Unintended Consequences of Teaching “Habits of Mind” for Algebraic Thinking

(This is a modified version of an article that appeared in Education News on January 28, 2013. )


The idea of whether algebraic thinking can be taught outside of the context of algebra has attracted much attention over the past two decades.  Interestingly, the idea was  raised as a question and a subject for further research in an article appearing in American Mathematical Society Notices which asks,  “Is there evidence that teaching sense making without algebra is more or less effective than teaching the same concepts with algebra?” I sincerely hope someone follows up on this question.

The term “habits of mind” comes up repeatedly in discussions about education — and math education in particular.  Teaching algebraic habits of mind outside of and in advance of a proper algebra course has been tried in various incarnations in classrooms across the U.S.

Habits of mind are important and necessary to instill in students.  They make sense when the habits taught arise naturally out of the context of the material being learned. Thus, a habit such as “Say in your head what you are doing whenever you are doing math” will have different forms depending on what is being taught.  In elementary math it might be  “One third of six is two”; in  algebra “Combining like terms 3x and 4x gives me 7x”; in geometry “Linear pairs add to 180, therefore 2x + (x +30) = 180”; in calculus “Composite function, chain rule, derivative of outside function times derivative of inside function”.

Similarly, in fifth or sixth grade, students can learn to use the distributive property to multiply 57 x 3 as 3 x (50 + 7).  In algebra, that is extended to a more formal expression: a(b + c) = ab + ac.

But what I see being promoted as “habits of mind” in math are all too often the teaching of particular thinking skills without the content to support it. For example, a friend of mine who lives in Spokane directed me to the website of the Spokane school district, where they posted a math problem at a meeting for teachers regarding best practices for teaching math.

The teachers were shown the following problem which was given to fifth graders.  They were to discuss the problem and assess what different levels of “understanding” were demonstrated by student answers to the problem:

The problem is more of an IQ test than an exercise in math ability.  Where’s the math?  The “habit of mind” is apparently to see a pattern and then to represent it mathematically. Another drawback is that very few if any students in fifth grade have   learned how to represent equations using algebra.

Presenting problems like the button problem above prior to a pre-algebra or algebra course will likely result in clumsy attempts at solutions that may or may not lead to algebraic thinking. For example, one student answered the problem as 1 x (11 x 3) + 1, which would be taken as evidence by some that the child is learning the “habit” of identifying patterns and expressing them algebraically.  Another student answered it as 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 34.

Rather than establishing an algebraic habit of mind, such problems may result in bad habits.  An unintended habit of mind from such inductive type reasoning is that students learn the habit of jumping to conclusions.  This develops the habit of mind in which a person thinks that discovering a pattern is the solution and nothing further needs to be done.  Such thinking becomes a problem later when working on more complex problems.

The purveyors of providing students problems that require algebraic solutions outside of algebra courses occasionally justify such techniques by stating that the methods follow the recommendations of Polya’s problem solving techniques.  Polya, in his classic book “How to Solve It”, advises students to “work backwards” or “solve a similar and simpler problem”.

But Polya was not addressing students in lower grades who are on the novice end of the novice-expert spectrum of learning.  He was addressing students who are well on their way to developing problem solving expertise by virtue of having an extensive problem solving repertoire.  For lower grade students, Polya’s advice is not self-executing. Telling younger students to “find a simpler version of the problem” has about the same effect as providing advice on safe bicycle riding by telling a child to “be careful”.

As an example, consider a student who stares blankly at a problem requiring them to calculate how many 2/15 mile intervals there are in a stretch of highway that is 7/10 of a mile long.  The teacher can provide the student with a simpler problem such as “How many 2 mile intervals are there in a stretch of highway that is 10 miles long?” The student should readily see this is solved by division: 10 divided by 2. The teacher then asks the student to apply that to the original problem. The student will likely say in a hesitant voice: “Uhh, 7/10 divided by 2/15?”, and the student will be on his way.  Note that in this example, the problem is set in the context of what the student has learned — not based on skills or concepts to be learned later.

