Abstract Learning and Fractions

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

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?

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

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 . . .

"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!

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

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

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?

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

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"

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

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

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

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?

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?

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

(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"

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

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

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.