Access to Algebra 1 in 8th Grade; the Never-ending Story

“When students take Algebra I matters, but many students do not have early access.”
The Department is encouraging both access to and enrollment in STEM courses. Both aspects are important because, as we will see through the story, even where access to Algebra I classes are available students do not necessarily enroll in them.
The story is mostly grounded in civil rights issues and equity for all, but ignores a key factor in all this: Proponents of the Common Core math standards take a “Common Core wanted it this way” attitude, citing that the standards call for algebra in high school, but not in eighth grade. They take this stance despite Common Core allowing for such option as addresed in the Appendix to the math standards:
A “compacted” version of the Traditional pathway where no content is omitted, in which students would complete the content of 7th grade, 8th grade, and the High School Algebra I course in grades 7 (Compacted 7th Grade) and 8 (8th Grade Algebra I), which will enable them to reach Calculus or other college level courses by their senior year. While the K-7 CCSS effectively prepare students for algebra in 8th grade, some standards from 8th grade have been placed in the Accelerated 7th Grade course to make the 8th Grade Algebra I course more manageable;
But such words do not matter. Algebra continues to be the forbidden fruit of education, reserved for those whose parents can afford to have their kids learn it outside of school–or have enough clout to get their kids in to 8th grade algebra programs when they are offered.  As I wrote about here, the San Francisco school district  did away with algebra in 8th grade. Jo Boaler and Alan Schoenfeld wrote an article in the San Francisco Chronicle, lauding this decision, and stating:
They  (i.e., San Francisco USD) found a unique balance that is now seen as a national model. They decided to challenge students earlier with depth and rigor in middle school. All students in the district take Common Core Math 6, 7 and 8, a robust foundation that allows them to be more successful in advanced math courses in high school.  
Again, an example of an inflexible interpretation of the Common Core Math Standards. And as I discussed in the referenced post, the San Luis Coastal Unified School District limits access to algebra in 8th grade by making it available to the “truly gifted”–a term that went undefined and which I heard  uttered by an official of that school district.  They determine the “truly gifted” by requiring students to receive high scores on two tests given in the 7th grade.  One test has been around for a while–a multiple choice test developed by two universities that did a good job in determining the students who were ready for algebra. 
With the advent of Common Core, the District decided to institute a second test, developed by an outfit called the Silicon Valley Math Initiative (SVMI). The test consisted of questions that in my opinion, were appropriate for formative assessments but not for summative. It did the job, however, and many students were suddenly deemed unqualified (i.e., not “truly gifted”) to take algebra 1 in eighth grade. (Assuming that one has to be “gifted” in order to take algebra in eighth grade; I do not believe giftedness is a necessity for it.)  In the 2015-16 school year only 17% of students took algebra in 8th grade: 88 out of 517, down from about 300 students in 2013.
The report from the Dept of Education is timely.  It is correct that civil rights issues are important, I think the problem goes beyond civil rights. Namely, one no longer needs to be in a minority to be stuck with inferior programs and goals.

The Flawed Approach of Traditional Math, Dept.


A May 29, 2018 article about Common Core from the Yale Tribune,  references another article that appeared in the Washington Post and Chicago Tribune by Jessica Lahey. It summarizes her views about the bad rap she and others feel is being given to Common Core’s math standards:

She believes that the gap between parents and students does not necessarily lie on the Common Core itself, but on the flawed approach of conventional math education where students were taught to memorize and dutifully accept axioms and mathematical rules without completely understanding its application and the principles at work.

This view is shared by many and has become the hobby horse of the math reform movement that gained significant traction with the National Council of Teachers of Mathematics’ (NCTM’s) math standards, first published in 1989 and subsequently revised in 2000.

I have written about the mischaracterization of conventional –or traditional–math many times.   My main message is that the underlying concepts were in fact taught, and students were then given practice applying the various algorithms and problem solving procedures.  I have provided evidence of such explanations in excerpts from the math textbooks in use in the 20’s through the 60’s. Yes, the books required practice of the procedures, but they also showed the alternatives to the standard algorithms.  These were presented after mastery of the standard algorithms as a side dish to the main course.  These alternative methods are by and large the same methods that are taught today under the rubric of “Common Core Math”.  The difference is that the alternatives are generally taught before the standard algorithm in the belief that teaching the standard algorithm first eclipses the understanding of the “why and how” of the procedures.  Delaying the teaching of the standard algorithm by requiring students to use inefficient and often confusing techniques (in the name of “understanding”) can result in a confusion of what is the side dish and what is the main dish. The beauty and simplicity of the standard algorithm is lost among a smorgasbord of techniques that leave students more confused than enlightened.

