There must be something deeply attractive about the idea that children’s math success can simply be forced upon them. On Aug 7, yet one more opinion piece promoting drill and memorization in children’s math education was published in the New York Times. By the morning of Aug 8th is was trending in the #1 spot. In “Make Your Daughter Practice Math, She’ll Thank You Later,” Barbara Oakley argues for the importance of math in the lives of children. Unfortunately, she does so in a way that is fundamentally misinformed about both the landscape of K-12 mathematics education in the United States and the research-based consensus on learning theory and cognitive development. Also unfortunately, when an opinion piece is written by a professor, even though it is an *opinion* piece, the assumption is that it is based in research. People take it as truth. The stakes of this misconception are extremely high when talking about the education and futures of of our children.

Dr. Oakley makes two points that are valid:

- Having a solid foundation in math can be fundamental to a child’s future, especially in the world of STEM. Mathematics is powerful as both gatekeeper and gateway when it comes to higher education, and employment. [I would add also, democratic participation in civil society as well as many creative endeavors.]
- Research shows that all children, regardless of gender (or race, ethnicity, or class, or anything else) have essentially equivalent innate potential in mathematics such that outcomes are shaped by societal inputs and expectations, not by any biological or innate differences between groups.

She makes a third point that is also correct, but not in the way she intended it. She says that the way we teach math in the United States is harmful to all students and especially girls. 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. However, 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. The United States is behind other countries on international measures of mathematics performance (including PISA and others) because of an *overemphasis* on procedural, drill-based approaches to math *at the expense of* conceptual understanding. It is conceptual understanding that is the basis for complex problem-solving, a critical component of 21st Century STEM literacy. Yes, mathematics educators and education policy-makers have been working to shift the way we teach math in the United States toward practices that emphasize conceptual understanding in tandem with procedural fluency. Unfortunately, as a country, the United States K-12 education system is still a far stretch from a system that “downplays practice in favor of emphasizing conceptual understanding.”

Yes, children should have opportunities to mathematize the world around them through conversations with teachers, parents, other adults, and each other on a consistent basis. But these conversations should be based in creative and flexible work with numbers that will build number sense, pattern recognition and fluency in composing and decomposing numbers. This is distinctly *not* achieved through drill or rote memorization. Number theory is not introduced through recitation of the multiplication tables. Furthermore, research shows that it is precisely these drill-type tasks that teach children that math is a dull, meaningless activity, and turn many away from math at an early age, including and especially girls.

Finally, the attitude that “foundational patterns must be ingrained before you can begin to be creative” is the root of inequitable mathematics education throughout the United States. The belief that children cannot engage in meaningful and creative problem-solving tasks until they have learned how to be compliant calculators leads to education that is training for menial labor, labor that was long ago replaced by calculator technology. “More drilling,” teaching your daughter that “all learning isn’t – and shouldn’t be – fun,” and that, as a girl, there are certain things she should simply suffer through “even if she finds it painful,” will *not* set her up for success. Anyone who teaches children that they need to silently comply through painful experiences before they will be allowed to let their brilliance shine has no intention of ever allowing that brilliance to shine, and will not be able to see it when it does.

EDIT: A number of people have been wondering about the “research-based consensus” that I refer to above. I am in particular referencing the extensive bodies of research that came out of the research program of Cognitively Guided Instruction as well as the research produced through the QUASAR Classrooms project. I have linked each to a relevant “sample chapter” or Google Book since many of the academic articles are behind paywalls. For people who have access to academic journals, either search term should produce dozens of relevant peer reviewed articles.

*Thank you to everyone who has commented, including the comments that I have not posted. I am closing comments at this point (8.30.18).

Hi, just stopping by to say thanks for this amazing post. The last sentence (and especially that last clause) is such a perfectly constructed rebuke to an entire dehumanizing school of mathematics instruction.

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Thank so much Dan for reading and for your comment.

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Reblogged this on Her Mathness and commented:

This post was written by my dear friend and former colleague, current Standford graduate student, Emma Gargroetzi. I’m glad she wrote it for me! ; )

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Yes! Well written! Sharing widely!

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This brought tears to my eyes! I want every educator, regardless of subject, to read this! Last year was my first year teaching math (technically a tutor). At our district we have an amazing coach. She introduced us to 3 act math, open middle, and productive struggle. I learned so much! Using these techniques allowed for students with any gap to access the mathematician within. I never knew that you could be creative in math. I believed it to be black or white; wrong or right. This approach only allowed students that were “good at math” to excel and grow. I am so glad to be enlightened so early in my career!

