Unpublished Comments on the Rebuttal to Barbara Oakley’s NYT Op-Ed

I submitted a comment to the blog post rebuttal of Barbara Oakley’s NY Times op-ed.  It was never published and comments are now closed on that particular post.  I know someone else who submitted a comment that went unpublished.  The comment provided contrary examples and evidence to what was stated in the blog post.

Well, it might be hard to publish evidence that goes to the contrary after Dan Meyer praised the post on his blog, stating that “Gargroetzi  [the author of the blog post in question] highlights two valid points from Oakley and then takes a blowtorch to the rest of them.”

He goes on to say:

“A math program that endorses drills and pain as the  foundational element of math instruction (rather than a supporting element) and as a prerequisite for creative mathematical thought (rather than a co-requisite) inhibits the student and the teacher both, diminishing the student’s interest in producing that creativity and the teacher’s ability to notice it.”

Dan is welcome to his opinion, but as I have shown in many articles about math education, traditionally taught math is often mischaracterized as rote memorization with no understanding of concepts, and no connection between prior mathematical ideas. A glance at math books in the past (as I have also illustrated in articles) shows that both procedures and concepts were taught.

Oakley did submit a comment which was published, but her latest–which addressed questions from another commenter–was not. I believe her response addresses the commenter’s question, and may also address the opinion expressed by Meyer above.  I have reproduced it below, along with the commenter’s questions. Of particular interest is her recounting of her experience in trying to obtain a grant from the National Science Foundation (NSF).  NSF, readers will recall, provided grants in the early 90’s to produce thirteen inquiry- and reform-based math textbooks, including “Everyday Math”, “Investigations in Number, Data and Space”, “Connected Math Project (CMP)” and “Interactive Mathematics Program (IMP)”.

Thank you for your thoughtful questions.  Here’s some feedback (I’ve put your original questions in italics).

1. Do you have evidence to support your claim that, “We’ve gone so overboard with the value of conceptual understanding that students think it’s the golden key–they don’t need to practice”? That is: is there evidence that wholly (or largely) foregoing drilling/practice is, in fact, what’s happening in a large number of classrooms? I do agree that *SOME* conceptual-understanding-focused approaches to math education seem to be too reactionary in their wholesale rejection of “rote” practice. But other programs – and, I strongly suspect, *many* teachers – are interested in finding a good balance between conceptual development and skills practice, seeing them not as antithetical to each other but rather complementary.

My own experience is that I created and co-teach, (with neuroscientist Terrence Sejnowski, the Francis Crick Professor at the Salk Institute), the online course Learning How to Learn. Because this has become the world’s most popular online course, with nearly 2.5 million registered students, I am annually invited to speak to dozens of universities and high schools around the world. (That’s why I was a little delayed responding—they’re keeping me pretty busy here in Norway).  This means I speak in front of (or get emails from, sigh) tens of thousands of students, teachers, and professors from around the world each year, and have the opportunity to field their questions, hear their concerns, and interact with them.  Learning of the value of “chunking”—that is, the value of creating sets of neural patterns of procedural fluency, is one of the aspects of learning that people often tell me has proven most valuable to their subsequent success in mathematics and analytical topics. Sometimes it is quite striking, how different US K-12 teachers are in their understanding of the value of procedural fluency and practice, and how those approaches are an important aspect of the development of conceptual understanding.

On a smaller scale, I volunteered for five years to help with math in the fifteen or so elementary schools an inner urban school district. The kids were great! The teaching methods used for math were hair-raising.  There was no such thing as practice or procedural fluency in class room—everything hinged on “conceptual understanding.”  In practice, this meant that teachers stood around explaining or having students do “group work,” without ever having to worry about grading papers.  Many fifth graders there were unable to perform simple mathematical calculations, like adding 5 + 3. But the teachers were happy because they felt that the students had a conceptual understanding of addition.

I once went to NSF headquarters in Washington DC preparatory to submitting a grant to study the effects of Kumon-style practice methods in elementary schools.  The program officer their warned me that I was foolish to try to submit a grant meant to develop procedural fluency or promote practice, since after all, all the professors on the review committees would be extremely unsupportive. Indeed, when I went to submit the grant, the Dean of my university’s School of Education refused to sign off on it, because she thought it was ludicrous to support procedural fluency or practice.  Getting her signature on a simple statement that said “I support this proposal” finally meant that I had to wait for 3 hours in her outside office on the day the proposal was due. When I caught a glimpse of her catching sight of me, she actually ran the other way down the hallway. I ran down another hallway that connected to her hallway, and in that way was able to finally corner her and get her signature only moments before the proposal was due.  The proposal, of course, was rejected, with the statement that everyone knows procedural fluency and practice are pernicious.

