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 I know someone else who submitted a comment that went unpublished.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 courseLearning 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.

I was the other commenter on the blog. Here is that comment:

I have three children who practiced math on the weekends because as a trained electrical engineer I could see that the math in school was not setting up my children for success. I have two boys and a girl. My youngest, the girl, had an extra math class with two other girls with me on the weekends from first to fifth grade. We used Singapore Math and in the early years, we focused on the drills and basics. We had fun with the drills. I know confidently that mastery of math skills frees children up to learn algebra, geometry, trigonometry and calculus. The drilling did not kill my daughter’s love of math. I know that the foundation set in those early years set her up for success in high school and college. My daughter is now a freshman at Cornell studying engineering. The focus on the basics in the early years gave her confidence in herself and her abilities.

Additionally, I am a math professor at a Community College. I see far too many students who struggle with Algebra I, Algebra II, Trigonometry, and Precalculus because they do not have mastery of the basic skills.

Students in every subject need to master skills that require practice so that they can be free to do other things. It is a known fact that practice at anything makes you better at it. Practice is not always fun. If you want to be a great soccer player, musician or math student, you will have to practice skills that may be boring. That practice does not kill the love of the subject. The practice enables the student to score a goal, play that challenging piece of music or solve interesting and challenging math problems.

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“practice does not kill the love of the subject”

This is their fundamental flaw. They think, however, that group engagement, creativity, and conceptual (basic) understanding in class magically transforms itself into the ability to do individual homework sets and the subtle and deeper understanding required to solve all different problem variations. It does not. That only comes from the enforcement of individual homework sets from a good textbook.

Everyday Math tells teachers to “trust the spiral” and to keep moving while not enforcing mastery of material at any one time. They believe that their process works and that if you are not successful, then it’s your fault. Skills as a “co-requisite?” No. They assume that skills are the automatic result of their process. The “requisite” and enforcement is not required. If you don’t get the skills, then they will spiral through the material again next year – repeated partial learning. The “co-requisite” onus is placed squarely on the student, parents, and tutors. My view is that this is driven by full inclusion in K-6. All of our home parental mastery help just provides cover for them. I’m sure his school uses my son as a poster boy for Everyday Math.

In music, students have private lesson teachers who enforce skills and understanding, and schools provide band and orchestra for group learning. All musicians who make it to All-State had private lessons for years. Those who just have group band in school do not. In our state’s Solo & Ensemble Honors Concert (selected by adjudication and audition), they list the students’ private lesson teachers in the program, not their band or orchestra director.

Is this the educational model that all schools and teachers want? It’s happening while they are telling us that it’s somehow better with more creativity and understanding. Good teaching is better than bad teaching and enforcing mastery on a grade-level basis is a school requirement, not something you dump on students and parents. Mastery with basic understanding can be fixed, but conceptual understanding without mastery cannot. It’s meaningless. In-class group fun and games that automatically drive mastery as a co-requisite? No. The process is NOT the product.

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Well said Steve

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“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.””

Pain? Strawman. Kids love mastery, and mastery always includes understanding. Besides, in the last 20+ years. I’ve never seen mastery of skills valued as anything close to a “co-requisite” status.

“…everything hinged on “conceptual understanding.””

That’s how it is. I might be more interested in some of the new ideas if they really stressed and pushed mastery of skills as a real “co-requisite”, but I NEVER see that, AND I don’t see how it would work. Where are the examples where students are prepared for a STEM degree program in something other that a traditionally taught math sequence? All of my son’s STEM-prepared friends got extra help at home or with tutors who focused on mastery of skills. They had this help mostly in K-6, not in our traditionally-taught math sequence high school. When my son was in pre-school almost 20 years ago, I wanted something more than what I had when I was growing up in the 50’s. Then I found out that the school used MathLand, a program so bad it’s been wiped off the face of the internet. They got it completely backwards. Then they moved to Everyday Math, which tells teachers to keep moving on and to “trust the spiral.” It doesn’t work. My son’s fifth grade teacher finally had to slow down and enforce mastery of skills.

None of the best math students get that way with a “creative mathematical thought” first process, let alone a co-requisite process. The AP and IB Calculus paths are virtually the only ways to get to a STEM level. One might argue that many other students would be better served with a different approach, but they still would not get to that same STEM-prep level. However, the main argument of these understanding first pedagogies is that it’s somehow better for all. Clearly, they show no examples of that, and reality points to the opposite. They don’t even offer an effective alternate path to those not helped at home in K-6.

The problem with alternate paths is that kids are never expected to achieve any appropriate STEM-level of mastery in K-6 no matter how much understanding is pushed. By high school, STEM is no longer an option and no amount of “creative math thought” will fix that. CCSS starts the slope to no remediation for Collega Algebra in Kindergarten, and Pre-AP math thinks that a proper algebra class in ninth grade will magically get these delayed students through 4 years of high school math in three. Only in dreamland.

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There is little to add here, as Barbara Oakley handled the salient questions extremely well, and Steve H. already addressed the mindless comment of Dan Meyer head on.

Yet I would like to add something to Steve’s comment. Something almost bordering on ad hominem so I apologize up front — it is not ad hominem, but it goes to Dan Meyer’s qualifications.

His comment says:

“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.”

The comment is not about his personal observations how *some* of his students learn, but rather he generalizes how the whole population learns based on his limited exposure as a teacher. Now, Dan has barely a fresh PhD in education, not in cognitive psychology. So why exactly should we listen to Dan with his shallow scientific record and not to accomplished educational and cognitive psychologists like David Geary, Paul Kirschner, or John Sweller, who have studied this issues for many decades and came to essentially opposite conclusion?

There is no question that Dan Meyer is more popular than those serious scientists. But that doesn’t mean he knows even a fraction of what they know. And we, as non-experts, should not forget this elementary truth.

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