The beautiful simplicity of saying what you believe to be true

Filling the pail

Deans for Impact have released a blog post that is intended help learning scientists influence education. As with many publications by Deans for Impact, it seems to have been well received. However, I always react badly to pieces that position the authors as the adults in the room and my problem with this article runs deeper still. I am going to have to dissent.

I am, of course, sympathetic to the aim of convincing teachers that learning styles are a myth. However, I feel much the same about the Deans for Impact post as I do about this piece, the aims of which I certainly do not share. So what is my issue?

I don’t believe that proponents of evidence-based education should be in the business of consciously trying to spin concepts in order to manipulate their readers. At a basic level, it is wrong because it treats the intended…

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Education Shock: Actually Teaching Students Math is Effective

This 2014 story published in the Atlanta Journal Constitution reports the shocking news that teaching first-grade students math using the dreaded worksheets, and traditional modes of education was more effective than ” group work, peer tutoring or hands-on activities that use manipulations, calculators, movement and music.”

According to Maureen Downey in her article, “This is an important issue as I increasingly see schools – including those my children attend – tout group learning activities. In many classrooms now, you will see students working at tables together on math. A friend who teaches in a Title 1 school lamented that her students didn’t do as well in the math CRCT as the classroom next door where the teacher used worksheets all the time. My friend’s classroom was a beehive of fun activities around math, but the worksheet class continually outperformed hers. These new findings help us understand why that might have been.”

What I find interesting is the conclusion that the direct, traditional instruction benefits those students with math difficulties, implying that those students without math difficulties do just fine with student-centered approaches. The possibility that difficulties with math may be a result of the student-centered approaches is something that is not discussed even though the study by Paul Morgan and George Farkas of Pennsylvania State University indicates that “a higher percentage of [students with mathematics difficulties, or MD] in the first-grade classrooms were associated with greater use by teachers of manipulatives/calculators and movement/music to teach mathematics. Yet follow-up analysis for each of the MD and non-MD groups indicated that only teacher-directed instruction was significantly associated with the achievement of students with MD (covariate-adjusted effect sizes [ESs] = .05–.07).”

I recall in an article I wrote called “Being Outwitted by Stupidity” I suggested that the increase in students being diagnosed  with learning difficulties in math raises the question of whether the shift in instructional emphasis over the past several decades has increased the number of low achieving children. I also question whether the learning difficulties came about because of poor or ineffective instruction and whether such students would have swum with the rest of the pack in previous eras when traditional math teaching prevailed. I stated “I believe that what is offered as treatment for learning disabilities in mathematics is what we could have done—and need to be doing—in the first place.”

This article garnered about 80 comments, many of them hostile, including my all time favorite which named me a “conservative simpleton fraud”.

I continue to maintain that many of the difficulties we see students having in math may be attributed to insufficient and ineffective instruction. To put it as simply as I can, they may not be learning math because they aren’t being taught math.  But the Morgan/Farkas study is being interpreted in the usual manner: “Teacher-directed instruction is also linked to gains in children without a history of math trouble. But unlike their math-challenged counterparts, they can benefit from some types of student-centered instruction as well – such as working on problems with several solutions, peer tutoring, and activities involving real-life math.”

Not mentioned is whether and to what extent such students receive additional help in the form of parents at home, tutoring, or learning centers.  We might have to wait a while for that kind of study to surface.


Some comments from our readers

The previous post (Articles I wish I never finished reading, Dept) stimulated several insightful and useful comments which I am reproducing here. The first, from Greg Ashman, a PhD candidate in education who resides in Australia, and is outspoken on the issue of the ineffectiveness of constructivist teaching strategies, commented at the article itself.  He writes:

A few points to note:

  1. Finland is declining in its PISA maths performance and has been doing so since 2006. If we therefore wish to look to Finland as a model then we probably need to examine what it was doing prior to 2006. If we do so then we see it was pretty traditional:  Reference. 
  2. Teaching for conceptual understanding is actually quite a strong theme in American maths education and not something that has been ignored. A comparative study of the attitudes of mathematics teachers in different countries found that American teachers prioritised the idea of understanding-first more than teachers in East Asia. East Asian teachers still thought understanding was important but were relaxed about whether it came before or after a knowledge of procedures (East Asian countries do quite well in international tests): Reference
  3. When teachers prioritise understanding-first then they are often drawn to ‘constructivist’ pedagogies where students are asked to design their own strategies for solving problems rather than learn canonical ones. These pedagogies are ineffective:  Reference. 
  4. Canonical procedures have advantages over student invented procedures, particularly for more complex work:  Reference  
  5. Jon Star has done a great deal of work on the fact that what we often call ‘procedural knowledge’ actually contains a great deal of conceptual understanding. Procedural and conceptual knowledge are bidirectional and iterative:  Reference 

