thought it would run its course and play itself out after a while. But it didn't. His mind was an unquenchable little sponge and the questions, always related to numbers and groups of numbers, kept coming.

But those simplle questions soon became "Daddy, what's 38 and 5?" "What's 57 and 8?" And when his daddy was tired, aunts and uncles and grandparents were the next victims. And all that we told him, he took in and somehow analyzed it correctly. And the questions became more precise, showing that he was honing his skill.

We started turning the tables on him. We started asking him questions and were just blown away with what happened. This little guy had somehow figured out sets of "tens" in his mind. He knew how to regroup and go past tens. He could answer anything we asked him. And he was right on the money -- every time.

He could subtract, too. He could add and subtract 8's or 7's just as easily as you or I.

He was not really any smarter than his siblings. He just had a bent toward math concepts. For some reason it was a challenge, it was compelling, it was satisfying to him. To him, the mental computations were enjoyable, and even something difficult didn't set him back. He kept wanting to know more.

His questions turned to money and how much things cost, and even multi-step procedures didn't make him waver.

One evening, my husband and I took him to a small airport in our area to watch the commercial planes come and go. He asked us a question (number related, of course).

"How much would it cost to fly to . . . (a nearby town)?"

My husband said, "Oh, about $5," mostly just to give an answer, rather than to be accurate. Immediately, my nephew shot back, "Well, I'm saving my money till I have $30, and I'm going to take you both to . . . . (the town)."

He had added and multiplied so rapidly, we were stunned. He even planned on our return flight!

If your child has such an appetite, such a natural bent toward math, nourish and encourage it. Provide opportunities for him to grow. Make sure he doesn't languish in a "fuzzy math", "non traditional" classroom. You may not think that the misguided approach of "new math" can have a detrimental affect him, but it can.

Or if your child doesn't show an early interest, provide opportunity for little baby steps of counting and of understanding numbers and make it fun. He/she could be a late bloomer. He can make good progress with just a little encouragement and interest from a parent. And make sure he doesn't languish in a "fuzzy math", "non traditional" classroom. He could likely be at the most risk because he is neither strong or weak and can easily slip through the cracks.

And if your child really does struggle and it never gets easier, just keep being your kid's biggest fan and biggest encourager. Don't ever show your disappointment in him, and don't give up just because he/she doesn't have a natural bent. Support him and praise him for even small gains. And for certain make sure he doesn't languish in a "fuzzy math," "non traditional" classroom. Be prepared for an early intervention with tutoring and evaluations. Make sure he learns to ask questions. And he must know he can come to you. And give him room to excell in something else -- art, or music, or helping people.

Money, the Logical Hands-on Teaching Tool

2008-06-23T14:34:00.010-05:00

Having taught at the same private school for many years, it only follows that I have taught several brothers and sisters of many families. I've never expected younger brothers and sisters to mirror their older siblings, but I've also noticed that they often do because of the parents' expectations. Parents who expect diligence and responsibility from their children usually have an entire household of diligent and responsible children.

One such case that puzzled me for several years was the inability of children from one particular family to count money, to make money exchanges accurately, to make change involving bills and coins, and to solve general word problems involving money. This was especially baffling considering I teach 5th grade and it is unusual to have one child, let alone several, who can't make small change accurately and easily.

And then I figured it out. These children apparently were never required to be responsible for the money they were given by their parents or for bringing accurate change back home. If one of them needed money for something, the student would be given a $20 bill with no thought for what he/she was expected to bring back home. In this one case, the lunch cost $4.50, and the student had no idea what change to expect. I discussed it with her and she wouldn't even try to reason through it. All she did was just shrug and say "I have no idea" or "I don't know" to every query I made.

__________________________________________________

So begin early and teach your children how to exchange 5 pennies for 1 nickel, etc. Have them count money (and remember to start at zero when counting). Give them small amounts of money to purchase something and discuss ahead of time overpaying and expecting change from the clerk.

Do you save pennies in a jar? This is a good way to involve your child in a family project. Count the money by the week or month, depending on the child's age. Have the child help you swap nickels for the pennies. Then later swap dimes for nickels, etc. This helps teach your child equivalent amounts.

One game that is fun for children can be played in the car as easily as it can at the breakfast table with coins. Say to the children, "How much money do I have? I have 1 quarter and 1 dime." As children learn to solve these problems, ratchet the game up a notch to this next level: "How much money do I have if I have 2 less than 3 quarters, 1 dime and 1 nickel?" Children in second and third grade can get these problems. Then try going past the dollar, but do not mention dollar bills. "How much money do I have if I have 3 more than 6 quarters, 2 dimes and a nickel?"

Give students reasoning problems, but call them "riddles". "Here's a riddle for you. Bill has 5 coins. What five coins did he have if he has 38 cents?" At first, this can be done with coins in hand, but as children get older, they need to be able to solve this type of problem mentally.

Use money to begin teaching tally marks. More on tally marks in another post, but money is an easy way to get your children to think of counting by sets of numbers. And remember counting is the beginning of solving abstract concepts in numbers. Make sure you do a lot of counting.

Have a pretend store at home. Tell your young children that you are practicing what to do at a real store. Teach them to make change. However, the true goal should be more. You want your children to recognize incorrect change. Explain to them that this is "Wrong Change" day at the store. Tell them you are going give incorrect change for a pretend purchase, and have your child figure out what is incorrect.

Expect your child to be responsible and accountable with any money, even his own. Do not pass it out to him/her like it was candy. Children will value its importance if you value it, and will learn to value it enough to want to know how to use it wisely.

Geometry of Today is Not Geometry of Yesterday

2008-06-23T11:25:00.010-05:00

I'm pulling up old stuff, I know, but you can't get better than Barry Garelick.

Barry posted this comment on the first Kitchen Table Math back on June 5, 2005.

"From NCTM's PSSM, here's what NCTM has to say about their geometry standard: 'Geometry: Geometry has long been regarded as the place in high school where students learn to prove geometric theorems. The Geometry Standard takes a broader view of the power of geometry by calling on students to analyze characteristics of geometric shapes and make mathematical arguments about the geometric relationship, as well as to use visualization, spatial reasoning, and geometric modeling to solve problems. Geometry is a natural area of mathematics for the development of stusdents' reasoning and justification skills.'"

Translation: High school geometry used to emphasize proofs. Now it just emphasizes shapes and formulae, with an occasional proof and in general is not much more advanced than the geometry presented in 7th grade, except for the fact that not much geometry is presented in 7th grade."

My observations and thoughts:

NCTM is the National Council of Teachers of Mathematics. They are a body of educationists (my word) who are responsible for writing the national math Standards which are supposed to define the expectations for students in each subject area at each grade level. I say "are supposed" because the expectations are so watered down and are so vague that no one can actually identify a specific expectation.

You have to hunt far and wide to find geometry in textbooks today. This "broader view" is part of that "1/8 inch deep and a mile wide" approach to teaching Math. The subject of Geometry is spread all through other textbooks and is no longer taught in a coherent fashion semester by semester.

A high school teacher commented to me 2-3 years ago how much he wished he could teach geometry as an isolated subject so he could concentrate the students' focus on geometry.

Your student is probably being robbed of the opportunity to learn to prove geometric theorems. It's no wonder our high schools students score so much lower than Asian students.

Expectations Need to Be Measureable and Concepts need Time

2008-06-23T02:06:00.016-05:00

Standards are "expectations". They are the targets. They are what you are shooting for. They define what is expected of the students at specific grade levels in specific subject areas. (In the United States, curricular expectations are defined as "standards".) In order for standards to be effective, they must be specific.

Specific expectations are easy to measure. If your expectation is that your child know the capitals of all of the states, that is specific and measurable. You can easily discern if a student has met the standards. How? Have him demonstrate that he knows the capitals.

And that is why so many people are concerned about "standards". It is the vagueness of the standards that troubles us.

I came across an article by William H. Schmidt entitled "What's Missing from Math Standards?" which was published at the American Educator website in the Spring of 2008. He discussed the findings of the Third International Mathematics and Science Study (TIMSS).

According to Schmidt, TIMSS found that "student performance is directly related to the nature of the curricular expectations." He explains that he does not mean the instructional practices, but rather "the nature of what it is that children are to learn within schools."

"The TIMSS research has revealed that there are three aspects of math expectations, or standards, that are really important: focus, rigor, and coherence."

Here are Schmidt's comments about all three of these aspects of expectations.

Focus:

"Focus is the most straightforward. Standards need to focus on a small enough number of topics so that teachers can spend months, not days, on them. . . [i]n the early grades, top-achieving countries usually cover about four to six topics related to basic numeracy, measurement, and arithmetic operations . . . In contrast, in the U.S., state and district standards, as well as textbooks, often cram 20 topics into the first and second grades."

