Don’t Tell Jo Boaler, Dept.


A group of students in Lake Charles, Louisiana is promoting knowing the multiplication facts.

Those who think that the traditional ways of teaching mathematics have been shown to be harmful for all students, will find this quote from the article to be heresy:

“The ability of an individual of any age to be able to multiply consistently and effectively can build confidence in other areas of life,” Nonnette said. “We as an organization will succeed in our mission to enhance the awareness of math education through one multiplication chart at a time.”

Now, if only we can find some way to make the progressives think they came up with this!

Reactions to Barbara Oakley’s op-ed: Revisited

I’ve noticed a spike in traffic at this site, looking at a post I wrote over a year ago. The piece I wrote addressed a blog post that criticized an op-ed on math education written by Barbara Oakley.

The blog post is here but comments have long been closed. I recall there was some flap with the blogger who wrote it, when I said that comments from Barbara Oakley and myself hadn’t been published.  She did publish Barbara’s comment, but I notice that my comment is still “awaiting moderation”.  Meaning she forgot about it, or didn’t want to publish it. I have no idea which may be true, but if you’re curious, here is the comment:

In your post you state: “It is true that traditional ways of teaching mathematics have been shown to be harmful for all students, and even more harmful for non-dominant populations, including girls. This phenomenon has been widely documented by professors of mathematics education such as Rochelle Gutierrez, Jo Boaler, and others.”

What research has shown this to be true for ALL students as you state? You cite Boaler who has gone on record as saying that learning times tables can be injurious to students. She believes that memorization does harm and undermines “understanding”. You also you cite Gutierrez who claims that minorities and girls don’t do well in math because of the way it’s taught. Even if we agree with Gutierrez’s claim, that would say that not all are injured, but just minorities and women.

I wrote an article some time ago about traditionally taught math and showed data from the Iowa Tests of Basic Skills (for grades 3 through 8 ) and the ITED (high school grades) from the early 40’s through the 80’s for the State of Iowa. (See

The scores (in all subject areas, not just math) show a steady increase from the 40’s to about 1965, and then a dramatic decline from 1965 to the mid-70’s. One conclusion that can be drawn from these test scores is that the method of education in effect during that period appeared to be working. And by definition, whatever was working during that time period was not failing. And this was at a time when traditional math teaching was the mainstay.

It is true that traditionally taught math can be done poorly. It can also be done well, and there have been many people who have benefitted from such instruction. Regarding memorization, I suggest you read “Memorable Teaching” by Peps McCrea which is an exploration of how memorization is an essential part of the learning process. (

Ch 13 of “Out on Good Behavior”

For those following the continuing series Ch. 13 is now up at Truth in American Education.


In planning my future classes during the summer before the upcoming school year I proceed from an undying faith in my expectations of how things will be. During the actual school year, I then deal with the reality. In the end, it is always astounding to me how some intuitions turn out surprisingly well.

My Math 7 class at Cypress my second year was the non-accelerated version. I had taught accelerated Math 7 the year before, but was now faced with a challenging group of students who I knew were disheartened about math and likely dreading the next year. While planning my lessons during the summer using the JUMP Math teacher’s manual, I had a vision that the students would upon succeeding and getting good grades on tests and quizzes, eventually discover that the math was actually interesting and that they could manage it.

The reality was slightly different as I was finding out and as I’ve written about in preceding chapters. I knew that something was happening. Just not in the manner I had envisioned.

Read the rest here.

Extended metaphor, Dept.


We frequently hear about how in math education we should engage students in “productive struggle”. While there is some value in having students synthesize prior knowledge from worked examples and scaffolded problems, this is generally not what is meant by “productive struggle”. Generally it means having students solve problems that are usually one-off types that do not generalize. What prior knowledge students may have to draw upon is in most cases very small and lacking in sufficient practice for students to be able to apply it efficiently. And if prior knowledge is absent, students are expected to obtain it via “just in time” learning, which would arrive without sufficient practice and mastery.

Students are expected to collaborate with fellow students, and dissuaded from asking the teacher for help. If the teacher is asked to help, the teacher is usually instructed to not give answers to students questions but to facilitate the student to answer their own questions.

The result is like throwing someone who lacks swimming skills into the deep end of a pool and asking him/her to swim to the other side. The result is generally a struggle to keep from drowning–which is not the same as learning how to swim.

PBL: A guide to the hype

Edutopia has advice for those math teachers who believe that Problem (Project) Based Learning (PBL) actually has something of value to offer.

I offer a brief commentary on their suggestions.

Address math myths: “Some teachers worry that PBL will take away time needed to practice math skills. Others insist that they need to “front-load” concepts before students can apply them, or worry about students encountering concepts out of the order outlined in their math curriculum.”