In a more poignant personal example, one of my US-raised engineering students once remonstrated with me about his failing test score.  “I don’t understand how I could have flunked this test,” he said. “I understood it when you said it in class.”  We’re so overboard in the US about the value of conceptual understanding that students think that’s all they need.  (I wrote about that incident here: https://www.wsj.com/articles/barbara-oakley-repetitive-work-in-math-thats-good-1411426037.)

I absolutely agree with you that many teachers are seeking the right balance between conceptual understanding and procedural fluency.  But the message that they are getting from some of the thought-leaders in mathematics education can be so one-sided in favor of conceptual understanding and antithetical to practice and the development of procedural fluency that it makes it difficult for them to find that balance.

2. (a) Your Op-Ed, and also your response here, emphasizes the idea of making math “fun” as a principle motivation behind the conceptual-first approaches you object to, contrasting it with your aims of making students successful. This feels like it’s straw-manning the position you’re arguing against. (Actually, to be totally frank, when coupled with a statement like “I would hope that educators in mathematics would open their eyes,” it seems outright dismissive.) (b) Speaking as a former math teacher who prioritized conceptual understanding and problem solving: it’s *not* always more “fun” than more mechanical practice/drilling. Having to think anew about each problem, as opposed to learning a procedure that lets you get into a “groove,” can be really exhausting and frustrating and just plain *hard* for students… but, to borrow from one of the researchers you cited, it’s a “desirable difficulty.”

It may seem that I’m “straw-manning” fun in math, but with the experiences I’ve had above, plus the hundreds of conference presentations I’ve been to in the US related to making STEM more fun (never a peep about the value of practice or the development of procedural fluency), that it makes it easier to come to the conclusion that for many reform thought leaders in mathematical education, creating a fun learning experience, rather than an educational learning experience, is their primary motivating factor.  Don’t get me wrong!  I truly believe there is great value in adding fun into learning math.  But far more so than learning other topics, for example, foreign language or reading, it seems clear that some of today’s important reform mathematical thought leaders focus so much on fun that they neglect or denigrate invaluable basic building blocks of mathematical thought, such as learning the multiplication tables. 

When I said “I would hope that educators in mathematics would open their eyes,” it’s because I really would hope that they would open their eyes to the findings of neuroscience. I believe it would change their siloed conceptions that they are the only ones who can understand how to teach math to kids, so they needn’t pay attention to findings from any other field, no matter how relevant those findings may seem to others who are not K-12 math educators. In my many interactions with pedagogical professors in schools of education over the past decades, I’ve been appalled at the frequent insular statements I’ve heard from them about how they don’t need to interact with or learn from other fields. 

On part b) of your question, when I’m discussing procedural fluency, I’m not just saying “have kids do rote problems over and over again until they get buggy with boredom.”  The problem-solving you describe above, where students think anew about each problem, getting plenty of practice as they are doing so, is part of what I feel is invaluable in using practice to help develop mathematical skills. Your students are lucky to have you as a teacher.

3. It’s wonderful that you, your children, your colleagues – and, for full disclosure, I, too – came to enjoy math after an early education that focused on drilling. But what about the many, many, many American adults who, if you mention anything relating math class, will say some variation of “oh, I’m no good at math” or “I hate math” or “wow, I sure don’t miss math class”? Sure, early drill-focused learning works great for some people; and it’s no surprise that those it worked for are the ones you’ll find now as successful engineers, scientists, etc.: that’s simple survivorship bias. The question at hand here is whether or not an approach that included more emphasis on the conceptual would produce more people like you and me.

I think it’s pretty clear from my experiences that I feel the conceptual approaches to teaching math that are so emphasized in the US are part of the reason that only 7% of the graduating high school population ultimately graduate in STEM topics, despite the overwhelming need for STEM graduates in this country.  When you don’t have those basic patterns of procedural fluency embedded, it’s tougher to want to go into any type of analytical field.  Metaphorically speaking, it’s like learning to ride a bicycle.  If you’re only taught conceptual understandings of how to ride a bicycle, and you rarely actually get on to practice—falling off and bruising yourself on those few occasions when you do practice—riding a bike seems no fun at all.