In short, the ideas expressed in the two articles referenced above represent the groupthink that pervades education schools and other forms of the education establishment. The prevailing mode of thought views drills, practice and the learning of procedures as “rote learning” and prevents true “understanding”. If students “understand”, then everything else follows–the corollary of which is that understanding must come before procedure.

What is left out of such pronouncements is the difference between novice and experts. There are levels of understanding as one goes through school, and depending where one is on the spectrum between novice and expert, the level of understanding may be deep, shallow, or in between.  Procedural understanding is a level of understanding, but students who are at such level are sometimes referred to as “math zombies”.  This term is is relatively current but is what Lahey and others think is the end result of “conventional math.”  And unfortunately, their view seems to rule the roost.

For more on the notion of “understanding” and traditional math see here, and here.  Tell your friends.  Then hire a tutor.



Spinmeister, Dept.

San Francisco’s Unified School District decided to eliminate access to algebra for 8th graders even if a student is qualified to take such a course.  The latest article to justify the action is one written by Jo Boaler (whose self-styled approach to math education in my opinion and the opinion of many others in education who I respect has been ineffective and damaging) and Alan Schoenfeld, a math professor from UC Berkeley whose stance is consistent with math reformers. I.e., “understanding” takes precedence over procedure, among other things.

The article states:

“The Common Core State Standards raised the level and rigor of eighth-grade mathematics to include Algebra 1 content as well as geometry and statistical topics previously taught in high school.”

This is not true. A high school level course includes rational expressions (i.e., algebraic fractions), polynomial division, factoring, quadratic equations, and direct and inverse variation. The 8th grade standards do not include these. I teach an 8th grade math class as well as high school algebra for 8th graders. The latter is far more inclusive. Elimination of access to algebra in 8th grade is certainly not strengthening math ability for those students who are qualified to take such a course.

The article also states:

They  (i.e., San Francisco USD) found a unique balance that is now seen as a national model. They decided to challenge students earlier with depth and rigor in middle school. All students in the district take Common Core Math 6, 7 and 8, a robust foundation that allows them to be more successful in advanced math courses in high school.  The key is conceptually rich courses that benefit everybody, including those who go on to STEM majors in college. In-depth instruction helps all students and provides a more solid base for later math courses. All students get a solid foundation, and acceleration is offered in the 11th and 12th grades.

Translation:  For those students who wish to take calculus in 12th grade, they can double up math courses in 11th grade, so they can take Algebra 2 and Precalculus.  As far as what they mean by “conceptually rich courses that benefit everybody”, it’s anybody’s guess. I work with the textbooks that adhere to the CC standards for 6th, 7th and 8th grades.  I supplement freely with a pre-algebra book by Dolciani written in the 70’s and other materials.  The emphasis on ratio and proportion in 7th and 8th grades is rather drawn out and can be done more concisely, rather than harping on what a direct variation and proportional relationship is. Traditional Algebra 1 courses present direct variation in a much more understandable way, rather than the “beating around the bush” technique that defines such relationships as straight line functions that go through the origin, and whose slope equals the “constant of variation/proportionality”.

So much time is spent on trying to make the “connection” between slope, unit rate, rate of change and constant of variation, that students think they are all different things and are largely confused.  While Boaler and Schoenfeld may say that the confusion arises because teachers don’t know how to teach it, I assure you I know how to teach it. I use an algebraic approach in an algebra class, when students have the algebraic tools with which to grasp the concept more easily.

But the real goal of San Francisco’s elimination of algebra in 8th grade is to close the achievement gap as evidenced by the last paragraph in the article:

Groups that traditionally underachieve — for example, students of color, female students, students of low socioeconomic status, bilingual students and students with special needs — have all experienced increases in achievement. We congratulate the district for its wisdom in building course sequences that serve all students increasingly well.