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The notion of mathematizing our lives in ways that celebrate culture and identity…just magnificent. I call this Eq-STrEAM educational responses: positioning equity and representation and creative arts in STEM education discourse is the only way we are truly going to get there. Paying attention to the language we use and the canons we select as significant (especially media outlets) is key. The NYT is among the many canons the whole world uses to get the news. This piece is wonderful! Thank you. I will be sharing in my little corner of the universe.

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A perfect rebuttal and argument to people who know a lot of math but not about how kids really learn. Thank you for taking the time to craft it. I will be sharing as well!

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Thank you so much for your response. I do hope others read it.

All I could do as I read this was nod my head and say thank you!

We had a brief but interesting chain of responses to the op-ed piece on MY NCTM. I think I will share your piece to see if we can stir up some more conversation as ALL math teachers K-12 should be involved in this conversation.

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Could you provide a reference for stating that there is currently an overemphasis on precedural vs conceptual? Conceptual is all the rave and all I see so called math gurus gush about on Mtbos

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Hi Chad,

Given my statement that Dr. Oakley’s understanding of the landscape of K-12 mathematics education was not informed by research, I appreciate you asking for a reference for my own claims. You can find a thorough overview of the landscape of mathematics education in the United States in this relatively short 150 page overview called “Mathematics Education in the United States, 2016: A Capsule Summary Fact Book” (Dossey, McCrone & Halvorsen with the Nationa Council of Teachers of Mathematics and the United States Commission on Mathematics Instruction). It’s accessible online here: https://www.nctm.org/uploadedFiles/About/MathEdInUS2016.pdf.

Here’s a quote from page 31: “According to an industry survey, 78.8% of U.S. schools in 2014–15 reported using a basal mathematics series that they either follow very closely (41.7%) or from which they pick and choose (37.1%) as needed. The combined 78.8% using the textbook at least as a source is significantly lower than the combined percentage for the same two questions in prior years: 94% in 2001, 93.4% in 2005, and 88.1% in 2011 (Resnick and Sanislo 2015).”

A “basal mathematics series” usually refers to a skills-based workbook series. This quote is in reference to grades K-2. Notice that in recent years, in the lower elementary grades, schools have gone from near complete reliance on these procedural workbooks to only ~80% of schools relying on these materials. Lower elementary grades are the place where conceptual understanding has been most profoundly taken up by teachers. The shift toward conceptual understanding is seen less and less as you go up in grade level. There is a lot more great (or not so great) information in the book itself if you have time to parse through it.

I think if you asked most people in the MTBoS community what the landscape of mathematics looks like to their left and right they would agree that it’s a constant and uphill battle to find ways to center meaningful, conceptual, and creative experiences with mathematics in their classrooms in the face of pressure from many directions to do more drill and procedurally oriented work with students.

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I learned Russian at the Defence 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 on trying to learn remedial high school algebra at age 26, it worked beautifully. As well it should, since the Defence 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.

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Professor Oakley,

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.

Best,

Emma

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Thank you, Emma. I think it helps to remember that people on the “practice helps make perfect” side aren’t suggesting that all math must be boring and repetitious. They are just suggesting that, as with any subject, acquiring the fundamentals is a necessary path to expertise. My question for you is, have you read the many references I’ve cited, including those of Ericsson, Bjork, Morgan, and Geary? (Another good reference is that of Dr. Wu at Berkeley, “”Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education.”) Because if you’re saying I’m not familiar with the literature, but you haven’t read those vitally important papers in a way that you can synthesize and cogently respond to their points, doesn’t that mean you might be doing the same thing you are saying that I’m doing?

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Professor Oakley,

A few thoughts:

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.

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.”

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.

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.

Best,

Anand

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It is indeed a mistake to see basic skills and conceptual understanding as a dichotomy, but there is no doubt that there is a dichotomy between instruction that relies heavily on drill and instruction that provides opportunities and guidance for students to make connections between mathematical ideas and understanding the logic behind mathematical procedures. As someone who has not only read most of the references cited but published in the field myself, I have to say that throwing out “supporting” citations without looking closely at what the results were and what they mean for instruction is misleading at best.

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It is completely incorrect to say that I have thrown out supporting citations without looking closely at what the results were and what they mean for instruction. I have carefully read each citation I have provided, along with many other related citations. I know deeply whereof I write. If you had indeed read the relevant citations, you would know that it is misleading at best to characterize them as one-sided proponents of drill without including understanding–a typical mischaracterization of traditionally taught math.

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