4. As you noted above, the Morgan, PL, et al. article concludes that teacher-directed instruction is more important than other learning activities specifically for students with mathematical difficulties (MD). You didn’t mention, however, that “for both groups of non-MD students, teacher-directed and student-centered instruction had approximately equal, statistically significant positive predicted effects.” The second article’s title (it’s behind a paywall) sounds like it suggests a similar result. The idea that the optimal balance between skills practice and conceptual development may vary depending on students’ current confidence and skill is quite a bit different than the claim that we should make all of our daughters practice some math every day, whether they like it or not.

Morgan’s excellent paper related to how reform mathematics approaches appear to hurt those most in need of help in mathematics.  This related to some of the claims of the blog poster, as opposed to my own original op-ed topic, which related to how to balance out the uneven skill set typically seen in little girls. 

More Update on Rebuttals

I submitted a comment to the blog post rebuttal of Barbara Oakley’s NY Times op-ed.  It has been four days and it is still under moderation, so rather than keep people in suspense, here is the comment I submitted:

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 https://www.educationnews.org/education-policy-and-politics/traditional-math-the-exception-or-the-rule/)

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. (https://www.amazon.com/Memorable-Teaching-Leveraging-learning-classroom/dp/1532707797) I also suggest you read some of the papers and articles written by a group of cognitive scientists who call themselves “The Learning Scientists”. Their website is here: http://www.learningscientists.org/

Update on Rebuttals Heralded as Truth, Dept.

It turns out that the blog that criticized Barbara Oakley’s NY Times op-ed has run Oakley’s comment, which was featured here a few days ago.  The blogger responded:

“Thank you so much for taking the time to respond to my post and also for including references as well. I have edited my original post to include reference to some of the more extensive research programs on children’s mathematics learning.”

It’s hard for me to tell what she changed in the blog post, but she did bold the following statement:

 “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.”

Really? All students? What research has shown this to be true?  She cites Boaler who has gone on record as saying that learning times tables can be injurious to students.  And she cites Gutierrez who claims that minorities and girls don’t do well in math because of the way it’s taught. Probably just me, but I don’t consider her sources to constitute strong evidence.

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.

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.

The blogger also states that:

“Oakley’s characterization of “the way we teach math in the America” is backwards. Whereas she says we have foregone drill and practice for conceptual understanding, our problem in the United States is understood by learning scientists to be precisely the opposite. “

I think one can point to the last almost-three decades as seeing more reform methods being used in math, thanks in large part to NCTM’s questionable math standards that de-emphasized practice and memorization.  Why is it that 30 or 40 years ago, students entering high school knew math facts and how to work with fractions and percents, whereas now we see many students who have significant deficits in such areas?  Because math has been taught traditionally?

Unfortunately, the math reform crowd seems to hold sway, even though they claim that poor math performance is due to traditional methods.  The reform thought leaders continue to ride this wave and have quite a lot of followers.

Rebuttals Heralded as “Truth”, Dept.

Barbara Oakley wrote an op-ed in the NY Times, that went against the current faddish and prevalent thinking that math taught in the traditional manner kills all desires to learn math. She advocates learning the foundational aspects of math through practice (derisively referred to as ‘drill’) and cites her own experience learning Russian at the Defense Language Institute–one of the best language learning facilities in the world.

A recent blog post offered a rebuttal to this, which is not unusual if you read the comments to Oakley’s NY Times piece.  People do not like being told that traditional methods of math teaching have merit–particularly when the current “narrative” about traditional math teaching is that it doesn’t work and turns students off of math.  It is portrayed as nothing but drill, rote learning, no connection between concepts and no understanding. Also, they tend to make questionable statements such as this: “Whereas [Oakley] says we have foregone drill and practice for conceptual understanding, our problem in the United States is understood by learning scientists to be precisely the opposite.”