Far from a challenge to the prevailing orthodoxy, this opinion piece is a good expression of it. It is thus part of the problem.

And one of our readers, Chester Draws, has this to say about the belief (or poster child of the progressivists/math reformers) that the distributive rule leads students to a lack of understanding in solving certain algebra problems. Specifically, he is referring to this part of the previous post: “When they see an equation such as 3(x+5)=30, they will distribute rather than divide both sides by 3 to get the simpler equation of x + 5 = 10. The author claims ‘but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.’ ”

Chester comments:

But both procedures are not equally useful.

If you have a standard 14 year old and you give them 7(x + 5) = 30, then they are going to stuff it up if they don’t distribute first. x + 5 = 30/7 is not something you want them to be attempting. Whereas 7x + 35 = 30 produces no such problems, despite yielding the same fractional negative.

The divide first method is usually more difficult and much more prone to error. Why would you even want them to know about it? I have nothing but derision for teachers that show students methods that aren’t universally applicable so they have “choice”. They don’t need or want choice, they need and want to get the answers correct with a reliable method. The time lost teaching a trivially useful technique would be much better spent getting the ones that they do need properly organised in their heads.

I teach all my students that normally the first thing you do in any algebra solving problem is get rid of fractions and brackets. Then you can see what you have. They then have their minds freed of what to do first — remove brackets and fractions — and that leaves more brain power for the hard bits.

Finally, we have SteveH, addressing this same issue:

In a traditional approach to algebra, you learn that there is no one way to solve anything, even though pedagogues really push ideas of order of operations. Learning this is not an understanding issue. It’s a practice issue, where mastery of problem sets give you plenty of chances to solve problems in different ways. Practice for SAT also teaches you to look for tricks and short cuts, but that is neither necessary or sufficient. Practice, practice, practice on problem sets is the solution. That level of understanding is only driven by individual practice on problem variations, not transference of a few in-class group projects covering general ideas. Words are not understanding.

There are ways to talk and provide proper and more abstract algebraic understandings, but most of these rote pedagogues don’t have a clue. In the end, the only way to create proper understandings is via lots of individual practice on problem sets. Practice is not just about speed.

Articles I wish I never finished reading, Dept.

Yet another in a long line of articles that caricatures traditional math teaching as rote learning, lacking conceptual understanding:

Traditional middle or high school mathematics teaching in the U.S. typically follows this pattern: The teacher demonstrates a set of procedures that can be used to solve a particular kind of problem. A similar problem is then introduced for the class to solve together. Then, the students get a number of exercises to practice on their own.

For example, when students learn about the area of shapes, they’re given a set of formulas. They put numbers into the correct formula and compute a solution. More complex questions might give the students the area and have them work backwards to find a missing dimension. Students will often learn a different set of formulas each day: perhaps squares and rectangles one day, triangles the next.

The article goes on to say that this “rote process” is seldom questioned, seems to make math easier, and students find such kind of teaching to be very satisfying.  I’ve written before about textbooks from previous eras that did indeed provide explanations for why various algorithmic procedures and formulas work.  Areas and volumes are explained in terms of multiplication for figures such as rectangles and triangles. Areas of quadrilaterals such as parallelograms and trapezoids are also explained; I recently showed the derivation of same to my 7th grade math class. To say that teachers fail to teach understanding is misleading.  Some teachers may not, but many do.  A math teacher I know clarified his position on understanding versus procedures

A few years back I started explicitly telling my students “I don’t care if you understand it, provided you can do it” when they complained that they “didn’t understand”. I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.
Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.
Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding because they never have (most kids really don’t want to understand very much). Every now and then I have a student who refuses to learn a new skill until they “understand” it — and that causes problems, largely because they learn so unnecessarily slowly as a result, which I find difficult as a teacher.