Those are my concerns as well. What the author calls “front-loading” concepts is what the rest of us call teaching. Many of us teach using direct and explicit instruction with worked examples and pratice problems.  We do this so that students can then put to use what they learn–with guidance.  The alternative is what I call  “just in time teaching”.  This is similar to throwing a kid in the deep end of the pool and instructing him/her to swim to the other side. The belief is that  he/she will be ready to absorb the instructions the teacher is shouting from the side of the pool on how to swim.

Fancher and Norfar rely on research from the National Council of Teachers of Mathematics (NCTM), among other sources, to overcome common concerns about PBL.

That’s their first mistake.

For example, developing what NCTM calls procedural fluency does not require having students labor over worksheets. Fancher explains, “It’s better to have students do four or five rich problems and explain how they solved them.”

Rich problems: What are they? Generally, they are very wordy, tedious one-off type problems that do not generalize. Often they are open-ended and ill-posed (like, the area of a rectangle is 48 sq ft; what are its dimensions?) and require students to learn procedures on a “just in time” basis.

Explain how they solved them:  Back to the kid who was tossed in the deep end of the pool. If by some miracle the kid makes it to the other side without drowning, he/she will likely say: “I don’t know how I did it, but I never want to do that again.”

“Don’t stop doing PBL if your project doesn’t go the way you dreamed,” Norfar cautions. Instead, reflect on what worked well and what didn’t, and consider how you can improve the project next time around. “PBL isn’t a cure-all,” she adds, “but it’s too powerful to give up on.”

My advice: Don’t waste your time, but if you do, your reflection should be on how you could have instead provided direct instruction on mathematical procedures. This could have been coupled with providing adequate practice with scaffolded problems. Finally, you could have provided instruction (and practice) on how to solve specific types of word problems. I believe that such tried and true methods are the ones too powerful to give up on. They are also the ones the education industry has discarded and from which the tutoring industry has greatly benefited.


The Prevailing Caricature of Traditional Math, Dept.

In a recent article about math education that ballyhoos the “latest approach” in how to teach math, this statement was made:

“If the teachers are telling students how to solve a problem, and then that problem isn’t exactly what’s on the test, it creates this disequilibrium for a student,” said Beverly Velloff, the math and science curriculum coordinator for the University City School District.

There is nothing new about the so-called breakthrough ideas the article discusses. Moreover, this quote is representative of how traditionally taught math is mischaracterized.  The notion is that students are taught by rote, given a set of problems that are all exactly alike, and thus leaves them flummoxed when presented with a problem that is even slightly different. Such a caricature may be true for traditionally taught math done poorly, but it makes no allowance for it being done well.

Looking at an example from algebra: there are many varieties of distance/rate problems. There are problems in which two objects are going in opposite directions, going in the same direction (i.e., playing catch-up), round-trip problems, objects being influenced by wind or current, and so on.  At some point students are given basic instructions for solving these various types.  In opposite direction problems, students should be taught that we are dealing with two distances that are equal.  That is, if two people are driving towards each other, then the distance each travels before they meet up is equal to the initial distance of separation between them.

In well written textbooks, such problems are scaffolded so that initial problems are solved by following the worked example. But subsequent problems might have some small variation.  Instead of two cars coming towards each other at respective speeds of 60 and 40 miles per hour, we might be told the speed of one car, the distance between them initially, and the time it takes for them to meet. For example, two cars are 200 miles apart, and one car goes 60 mph. It takes 2 hours for the two cars to meet. What is the speed of the other car?  We know that in two hours the 60 mph car has travelled 120 miles. The distance of the second car added to 120 miles equals 200 miles. That’s the “distance = distance” relationship so that if x equals the speed of the other car, then 2x+120 =200. The speed of the second car is 40 mph.

Students will need some guidance in going through these problems, but after given practice with these types, they learn what to look for.

Math reformers may look at this as spoon feeding and rote. They would rather give students problems for which they have not been given specific instructions, and need to synthesize prior knowledge. Alternatively, they are expected to learn in a “just in time” manner what is needed to solve problems.  Thus, problems are given in a top down form in the belief that over time, students will develop a problem solving “schema”.

An article by Sweller et al (2011)  states that such notion is mistaken:

Recent “reform” curricula both ignore the absence of supporting data and completely misunderstand the role of problem solving in cognition. If, the argument goes, we are not really teaching people mathematics but are teaching them some form of general problem solving then mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general and that will make them good mathematicians able to discover novel solutions irrespective of the content.

We believe this argument ignores all the empirical evidence about mathematics learning. Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach was never the most effective or efficient way to learn mathematics.

Nevertheless, reform/progressive math ideas rule the roost in education. Articles such as Sweller’s are thought of as fluff, not proven, no evidence to back it up, or dismissed in light of arguments such as “It has worked in my classrooms”.