I have had the pleasure of being in contact with Ms Oakley over the past two years and alerted her to the blog.  She tried to post a comment on it but it has not yet appeared.  I don’t wish to assume that the blogger refuses to post it, but instead will just speed things along and post Oakley’s comment here.  As indicated in Oakley’s comment below, there are learning scientists that support her view in contrast to the above quoted statement.

I learned Russian at the Defense Language Institute–one of the best language learning facilities in the world.  The learning practices there included plenty of drill and yes, rote memorization, along with conversation and application.  When I applied those same techniques to learning math when I started trying to learn remedial high school algebra at age 26, it worked beautifully. As well it should, since the Defense Language Institute approach involves virtually all aspects of what we know about how to learn a language–or any subject–well. 

My own daughters received plenty of drill practice of math in the ten years I had them in Kumon mathematics, from ages 3 to 13.  According to the precepts of the above blog post, my daughters should hate math.  Instead, their early wobbly dislike for math turned into expertise and enjoyment–just as with a child who spends not-always-fun time with piano practice can grow into an adult who treasures her ability to play the piano.

The vast majority of my colleagues in engineering are from countries that emphasize rote approaches to learning math.  Their early rote training certainly didn’t kill their desire or ability to excel in analytical topics. Virtually every subject we know of, from language learning to playing an instrument, to learning in math and science, proceeds from a basis of solid mental representations developed through practice and procedural fluency that minimizes cognitive load while maximizing access to neurally embedded information. (See Anders Ericsson’s seminal work on the development of expertise.)

Many schools in the US been strongly discouraged by previous policies of the NCTM from encouraging students to develop any type of procedural fluency or deeply embedded sets of concepts, such as multiplication tables. (Yes, even memorization of multiplication tables helps children to develop pattern as well as number sense.) I still remember the student who came up to me after flunking a statistics test, saying “I just don’t see how I could have flunked this test–I understood it when you said it in class.”  We’ve gone so overboard with the value of conceptual understanding that students think it’s the golden key–they don’t need to practice to build, maintain, or enhance their “understanding.” And of course, what a student thinks they understand is often only a glimmer of the real understanding that comes from plenty of interleaved practice. In reality, procedural fluency and understanding proceed hand-in-hand.  See Rittle-Johnson, B, et al. “Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics.” Educational Psychology Review 27, 4 (2015): 587-597.

Note that the above blog post cites only one-sided articles in support of the author’s message–no mention is made of the researchers and results cited in the original op-ed, or other meaningful, solid research such as the following, that rebuts the author’s assertions. See, for example:

Morgan, PL, et al. “Which instructional practices most help first-grade students with and without mathematics difficulties?” Educational Evaluation and Policy Analysis 37, 2 (2015): 184-205.

[Morgan notes that music, movement, and manipulatives are fun, but the basics, with explicit instruction and plenty of worksheet practice, are best for struggling math students.]

Geary, DC, et al. “Introduction: Cognitive foundations of mathematical interventions and early numeracy influences.” In Mathematical Cognition and Learning, Vol 5: Elsevier, 2019.

Geary, DC, et al. “Developmental Change in the Influence of Domain-General Abilities and Domain-Specific Knowledge on Mathematics Achievement: An Eight-Year Longitudinal Study.” Journal of Educational Psychology 109, 5 (2017): 680-693.

Neuroscience and cognitive psychology are doing a great deal to advance our understanding of what is necessary to excel in a given subject.  I would hope that educators in mathematics would open their eyes to new and relevant insights from these disciplines, and realize that their desire to always make their subject fun (something that teachers of virtually every other subject realize just isn’t possible), results in disempowering the very students they mean to help.

Stop worshipping conceptual understanding

Greg Ashman also writes about conceptual understanding in math and does a good job of it.

Filling the pail

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There is a story often told about maths teaching. It is a story of how, in olden times, children were taught rote mathematical procedures. They were never taught conceptual understanding of the principles involved. These days, we have computers to perform mere procedures for us and so, instead, we should focus on conceptual understanding.

This is flawed logic.

Take the principle of equivalence. This is an idea that is often investigated in educational psychology experiments as an example of conceptual understanding. When children first meet mathematical equations, they are of the form 2 + 3 = ?. This means that they reasonably, but incorrectly, infer that an equals sign (=) is a command to write a correct answer. In fact, an equals sign means ‘the same as’ and a failure to grasp this may cause problems later when students have to solve problems of the form 2 + ? = 5.

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