Wish List, Dept.

I have written a number of entries regarding “understanding” in math. I have discussed various misunderstandings about understanding in math.  There are two statements I haven’t addressed, which for me raise many questions.

I have heard many people express the thought that “Calculation is the price we used to have to pay to do math. It’s no longer the case. What we need to learn is the mathematical understanding.”

And often on the heels of this statement I will be told that they had done well in math all through elementary school, but when they got to algebra in high school they hit a wall.  Or, similarly, they did great in high school, and hit a wall with calculus.

There is much information that we do not have from such statements.

  • Was the education they received really devoid of any kind of understanding and all rote? 
  • Are there people who get A’s in math in high school who are really math zombies and cannot progress to the next level?
  • Are these complaints limited to those who were educated in the era of traditional or conventionally taught math?
    • And of those, how much of what they experienced is due to concepts not explained well, emphasis on procedures only, and grade inflation?
    • Are there gaps in their math education which compound on themselves?
    • And to what extent are these problems the result of the obsession over understanding?

Considering these questions, I have listed some ideas for future studies based on communication I’ve had with people in math education:

  • To what extent does success in high school math programs correlate with success in higher level math and science courses in college? (Differentiated by regular track vs AP/IB/honors track)
  • For successful math students in high school, and college math what did they do that’s different than those who were successful in high school but did not do well in college math?

And a corollary of such a study would be:

  • What percent of the student population has had math tutors, or been enrolled in learning centers? 
  • And for such students what are the basic teaching techniques used for math?

Finally, two more:

  • What effect has the emphasis on understanding been on students who have been identified as having a learning disability?
  • And a more difficult question, is there evidence that such emphasis has resulted in students being labeled as having learning disabilities?

I of course am interested in any studies you may know of that would shed light on these questions.


How is Understanding Measured?

I have written about understanding in math, and the education establishment’s view of it. With all this talk about how it is important for students to “know math” and not just “do math” the question arises: “How do we measure a student’s understanding as opposed to their ability to go through procedures?” That is, how do we differentiate someone who truly understands from that of a “math zombie”.

In my opinion, the most reliable tests for understanding are proxies involving procedural fluency and factual mastery but which involve some degree of mathematical reasoning.  Here’s an example.

On a placement test for entering freshmen at California State University, a single item on the exam correlated extremely well with passing the exam and subsequent success in non-remedial college math. The problem was to simplify the following expression. (Multiple choice test):  

Without verbalizing anything or explaining one’s answer, simply recalling the arithmetic properties of fractions along with being fluent in factoring well enough to complete the task correctly was enough for a reasonable promise of mathematics success at any CSU campus. For those who are curious, the answer is (y+x)/(y-x).

Yet, the education establishment often proceeds from the belief that “Getting answers does not support conceptual understanding.” In the teaching of math in K-12, we are seeing more  interest in the process by which students obtain the answers to “authentic” problems. If students really understood, the thinking goes, then they could apply prior knowledge to problem types they have never seen before.

But we have to be aware of level of understanding. Novices don’t think like that. Novices learn how to solve problems from worked examples.  Subsequent problems are varied slightly beyond the initial worked example, forcing students to make connections to prior knowledge. This process is called “scaffolding”.

For example, if we ask what is the perimeter of a rectangle, with sides that are 5 and 7 inches, the student applies the formula he has (yes) memorized: 2W + 2L = perimeter, where W and L represent width and length and comes up with 24 inches. Subsequent problems are variations on this theme:   A rectangle has a side that is 7 inches with a perimeter of 24 inches. What is the length of the other side? … and so forth.

But such scaffolding is sometimes held in disdain, viewed as rote memorization of procedure. To counter this, we have students working on problems that can’t be readily solved by formulas or previous learned procedures. These are called “rich problems”.The best I can do at a ormal tion of “rich problems” comes from someone who disliked “algorithmic” problems: “It’s a problem that has multiple entry points and has various levels of cognitive demands. Every student can be successful on at least part of it.”