It is Schmidt' s opinion that this number of topics is far more than any primary grade student can absorb.

Rigor:

"Rigor is also pretty straightforward -- and we don't have enough of it.... .[I]n the middle grades, the rest of the world is teaching algebra and geometry. The U.S. is still, for most children, teaching arithmetic. . . [O]ther countries outperform us in the middle and upper grades because their curricular expectations are so much more demanding, so much more rigorous."

Coherence:

Coherence may not be as easy to grasp as focus and rigor, but according to Schmidt, "it is the most important element." He explains that there is a formal academic body of knowledge that has been parsed out and sequenced from kindergarten through 12th grade, and he describes how especially important this parsing and sequencing is in the subject of math.

"Topics in math really need to flow in a certain logical sequence in order to have coherent instruction. If you look at the math curriculum of top-achieving countries, you see a very logical sequence. The more advanced topics are not covered in the early grades. Now that seems obvious -- until you look at state and district standards in the U.S. Everything is covered everywhere. Far from coherent, typical math standards in the U.S. often appear arbitrary, like a laundry list of topics."

Some of you may want to consult the entire article to see why our country has such unfocused, undemanding, and incoherent math standards.

Two related articles can be found here and here.

____________________________________________

Some additional thoughts . . .

By spreading topics all through the curriculum, nothing is covered in depth. Teachers are expected to cover so much material during the year that they must fly quickly to the next topic, meaning that there isn't time for a student to grasp the coherence of one concept with another. There is rarely time for the feeling of "Aahhh! I get it!"

Depth is better than shallowness. Kids can get their thoughts around a concept and understand connections when a topic is covered deeply and thoroughly.

For some reason this picture is going through my mind right now: I'm thinking of trying to drink lemonade through a straw -- after we have spread it 1/8 inch thick all over the table or counter top. And we only have one minute to do it!

That might be how many of our children feel when they are trying to "get hold of a math concept". This produces frustration and a feeling of "there's something wrong with me -- I didn't get it."

Laying the Foundation for the Abstract Concept of Numbers

2008-06-22T23:02:00.013-05:00

Learning the abstract concept of numbers starts long before the usual introduction of pre-algebra and algebra. It begins when your child is first introduced to the meaning of numbers by counting. Yes, counting is where the development of number sense all begins.

I remember when our son was two and I was participating in an exercise class. We all took our children along and they played next to us while we exercised. Our son, interested in numbers from early on, was fascinated by the class leader's counting out loud, "One, two, three, four; two, two, three, four; three, two, three, four; four, two, three, four; ("again") one, two, three, four; two, two, three, four; three, two, three, four; four, two, three, four . . ." And our son was fascinated by the words. We adults understood that there were two sets of counting patterns being used at once, but to our son, these were just words, recited along in a pattern, with a nice rhythm, and he was intrigued, taking it all in.

A few days later, I heard him repeating the number words, in random order. He had no sense of counting; to him the words were meaningless. Ooops!

Children love counting. Counting games, counting songs, and counting rhymes all serve to draw our children's interest to the fun of counting, and at an early age.

Counting is where the understanding of the concepts and the meaning of a number begins. And it is important to know that there is much more to counting than merely calling out the numbers.

Here are a few ways to properly start your child counting so that early on, he/she will not only begin to grasp the meaning of numbers and will also apply "counting" to objects around him

Assemble a set of objects in a plastic bowl or tub. They may be pennies, blocks, craft sticks, toys, anything. You also need a second empty container. Have your child pick up an object, place it in the empty bin, while saying its number, beginning with "one". (Demonstrate this yourself. Children love to copy.) It is important that your child actually picks up the object and moves it, rather than sliding it across to another pile. Begin with 10 objects or less. Be sure that your child only picks up one object at a time.

After your child has moved the last object to the other bowl, ask, "How may trucks are in your other tub?" Your child should learn to properly say the last number he counted. Vary the number of objects in the tub. It is also important to vary the objects you use so that he/she understands that numbers are used to refer to different types of objects.

This early activity of counting, using concrete objects, though simple, is part of the learning of the abstract number concepts to come later.

And this is important: have one tub on one side of his body and the other on the other side. We want his/her hand to cross the midline of the body with each motion. We know that the brain cells fire away each time an arm or leg crosses that midline.

As your child gets older, and after he/she masters this first counting activity, there is something else very important that you must do. You must develop a sense of "zero". So this is the next step in the counting exercise: Teach your child to start counting with "zero". Before any objects are moved, teach your child that there are "zero" objects in the second tub. Teach him to say "zero" before he begins, followed by "one" as the counting and moving of objects begins.

Next, using blocks that are numbered, or making your own on pieces of paper of cardboard, teach your child the symbols (digits, numerals) that stand for each number from 0 to 20. Line them up in order and point to them as you recite the numbers. Have your child recite them with you as you point to the symbols. Then have your child recite them alone.

Later on, with the blocks still in proper order, point to the symbols in random order and have your child name them.

Much later, mix them up and have your child find them in order and say them as you and he line them up in order.

As your child learns, you can gradually increase the number set up to 20.

Books that have counting pictures can be useful. Point to the symbol and ask your child what it is. Sing counting songs with him/her. Avoid songs that count backwards at first.

Count the silverware as you place it by his plate, or as you remove it from the dishwasher, remembering to begin with "zero". Count his/her socks as you place them into his drawer. For an older child who has mastered the counting of single objects, count the socks in groups of two: "one, two," "three, four", etc., which will lead to counting by multiples of two.

Where to Start

2008-06-22T04:26:00.004-05:00

OK, Time to discuss your child's needs.

Does you child struggle in math? Is he in the 4th - 6th grades? OK, then something needs to be done and something can be done. It needs to be started now and it can be started now.

What you know, you can address. What you don't know or aren't sure about, you must find out. You must delve into until you do know. You must identify what it is, or hire someone to tell you what it is that your child needs help on. But I think parents are smart enough to figure some of it out.

With what does your student struggle? Ask yourself and be specific. Or ask him/her. They can usually tell you something at this age.

Is it number facts? You, the parent, can do something about that and you must. It is a lie to say that memorizing facts isn't important and doesn't really matter. It matters, and you can and must do something about that. A child in intermediate grades that doesn't know the facts, all of them, is like a kid trying to ride a bicycle with two very flat tires. He takes so long just stuck at a stand still at the starting gate, never getting very far. Knowing facts gives him a jump start in solving the more difficult, multi-step problems.

There are some very simple methods to use that will help your child learn the facts, and there are some methods that will do very little good. If this is what your child needs, speak up! I'll help you NOW.

Is it 2-digit multiplication? If so, I would bet your student has a gap somewhere in understanding the concept of place value. If you can't help him/her with that, get assistance from someone who can. Two-digit multiplication is usually taught in the 4th grade and reviewed in 5th before 3-digit multiplication is taught. Work on place value with them now. Or hire someone to do it. Students need to be able to correctly solve 2-digit multiplication quickly. And if your student hasn't mastered multiplication facts, back up and work on that with them. And then tackle the multiplication. And if it's one-digit multiplication, start there.

Is it long division? It could be because your student hasn't mastered multiplication facts once again. Division is really just searching for the missing factor. FACTS -- FACTOR -- get the connection? And long division should be mastered in the 5th grade. Two-digit long division should be mastered in the 5th grade. And it can be.

Is it fractions? Is it decimals? Is it percent? If your student is moving into 6th grade in the fall, a good, clear understanding of these 3 is a must. Get help now, over the summer. If your student struggled with any of these, hire a tutor, check out some on-line sites that will help you help them. A good understanding of fractions is the one single determiner of success in pre-algebra and algebra.

And parents, if you don't understand the four operations of fractions, converting fractions to decimals, decimals to fractions, and converting both to percent, DO NOT TELL your child that you won't be able to help them because you were never good at math.

Tell them that even though you were not good at math, you are going to learn it now so you can both work on it together this summer. Show your student that you are willing to do that. You are not too old to learn it. Go to the tutoring class with your student. Your student will take note of your decision. It will speak volumes. It will say, "Math is important in our family and because it's so important, I'm going to make sure we all know it, and that includes me."

So . . . Reality Check right now. Math troubles are real. Admit and identify at least something, something small perhaps, but something that you can work on. And do it. Don't wait until another school year starts. If you wait, then you'll be tempted to put it off another year. It will not be easier come fall. You student will begin next fall at a lower level than he finished at the end of this past spring. It just happens -- a month or two off can cause a set back, especially for a struggling student.

(This is where I wish I was your next door neighbor. Or your child's aunt or uncle. I would only need to watch him/her work for about a half an hour and I would know enough to be ready to start. Yes, it frustrates me to know that your child is there somewhere and perhaps you don't know what to do. But please try.)

Properly Building Self Esteem

2008-06-21T09:26:00.008-05:00

Self esteem.

