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.

"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?

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.

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.

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.

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

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 II

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

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

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

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.

Pencil and Paper; the Benefits

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".)

Some Math Stories, Same Troubles

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.

Walter Williams on 'Fuzzy Math'

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)