It would indeed be amazing if we could teach students algebraic thinking skills devoid of the content that allows such thinking to occur. I tend to believe, however, that a proper study of this will show what many have known since the time of Euclid: there is no such royal road.

Count the tropes, Dept.

With respect to the poster below, questions abound.

Why is scaffolding only for “diverse learners”? And what IS a diverse learner? And must classroom routines be co-created? Why is teacher authority a bad thing?

Why are students now called “learners”? There really is no need to invent a whole new vocabulary. It may make you think you’re important, but most people see through it. And in the schools I’ve taught, we talk about the students or kids.

And what is with “learner agency”? It used to be “ownership” which was bad enough–now we have “agency”. Of course it pertains to “self-directed learning”. Nothing wrong with students doing things on their own but they do have to receive instruction somewhere along the line. Novices are not experts–but the chart makes no allowance for where a student (or learner) might be on that spectrum.

Finally there’s “productive struggle”. Yes, students should have to stretch beyond initial worked examples. But if they’re struggling, give them some help. Though the chart makers would cringe at this, sometimes students are ready to absorb a direct answer to a question. If so, then just tell them!Edugraphic

Edu-Soap Operas, Dept.

The first in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” is now appearing at Truth in American Education.

Here’s an excerpt:

Various Narratives, Growth Mindsets, and an Introduction to One of my Parole Officers

If you are reading this, you either have never heard of me and are curious, or you have heard of me and have pretty much bought into my “narrative” of math education.

I tire of the word “narrative” (almost as much as I tire of the word “nuance”) which I see in just about everything I read nowadays. I thought I’d charge it rent, so to speak, since it seemed appropriate for the teaching experiences I’m about to describe. I’m currently teaching seventh and eighth grade math at a K-8 Catholic school in a small town in California. Prior to that, I taught seventh and eighth grade math for two years at a K-8 public school in another small town in California, which is where I will start this particular narrative.

It is a one-school district so superintendent and principal were always close by. After receiving praise from the superintendent both formally and informally, I received a lay-off notice. Such notices are common in teaching, with the newest teachers receiving such notices and usually getting hired back in the fall.  Nevertheless mine was final.

It is tempting to make my termination fit various narratives pertaining to the kind of teachers the teaching would like to see less of. Specifically teachers like me who choose to teach using explicit instruction; who use Mary Dolciani’s 1962 algebra textbook in lieu of the official one; who believe that understanding does not always have to be achieved before learning a procedure; who post the names of students achieving the top three test scores; who answer students’ questions rather playing “read my mind” type of games in the attempt to get them to discover the answer themselves, and attain “deep understanding”.  However logical, compelling and righteously indignant such narrative might be, my termination will have to remain a mystery.


Read the rest here.  

The So-Called “Instructional Shifts” of the Common Core and What They Mean

Long-winded Introduction/Preamble

The San Luis Coastal Unified School District is in the central coast area of California. It includes schools in San Luis Obispo and the nearby towns of Morro Bay and Los Osos. The district, under the direction of the current superintendent, follows the trend of  teaching that adheres to constructivist-oriented approaches; i.e., inquiry type lessons, with teachers facilitating rather than teaching. The math text used starting in middle school through sophmore year in high school is CPM, an inquiry-based program.

The district is so much beholden to this philosophy that part of the interview procedure for teaching jobs entails giving a mini-lesson to students, which is in turn rated according to criteria in the Danielson Framework.  The Danielson Framework is (according to their website) “a research-based set of components of instruction, aligned to the INTASC standards, and grounded in a constructivist view of learning and teaching.”

In other words, traditional-minded teachers need not consider applying for a teaching job in the District.  (For followers of my writings, I wrote about two teaching assignments in the District in “Confessions of a 21st Century Math Teacher”.)

I am always curious about math teaching positions that are advertised in the District.  As of this writing, two positions are open. The application always asks for the same essay-type question which I’ve always found intriguing: “Describe your knowledge of the shifts occurring in Common Core State Standards.”