We are told, at every turn, that our children need to have self esteem. You want that for your children. I want that for my children. I want that for your child when he/she is a student in my classroom.

Question: How do we build self esteem? Or better, How do we build proper self esteem? Well, I'll tell you how NOT to build self esteem. Give your child NOTHING to do that requires effort on his/her part. And then, tell him how wonderfully he did it!!!!!

Here are some examples:

Give your 5th grade student 2nd grade spelling words
Have your upper grade student work a 1st grade math story problem.

You get the idea.

Will he succeed? Of course, but you have required no effort on his part. You've guaranteed that he will succeed, yes, but you've really required nothing from him.

Why am I mentioning this? One of the trends of modern education is to "dumb down" everything. It is pervasive in all subject areas, but I'm particularly addressing math in this post. They feel they need to make math less stressful and so the tests are made extremely easy or students are given art projects, on which children are given inflated grades, all in an effort to "even out" the grades.

A primary aim of these educators is to have a completely noncompetitive environment in the classroom. It must be, according to them, a place where all students are judged equal, and judged to be equally successful. And the only way to make all children equally successful is of course, to set the bar so low that no one appears better/smarter than others. All students can then feel good about themselves because no one is better. And this is supposed to give each student "self esteem" -- good "self esteem".

We see this effort all across the country when math teachers do not grade homework. Getting the correct answer on a problem is not required. In fact, if students don't know how to solve the problem, all they need to do is show an effort, just try to solve it. Everyone gets the same credit.

Let's back track to when your child was very young. Remember how he/she loved to help with little chores such as unloading the dishwasher, or putting the silverware on the table, or taking the folded clothes to each bedroom. Why? Because it showed that he/she was able to do a "big" person's job. And then, Moms, what did you do? Perhaps you said to your husband, "Daddy, look at the table. Guess who helped Mommy put the dishes on the table!" And then you and Daddy, made a big "to do" and praised your child for his efforts.

I remember teaching my daughter how to use the dust mop up and down the tile hallway to get all the dust. It took several trips down and back, but she quickly learned to do it without missing a speck of grass or dirt. She was really little and it wasn't easy for her, but she learned it and was so pleased that she had done it properly and well!!

Now back to the classroom. How do we build a student's self esteem? We DON'T give them assignments that are below grade level and tell them how well they've done. We give them something hard to do, something that is a challenge for them. And then when they do it, we praise them for it. This is what builds self esteem. And this is what makes them feel successful.

We need to set high goals and push them to reach those goals. Tell them that you know it won't be easy, but that all other 5th graders have been able to do it and that you know they can do it too, and that you will help them till they get it.

Dr. Wickelgren, in his book Math Coach, A Parent's Guide to helping Children Succeed in Math, confirms this approach. And it is the lack of high goals which causes "new math" methods to be so harmful, according to him. The new way doesn't encourage students properly. It certainly doesn't encourage a student who has put forth an effort to do the work and to solve a difficult problem.

I have seen first hand some of these methods used as my own children came through school. They were discouraged when students who had done no work got equal credit as students who had worked hard. What incentive does that give a diligent student to work hard next time? (We all grew to hate group projects where all students got the same grade, regardless of the effort put forth by each student.) Rewarding a student for no effort does damage, not only to that student, but to the entire class.

Dr. Wickelgren also believes that judging all students the same can have "deleterious psychological consequences for children." Why? Because it isn't the truth and it can cause those children to "have an unrealistic view of their abilities." According to Dr. Wickelgren, this can mislead children "to think they have skills they lack" because they have acquired the reward without doing the work. "Such children may not do the hard work necessary to succeed later in life." They have been trained to believe they will be rewarded irregardless of the work or effort.

So giving students hard tasks is important. It builds self esteem because the children know they have succeeded at something difficult. They know the difference. They know when they have worked hard and when they have not.

It also develops in them the character quality of perseverance, which they will need later in life, not only in high school, but also on the job.

Will all children be able to be successful on all of the work I assign in class? Maybe not. Probably not, because there will always be some things that are hard for some children. There are problems and concepts that I know some children struggle with.

I consider it part of my job to know what each student finds difficult in math. And I must be careful to praise, praise, praise when I know they have worked hard and solved a problem that was difficult for them. I need to give verbal praise immediately. In addition, I also like to write words of praise on a paper as I grade it -- "Good for you!" or "Wow! This is wonderful!" or "Yeah!!! You got it!" This is something that a parent will later see and gives them an opportunity to add their own praise

Even if a student fails to get the correct answer, if they remembered and used the proper procedure, I will write, "Thank you for working so hard on this." or "Keep trying. You'll get it." or "Come see me. I'll help you." This provides encouragement to the struggling child. It tells them they you are noticing their efforts and that their effort has not been wasted.

Every child needs encouragement. They also need to be challenged and rewarded when they have worked hard. Students can be/ should be recognized for excellent and perfect math work. They should also be recognized for improvement.

For a struggling child, recognition for improvement is HUGE. And it can build self esteem because the child knows the task wasn't easy for him. And it gives him hope. Hope that things are getting better and hope that he can do it and hope that he will succeed.

That builds self esteem. Proper self esteem.

A Few More Gleanings: Dispelling Myths

2008-06-20T07:46:00.008-05:00

Here are more snippets from the Math Coach, A Parent's Guide to Helping Children Succeed in Math, responding to several "myths" being promoted by the "new math" proponents:

"Concrete and abstract ideas are not separated in the brain, but lie on a continuum. So the implied mental leap necessary to cross from concrete ideas to abstract ones is fictitious." (written in response to "new math" supporters' claim that children under twelve years of age are not capable of learning abstract concepts and operations)

"Object-oriented activities are very useful for young children who are just getting a grasp of the concepts of number, addition, and subtraction. But kids get a grip on such abstractions quickly, and they rapidly outgrow the need to manipulate beads and lay rods end to end every time they are asked to add and subtract. Once children are ready to start doing math on paper, such activities are a tedious waste of time." (responding to the overuse of exercises that involve manipulating objects to teach arithmetic)

"Age is not an important factor; knowledge is." (responding to the myth that the brains of middle school students are too immature to learn algebra, and elementary students have limited ability to understand story problems because they have trouble understanding the meaning of addition or subtraction)

"Over the past two decades, cognitive psychology researchers have repeatedly shown that problem-solving methods and other higher order thinking skills can be used effectively only when a person has a large body of knowledge on which the thinking skills can operate." (responding to the big push to teach students general problem-solving methods)

Dr. Wickelgren points out that these general methods are not really beneficial because they are replacing math facts and specific problem solving skills.

"Creativity is an outgrowth of learning, and a lot of it. . . the more a person knows about a subject, the more creative he or she can be in it. . . A student's ability to be creative in any area of knowledge increases with his or her knowledge of that area. . . Thus, a desire to enhance creativity should not move a curriculum away from the math basics -- but closer to them." (responding to the idea that developing creativity frees a child to let his/her thoughts blossom and thus allows him/her to gain more knowledge)

. . . . . . . .

In response to a comment by Concerned, I need to make clear that Dr. Wickelgren's book is not only about identifying "new math" teaching. The book is divided into two parts.

Part 1 covers such topics as Setting Goals for Your Child, Evaluating Schools, Strategies for Excellence. In Part 2, he covers such topics as Teaching Tips for Parents, Basic Arithmetic, Basic Story Problems, Fractions, and Algebra.

Hangin' It All Together

2008-06-20T05:25:00.007-05:00

In a previous post, I referenced Dr. Wickelgren's comparison of the traditional approach and "new math's discovery" approach to teaching math concepts. In his book, Math Coach, A Parent's Guide to Helping Children Succeed in Math, he made the following observation about the traditional approach:

"The traditional approach, in which classes are arranged by topics such as arithmetic, fractions, algebra, and geometry that build from one level to the next, has been used for decades for good reason. The material within each subject hangs together in logical ways, and is typically broken down into smaller units within which knowledge is even more tightly linked. Teaching students to hang together closely related pieces of knowledge makes sense and produces a deep understanding of a subject."

I've been blessed to teach a very structured, incremental math program. Students who are used to struggling in math tell me over and over that they "get math" for the first time. Now, I see why. Closely related pieces of knowledge "hang together" easily and make sense to them. The slow, incremental steps mean that they practice and master the small pieces, one at a time. Then those pieces are carefully linked to other small pieces and the students see the logical relationship and how they are connected.

Contrast that to a sixth grade book I was using recently to tutor a student. All of the definitions and all of the formulas were thrown together in quick succession, so that my student was thoroughly overwhelmed and confused. (The students are given only that chapter to memorize them all, while trying to learn how to use them, all at the same time. ) It's madness!