Students need instruction, worked examples, scaffolding, ramp-ups in difficulty of problems, guidance, and much practice. Reformers view such steps as “inauthentic math” that produce “math zombies” who do not have “deeper understanding”.  Ignored in all this is the fact that the so-called math zombies are the ones in college who by and large are not in need of remedial math classes.

Bad PD # 517

At a six-hour PD I had the misfortune of having to attend, the moderator put this slide on the screen in a defense against the call for evidence that certain teaching practices are effective. It was a slide from a presentation by David Theriault, who teaches English and has a blog:


Essentially, Mr. Theriault felt that the question about having research to back up a practice was irritating. What he calls research is what he sees in the classroom. I’ve heard it many times before in a “It works for me” type defense.

Well, traditionally taught math worked for me, but I’m fairly certain the moderator of the PD as well as Theriault and others would not find that acceptable.

The PD was full of the usual platitudes that “worksheets are bad, experiential learning is good”. The penultimate task of the day had us drawing specific geometric figures using a computer language that we had to figure out without instruction.  The person next to me was familiar with the language so he taught it to me, using direct and explicit instruction. Others learned in the same way.  The moderator was very pleased with the fact that we learned from each other rather than from him since he seemed to think it proved his point that teachers should facilitate what is supposedly innate in students. Except for the fact that this wasn’t innate; the person who taught it to me learned it the old fashioned way.

We then had to construct a one-page lesson plan on any topic.  My partner for this task was a woman from a neighboring school who, like me, taught math in middle school. I suggested a plan to teach why the invert and multiply rule works for fractional division.  She got hung up on providing a context for division. I couldn’t get beyond her hang-up and reminded her we only had a few minutes left. Time ran out, and she got into a conversation with the moderator about her hang-up and I left.

The teachers at my school never talked about that PD, and I notice they never asked the moderator back.

I offer you this as one of a continuing series of bad PD that infiltrates our education system.

Arrogant jerks, Dept.

In a recent column in the Washington Post, Jay Mathews has written what has become the emblematic anthem against algebra II in high school

I can understand the argument against requiring algebra II for graduation, since at one time that was the case. Students only needed two years of math, and that usually consisted of algebra I and geometry.  But his argument seems to be to get rid of it altogether and in its place have courses that are more relevant like statistics.

He ignores the fact that if you really want to pursue statistics, you will have to have some facility in the topics taught in algebra 2.  So what is he suggesting? Students should take that in college?

But there are ways to ease algebra II out of high schools. Gregg Robertson, longtime principal of Washington-Liberty High School in Arlington, Va., noted that his math department has courses in probability and statistics, both regular and Advanced Placement, as well as a dual-enrollment quantitative reasoning course through Northern Virginia Community College.

I think I’m reading that right. Easing algebra II out of high schools means it isn’t an option for anyone. Unless he wants to walk it back and say “What I meant was ‘easing it out of graduation requirements’ “.  But he didn’t say that.

Mathews relies on the tired old arguments that supposedly give him credence: He took both algebra II and also calculus. He’s never used it in his life. There; that’s proof of its uselessness for you.  Many people are not scientists, engineers, or mathematicians. Why not ask them if they’ve ever used these courses that are deemed so useless?  Maybe such information would propel journos like Mathews to write more relevant columns.

Puff Piece, Dept.

Just read a rambling article in the Atlanta Journal Constitution by Maureen Dowd that points fingers and doesn’t come to any conclusions. Main point: math ed has been bad in GA for many years so why blame Common Core.

Citing parents’ laments that they wish math could be taught as it was 30 years ago, Dowd asks whether this is really a solution. She states: “But did students learn math more effectively a generation ago? When the Program for the International Assessment of Adult Competencies evaluated numeracy skills of adults in 23 countries, 20 outperformed the United States.”

The study cited aggregates populations from ages 16 to 65. Thus, there are different types of math teaching people were exposed to based on age. However she leaves out this finding, stated in the study: “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.”

She also cites Elizabeth Green’s NY Times article from a few years ago (Why American’s Stink at Math) and pulls this quote:

“The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices. The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them.”

In other words, the methods of math reform would work if they were only done right–but they’re never done right. There is much to contest there.

Lastly, she talks about Japan’s “integrated approach” to math, which Georgia will emulate. Japan isn’t the only country to take an integrated approach to math in high school; many European countries do this also. But they do it fairly well. The U.S. has done a horrible job of it; one need only look at the integrated approaches used here: IMP, Core Plus, MVP math.

In short: A typical puff piece that refuses to look at research that would upend the opinions expressed in the article.

Selective Reporting, Dept.

Dana Goldstein, a New York times reporter, asked on Twitter if there were any teachers willing to be interviewed for an article she was writing. I responded (I was on Twitter at this time), suggesting she view the video of a talk I gave in which I mention some of the problems with Common Core. (I even told her to start at minute 19:24 to save her some time).