The “shifts” in the Common Core State Standards (CCSS) are not something that are stated as standards. Rather, people who subscribe to the view that the CCSS are game changing, refer to the change of the game as the “shifts”–a change in how math is being taught because of the standards themselves.

Inside Common Core’s “Instructional Shifts”

The “shifts” in math instruction are discussed on Common Core’s website. There are three shifts defined: 1) Greater focus on fewer topics,  2) Coherence: Linking topics and thinking across grades, and 3) Rigor: Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity.

The first shift is a nod to the notion that previous standards and what they covered resulted in curricula “a mile wide and an inch deep” which has been the prevailing criticism of how math has been taught for the past several decades.  It suggests that math has/is taught “without understanding” and succumbs to rote memorization.

The second shift is another attack on how math has been taught, stating that there has been no connection between mathematical ideas, and that topics are taught in isolation–again “without understanding” and using rote memorization techniques.

Which brings us to the third shift, “rigor” to which I want to devote the most attention and focus.  The website translates “rigor” as “Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity.” The site also mentions that students should attain fluency with core functions such as multiplication (and by extension, multiplication of fractions): “Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.”  Again, a nod to the notion that before Common Core, math was taught as a set of procedures “without understanding” using, yes, rote memorization.

This shift has been interpreted and implemented by having students use time consuming procedures that supposedly elucidate the conceptual underpinning behind things like multidigit multiplication, fraction multiplication and other topics.

I learned of the connection between these “instructional shifts” and the current practice of drilling understanding in a conversation I had with one of the key writers and designers of the EngageNY/Eureka Math program. EngageNY started in New York state to fulfill Common Core and is now being used in many school districts across the United States. I noted that on the EngageNY website, the “key shifts” in math instruction went from the three on the original Common Core website  to six. The last one of these six is called “dual intensity.” According to my contact at EngageNY, it’s an interpretation of Common Core’s definition of “rigor.” It states:

Dual Intensity: Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year.

He told me the original definition of rigor at the Common Core website was a stroke of genius that made it hard for anti-intellectuals to speak of “rigorous” in loosey-goosey ways. He was able to justify EngageNY/Eureka’s emphasis on fluency. So while his intentions were good—to use the definition of “rigor” to increase the emphasis on procedural fluency—it appears he is taking the reformist party line of ensuring that “understanding” takes precedence and occurs before learning the standard algorithms or procedures.

In our discussion, I pointed to EngageNY’s insistence on students drawing diagrams to show place value in adding and subtracting numbers that required regrouping (a.k.a. “carrying” and “borrowing”—words now anathema in this new age of math understanding). I asked if students were barred from using the standard algorithm until they acquired “mastery” of the pictorial procedure.

His answer was evasive, along the lines of “Of course we want students to use numbers and not be dependent on diagrams, but it’s important that they understand how the algorithms work.” He eventually stated that Eureka “doesn’t do standard algorithms until students know the prerequisites needed to do them.”

Thus, despite Common Core’s proclamations that the standards do not prescribe pedagogical approaches, it appears their definition of “rigor” leaves room for interpretations that conclude understanding must come before procedure.

What Does This All Mean?

What this means for me is that I do not subscribe to this philosophy. I believe it is injurious to students and defeats the purpose of providing understanding by burdening their overloaded working memories.

I am essentially providing this essay as a public service to anyone who is thinking of applying for the various teaching positions in the San Luis Coastal USD.  If you do apply for the positions, resist the temptation to provide a link to this page when they ask you about the Common Core shifts.

But I think you knew that going in.


Tales of Professional Development, Dept.

Last year, the principal of the school where I taught wanted me and the other math teacher to attend six all day professional development sessions over the course of the school year. According to the flyer advertising the PD the sessions encouraged “collaboration” amongst the math teachers in the county where I taught. It was to be  facilitated by someone who believes that students who are faltering but need just a little more time to get it are lacking some key bit of information. Her solution is “just in time” learning in which the problem dictates what the student needs to know in order to solve it.  I don’t think much of “just in time” learning and have written about it elsewhere so will spare you any rants about it.