For those students whose parents can afford it, they take algebra elsewhere in 8th grade and circumvent the system.  Those whose parents cannot afford outside help are stuck with what Boaler and Schoenfeld, and the SFUSD think is equity for all.

Scaffolding Dept.

The late Grant Wiggins was adamant about “authentic problems” and “authentic problem solving”. He felt that scaffolding problems was a cheat and that it short-circuited understanding. That is not the case.

In solving word problems, worked examples provide students a direct access to solving problems that are similar, and in the same category. By scaffolding such problems–that is, varying the problems slightly beyond the initial worked example–students are forced to stretch and to make connections.

Students do best with explicit instruction, starting with simple problems. They then begin to develop the knowledge and skills to solve increasingly more difficult problems with novel twists. Without explicit instruction in problem solving, many just give up and don’t try the problems. Students benefit by seeing how to think about the problem before actually working it. Imitation of procedure therefore becomes one of imitation of thinking.

While people may criticize this as mere imitation and rote learning it is not. As anyone knows who has learned a skill through initial imitation of specific techniques such as drawing, bowling, swimming, dancing and the like, watching someone doing something and doing it yourself are two extremely different things. What appears easy often is difficult–at first. So too with math.  Imitation of thinking is a level of understanding as one goes up the scale from novice to expert.

For example, students may be shown how to solve this type of problem: Two trains, 360 miles apart, head toward each other, one going at 100 mph and the other at 80 mph. How long will it take them to meet? The student can be shown that the sum of the two distances represented by 100t and 80t (where t is the time traveled by each train) makes up the initial 360 miles. A variation of this problem is: After the trains pass each other, how long will it take for them to be 90 miles apart? In this case, the same concept is at work: the sum of the two distances represented by 100t and 80t makes up the future distance of 90 miles.

In the words of Dylan Wiliam (Emeritus Professor of Educational Assessment at the University College of London Institute of Education): “For novices, worked examples are more helpful than problem-solving even if your goal is problem-solving”


Previously, about Common Core

I wrote a series of articles for Heartlander, that over the years have changed URL locations. My dedicated readers (as well as those who intensely dislike what I have to say) therefore cannot find what was once readily accessible.

For those of you who wish to revisit what I have said about Common Core, the articles are here:

And for those of you who missed it, my talk on math ed in the US is here, with my comments on Common Core starting at minute 19:24.

Conversations I’ve Heard Enough Of, Dept.

The Kansas City Star published a column in defense of Common Core’s math standards, containing the usual rhetoric–to wit:

I was recently part of a conversation about education. It was a social media conversation, intended to bash the alternative strategies of teaching math.

The strategies have been caught up in the term “common core,” but are actually teaching methods designed to help kids reach common standards.

I offered an alternative viewpoint to the woman’s outrage. I was once told by the faculty at my kids’ school that these learning strategies aren’t designed just to teach the material, they teach the kids to learn. How to analyze. How to understand why math works, not just how to solve a problem.

My positive input was unwelcome in the echo chamber this “new math” naysayer had created, and she formally dismissed me from the conversation. She wanted solidarity in her outrage, not to see the information in a new light, from a different angle. She did not want to acknowledge the benefits. She did not want to learn synonyms for her limited math vocabulary.

Interestingly, the writer feels that she is on the outside looking in, that her views on math education hold validity, while the “conversation” about Common Core math is dominated by the unenlightened. (I put “conversation” in quotes because it has become one of those trendy words like “relationship”, “narrative”, and “nuance”;I would be all too happy to see journalists given jail time for using it.)

The points she makes are easily dismissed. Some might think I’m wasting my time dismissing them and I would tend to agree but for one thing.  She isn’t the only one who thinks this way, and I have met many in education who espouse such views.  And in particular, such views are not only espoused but taught in schools of education to future teachers who then embrace and implement such thinking.

First off, she bolsters her thesis with the tired old arguments that the future may require knowledge that isn’t being taught–which is analogous to the bromide that the future consists of jobs that haven’t yet been created. Those who make such argument posit that basic knowledge is apparently useless in the face of what we will be required to know in the future–and if you need to know something, just Google it.  To wit, again:

What’s imperative is that future generations must be adaptable. What they learn today may or may not (leaning heavily toward the may not end) apply. … We were raised by people who cling to The Correct Way of solving an addition problem. History matches the books supplied by our schools. Spelling must be memorized and written out in cursive. And now we raise our kids in a technology-rich environment that changes on the fly. Kids don’t need to solidify a bunch of facts in their minds spanning myriad topics — in case they need that information in the future. They merely need to learn how to learn and leverage the tools they have. A generalist who knows how to find specific information will be as effective as a specialist with a narrow body of knowledge.