And sadly there's not usually much practice of each small piece of knowledge, isolated from all of the other pieces. (Remember lack of practice is a characteristic of "new math.") Here's the line up: Teach one formula, and maybe do a little work. Then the next day, along comes the next formula, and then the next day, another formula. The students are soon doing problems of each formula, before they have mastered and are comfortable with the first one. This is NOT how kids learn.

There was nothing to "hang" the second , or the third, on. It was as if there were pieces floating around in the air and the student saw them all and had no idea which of them was related to what he was being expected to to next. There were no hooks to hang anything on because there had been no time for mastery.

Teachers probably feel that they don't have time for mastery because of the number of chapters they know they must cover. This book was huge!

And if smart kids have trouble with it all hitting them at once, imagine the weaker student's response. They are overwhelmed.

This is not how kids learn!!

Students must see how the pieces fit and hang together. And traditional math helps them do that.

Traditional math is wonderful! It's beautiful!! It's joyful! It's exciting!! It's liberating!! It's confidence-building! It's knowing I can succeed!!

It's like a kid who can finally ride a bicycle by himself!!

Traditional math helps kids do what they all want to do -- learn! I've never seen a kid who didn't want to learn. Learning new things is fun. Learning how to do something hard is even better!

Using "Guess and Check" rather than the reliable "long division" algorithm

2008-06-19T18:40:00.010-05:00

I came across an article I had discovered three years ago, written by C. Bradley Thompson on the Teach Math site. Thompson is currently a professor at Clemson University.

In this article entitled "Cognitive Child Abuse in Our Math Classrooms", Thompson discusses the cause of the dropping math scores in the United States. While discussing "whole-math", he makes these two observations about its proponents . . .

Advocates of 'new math' reject the idea that there are right and wrong answers and that . . . there are basic skills that students must be taught.

Advocates also believe that each student should invent his or her own math "strategies" by using the "guess and check" methods tauted by the "fuzzy math" supporters.

Here's the entire article.

In his descriptions of the activities in the "whole-math" (or "new math") classroom, you'll see students making piles of marshmallows to multiply, counting a million birdseeds in order to grasp the concept of "a million". And rather than have six-graders use multiplication or division facts to solve a problem, students are told that their strategies of "guess and check" are just as good as the logically proven principles of long division.

Would you want to be treated by a surgeon who learned his procedures by "guess and check"?

I'm amazed every day at the lack of grade school math being taught, when proficiency in the basic algorithms are desperately needed. It is these algorithms that form the foundation for higher forms of math knowledge.

. . . . . . . . .

Sadly not much has changed since this original article was published. School districts are sill adopting "fuzzy math" curriculum. Students are still struggling and math test scores are still low. Congress is still trying to decide how get our country's math scores up with the rest of the world. And the writers of curriculum are still resisting making needed changes because they refuse to acknowledge that the blame rests with their methods.

And any changes in Standards that thankfully are now being recommended will be slow in coming. It could be years before it trickles down to students. And in the meantime, another generation of students is seen their hopes of getting into engineering school dashed.

Perils of Discovery Learning, Part III: 'Interdisciplinary' Activities

2008-06-18T19:07:00.024-05:00

One of the big "buzz words" in mathematic circles today is "interdisciplinary" activities and projects.

If I'm teaching a unit in Science, students might do research in my class or the Library, write papers using skills taught in Writing class, and then generate the paper in the Computer Lab. Or if students are learning about Indian Villages in History, they might work in groups, or individually, to make a village in Art Class.

I often feel that teachers are evaluated (unofficially perhaps) on how much content crosses over into other disciplines, although no requirements have ever been made of me in my private school. It has been "suggested" that I find ways to involve other disciplines, but that's been the extent of it.

I have seen instances when I feel an interdisciplinary project has been very effective and where students are completely immersed. If the other discipline is a favorite of a student, if he loves art, or if she loves to write, he or she will really be engaged. I just don't like the pressure of forcing the project where it doesn't naturally go, where time is lost, all for the sake of "show". We can now brag at how many other disciplines were involved!!!

Dr. Wayne Wickelgren has made studies of interdisciplinary projects and I respect his opinion. In his book, Math Coach, A Parent's Guide to Helping Children Succeed in Math, he contrasts the traditional approach with the interdisciplinary approach:

"The traditional approach, in which classes are arranged by topics, such as arithmetic, fractions, algebra and geometry that build from one level to the next, has been used for decades for good reason. The material within each subject hangs together in logical ways, and is typically broken down into smaller units within which knowledge is even more tightly linked. Teaching students to hang together closely related pieces of knowledge makes sense and produces a deep understanding of a subject."

[That is just beautiful! And it makes such good sense.]

Hangs together in logical ways.

Now for Dr. Wickelgren's assessment of the interdisciplinary approach:

"Teaching across subject boundaries lacks depth. It may be fun for the students, but it doesn't help the mind organize the knowledge in a logical way, making it harder to remember."

There is also the likelihood that a teacher will overlook an important basic fact or principle that would usually be included in an incremental, structured approach.

I just have to say this one more time:
Hangs together in logical ways.

Perils of Discovery Learning, Part II

2008-06-18T17:19:00.006-05:00

Dr. Wickelgren, author of Math Coach, A parent's Guide to Helping Children Succeed in Math, explains that it is not possible for children to "discover" much of what is important for children to learn in elementary grades. How, he asks, can a student deduce, through discovery, the meaning of "3 to the 4th power" or that there are 5,280 feet in a mile. There are many things which must be taught by the teacher and then memorized by the student. Now you may avow that these are things that a student could conceivebly discover, so let Dr. Wickelgren describe how a typical "discovery" session works.

"Using the discovery method, students are given very little of the information necessary to solve problems. Solving problems thus requires gigantic leaps of intuition that virtuallly no students possess, so they flounder. A class typically spends half an hour -- and sometimes as long as two and a half hours -- on a single problem."

Because the "discovery" time is largely unsupervised by the teacher, it proves to be a very inefficient use of students' time.

Dr. Wickelgren tells of a student who "complained to me that his class spent an entire week on one problem before the teacher told the students how to solve it."

And remember that in addition to the inefficiency of this procedure, students are also asked to spend time discussing what to do, and then to spend more time writing about how they arrived at their solution. And what if their solution isn't even correct??!! Remember, the teacher is not involved, is not guiding students back on track when they wander off on a rabbit trail.

As a result of all of the wasted time, very few problems actually are solved by the students. Teachers are giving little or no information, and student proficiency takes a nose-dive.

Contrast this scenerio to the traditional classroom, where children are taught by direct instruction. Students are given most, not all, but most of the information needed for solutions. Smaller amount of insight is needed and thus some students, though not all, solve the problem quickly. Several of these problem-solving activities are given throughout the course of the lesson, affording students additional opportunities to use small bits of insight to solve other problems. (And a teacher steps in to help, giving additional information when needed, if the problem takes too long.) Many students gain confidence because of the successes they make each lesson. And they see the point in what they are doing. The goal is short ranged.

Solutions are found quickly, not hours later, and students do not lose interest because the problem's answer is so long in coming. I'd rather have students engaged for several small problems they have hopes of solving than disengaged for an hour because they see no point in what they are doing.

Perils of Discovery Learning, Part I

2008-06-18T16:30:00.005-05:00

I love to watch children learn. When people ask me why I like teaching, my answer is always the same: "I love to see that light bulb go off, to see that elation on a child's face when he 'gets it'". So there is a part of every math lesson I teach where I give students a chance to discover on their own.

But these are not huge, time-consuming tasks which involve groups of students. (I never figured out how you know which child is making the discovery when students are working in groups!) Instead, these are short moments built into every lesson where children begin with a known concept and are led to discover and figure out the new concept.

Later into the lesson, for the students who have failed to make that connection to the new, teacher intervention in a one-on-one basis with that student can pinpoint where the connection failed and the student has another opportunity to discover. I have many memories which I prize, memories of struggling children smiling with great joy. They got it. They figured it out without my actually telling them. I love these "light bulb" moments.

Dr. Wickelgren also acknowledges that students need these moments, but he also describes the perils of great units of time spent on "discovery learning projects".

The troubles of "fuzzy math" failures, which have created an uproar and lead to "math wars" around the country throughout the 90's and since the turn of the century are well documented. Parents have risen up to protest and to request changes in curriculum because of the failures of the programs to teach their children rigorous math.

Dr. Wickelgren describes the failures linked to Standards math (fuzzy math, new math) as follows:

"It dumbs down class content and lowers expectations for all kids. It doesn't adequately tests kids' knowledge. It wastes far too much time on activities that have little to do with math. And despite good intentions, it can actually decrease student participation.