I was distressed about having to attend the PD.  My distress was not only because of missing six days of teaching. It was the idea of sitting around with other teachers sharing dubious ideas, including but not limited to the virtues of working in groups, “just in time” learning, differentiated instruction and other ineffective practices that pass as superior to the traditional methods that are derided as “having failed thousands of students”.

So when I learned the PD had been cancelled because I and the other math teacher, James, at my school were the only two people who had signed up for it, I was delighted.

My delight was rather short-lived, however. The moderator was one who didn’t give up easily.  She met with our principal and came up with the alternative of having a two hour meeting with the two of us at our school in the early part of the day before our classes started. To her this was a win-win since she got to deliver her PD and we wouldn’t have to miss any class time.

We tried it out one time. I happen to know a bit about her background because I took a look at her blog.  She’s a fan of Phil Daro, who is largely responsible for the Common Core math mess, and getting California to adopt the standards. He talks a lot about how in traditionally taught math, students are taught “answer getting” but not understanding in math. It was evident that she bought in heavily to the idea of “answer getting” vs “understanding” during our confab with her.

She began our two-hour collaboration by talking about how the state tests that are aligned with Common Core in California are not about “answer getting” anymore—rather students must explain their answers. The tests now evaluate whether students are able to see problems in more than one way. Which raises the question of why a student is deemed to lack “deeper understanding” if they get the answer in one way, but cannot show additional ways. She said the tests aim at certain “targets” which are more the Common Core Standards of Mathematical Practice (SMP) than they are of Common Core’s content standards. The SMPs are generic competencies like “persevere in solving problems”, “find structure and repeated reasoning in problems”:  things that would come about anyway from practice of content, rather than trying to develop “habits of mind” outside of the context of content.

Given that the focus of the test is on vague and largely immeasurable competencies, she went on to say that on the state test, students can get full credit on problems where they have to provide explanations even if they get the numerical answer wrong–provided the reasoning and process are correct. (Full confession: I give partial credit to my students if they set up a problem correctly, but I do take off points for numerical mistakes.)

But now she was warming up to what she really wanted to talk about. She said that explaining answers is tough for students and for this reason there is a need for “discourse” in the classroom and “rich tasks”.

I was doing a good job of keeping my mouth shut, but at this point I could contain myself no longer and asked “Could you define what a ‘rich task’ is?”

She answered as follows: “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.”

This, of course, says nothing very eloquently.  I had had some experience with rich problems so I knew exactly the type of problem to which she was referring; problems like “A rectangle with an area of 20 has what dimensions?” or something similar.

The one-sided conversation she was having continued for a few more minutes.  Apparently she loved math while in school but was doing what she described as “following the rules and getting an answer”.  (And she had given us forewarning that “answer getting” is not a desired outcome.)  Later when she taught math, she found she couldn’t explain to students all the time what were the underlying reasons.  I found this interesting given that I do understand a lot of the underlying reasons, and I had the same traditional math background she described—and she was a math major like me.

At this point James could take it no longer. He said that meeting for two hours for five sessions was superfluous if it was just the two of us. “I teach three different math classes plus doing IT for the school and don’t have time to delve into alternative approaches other than to follow the script and curriculum as laid out in the book.”

She took this as another “entry point”—the two of us must have seemed to her like a rich problem.  “Books are just tools,” she proclaimed.  “They may be strong in one area but weak in another. Traditional textbooks tend to be lacking in opportunities for conceptual understanding and are old school in their approach.”

She sensed that both of us were more than willing to let her dig her own grave here.  She quickly added, “Though there’s nothing wrong with old school.”

I saw no need to tell her I use a 1962 textbook by Dolciani for my algebra class.

She asked if we relied on our textbook for a “script”, meaning scope and sequence.  She turned to me and asked “Do you read just one textbook?”

“I read lots of textbooks,” I said.  She looked surprised.

“He’s also written books,” James said.

“Oh, how nice!” she said and feigned an interest by asking me what they were about. I gave a “rich” answer. “Math education,” I said.

“Wonderful!” she said.