Whether she watched it or not, I don’t know but the article she wrote does not seem to reflect any of the insights I provided. So I’m assuming that she was under a tight deadline and couldn’t be bothered with messy details.

She focuses primarily on reading, but does give a nod about the math standards:

On social media, angry parents shared photos of worksheets showing unfamiliar ways to solve math problems. One technique entailed “unbundling” numbers into multiples of 10, to help make adding and subtracting double- and triple-digit figures more intuitive. Another, called “number bonds,” required students to write the solutions of equations in stacked circles.

Both methods are commonly used in high-achieving nations. But to many American parents sitting at kitchen tables and squinting at their children’s homework, they were prime examples of bureaucrats reinventing the wheel and causing undue stress in the process.

What she reports is true; high-achieving nations do indeed use these techniques. What she leaves out however, might help us to understand what’s going on.  The techniques to which she refers are nothing new and were used in US textbooks in earlier eras, like the 60’s, 50’s, 40’s and so on. The difference between then and now, is that the textbooks taught the standard methods or algorithms first—as an anchor—and students were then given problems (deemed “drill and kill” by those who hold practicing as mind numbing and counterproductive). Alternative techniques such as the “unbundling” of numbers (e.g., representing 235 as 200 + 30 + 5, or 2 x 100 + 3 x 10 + 5 x 1) and number bonds are nothing new. Number bonds were even part of the textbook I used in elementary school, shown below:


Source: Brownell,et al; 1955. Ginn and Company; New York.

Unlike the current practice in the US, the high performing countries alluded to in Goldstein’s article do not dwell upon these techniques for months on end. Nor is the teaching of standard algorithms withheld until students show understanding using alternative methods. Looking at Singapore’s Primary Math series for example, students are expected to be familiar with number bonds and development of mental strategies for ease of computation. But the difference between what we are seeing with textbooks aligning with Common Core and those used in Singapore is that mathematical development will evolve naturally with the build-up of levels of understanding. The Singapore textbooks do not insist that students use the specific methods that are embedded in Common Core’s standards.

For example, while the process of “making tens” is used in Singapore’s “Primary Math” first grade textbook, it is one of several strategies presented that students may choose to use. (See figure below for how “making tens” was explained in my 3rd grade textbook—and introduced after mastery of the standard algorithm for addition).


Source: Brownell, et al. (1955); Ginn and Company, NY.

There is no requirement that I’ve observed in Singapore’s textbooks that forces first graders to find friendly numbers like 10 or 20. This is probably because many first graders likely come to learn that 8 + 6 equals 14 through memorization, without having to repeatedly compose and decompose numbers to achieve the “deep understanding” of addition and subtraction that standards-writers feel is necessary for six-year-olds.

The Common Core grade-level standards are minimum levels of expectation and goals, and are to be met no later than that particular grade level. Thus, the standards do not prohibit teaching a particular standard earlier than the grade level in which it appears. But publishers have interpreted the standard to mean that the standard algorithm not appear until fourth grade. And in compliance with the Common Core’s “Progression Documents” for those publishers that introduce the standard algorithms earlier than the grade in which it appears in the Standards, the publishers’ test materials do not include questions on standard algorithms before their time.

The math reform influence upon Common Core’s standards manifests itself most obviously in the delaying of teaching standard algorithms. Delaying teaching of the standard algorithm for addition and subtraction of multi-digit numbers until fourth grade is thought to provide students with the conceptual understanding of adding and subtracting multi-digit numbers. The interim years (first through third grades) have students relying on place value strategies and drawings to add numbers. The means to help learn, explain and understand the procedure becomes a procedure unto itself to become memorized, resulting in a “rote understanding”.

Students are left with a panoply of methods (praised as a good thing because students should have more than one way to solve problems), that confuse more than enlighten. The methods are side dishes that ultimately become indistinguishable from the main dish of the standard algorithm.  As a result, students can become confused—often profoundly so. As Robert Craigen, math professor at University of Manitoba describes it: “This out-loud articulation of ‘meaning’ in every stage is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill.”

None of this is discussed in the Times article—it would take too many words, I suppose. What readers are left with is the longstanding impression given by the bevy of education journalists that parents are a bunch of whiners and complainers who don’t realize a good thing when they see it.

She hints that perhaps Common Core wasn’t done right (albeit with respect to reading). “Still, not everyone agrees that the Common Core was faithfully implemented at the classroom level.

I implement it by exercising my mathematical judgment and teaching it in an efficient and effective manner. That is, the way that students from affluent backgrounds may be learning it thanks to tutors and learning centers.

Reference: Brownell, Guy T., William A. Brownell, Irene Saubel. “Arithmetic We Need; Grade 3”; Ginn and Company. 1955.