Barbara Oakley, a professor of engineering at Oakland University in Michigan, wrote a book called “Learning how to Learn”  in which she describes techniques one can use to succeed in difficult subjects. She does not hold memorization in disdain, nor learning facts, nor practice. Recently she wrote an op-ed that appeared in the New York Times about the value of practice and memorization in becoming proficient in math–and was castigated in comments that followed as well as in blogs for what was characterized as narrow-mindedness and resistance to innovation in education. The criticisms bore a resemblance to the Kansas City Star columnist’s view of having to deal with the great unwashed.

The Star’s column is typical of the “conversation” about math education in the US. As such, it is nonsense. One doesn’t learn to think critically, or to be creative, without some basic knowledge. And that basic knowledge isn’t something that is dug up on a just-in-time basis. Knowledge is the basic tool with which to think critically; without it, you have nothing to think critically about. Learning how to learn requires some amount of memorization–and memorization allows one to reason with that information. It is not “rote learning” as traditional education is frequently mischaracterized.

As far as Common Core is concerned, one can interpret the standards in various ways, but the prevailing interpretation seems to be dominated by those who believe as the columnist does that students must “understand” math — otherwise they are “math zombies” who “do math” but do not “know math”.

My comments on math education in general and Common Core in particular can be found here: . If you don’t wish to suffer through a half hour of someone whose belief system may be opposed to yoursyou might want to skip to minute 19:24 where I talk about Common Core. I can assure you, however, that if you don’t agree with my views, you won’t like my comments about Common Core either.

Zombie arguments

Worth reading and commenting on

Filling the pail

At the start of August, Barbara Oakley, a Professor of Engineering, wrote an op-ed for the New York Times about maths teaching. It is a largely sensible piece about the fact that improvement in maths requires practice, not all of which is particularly pleasant.

Oakley’s article is still creating a stir in the maths blogosphere. Dan Meyer, a education software designer*, has now devoted two blog posts to attacking it. This is notable because Meyer has recently appeared to withdraw from the maths wars, having once been a constructivist/reform/inquiry/fuzzy maths partisan (see his 2010 TED talk and his blog motto, ‘less helpful’).

Meyer’s latest blog post drags up ideas about ‘conceptual understanding’ and ‘maths zombies’ that will be familiar to those of us who have been around the block a few times. I have written a lot of posts about conceptual understanding and it’s actually quite…

View original post 797 more words

Count the Tropes, Dept.

Another in one of an infinite series of articles that has appeared at Forbes and other like publications about how the Common Core math standards are being maligned has surfaced. I gave up counting the tropes in this one.  The author makes the assumption that because students were unable to apply standard algorithms or “rote” procedures to “advanced math” (left undefined in the article), that teaching procedures is tantamount to “no understanding”.  There is no mention of the difference between how novices or experts think–and in fact, going out on a limb here–I would venture to say that the author of this piece probably learned things in the manner that he now derides.  He makes this statement:

There is nothing in the standards that specifically requires any instructional strategy to be taught. There is a great deal of academic freedom for teachers to teach in whatever manner they wish. The CCSS does not require this kind of new “number sense” methodology.  What it does require is that students learn multiple ways to attempt to solve a problem, to promote strategic critical thinking and creative problem solving.

I might ask where in the Common Core standards does it require that? The Standards of Mathematical Practice?  Maybe, if you interpret it that way.  The CC standards also have a standard requiring that students learn the standard algorithm for multidigit addition and subtraction by 4th grade. This has been interpreted to mean that the standard algorithm should not be taught until then, but both Jason Zimba and Bill McCallum, lead authors of the CC math standards have stated publicly that the standard algorithm can be taught earlier than 4th grade.  In fact, Zimba even recommends that students in the first grade be taught the standard algorithm.