I happen to teach the same way, and I suspect others do too but cannot be too vocal about it given the current atmosphere surrounding math education these days.  The author of the paper, however, seems to think that understanding is rarely taught.He provides what he believes is a telling example: Students are taught how to distribute multiplication across addition (e.g., 3(x+5)= 3x+15, with the result that it becomes so ingrained (because of the “rote” nature of the procedure) that when they see an equation such as 3(x+5)=30, they will distribute rather than divide both sides by 3 to get the simpler equation of x + 5 = 10.  The author claims “but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.”  

Actually we do show them when we come to such equations so they can recognize when it’s advantageous to divide, and also (not mentioned by this author) when it is not.  But he would rather disparage the teaching of the distributive rule because it may interfere with some later “conceptual understanding” of equations which is easily handled by–dare I say it–direct instruction.

He concludes his essay with

If we really want to improve America’s mathematics education, we need to rethink both our education system and our teaching methods, and perhaps to consider how other countries approach mathematics instruction. Research has provided evidence that teaching conceptually has benefits not offered by traditional teaching. And students who learn conceptually typically do as well or better on achievement tests.

This song has been sung in the US for the past 30 years; he’s not the first person to sing it and he likely will not be the last.  The US has floundered with various types of teaching conceptual understanding in the lower grades. How’s that been working out for everyone so far? Oh, he has that one covered: It’s because we’re doing it wrong:

As an education researcher, I’ve observed teachers trying to implement reforms – often with limited success. They sometimes make changes that are more cosmetic than substantive (e.g., more student discussion and group activity), while failing to get at the heart of the matter: What does it truly mean to teach and learn mathematics?

What he and other writers of this type of math ed pabulum never seem to get around to doing is seeing how students who do succeed in math have done it.  No mention is made of the practice they get at home, or of tutoring they might receive and how such tutoring is conducted.  Rather, we get articles that extol programs that challenge students with difficult “rich” problems such as what is done in the Russian School.  But rarely do we get a glimpse of the practice such students do using what he would call “rote methods”.

And of course parents want the traditional methods it’s because they just want what we had when we were growing up or that our kids are somehow different than others.

Yeah, OK, whatever you say.

Nothing to see here, Dept.

Right; not much to see here:

How do teachers prepare students for a world they do not know?

Well, for starters, learning foundational facts and procedures so that they are second nature. But that’s been deemed old school and blamed for failing “thousands of students”.

Graham Fletcher, former classroom teacher and math specialist, led of series of workshops for educators at Greenwich Country Day School on Monday and Tuesday aimed at rebuilding math education for that future.  “Math is so much more than just answer-getting,” said Fletcher. “When a student has self-intrinsic motivation to do something, they’re going to go far and above what you want them to do.”

“Certainly, the way we were learning when we were in school was very procedurally-focused, very answer-driven,” she said. “It wasn’t really supported by a lot of conceptual understanding, so the shift in mathematics education now is to develop student mathematicians who understand why algorithms work, why they are executing a particular formula in the way that they are.”

Of course; teachers never taught understanding, everything was done by rote, and the only students who succeeded, were gifted who would have learned under any method. Heard it before.  And how has the “understanding before procedure” regimet been working out for the last 30 years?  Freshmen at high school using calculators for the simplest of computations, and having difficulty with fractions, decimals and percents. But if you repeat something long enough, it gets taken as gospel, as evidenced by the plethora of news stories like this and conversations one hears at edu-conferences and PD sessions.


PBL: Problem vs Project Based Learning –another distraction

In an Edutopia article  the difference between the two PBL’s: Project- and Problem-Based Learning was addressed. It included a nice definitional list of what is involved in Problem-Based Learning:

Problem-based learning typically follow prescribed steps:

  1. Presentation of an “ill-structured” (open-ended, “messy”) problem

  2. Problem definition or formulation (the problem statement)

  3. Generation of a “knowledge inventory” (a list of “what we know about the problem” and “what we need to know”)

  4. Generation of possible solutions

  5. Formulation of learning issues for self-directed and coached learning

  6. Sharing of findings and solutions

Nothing about this is new. Much has been written about the debate on how best to teach math to students in the K-12grades—a debate often referred to as the “math wars”. I have written much about it myself, and since the debate shows no signs of easing, I continue to have reasons to keep writing about it.