"But the most important downfall of the approach is that it often results in only cursory knowledge of the nuts and bolts of math -- including basic aritimetic facts . . . and how to solve a variety of problems. This severely weakens the math curriculum because basic mathematical knowledge and problem-solving skill are the key ingredients of math proficiency. Mastering basic facts early is critical becase they form the basis for a huge amount of mathematics that follows. A child who doesn't know those facts by heart -- and how to use them in problems -- is at a serious disadvantage, even if he or she understands the concepts of addition, subtraction, multiplication, and division."

And this stunning statement from Dr. Wickelgren:

"The primary reason for the downfall: excessive reliance on student discovery of facts and principles instead of explicitly teaching them."

And another:

"Discovery sounds good on paper. In practice, it is time-consuming, inefficient, and results in little learning."

Discovery Learning; More from Math Coach

2008-06-18T15:04:00.010-05:00

Dr. Wayne Wickelgren, author of Math Coach; a Parent's Guide to Helping Children Succeed in Math, has much to say about "discovery learning" and the "new math", often called "fuzzy math" (and which he calls "Standards math").

Here are some of his insights:

"The most pervasive theme echoed throughout the Standards is their emphasis on student exploration and discovery. Instead of presenting information to the class, Standards math teachers ask their students to discover mathematical concepts while solving math problems. "Typically, students break up into small groups of four or so to solve a problem. The teacher circulates among the groups to observe the discussions but otherwise does not interfere with their learning by providing too much information."

"This framework is intended to let each child's natural creativity in math blossom, enabling children to discover important concepts and problem solving methods on their own. . . Teachers encourage students in the groups to do a lot of talking and writing about their thinking process that led to a solution . . ."

"To facilitate discussion, the groups are often working on math problems that are somewhat different from traditional computational or story problems. . . grounded in real-world situations. They are open-ended and contain many parts and many possible answers. . ."

"One example: A middle school math teacher demonstrates a pendulum made from a string and a weight and asks students to construct a pendulum, investigate how it functions, and formulate questions that arise."

"Traditional problems by contrast, have a single correct answer and focus on closely related mathematical ideas and facts."

Dr. Wickelgren explains that hands-on, real-world activities dominates "Standards" math classrooms. Students are never encouraged to memorize addition, subtraction, multiplication, and division facts, but rather to manipulate objects to find the answers, rather than using pencil and paper, repeating the activity each time a set of facts are presented.

"Memorization is regarded as dull for kids and also ineffective as a learning method, as it seems at odds with really understanding the material."

All teachers know the value of manipulating objects to teach the four basic operations, but for a child to have to repeatedly group objects to solve 8 X 6 is standard operating procedure in a "fuzzy math" classroom.

Dr. Wickelgren also describes the organization of the textbooks:

Rather than dividing a year by topics, such as two-digit multiplication, fractions, long division, and decimals, the year is organized into group projects which link to other subjects to create interdisciplinary studies. Rather look at how the textbook is organized:

"A year's worth of mathematics might look like this: A Wagon Train's Journey West, A Genetic Study of Fruit Fly Reproduction, Managing a Supermarket, A Month in the Life of a Real Estate Broker, A Voyage to Mars."

The "fuzzy math" crowd design these activities because they believe that prior to adolescence, children's minds can handle such "concrete" mathematical concepts, mentioned above, "but are not mature enough to handle abstract numbers and operations."

And so your child, being too immature to memorize facts, may be asked to design and create a pretty portfolio with attractive bindings to hold their writings about how they feel about math. All of this is during math class!

Dr. Wickelgren acknowledges such positive features of Standards math as "emphasis on understanding, solving more challenging problems, enriching math curriculum with more probability. . . early study of coordinate geometry". But he also warns of the perils of the "discovery learning", which will need to be discussed in a later post.

[Dr. Wickelgren also acknowledges that the Standards have been modified, but warns that the changes are slow to make their way down to the classroom, especially since there are so many who refuse to see the error in the "new math" approach. Curriculum writers take it very personally. Their "works" are "their babies". ]

More from Math Coach . . . Know the Warning Signs of Inferior Education

2008-06-17T19:15:00.010-05:00

Back to reading more from Math Coach, A Parent's Guide to Helping Children Succeed in Math, by Wayne Wickelgren.

Wickelgren, bemoaning the new math curriculum which had been introduced just before his daughter entered sixth grade, said he was able to identify the problem quickly because he knew the warning signs of inferior math education "...despite the claims of the teachers that this new program would be superior to the previous one."

At first, according to Wickelgren, he tried "to counter the new math agenda."

When that failed, he "implemented a plan to teach (his daughter) the math she would suddenly not be taught in school."

Wickelgren went on to say that their "stopgap measures prevented any long-term damage."

Here is one last paragraph written by Wickelgren on the subject of inferior math education and preventing its damage:

"Oddly, the methods that so frustrated (his daughter) and left (his son) far behind are part of the latest fashion in math education, one based on the 1989 Standards of the National Council of Teachers of Mathematics (NCTM) -- which I call Standards math. Virtually all parents of children going to school in the present era will encounter teachers and other educators influenced by Standards math. Thus, it is critical to understand these methods and how they might shortchange your child's math education."

Three things that I want parents to notice:

1. Wickelgren was able to quickly identify the problem because he knew what "Standards math" involved. This is where most of us as parents unintentionally drop the ball (I was one many years ago myself.) and this is when most of our children "get off the track." And sadly, we might not even know they are off the track because we are lulled by the teachers' promises of a better way coming.

2. He tried to counter the new math. (more on this later) I think this is what most of us do, if we are knowledgeable enough to be able to identify the problem at the start.

3. He identified what was not being taught and intervened. He implemented a plan which taught his daughter what she would fail to learn.

My first encounter with "new math" (but I didn't know it):

Our own son, coming through high school in the late 80's and early 90's, was very smart in math, and was on a fast track. He started geometry, and quickly told me "Mom, I'm not enjoying math at all."

I looked through his geometry textbook. Geometry had been about my favorite math, and I was incredulous at what I saw, or more specifically, what I didn't see. This math did not even resemble what I expected! Students were given problems to solve but had not been taught theorems nor anything about using theorems in the proofs. Folks, that's what plane geometry is -- using/listing theorems needed to solve the problem and then (proofs) why that theorem helps get the answer. The theorems and proofs take you step by step through the problem until you get the desired solution.

I tried to explain theorems and proofs to our son and how they worked, (I guess I was at step 2 above -- trying to counter what he was learning -- but I made the mistake of using the school's textbook and I just couldn't help him. I knew our son was terribly off track. It was week two and damage had already been done. His self confidence was zero.

So I had identified a problem, although I knew nothing of "new math". And after trying to help, I had been pretty quick to identify what was NOT being taught (step 3), but my mistake was trying to counter what was being done by using his textbook as it was being taught. It just didn't work at that point because there were gaps and I couldn't fill in the gaps using that "new math" system.

And so from my experience, I will agree with Wickelgren that trying to counter what a child is learning probably won't work by following what's in the "new math" book. I needed to start all over again and teach him plane geometry the "traditional" way.

(Let me call your attention to this: It is typical of "new math" curriculum to require students to solve problems before they are exposed to the information and concepts they will need -- to give them the chance to "discover" and to build their own methods and plans. All it did was upset and confuse my smart son.)

I called the teacher, who was very nice and assured me that our son was not alone, that it was typical for students to be confused at this point. She also asked that I give them a few more days to get through the chapter and then it would get better. It did get better, thank goodness, and we had our old son back! Remember, he was a good math student. I wonder now what happened to students for whom math didn't come easily.

I'm convinced in hindsight, that this was one of the very first "fuzzy math" textbooks. However, I had never heard of that, and it never entered my mind that anyone would be so foolish as to skew the math I had loved so much into a twisted bunch of disconnected ideas.

It is my desire to help parents quickly identify bad math teaching and implement a plan to correct it and fill in the gaps. No matter how old your child is, it's not too late. It's doable.

Helping Children Succeed

2008-06-17T12:51:00.009-05:00

I'm reading Math Coach, A Parent's Guide to Helping Children Succeed in Math, by Wayne A. Wickelgren, Ph.D.

This is a wonderful book, directed at parents, on how to help children who need more instruction, who need to get caught up, or who need to be challenged.

He says it's not too late, even if they are in 5th or 6th, to catch them up by helping them at home in time for Algebra in 8th grade.

Here are some quotes that are especially appropriate for our intermediate grades:

"Most children can learn math much faster than the school system prescribes. That means that even if your child is currently behind, there is plenty of wiggle room for catching up, especially if you start early. Math education proceeds slowly in school before eighth or ninth grade, so those are the ideal years to improve your child's mastery of math or speed him or her ahead through the curriculum (though starting later is okay too)."

" . . .{by taking prescribed steps}, your efforts will change your child forever. Investing time in your child's education -- whether it's tutoring, consulting with educators, or helping your child set goals -- will show your child that you think education is important. Even when your child complains about doing extra schoolwork, rest assured that he or she is lerning, however, unconsciously, to value learning. It is a value your child will carry throughout life and, in all likelihood, into the next generation."