I then tried to summarize our feelings by saying that the collaboration idea seemed superfluous. Neither of us teaches in a vacuum. I read lots of textbooks and talk to lots of teachers.  James had a lot more experience than I do (he’s been teaching for 22 years) so he has acquired knowledge as well.  I didn’t think that this 2-hour collaboration every month was going to add much more. In addition I said I was getting mixed messages.

“On the one hand I’m told by the administration that I’m doing great, and I hear from parents that I’m doing great,” I said.  “But then I’m told that I MUST attend this PD. Is there something about my teaching that’s lacking?  What is this about?”

She assured us that there’s nothing lacking in our teaching and that she’s sure we are both fantastic teachers.

I said “What is it then?  Is this about test scores? Is that it? They think this will raise test scores?”

She had no answer for this except something that I can’t remember.

She saw the handwriting on the wall and said “No use beating a dead horse” and said she would talk to the administration about it.   I felt a bit sorry for her, but not that much.

Later on in the day I met with the principal on another matter. Normally she is quite cheery but she didn’t look happy to see me, so I knew she was disappointed.

But the next day she was cheerful again, life went on as normal and another of life’s disappointments had passed.




Revisit of “rote understanding”

I originally wrote about this in an October 2014 article published in Heartlander.  I re-read it recently and decided it’s still true.  I have reprinted it here with minor updates.

During a course in math teaching methods I took in ed school, I watched a video of a teacher leading his students to do a variety of tasks, ostensibly to teach them about factoring trinomials, such as x2 + 5x + 6. But rather than teaching factoring techniques, as is done in traditionally taught classes, the session was a mélange of algebra tiles (plastic squares and rectangles used to represent algebraic expressions) and a graph of the equation being factored (a parabola).

The teacher “facilitated” the class into making connections between the factored equation and where the graphed parabola crossed the x-axis. The class had not done factoring, quadratic equations, nor a host of other things that would have qualified as “prior knowledge” which is generally deemed by traditionalists as important for building onto, making connections and progressing to newer material.

After the video, our teacher asked for our reactions. I said that rather than teach students factoring first and having them practice it, they were doing things that generally came after such mastery. “There’s so much going on, that I’m not sure what they’re learning or if they’re learning anything at all,” I said. My teacher’s face went into a frown, and she called on another student.

What I Saw During a Recent Classroom Visit

I recount the above because of an eighth-grade math course I observed as an instructional assistant (a role which one assumes after doing countless hours of subbing, and is generally the last step before being hired as a teacher). The math course was given just before full-fledged adoption of the Common Core standards in California and was piloting lessons that aligned with Common Core math standards. The teacher was quite good, and I do not hesitate to say she is excellent at what she does. But I also add that one can be very good at implementing things that are horrible. Her sessions were a mixture of letting the students “struggle” with a problem and then providing some explanation through limited direct instruction and questioning. During one session the class was learning about linear equations, graphing and functional form.

I watched as the teacher explained that an upcoming test would require students to write two sentences describing how to find the slope and y-intercept from a) a graph, b) a table of values, c) an equation, and d) a word problem.

She explained to the class why they were doing the lessons in this manner.  It struck me as an almost verbatim litany of what I had read in literature of why traditional math has failed thousands of students:

“You’ve learned about slope before, in seventh grade,” she told the class. “You were told, ‘Here’s the procedure, now let’s do the procedure, now you do it alone’: Wash, rinse, repeat—and repeat and repeat and repeat. Lots of practice, practice, and more practice. Now you are being asked to analyze, not just ‘plug and chug.’ You will have to explain what you’re doing, not just perform the procedure. Common Core is about thinking and understanding, not just doing. The goal of this school is to work on improving your writing skills, so you have to be able to explain what you do and why. Don’t just say ‘The slope is 4.’ Tell me how you got the slope, how did you find the intercept. Don’t just tell me ‘because when x is 0, y is 3.’ Tell me why that’s the intercept.”