While the debate is complex, the following two math problems provide a glimpse of two
opposing sides:

Problem 1: How many boxes would be needed to pack and ship one million books collected ina school-based book drive? In this problem the size of the books is unknown and varied, and the size of the boxes is not stated.
Problem 2: Two boys are canoeing on a lake and they reach a point where the lake joins a river.  They are paddling to get to a little cabin upriver that happens to be the same distance from where they are, as the distance across the lake they just crossed. They have an argument.  One boy says it takes longer to go upriver to the cabin and back to where they are now, than it would to go across the lake and back, assuming they row
at the same constant rate of speed in both cases. The other boy says it would take the same amount of time.  Who is right?

The first problem is representative of a thought-world inhabited by education schools and much of the education establishment. The second problem is held in disdain by the same, but favored by a group of educators and math oriented people who for lack of a better term are called “traditionalists”.

In the spirit of full disclosure, I fit in with the latter group.

I have heard various education school professors distinguish between “exercises” and “solving problems”. Textbook problems are thought of as “exercises” rather than “problems” because they are not real world and therefore, in their view, not relevant to most students. Opponents of such problems contend that they contain the quantities students need to solve the problems and therefore do not require students to make value judgments. Such criticisms of traditional approaches in mathematics have led to the “problem-based learning” approach.

The first problem presented above is an example of a PBL type problem which
appeared in a paper called “Teacher as designer: A framework for teacher analysis of
mathematical model-eliciting activities”, by Hjalmarson and Dufies Dux. The problem is called “The Million Book Challenge”. While it may be engaging, students will generally lack the skills required to solve such a problem, such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification.

Problems such as the million book challenge are predicated on the idea that by repeatedly confronting students with new situations as well as problems with which they have little to no experience, they will develop problem solving schema. The open-endedness of the problem is seen as a means to engage students in the “process” of critical thinking. In my opinion, however, such approach is like learning German by practicing particular sentence constructions in English (e.g., “I know that he the book read has”) in the hopes of building up a structure that then only needs vocabulary to complete the learning process.

Let’s turn now to the second problem—a problem appropriate for an honors algebra 1, or regular algebra 2 class. Unlike the million book challenge, it allows students to rely on prior knowledge, it is well-defined, and has specific mathematical goals. I think of problems such as the canoe problem as going down a well traveled road—you know where it is taking you even though it may have a few twists and turns and detours in unfamiliar territory.

Proponents of PBL tout problems such as the million book challenge as demonstrating that the real world provides more meaningful and useful mathematics than traditional problems.  But problems like the canoe problem demonstrate that mathematics is needed to describe what is going on—and can lead us to a conclusion that may contradict what one intuitively believes. Many students will assume that in the canoe problem either route will take the same amount of time. They reason that if the boy travels upstream and downstream for the same distance at the same rate of speed, the amount the canoe is slowed by the current when travelling upstream is cancelled by the additional speed the current imparts when travelling downstream. But the mathematics shows otherwise.

It is also interesting that despite the criticism of math reforms that traditional problems lend themselves to formulaic solutions, the canoe problem does not lend itself to a “plug-in” solution.  The proof that the time it takes to go across the lake is less than going up and downriver the same distance is not easy for beginning algebra students. Proofs of inequalities do not easily lend themselves to algorithmic solutions–in fact, they require the critical thinking and analytical skills that many believe problems like the Million Book Challenge develop.

According to Vern Williams, a middle school math teacher, and who served as a member of the President’s National Mathematics Advisory Panel, “By taking math that has been taught to them and attempting to solve difficult problems, they will discover relationships between content and methods that they already have in their arsenal even if they don’t solve the problem or arrive at the correct answer.”

The issue of PBL versus traditional type problems has particular significance
in light of the recent interest in developing assessments for math that are considered to measure “authentic” reasoning and skills. Critics of traditional type problems believe that assessments should evaluate the “critical thinking” skills of students rather than having students solve “exercises” that lend themselves to applications of previously learned problem-solving procedures. Many such critics also believe that students in
other countries that surpass the US in math on international tests are being taught only how to take tests.