My definition might be a bit clearer: “One-off, open-ended, ill-posed problems that supposedly lead students to apply/transfer prior knowledge to new or novel problems that don’t generalize.” (See figure)

For example: “What are the dimensions of a rectangle with a perimeter of 24 units?” A student who may know how to find the perimeter of a rectangle but cannot provide answers to this (and there are infinitely many) is taken as evidence of not having “deep understanding”.  In their view, the practice, repetition and imitation of known procedures as illustrated in the original example about perimeter of a rectangle, and variations thereof, relies on “imitation of thinking”. 

But imitation is key as one goes up the scale from novice to expert. As anyone knows who has learned a skill through initial imitation of specific techniques, such as drawing, bowling, or learning a dance step—watching something and doing it are two different things. What looks easy often is more complicated than it appears. So too with math. The final accomplishment often does not resemble how one gets there. Like playing a game of football or running a marathon, the building blocks of final academic or creative performance are small, painstaking and deliberate.

As the cognitive scientist Dan Willingham has said, only experts see beyond the surface level of a problem to its deeper structure. “But if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots.” Rather, they are making the small steps necessary to develop better mathematical thinking. Simply put, no one leaps directly from novice to expert.

Blast from Past, Dept.

I wrote this 4 years ago regarding a column in USA Today. I commented and got into an argument with Linda Gojak, former president of NCTM. She presents the usual obfuscation and claims as evidence that students lack ‘understanding” because they cannot apply procedures in a variety of different problem solving situations. Well, if you ignore the novice-expert spectrum and put an expectation of expert thinking on novices, then yes, there’s your evidence I guess.

Here’s what I wrote four years ago:

Linda Gojak, former president of NCTM, decides to answer my comment on a comment she made in response to someone else and … Where was I? Well, it was a USA Today article proclaiming that Common Core math is not fuzzy.

Here’s what I said: “Linda Gojak Some understanding is critical, but not all. Sometimes procedural fluency leads to that understanding. It works in tandem. “

Her response: “Barry Garelick : I never implied that students need to understand all methods…they understand and use the strategy that makes most sense to them. Kids who struggle in math tend not to develop understanding through procedural fluency — my mathematically talented students sometimes did — and sometimes did not. What they had in common was that they couldn’t apply the mathematics in a variety of problem situations if they didn’t understand it (at the middle school level) The common core calls for a balance of conceptual understanding, procedural fluency/skills and applying mathematics to a variety of situations which is described quite explicitly in the introduction. Past mathematics instruction has focused too much on drill and kill. plug and chug — and the reality is that we have left too many kids hating math and parents bragging that they were never very good in math.”

Good thing I said “sometimes”. Anyway, I find that the weaker kids who understand procedures can actually do problems. As far as applying mathematics in a variety of problem situations–that’s difficult for all students, even ones in Singapore. But shhhh. Don’t tell Linda Gojak that. Let her “discover” it. It’ll sink in better.

Beliefs About “Understanding” in Math, Dept.

Here are some of many beliefs about “understanding” in math.  It was hard to choose from so many candidates, but feel free to add some of your own.  

We shouldn’t be teaching kids algorithms before they have the conceptual understanding.

The belief is that standard algorithms for mathematical operations (like adding/subtracting multidigit numbers, multiplying and dividing multidigit numbers, multiplying/dividing fractions, etc) eclipse the conceptual underpinning.  That is, why the algorithm works.

The standard way used to be taught first, and alternate ways later, after mastery of the standard algorithm. Now it’s other way around in the belief that std algorithms eclipse “understanding”. Side dishes now become the main course and students grow confused—sometimes profoundly so. 

Problems are to be solved in more than one way, in the belief that doing so imparts and gives evidence of “understanding”.  You have students being required to solve simple problems in multiple ways supposedly to enhance discovery and impart understanding.   You have students drawing pictures for much longer than necessary, serving as both a means to simultaneously understand and explain an otherwise simple procedure.

Many of these are the same methods for operations that are taught in the traditional manner. But these alternative methods were taught after students had achieved mastery with the standard algorithms—and not for weeks on end. And many students discovered these methods themselves.  Adding 56 +68 and being made to say 60 +60 instead of 6 +6 when carrying and adding the tens column for example defeats the purpose of the algorithm which is to free up working memory.