In finding equations from a table of values, students were instructed how to find the change in x and y values (delta x and delta y), then to “make a fraction” of delta y/delta x, and that was the “rate of change,” or slope. They were made to “struggle” with some problems to eventually see what the y-intercept is, and then to be able to “work with patterns” from a table of values to find the y-intercept. They had to do this using logic, the teacher told them. First, they had to calculate the rate of change. If the zero value was not listed for x, the other values could lead them to it. For example, if x = 2 and y =4, and x = 4 and y = 8, students were to see that the rate of change is 4/2; that is, the “pattern” is that as x increases by 2, y increases by 4. Going the other way, when x decreases by 2, then y decreases by 4. Going backwards from the point (2, 4), the pattern tells us that at x = 0, y = 0.

As was the case with the video I had seen in ed school, these students were being given multiple concepts all at the same time, and expected to make and “explain” the connections between them. Unlike the students in the video, they had learned some of the material previously, and had been working on this unit for about five weeks. Nevertheless, based on the questions the students asked, it was clear they were confused: “How do you explain this in writing?” “How do I find the equation from the table?” “How do I find the y-intercept from a word problem?”

What Do the Standards Say?

For reference purposes, here are Common Core standards for functions in eighth-grade math:

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Nothing about these standards is different from what would be in any conventional algebra 1 course. A look at the algebra books I used in high school (see Figures 1 and 2) illustrate how to obtain an equation from a table of values. They also illustrate the same four ways of expressing a function that the CC standards require.

Figure 1 (Aiken, et al., 1957)

Four parts

Figure 2 (Aiken, et al., 1960)

Unlike what I saw in the classroom, the traditional approach presents each concept in order—not all at once—and presents exercises for students to master each successive skill. Mathematics is relentlessly hierarchical; one concept serves as the foundation for the next to build upon. In the book I used in high school, the topic started with deriving an equation from a table of values. Then came how to graph a table of values on an x-y Cartesian coordinate plane, how graphs represent relationships that words can also express, the slope of a line and how to determine it from a table of values, the slope intercept form of the linear equation (y = mx + b), what is the y-intercept and how to determine it, and, finally, how to construct a graph from an equation without using a table of values. This was accomplished in about three or four weeks, with relative ease.


Drilling for Understanding: The Road to “Rote Understanding”

Those who view the traditional approach as strictly procedural believe it does not result in a “deep understanding.” They believe students do not learn the “why” of a procedure, just how to do it through “meaningless” drills. Yet the teacher I observed was still giving the students instruction after allowing them some time to “struggle.” She also gave them opportunities to practice. The difference was that her exercises included having to explain (orally and in writing) various connections—i.e., how the table of values related to the equation of the line, how the line related to equation, and so forth. The writing was also instructed: she gave them examples of sentences that she would say aloud but with blanks where key words would go.

This teacher’s method is consistent with what I’ve described in my previous articles. That is, students are being made to do what amounts to “rote understanding.” Instead of adding 9 + 6 in any manner students want, students must do a number of problems by “making tens.” Instead of adding multi-digit numbers using the standard algorithm, students must do problems using inefficient strategies that purport to show what is happening when doing such operations. And in the case of functions, instead of being taught concepts and skills sequentially, they are required to “show the connections” between concepts–something that would likely happen on its own. They are taught to reproduce explanations that make it appear they possess understanding—and more importantly, to make such demonstrations on the standardized tests that require them to do so. And while “drill and kill” has been held in disdain by math reforms, students are essentially “drilling understanding.”

In previous articles I’ve written about the Common Core standards, I have taken the view that the standards can be met with traditional approaches in math education. I have purposely highlighted examples from traditional era textbooks (i.e., 1940’s to 1960’s) to show that meaning and understanding were always integral to accepted methods of instruction that critics of traditional methods have denigrated.

It is both intriguing and disturbing to see that in the name of Common Core not a few school districts have made it much harder for a student to qualify for a traditional algebra 1 course in eighth grade. Because of this, some students who would have qualified are being forced into courses conducted in the manner I have described here. The take-away message from such policies is: “You are not bright enough to benefit from traditionally taught math; you would just be taking part in rote memorization with no understanding.” I hope such policies are challenged and reversed, and that common sense will once more prevail in math instruction—Common Core standards or not.