Ironically, the problem with a test that emphasizes the ill-posed “rich”  type of problems is that it accommodates students’ learning how to answer open-ended questions. Also ironically, the U.S. is not achieving as high scores on such tests as Asian nations.  (The PISA exam given every three years contains some of these type of problems).  Which begs the question of what mathematics are U.S. students exposed to PBL type approaches really learning? In the end the problems which students in Singapore, Hong Kong and Japan excel at solving are still likely to be off the script for many US students.

In a world in which problems that have a unique answer obtained through systematic application of mathematical skills and principles are deemed “mere exercises”, students are heading down a PBL approach to learning that leads more to math appreciation than math proficiency.  If, however, students are taught the skills and concepts necessary to solve well-defined and challenging problems, they will learn to surmount what a disheartening number of U.S. students now consider to be insurmountable.

Engaging one’s way through math, Dept.

It seems that one of the many catch-phrases and beliefs in education circles is “engagement”. The thinking is that if we can just get kids to engage in the material, then they would be interested enough to learn what needs to be learned.  I’m a bit old-fashioned when it comes to education as many well-meaning, outspoken people have told me, but I continue to believe that if students know how to do things in math, they find it more enjoyable.  If they are struggling because they haven’t mastered basic foundational skills and facts, then despite what experts say, they are pretty much not going to like math all that much.

So it wasn’t surprising to read the latest article about some enterprising people (from U of Arizona and U of Delaware) with degrees in math education and a $1.3 million grant from the National Science Foundation who believe that if mathematics is proven to be “interesting and useful” to students, that’s one key to engagement. Their grant would explore what’s working to motivate students to learn math, recording reactions on a real-time basis.

“This ASU-UD team is the first to research moment-to-moment experiences of high school students studying mathematics over time as part of their three-year, $1.3 million National Science Foundation-funded study, “Secondary Mathematics, in-the-moment, Longitudinal Engagement Study.”

” ‘We recognized a lack of research that could address, methodologically, how to investigate students’ experiences in the moment to understand the nature of their engagement with mathematics in a way that could reveal more general trends,’ Jansen said.

“By understanding these processes, they plan to help teachers to encourage more students to engage deeply, work hard, persist and become more mathematically capable.”

I’m hoping that such study will indicate deficits students may have (like basic foundational skills and facts) that may hinder their progress on even the most interesting of problems.

I found this statement to be rather intriguing by the way:

“It is entirely likely that there are many good ways to teach,” Middleton says, “but that some of those ways may be optimally effective in limited contexts.”

Just a guess, and I know I’m sticking my neck out here, but given that NSF is funding this baby, I’m thinking she means that direct, whole-class, teacher-centered instruction is limited to smart students who would get it no matter how it is taught.

Who’d-a thought? Dept.

For those of us who embrace traditional/conventional teaching methods, this story was not surprising.  A school in western Australia which had 31% of its students at or below the benchmark for numeracy, increased performance by using explicit instruction as the dominant teaching strategy.

To boost student outcomes, the school determined it needed a consistent pedagogical approach, greater support around basic skills and knowledge and to meet its own benchmarks for minimum student performance. …

Workman said that through applying explicit teaching throughout the school, students have been able to develop a better grasp of what they are being taught, which has improved both outcomes and engagement during lessons.

This story is tantamount to saying “Students learn when given instruction” and should not be headline news. Sadly, it is.  And even more sadly will be the number of people who, like the denizens of Garrison Keillor’s Lake Woebegon, “look reality squarely in the eye–and then deny it.”

However, it is heartening that parents know what the story is and will respond (and have started responding) accordingly:

“I believe that we have become the school of choice in our area. Our Education Department’s motto is ‘high achievement, high care’. I think this is what parents are looking for, and that’s what they’re getting.”

The Squeakiest Wheel: The Saga Continues

In an earlier post, I wrote about students at the Sobrato High School filing a petition protesting how math is taught at the school. They specifically protested against being taught in groups, lack of instruction, and “the math department does not tailor its teaching needs to every learning style of its students.”  The school board responded to the petition as reported in this article.