Robert Craigen, a math professor at University of Manitoba describes this approach as “the arithmetic equivalent of forcing a reader to keep his finger on the page sounding out every word, with no progression of reading skill. It amounts to little more than a “rote understanding” for procedures that unfortunately students probably cannot perform for problems they cannot solve.”  

A blogger recently summarized it this way:

Next year’s teachers that are used to students using an algorithm for multiplication are aghast when students use unsophisticated strategies like counting by ones by drawing pictures or partial product by drawing boxes, or when the students seem to not have any idea what to do. “What do you mean, just multiply!” But to “just multiply” by mimicking an algorithm isn’t part of what students had been doing. These teachers shrug in frustration and teach “the only right way”. Students are left feeling either shafted by the previous teacher or, most likely, that they must just not be “good at math”. “

“Students who fail to understand a concept are unable to know how to use it or build upon it. They will end up with misconceptions that can go undetected for months or years.”

How is “understanding” defined?  And what do they mean by failing to use it or build upon it?  Yes, if a student only knows that 3 x 2 = 6 but does not know what the multiplication represents or what types of problems can be solved, then that student will not be able to use multiplication. 

But does lack of conceptual understanding always have this result?  While progressivists all want to teach for “understanding”, they do so without fully defining what understanding is. A definition of “conceptual understanding” does exist; it appeared in the National Research Council’s 2001 report “Adding it Up”:

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

But this has been interpreted to mean that “procedural understanding” is rote memorization and does not entail connections to mathematical ideas.  An example of a student having procedural fluency but lacking conceptual understanding was given in a popular blog about math:

[The student} can accurately compute the area of a triangle, but doesn’t recognize how its formula was derived or how it can be extended to other shapes. (eg. trapezoids, parallelograms, etc.)

and elsewhere, someone referred to calculation as being only procedural:

“Calculation is the price we used to have to pay to do math. It’s no longer the case. What we need to learn is the mathematical understanding.”

Does this mean that a student cannot solve problems that involve areas of various shapes because he or she does not know how the formulas are derived? Is a student who does not know the derivation of the invert and multiply rule for fractional division unable to solve problems involving such operations?  

There is a difference between a novice and an expert. A student who knows a procedure but not necessarily the conceptual underpinning may later gain more understanding as they work with such procedures in solving problems.  And some students may never understand it.  What level of understanding are we talking about here?  Do we expect students to acquire expertise in a top-down fashion, with understanding first and procedure and application later?  Is it wrong to let them solve problems using the standard procedures. Or must we always sacrifice proficiency on the altar of the often undefined but cherished “conceptual understanding” ?

And last but not least, this old chestnut which has appeared in just about every math textbook written from every era you can imagine:

“In the past, math classes were about teaching facts, skills and procedures with no understanding,and mechanized drills.”

Despite this claim, it’s interesting that many adults who were educated in the eras caricatured as “failing thousands of students”, are much more capable at solving the arithmetic problems that today’s students struggle with, even those entering high school.  They are even capable of solving the open-ended, “rich”, depth of knowledge questions that are generally ill-posed one off problems that do not generalize and are assumed to lead students to “deep understanding”.

One example of such a problem is “What are the dimensions of a rectangle with a perimeter of 24 units?” I’ve seen adults who claim they are not good at math get further with such a problem–as ill-posed as it is–than the students judged to lack understanding because they cannot solve them. Apparently, knowing how to calculate the perimeter of a rectangle given its length and width is viewed as “mere memorization”. Subsequent scaffolding of such problems, like “What is the length of a rectangle with a perimeter of 24 and a width of 8?” are viewed as just more memorization. It evidently helped the adults able to come up with answers to the “rich” problem.

Again, we are dealing with levels of understanding along the spectrum of novice vs expert–a spectrum that is conveniently ignored as students are forced to endure a top-down approach to understanding with–it goes without saying but I’ll say it–disastrous results.

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: https://www.youtube.com/watch?v=RlLbXZOoAMU . 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.