Once the standard algorithm is taught, then alternatives to it make more sense than the other way around. In that vein, multiple methods can occur if the sequence of topics is done in a logical order.  I’ll go out on another limb and say that this is mainly because procedural mastery is more easily obtained via the standard algorithm than by pictorial or inefficient methods.  Ironically, students who have mastered the standard algorithm often find short cuts themselves.  Or they are taught the short cuts after mastery of the standard algorithm. In any event, teaching “number sense” and top down understanding before the foundation of procedural mastery is obtained seems to be the go-to advice of reformers.  Such approach makes the same mistake twelve-year-olds make when they assume they can achieve stardom in whatever field they daydream about, without the requisite practice and work.

Greg Ashman gives a good example of this in his blog:

If you have been practising column subtraction and you are presented with 2018-1999 then, even if you could work it out by counting up, you will probably draw on the most familiar strategy because you lack the bandwidth to analyse it at the meta level. If I really felt it was important for students to be able to determine when counting up is more efficient than column subtraction then I would teach this to students explicitly. I would use contrasting cases where I presented two examples such as 3239 – 2747 and 2018 – 1999 and I would work both problems, both ways, drawing a distinction between the two.

…[T]he idea that there are teaching methods that are better for conceptual understanding than explicit teaching is equivalent to the idea that there are teaching methods that can short circuit the novice-expert continuum and teach expert performance from the outset. This is the progressivist dream and one that, after a couple of hundred years, is yet to be realised.

This article originally appeared in Quora, and was picked up by Forbes. If you are interested, I wrote a comment on the Quora piece which addresses other aspects of his arguments.

Comments Gone Missing, Dept.

SteveH wrote a long comment addressing many of the statements Dan Meyer and others made in comments on a blog post that I talked about in a previous post.  

I saw SteveH’s comment this morning, but then this afternoon it was gone.  I don’t offer any theories for its disappearance, but instead offer my readers the original comment to digest and think about  He opens his comment quoting something Meyer said in an earlier comment and then responds:

The quote from Meyer:

“They’re making an argument about what students can’t or shouldn’t do before submitting to pain. They’re suggesting a precedence and a prerequisite. That’s the argument that needs addressing and dismantling.”

SteveH’s response:

Can we ever get past this cherry-picked strawman? You know that’s not the position that many of your critics take. All traditional math textbooks and teaching methods introduce concepts first in a carefully-built scaffold. Then comes the  homework to get individuals to better understand the subtle variations that are far more mathematically meaningful than basic concepts. Even basic skills require subtle understandings. Very few things are ever rote. This deeper level of understanding has to be carefully constructed on a unit-by-unit basis over years using individual problem sets. That’s what all STEM-prepared students get with AP and IB math sequences.

I’m open to seeing other opt-in sequences for those who might have other beliefs, but drill-and-kill, if it ever existed, has been gone from K-6 for decades without any opt-out options. Where are the results? I had to help my son at home with math when his schools foisted MathLand, and then Everyday Math on him, but once he got to high school, I didn’t help one bit. He just got a degree in math. All of his STEM-prepared friends had to get help at home or with tutors in the early grades. When I grew up, I got to calculus in high school with absolutely no help from my parents. That’s no longer the case with full inclusion and curricula like “trust the spiral” Everyday Math.

All properly-taught math sequences start with concepts, and engagement and curiosity are no magic wands for ensuring mastery. What works is what we see for AP and IB math. I don’t see STEM-prep success cases any other way. The problem now is that students in lower grades are stuck with a CCSS slope to no remediation in College Algebra and that the only ones who make the nonlinear transition to proper high school math have to get help at home and with tutors in the lower grades. If that doesn’t happen, it’s all over and no amount of engagement and concepts or “Pre-AP” math will fix it.

This has never been a question about basic concepts. it’s a question about eliminating low expectations and ensuring proper mastery and mathematical understanding on grade-by-grade basis that keeps all math doors open for each individual student for as long as possible. With curricula like Everyday Math, schools trust the spiral and have abdicated all responsibility of mastery beyond the low CCSS slope. Just ask us parents of your best students what we had to do at home. Engagement and curiosity was not my main focus, but that didn’t stop him from playing with GeoGebra for hours at a time – something that was neither necessary or sufficient. Mastery increased his curiosity, not the other way around.

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:

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.