". . . even if your child is behind in math now, you can get him or her back on track with only moderate effort. If your child is not behind, you can propel him or her substantially ahead in math."

"It is the math prior to algebra that is taught especially slowly in the U.S. Thus, the pre-algebra years are the ones in which you can most easily speed your child ahead, or help him or her catch up in math, giving the child a chance to learn as much math as possible."

Wickelgren has some fascinating studies that show how soon children can grasp concepts of adding and subtracting from a group. And there is great information here on when children can begin sounding out words, although that is a different subject.

He discusses age-related limits to math learning (I was surprised to find that there are very few.) and biological barriers which are connected to processing math, and (one that I've always heard thrown out here) maturational limits.

Here's a great quote from Wickelgren, related to that last statement:

"Of course your child may have temporary trouble understanding certain math concepts, but the reason for that is unlikely to be a maturation issue. It could be lack of verbal skills."

". . . it is the verbal encoding of addition and subtraction that is often lacking in young children and not the nonverbal understanding of these concepts."

Did you get that?

"NOT THE NONVERBAL UNDERSTANDING OF THESE CONCEPTS."

Children can understand math if we find another way of showing students what the digits and symbols actually mean.

I'm loving this book!!

Real World Math

2008-06-16T17:06:00.007-05:00

There is as great thread discussing "Real World Math" on Kitchen Table Math. If you are confused as to how "real world" activities can be detrimental, see the comments posted by Barry Garelick and subsequent discussion.

Intervention, Part 2

2008-06-16T14:49:00.007-05:00

Should you decide to "intervene" on behalf of your student, do not be surprised to find that some people in your school may not be happy about it.

Schools often design their own plans for intervention. There may be a distinct intervention "order" which you and your student should/must follow: You start at point A (probably your student's math teacher); if that isn't helpful, proceed to point B (another math teacher), etc.; then on up the ladder to perhaps the head of the math department. This is how the school has decided you must do it.

I know many teachers who gladly help students before school, after school. To them, the goal is student understanding and they are willing to help.

And don't misunderstand; I commend schools for becoming involved in the tutoring process to help students.

But let's consider that your student's math teacher is not competent in teaching new concepts or in identifying gaps in reasoning. What happens if your student goes to this teacher for tutoring (because that is where you must start)? This teacher will use the same approach used in class. What didn't work in class will be tried during the tutoring and your student will probably not get much benefit.

And how does that make your student feel? I'll tell you (but you know). He feels no better, maybe worse! Now he sees that even with extra help, he still doesn't get it!! He may start saying that he is "dumb".

So you desperately attempt to find alternative support and help. Woe be to you, if the school Math Department finds that you have, on your own, asked another teacher to help your child. You could be soundly scolded for not taking advantage of the proper recommended "line" of help.

I know because I was once an "alternative" choice by parents. They had tried the school's plan for after school tutoring by the student's math teacher. The student continued to fail tests, and the parents were desperate for some help. (He had gone from A's to F's in less than a year.) They pleaded with me to help. When I agreed, the parents asked for an extra book for me, but were chastised sharply for not following the "school's plan".

"You should have gone to . . .", the parents were told, and the plan of choice was explained.

This was so baffling to the parents. They just wanted help for their student.

Wouldn't you think that schools would appreciate a parent's desire to get help for a child? Well, schools may tell you that they appreciate it, but you may find out that they are grateful only if the school's "plan" is followed.

(By the way, I did tutor the above-mentioned student, though only for a couple of weeks, and I was able to help him. He passed his first test of the entire year (and this was April) after only two tutoring sessions.) The most important thing, however, according to his parents, was his self image. He knew he "got it", he understood it, and his confidence soared. This student wasn't used to failure. He needed to feel successful again.

All kids want to feel successful. Success give confidence. If your student needs intervention, take the steps. Go to battle, if necessary, for your kids.

Intervention

2008-06-16T14:24:00.006-05:00

Perhaps you have identified a few concepts that your student hasn't nailed down (maybe it's learning simple facts, or perhaps it's more complex like long division or multiplying fractions) and you have decided it's time for you to step up and try to do something. Where do you go? What do you do? Is it something you can do yourself? What if the school tells you that in a few months they will cycle back through and that perhaps your child will get it then. They tell you not to worry. Should you even try?

Yes, please try. You must intervene and the longer you wait, the harder it will be. There are so many web sites to help now, and so many families who have gone before you. Help is all around and I urge you to start with sites and blogs, such as Kitchen Table Math so you can have support and help. Locate a few websites, if you need a little review, or even hard copies of helpful books, such as Math on Call, which will give you procedures and confidence.

Be willing to try. Then, tell your child that you know it's been hard but that "we" are going to do something now to help.

Decide on an amount of time. If it's a tutor, the time will be decided by the tutor, or by your budget. (See below on suggestions for tutors.) If it's your time, set aside a certain amount each day. Explain to your child the time requirements so that it will be clear. Stick to it.

Decide on a short range goal, one concept, that you can work on, and go for just that. Do everything to ensure that your child experiences immediate success of some sort, even if it's something small, the very first day. Success builds confidence. I assure you your child wants to feel and be successful. So make sure that your student sees that he/she has made progress that first session.

Be willing to make this a long-term commitment, if it is needed. I mean for a year, or two, maybe more. Catherine Johnson, from Kitchen Table Math worked with her child, using an entirely different textbook in the afternoons, in addition to the school assignment. If you're looking for how much she did a week in the other textbook, I'm sure she'll help and encourage you.

Your child is looking to you for help. You can do it. At least try. Many have and you will be able to. Whether you do it yourself, or whether you support a tutor. Do it.

Many times I intervened, short term, for our own children when they needed a little help. I always looked over all of their work. I did that early on in primary grades and they got used to it and accepted it. If I saw trouble, I made up some problems right then and there on the spot and watched them work. I could tell by watching where they got off the track. Once the problem areas were identified and addressed, we worked a little more, maybe several nights. I usually had one rule: show me that you can do 5 problems correctly without any mistakes, and we will stop. What does that do? It makes them focus on all the little things you've identified and it makes them work carefully. A few days was usually all it took.

Now for tutoring: Any time I've agreed to tutor a student, I explain to the parents that I want more than one session a week. I will agree to tutor only do it if I can have a minimum of 2 times a week, but 3 is even better. Why? Well, usually, I find that some of the trouble is procedural and simple carelessness. And what good can I do if a student has one hour with me to work on correct procedures and then the rest of the week he reverts back to incorrect methods? It is said that a person has to do something about 34 times for it to become a habit. So practice becomes very important, but it takes not just practice, but the correct kind of practice, and consistency. The work must be done properly every time. So keep this in mind, whether you are the tutor, or you are considering hiring someone. I usually find that after about 6 weeks, students may not even need me anymore because they have corrected the bad habits and see the value of working with care.

Pencil and Paper; the Benefits

2008-06-15T17:16:00.013-05:00

One of the usual identifiers of a "new math" program/teacher is that there is very little pencil work. Fuzzy math people have been all over the map on this idea, straddling the line, trying to appear neutral, denying that they are against pencil and paper work, denying that they are opposed to learning and drilling on math facts. Now, some are making a turn around and publicly say they do not oppose use of pen and paper practice, but look closely and they provide little to none.

Now, I must explain that "new math" proponents are not opposed to pencil and paper work. They assign students work involving lists after lists after lists and all sorts of writing assignments. It's pervasive in every chapter throughout the books they love. But that use of pencil and paper is OK, because the "new math" teachers are doing it. Not only is it OK, it's desired! It's just not OK if I do it in my traditional classroom, using traditional books and traditional methods.

It's been several years now, but I took a little class called "Teaching in Texas Today" offered by a local school district. It was designed for people who were teachers but who had been out of the classroom 10 years or more or for people who were interested in becoming certified. It sort of brought you up on things that were new in education and each week we had a different instructor. One instructor showed us new ways to be sure that students were learning new concepts. (She used math as her example) Students were all given cards to raise in the air to give their answers. Teachers could easily identify the students who had the correct answer by looking at the card the child held up. No pen or pencil work was required. Students were encouraged to do the work mentally. And I'll never forget her saying, "Once you see that the student has done several correctly, you don't need to burden them by requiring that they do any more." (I'm not opposed to mental math -- I love it, my children love it and some even thrive on it. I use it every day, but not at the expense of proper pencil and paper practice.)

Soon afterward, I returned to teaching, and I've thought of the instructor's words many times, primarily, when I see how quickly some students forget newly taught material. Even with practice and practice, there are those students who will still struggle. All it takes is a 2-3 day sick spell, and they're lost. (I'm not talking about that student who can think it through, figure it out mentally in 10 seconds. If that is your student, you are not reading this blog. I'm talking about the students who can't figure it out instantly, but who need to be given a method to use to get the answer until they can figure it out by themselves.)