I had thought that this comment was likely to get attention, since the concept of learning styles is near and dear to most educationists and administrators.  My thinking was that the school board and administration would go the opposite way the students wanted and institute even more ludicrous practices in the name of personalized learning, and learning styles. I had not considered that the complaint of being put into groups could be considered a practice that went against a so-called learning style of some students.  Administrators would thus be forced to either 1) acquiesce and accommodate those students for whom group learning was not their “learning style”, or 2) admit that learning styles are tantamount to the old medical practice of “blood letting” and stop believing in such nonsense.

The latter did not occur.  What did happen was that a school board member applauded the students for circulating the petition and making their concerns known.  He complemented the school for providing an environment in which students felt safe doing so. And one of the trustees said it was not likely a fault of the teachers nor the curriculum they were using, per se but “the curriculum that the state requires the teachers in California to teach.”

This is a good start, but as much as I dislike the CCSS for their embedded pedagogies and math-reform dog whistles that cause interpretations along reform math ideological lines, nowhere do the standards require that students be taught in groups.  (If so, please correct me and provide a citation of the CCSS where it says so.  All I could find was the discussion in Frequently Asked Questions on the CCSS webpage:

Do the standards tell teachers what to teach?

Teachers know best about what works in the classroom. That is why these standards establish what students need to learn, but do not dictate how teachers should teach. Instead, schools and teachers decide how best to help students reach the standards.

In fact, it is the curriculum that is directing such practice.  A look at the Sobrato HS website shows that they use the CPM series of textbooks (College Preparatory Math) which is discovery-based and advocates/instructs teachers to organize classes in groups, with teachers being facilitators.  I am familiar with CPM since I had to teach algebra using that book when I was a student teacher. (Readers curious about this can read all about it in my book “Letters from John Dewey/Letters from Huck Finn” which talks about ed school, my student teaching experience and other facets that define the ideological and cultural divide between the math reformers (progressivists) and traditionalists).

Of interest is another statement by a board member regarding the situation at Sobrato:

“Trustee David Gerard did not want any board discussion on the subject to single out Sobrato, but agreed that some sort of discussion needed to happen in the future. ‘It’s a broader question, not just Sobrato. It’s not a question of blaming teachers. It’s something that requires a lot of thought because it is complex,” Gerard said. “Obviously there’s some concern and we want to show that we are addressing that as a board.” “

Indeed.  I’d be glad to furnish them with details. Watch this space for further developments.

Exceptional Arguments

For those of you who have been following my writing, you are aware that I teach math in middle school and that I am extremely interested in the most effective ways of teaching the subject.  I majored in mathematics and am pursuing the teaching of it as a second career after having retired several years ago. I balance my teaching by writing articles that address the problems with math education in the U.S.  You will also know that I was educated in the 50’s and 60’s, and thus obtained my math education via what is called “traditional” math teaching.

Because of my educational background and my beliefs in how math should and should not be taught, I often find myself engaged in the following dialogue, either in person or on the internet:

Someone: The traditional method of teaching math failed many students.

Me: The traditional method seemed to work well for me and many others I know.

Someone: You’re the exception.

Regarding my claim of traditional math working for me and others, I am mindful of the advice given by David Didau (author of “What if Everything you Know About Education is Wrong?”) who points out the following with respect to educational debates:

“If, in the face of contradictory evidence, we make the claim that a particular practice ‘works for me and my students’, then we are in danger of adopting an unfalsifiable position… We can insulate ourselves from logic and reason and instead trust to faith that we know what’s best for our students and who can prove us wrong?”

I will say in my defense, however, that “You’re the exception” is not much of an argument either. It is usually offered with either anecdotal evidence or none at all, and is based on a largely mischaracterized view of what traditional math is and was.  I offer here some rebuttals to three of the most popular “you’re the exception” arguments. Such arguments occur at school board meetings, in casual conversations, on the internet, and —disturbingly — in newspapers and on television. You are free to use them at edu-conferences, cocktail parties, or at troll fests on Twitter.

 1. If traditional math teaching were effective, the U.S. would be at the top of the world in math.

This argument ignores that in countries doing well on international tests, students learn math mainly via traditional means — and over the past two decades, increasing numbers of students in the U.S. have learned math using the reform-based methods. Reformers are quick to point out that Japan and perhaps other Asian countries actually use reform methods, ignoring the fact that many students are enrolled in “cram schools” (called Juku in Japan) which use the drilling techniques and memorization held in high disdain by reformers.