There is something about the "doing of the problem", about the "using of the pen or pencil" that helps children learn. I don't meant that "helps them 'remember'". We've all known that for a long time. But now we know that "the doing" actually helps children "learn". We now know that there are some children who learn by doing, by moving, by writing, by some sort of physical motion. And for them, it is in the actual doing of the work that they "get it". Using and moving concrete items (blocks, counting sticks, pennies) to solve problem is a must for these kids. Walking, physically moving, through the steps in the problem is also important for these children. But along with these actions, we know that there are those children, for whom it it a terrible mistake to restrict the pencil and paper. We rob them of their way of learning. For these students, there is a need for clear step by step procedures to follow and practice, practice, until they figure it out because this is the way they figure it out.

Oh, I forgot! I'm not supposed to be teaching them procedures or methods. We should be allowing them a chance to figure it out and discover their own methods of getting the right answer. You, the parent, may have been given such an answer when you've explained that your student doesn't understand what's being taught. You may have even been told NOT to teach them your old methods because children need to figure it out. Never mind that it might take them months, and that the rest of the class has long gone on to something else. So you're told not to help yor student and your student may even be publicly embarrassed in class if you do. So you are told NOT to intervene. Teachers are told not to intervene.

But I can't do that to your student. And as a parent, in your heart of hearts, you don't want to do that either because you see what it does to their confidence. You see the frustration and the discouragement and you want to help, NOW, not next semester, or next year.

And I do not agree that only if a person is mentally thinking on his own, deciding on his own methods to try to use solve a problem, that only then will he be engaged. Students who are going carefully through a step-by-step method can be very "engaged" because that is the method that gets their brain going. You will see them whispering, talking through the steps, and that is good. Yes, they can be very engaged. And they are learning.

(I'm going to cut my original post here and post the remainder on a new page, "Intervention".)

Is Your Student Being Taught "Fuzzy" Math?

2008-06-13T14:26:00.012-05:00

Parents may wonder if their student's math curriculum is "fuzzy" math. Here are some general identifying marks of "fuzzy" math:

1. Very little practice (You might have someone tell you that new studies show that too much drill kills interest in math. That is not true -- it makes students efficient and confident.) Lack of pencil and paper work -- in the "fuzzies' minds it's bad to do a lot of pencil and paper work.
2. Individual problems on the homework are not actually graded. Students are given credit if they show that they "tried" to do it. It doesn't even matter if they successfully solved the problem.
3. Calculators are used to do what children should be learning to do mentally. Some people may even be so bold as to claim that there are students who will never learn all of these math facts anyway. Calculators are said to be sufficient.
4. (Almost exclusively) student-centered activities. Students work in groups to figure out something, and the teacher may not even be involved at all.
5. There will be little or no instructions, certainly no explicit instructions. Activities are student directed and teacher input is lacking. Students are sent home to figure it out by themselves and the next day the class is asked by the teacher if there were any questions.
6. "Real world math" is what it's all about. It's one of the buzz words. After all, who doesn't want students prepared for the "real world".
7. Students are told to write paragraphs explaining all of the steps that are used to solve the problem. Or students are expected to write about how they feel about math. Or as Walter Willaims referenced, students write on such topics as "If Math were a color, what what that be?" In other words, students are writing their math. So much for the benefit of symbols and digits to help solve a problem efficiently, in the least amount of time.
7. Math books are huge, thick books, and pages are filled with visual "clutter" which distract students' thoughts. (For some "adhd" or "add" students, these pages can be a disaster. It is no wonder it takes them 50 minutes to do a 15 minute assignment.) There is a lot of "non-math" content such as photographs in color and motivational stories which are meant to "inspire" kids to greatness, I suppose, but which have no place in the middle of a math assignment.
8. Patterns, patterns, patterns. Looking for patterns, drawing patterns. Probability will be prevalent by 4th or 5th grade. And much time spent on data analysis projects (finding the mode, the mean, the median) starting in elementary school. What is appropriate for a high school course is inappropriate for elementary chldren. The time spent on one "projects" may be days and will indeed use lots of pencil and paper. Notice here: It's OK for them to do pencil and paper work. It's just not OK for me or you to do pencil and paper work on traditional math practice.
9. And here's one dead give-away that the curriculum takes a "fuzzy" math approach:
Students are given problems in one lesson, for which they are not at all prepared. Students spend much, much time on these problems, only to discover that the concept is taught a lesson or two later So if your student has no idea what is being asked, look ahead and you'll probably find that the concept is coming up. This is called "discovery learning" because students have a wonderful opportunity to "discover" the concept on their own. Imagine your 4th grader being forced to "discover" how to do long division with no input from anyone else!!

I'm sure there are more I've overlooked. One more thing that is obvious: Your student is (perhaps suddenly) discouraged, thinks he/she is not smart, gives up trying. I know one student who had been in my 5th grade class and was an excellent and a very diligent student. What didn't come naturally, she learned by shear determination and perseverance. How sad I was to hear that she was threatening to kill herself because her high school math was so hard for her and she didn't think she was ever going to get it. Her teacher's approach was to assign the algebra/geometry lesson, forcing the students to teach themselves, and then ask if any of the stusdents had any questions or problems. If there were no questions, they proceeded on to the next lesson.)

So, do any of these ring a bell to you? Well, if so, you must rescue your student as fast as possible. Go to Kitchen Table Math for insights how a family rescued their own student. It took lots of hours after school, but the results made it worth it.

Some Math Stories, Same Troubles

2008-06-13T12:02:00.007-05:00

A year or so after I had first discovered that not all math was the traditional math I had been taught, it began to hit close to home. I was celebrating Thanksgiving with my husband's family and sadly not all family members were there. In talking to my brother-in-law about my new found information on "fuzzy" math and "math wars", I was told, "Well, it's interesting that you should mention that." He then told me that was why his daughter and her husband were not with our family. They had used that extended weekend to visit her husband's family 200+ miles away because his young sister was suddenly having a lot of trouble with math and they were going to see if they could help her. And, he continued, this young sister was really smart and had always made really good grades in math and scored well on tests and the parents couldn't figure out what was wrong. How I wished I had been able to help this young girl or at least talk to her family. I didn't really know her, but I felt empathy for her and her family. And this is what often happens -- the whole family becomes involved trying to help, but because terminology is different, because parents are told NOT to try to help students by using the parents' old, out-of-date methods (because you're going to confuse them), parent involvement is often delayed until much damage has been done. And another thing, the student get's "down" on themselves and start telling themselves "I'm dumb".

In a second "family" story, I was able to talk to parents, but not until the students were in college. The mother was bemoaning the fact that they didn't know what they were going to do with their boys, my nephews, because they both were having so much trouble with college algebra. I knew they had had trouble in high school. It had been a source of bits of discussion off and on as their troubles were mentioned, as tutors had been hired, etc. We all caught on that, but for the tutors, those boys might not have made it through high school algebra. But now here they were in college, still struggling with algebra. I told the mom, "Well, it may not be your boys fault. It may be that the boys were taught "fuzzy" math." She went on to describe how they themselves had tried to help them during middle and high school, but the math didn't make sense and so they had to use tutors. She didn't know she was describing a situation that was repeating itself all across the country.

Has anyone noticed the number of Sylvan, Kumon, Huntington, etc., Math Centers popping up all around the countries. Parents used to be able to help children, prior to the first "new math" intrusions of the 60's and 70's. My own father helped me occasionally, but it didn't take much because the base/rate/percentage problems, or the algebra were the same as he had learned. My 7th and 8th grade "math" teacher was really an "art" teacher who had been asked to teach math for a time and unfortunately for me, it was during my time. No one in the class was getting it, including me, and I had to go home every evening and have my dad help me until I got it. Then, before school the next morning, we students would all meet together to go over math and I'd reteach the lesson, as my dad had helped me. We all made it through middle school math because of my dad. That was before the days of "tutors". Family could help family then. But not any more. At least, it's not easy to do now. Thankfully there are parents who will not give up on their kids, who try to find help when they don't know what to do. But my dad got me ready for algebra and I never had a bit of trouble with algebra (I'm sure it was "traditional algebra").

There are several variables: is it the math curriculum? is it the math teacher? is it the student? It could be more than one. If your student has a young math teacher, chances are he/she came through college having been taught to use the "new math" approach (Don't teach the students methods or algorithms! Make them figure it out by themselves.). But it could also be the curriculum and I'm going to add a post shortly with some identifying marks of "fuzzy" math so you can determine if that is the type of curriculum your student is using.

How Can It Happen?