The argument also fails to consider that traditional math can also be taught poorly. There have always been good and bad teachers, as well as factors other than curriculum and pedagogy that influence the data.  In order for such arguments to work, one would have to evaluate how achievement/scores vary when factors such as teaching, socioeconomic levels and other variables are held constant and when pedagogy or curriculum changes.

Studies have been conducted that examine how math is taught in specific areas of North America, as well as looking at the common traits of high-performing systems across the world.  One such study indicates that when both conventional and non-conventional (i.e., reform) math are taught by well-trained teachers, students learning under traditional mathematics instruction show much higher achievement than those learning under the reform math methodology.

2. If traditional math worked, the knowledge learned in school would stay with us.

That people do not maintain proficiency in math as they age says less about traditional or reform math than about the way in which a population’s knowledge and skill base is maintained over a lifetime. It is not evidence of failure of traditional math.  The results of not using math on a consistent basis can also be seen in a study conducted by OECD.  In the study, people from ages 16-65 in over twenty countries, including the U.S., were given the same exam consisting of math computations and word problems.  According to the study, “the percentage of U.S. adults between 55 and 65 years old who scored at the highest proficiency level (4/5) …was not significantly different than the international average for this age group.” These findings can be used in tandem with the first argument above since people in the U.S. in the 55 to 65 age group learned math via traditional math teaching—and the differences in proficiencies between the U.S. and other countries is not significant.

3.  Traditional math failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval

This argument relies on the tracking argument, when many minority students (principally African Americans) were placed into lower level math classes in high school through courses such as business math. It is based largely on the following premises:  “Most students did not go on in math beyond algebra, if that, and there were more than enough jobs that didn’t even require a high school diploma. Few went to college.  Now most students must take advanced math, so opting out is not an option for them like it was for so many in the past.”

First, in light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place — the instructional method had nothing to do with it.  Also, I don’t know that most students must take advanced math in order to enter the job market. And I don’t think that everyone needs to take Algebra 2 in order to be viable in the job market.

Secondly, many of today’s students entering high school are very weak with fractions, math facts and general problem solving techniques. Many are counting on their fingers to add and rely on calculators for the simplest of multiplication or division problems. In the days of tracking and weaker graduation requirements, more students entering high school than now had mastery of math facts and procedures including fractions, decimals and percents.

Some blame the “changing demographics” on the decrease in proficiency, but this overlooks variables like poor curriculum and the reform-based approach to math which views memorization “workarounds” as deep understanding. Also frequently overlooked is the fact that students in low income families who make up the “changing demographic” cited in such arguments do not have access to tutoring or learning centers, while students in more affluent areas are not held hostage — dare I say “tracked”? — to poor curricula and dubious pedagogical practices.

What’s Next?

The debate over traditional versus reform-based math has been going on for some time—for so long, in fact, that some on the reform side are saying that there’s nothing to discuss, it’s boring, just let teachers teach. I agree that we should let teachers teach, and that parents be given choices of what type of math they want their children to have. That doesn’t appear to be happening any time soon.

I believe that the debate should continue and that there is plenty to discuss. People may choose to use the information I’ve presented here — or persist in ignoring it. I don’t expect that I’ve changed anyone’s mind about anything, but I am always hopeful that there are some exceptions.

I also do not think that I am alone in drawing a distinction between reform and traditional modes of math teaching. While traditional math can be taught properly as well as badly, I believe that poor teaching is inherent in most if not all reform math programs. I base this on having seen good teachers required to follow programs that present content poorly, lack a coherent logical sequence and rely on questionable pedagogies.

I would like to see studies conducted to document how U.S. students who do well in math and science and pursue STEM majors and careers are learning math. The chances are fairly good that such investigations would show that in K-8, many students are getting support at home, from tutors, or from the many learning centers that are springing up all over the U.S. at rapid rates. Since tutors and learning centers (and parents) tend to use traditional methods for teaching math, I doubt that the clientele are exceptions to some ill-defined rule.  In my view, as well as the view of many parents and teachers I’ve met, there are few exceptions to the educational damage reform math programs have caused, even when such programs are taught “well.”

Epilogue and Sales Pitch

Finally, if you have enjoyed this piece, there are others of comparable or better quality in my book “Math Education in the U.S.: Still Crazy After All These Years”. Available here.