2008-06-11T16:54:00.009-05:00

A few years back I had the blessed privilege of teaching one boy in particular. Now, I have to say that I could say that about every student I teach, but this is a particularly sweet story for me to remember. If you knew his family history, you'd understand why I say this. I was told ahead of time what his limitations would be and what troubles to expect. (And indeed those were descriptions of what his 4th grade year had been like after he arrived.) I will tell you now, before I relate some of his story, that he was one of the smartest students I've ever taught.

As the first few days went by, I noticed that none of the anticipated problems were presenting themselves. None, at all! His grandmother/guardian asked me how he was doing, and I remember my answer to her. "He is knocking himself out to please me. He is working so hard, not complaining one bit. Nothing is difficult for him, he's really smart!" And it was true. I'm still convinced he was brilliant. His young life may have had a hard beginning, but as his confidence grew, his self-image grew and he just blossomed that year.

About 3 weeks into the year, we had our first Math Test. The next day, as I returned the tests for the students to look at, his response was one I'll never forget. As I gave him his paper, upon seeing the grade (95), he said out loud, "I've never made an A on a Math Test before!" That was the first of many "A's". In fact he made an A on his report card every reporting period throughout the year. Math wasn't hard for him, he didn't struggle to "get" a single concept, and he wavered only once, when he had been sick (and even then a little intensive input from me was all it took). He was so confident as he worked, and his family and I watched him grow and bloom even more. He left my classroom a different child, with his head held high. He knew he was smart. (I told him I thought he was brilliant!)

So, fastforward one year. Imagine my shock and surprise when I discovered that he was failing Math the following year. He was the same student, with the same brain, same brilliant reasoning abilities. So what happened?? I only have some ideas because I wasn't there with him each day, but I knew enough to believe that something was wrong. Terribly wrong. And I wished I had known earlier. I wished I could have done something to help.

As I enter this post, I am thinking that perhaps this might be happening in your family, to one or more of your children. I read the stories on other Math blogs of students who suddenly seem to fall apart in Math. I read the words of college math professors who bemoan the fact that students are ill prepared for college. (I do want to write more about this in later blogs.) I believe that if there is a predictor of which students will struggle with higher maths, it would be how well they transition into the abstracts and unknowns of pre- pre-algebra in 5th and 6th grade. (Yes, I mean pre- pre-algebra.)

On another matter, and in closing this post, I want to state that all students need to see their tests and have an opportunity to discuss the tests, within a day or two. And I believe that parents should be able to see the tests. I'm always baffled when I hear that a teacher won't let a test out of the room. How can a school say, "we want you, the parent, to help us, to support us with your child" but then not allow the parent to see a test!!?? This makes no sense to me. I'm a teacher, but I'm a parent also. I'm the one who might/will/can be helping him/her prepare for the next Math test. Do I want him to still be "unclear" on why he missed a problem? (I tutored a student this past spring semester, one who was failing math, one who had not passed a single Math test the entire year. I asked the parent if they had any tests I could look at, and they had never seen a test because students were not allowed to bring them home. The dad was trying to help him and reviewed with him each evening. But they were at a loss because they didn't know where they or he had failed.)

So what do you do when tests aren't coming home and you, the parent, are not seeing them. Here are a couple of suggestions. First go to the teacher, respectfully, and ask if the tests are being returned. (Students are notorious for "losing" papers.) If you discover that the teacher doesn't allow them to be taken home, ask about how much time a student has to study over the mistakes on that test. And then ask if the teacher would consider changing the policy of keeping tests. (Anticipate that you will probably be told something about needing to use the test the next year and that siblings coming up could have an advantage. If you get this excuse, respectfully tell them that you understand and that you don't want to compromise the test at all.) Sometimes a teacher will reluctantly, quietly, give you or your student the test. If they do, know you are blessed. But what if you try and still get no response? . . .

There is one last step that I used once when my son was in middle school. After having a very pleasant phone conversation with the teacher during which he explained why he didn't return tests, I felt that I still needed to see the test, if only for a brief while. So I calmly called the principal, and asked about the school policy. He explained that some teachers had that policy and others didn't, but it wasn't a school policy. I explained to him that I wanted to respect the teacher's desire to keep the test confidential, and so I requested that he, the principal, get the test and allow me to come up into his (the principal's) office to look over it in his presence. Guess what! It worked. I got to look over the test, I saw how my son had studied improperly, and got a glimpse at what the test looked like. And better than that -- I didn't make the teacher or the principal angry. I hope this works for other families.

Look, it's all about trying to help a student.

There are many reasons why a student's grades can drop, but if the parent can eliminate one of them, a parent should be and usually is willing to do it. The test will be used by the teacher to determine a grade. Hopefully, it is also used to help determine teaching strategies which can improve students' understanding of concepts. In other words, the test can be a tool to help the teacher learn better methods for teaching concepts. And if teachers can learn anything from the test, why not allow the student to learn from the test? Or the parent? Wow, that would be a new approach -- teachers and parents and students all benefiting from the test!

As I said, it's all about trying to help a kid!

Getting Started

2008-06-11T09:45:00.007-05:00

First things first. Let's get this out in the open. I'm new at this -- blogging that is.

But, I'm not new to teaching. And I'm not new to teaching Math. And I'm not new to children being frustrated and scared of Math. But I can tell right now, that it's going to be very different trying to help them by way of "keyboarding" rather than by "words, chalkboard, paper, counting sticks, pie pieces, manipulatives, etc." And yet I will be doing just that. At least I'm going to try to do just that.

Second, I know that I can help children lose their fear of Math in just a couple of days, maybe two weeks at most, if given a chance. The challenge will not be to help them. The challenge will be to take what I've learned in ways of helping and transfer that to the keyboard so that it comes out to "you out there." But I'm not afraid to try this unknown because I've seen fear and tears in children, become smiles, even exhilaration and exuberance, and helping them is in my bones! It may take some trial and error, but I'm willing to try. So come on in, be willing to grab a handful of pennies, or craft sticks, or whatever you may have handy and let's help kids.

Third, I'm going to aim this blog at the intermediate math student primarily because that's what I teach and that's where I see math and kids together. I'm a fifth grade teacher presently, although I've taught primary grades and tutored many middle school students in math. If your student is in 4-6 grades and struggling with math procedures or concepts, that's who I'm looking to help.

I pledge you my best as I try to balance teaching during the daytime and entering the blog world in the evenings. Be patient as I try to get this blog started and sort of get my balance. I feel, right now at least, that I'm looking at a huge, huge map and I'm trying to find a few grid lines so I can get my bearings. But the thought just came to me -- that's just what children are doing today, and if I can help them find their own "longitude and latitude lines", I know that will make them smile and that will make my day.

Walter Williams on 'Fuzzy Math'

2008-06-01T06:58:00.006-05:00

American education will never be improved until we address one of the problems seen as too delicate to discuss. That problem is the overall quality of people teaching our children. Students who have chosen education as their major have the lowest SAT scores of any other major. Students who have graduated with an education degree earn lower scores than any other major on graduate school admissions tests such as the GRE, MCAT or LSAT. Schools of education, either graduate or undergraduate, represent the academic slums of most any university. As such, they are home to the least able students and professors with the lowest academic respect. Were we serious about efforts to improve public education, one of the first things we would do is eliminate schools of education.

The inability to think critically makes educationists fall easy prey to harebrained schemes, and what's worse, they don't have the intelligence to recognize that the harebrained scheme isn't working. Just one of many examples is the use of fuzzy math teaching techniques found in "Rethinking Mathematics: Teaching Social Justice by the Numbers." Among its topics: "Sweatshop Accounting," "Chicanos Have Math in Their Blood," "Multicultural Math" and "Home Buying While Brown or Black." The latter contains discussions on racial profiling, the war in Iraq, corporate control of the media and environmental racism.

If you have a fifth-grader, his textbook might be "Everyday Math." Among its study questions are: If math were a color, it would be (blank) because (blank). If it were a food, it would be (blank) because (blank). If it were weather, it would be (blank) because (blank). All of this is sheer nonsense, and what's worse is that the National Council of Teachers of Mathematics sponsors and supports much of this nonsense.

Mathematics, more than any other subject, is culturally neutral. The square root of 16 is 4 whether you're Asian, European or African, or even Plutonian or Martian. While math and science literacy among white 15-year-olds is nothing to write home about, that among black 15-year-olds is nothing less than a disaster.

Few people appreciate the implications of poor math preparation. Mathematics, more than anything else, teaches one how to think logically. As such, it is an important intellectual tool. If one graduates from high school with little or no preparation in algebra, geometry and a bit of trigonometry, he is likely to find whole areas of academic study, as well as the highest paying jobs, hermetically sealed off from him for his entire life.

(An excerpt from 'Academic Slums', by Walter E. Williams, 17 Dec. 2007)