Aiken, Daymond J., K.B. Henderson, R. E. Pingry; “Algebra: Its Big Ideas and Basic Skills, Book 1”; McGraw-Hill, New York. 1957.

Aiken, Daymond J., K.B. Henderson, R. E. Pingry; “Algebra: Its Big Ideas and Basic Skills, Book 2”; McGraw-Hill, New York. 1960.

Misunderstandings about Understanding

I had the great fortune to attend the researchED conference in Vancouver on Feb. 9, 2019. I was also honored to give a presentation there.  Here is a summary, followed by a link if you are interested in the presentation.  It is a PowerPoint which when viewed in “Notes” format contains the script that accompanies each slide.


Misunderstandings about understanding w notes

Effective Math Instruction: Hiding in Plain Sight

This article originally appeared in Education News in 2015. It received about 80 comments, most of them indignantly negative, which told me I was probably on the right track. I revised the article to bring it up to date and it is now included in my book “Math Education in the US: Still Crazy After All These Years”, which I heartily encourage you to purchase–in fact get several copies and send them to people who disagree with you on educational matters. They’ll never forget you.

I include this article here because there has been much talk lately on Twitter about how reading is not taught properly (due in large part to a great article written by Emily Hanford in the New York Times). The first line of my article echoes some concerns being discussed now. I extend the argument to math.

In a well-publicized paper that addressed why some students were not learning to read, Reid Lyon (2001) concluded that children from disadvantaged backgrounds where early childhood education was not available failed to read because they did not receive effective instruction in the early grades. Many of these children then required special education services to make up for this early failure in reading instruction, which were by and large instruction in phonics as the means of decoding. Some of these students had no specific learning disability other than lack of access to effective instruction.

This phenomenon has been observed by others, which was documented in a blog which recounts a British teenager’s achieving reading fluency over 18 months (after having struggled for years to read). The incident demonstrated that “correct methods” can result in reading success and in the case documented in the blog, greatly reduced the teenager’s special education needs.

These findings are significant because a similar dynamic is at play in math education: the effective treatment for many students who would otherwise be labeled learning disabled is also the effective preventative measure.

In 2013 approximately 2.3 million students were identified with learning disabilities — about three times as many as were identified in 1976-1977. (National Center for Educational Statistics, 2015)  These numbers raise several questions:  1) How many of the students identified with learning disabilities were related to math?  2) Of those, how many students were so classified because of poor or ineffective instruction?  and (perhaps most importantly) 3) How many could have kept up  with classmates if they had been taught using the more traditional math teaching methods that had once prevailed? 

In my opinion, what is offered as treatment for learning disabilities in mathematics is what we could have done—and need to be doing—in the first place. While there has been a good amount of research and effort into early interventions in reading and decoding instruction, extremely little research of equivalent quality on the learning of math in the United States exists. Given the education establishment’s resistance to the idea that traditional math teaching methods are effective, this research is very much needed to draw such a definitive conclusion about the effect of instruction on the diagnosis of learning disabilities.

Some Background

Over the past several decades, math education in the United States has shifted from the traditional model of math instruction to “reform math”. Although the shift has not been a uniform one, evidence of such transition is indicated by perennial articles in newspapers and the internet featuring parents who question and protest the methods being used to teach their children math.[ii] The traditional model has been criticized for relying on rote memorization rather than conceptual understanding. Calling the traditional approach “skills based”, math reformers deride it and claim that it teaches students only how to follow the teacher’s direction in solving routine problems, but does not teach students how to think critically or to solve non-routine problems. Traditional/skills-based teaching, the argument goes, doesn’t meet the demands of our 21st century world.

The criticism of traditional math teaching is based largely on a mischaracterization of how it is being and has been taught: rote memorization and procedures being the main focus of instruction, with little or no conceptual understanding. It is often described as having failed thousands of students in math education despite evidence of its effectiveness. Reacting to this characterization of the traditional model, math reformers promote a teaching approach in which understanding and process dominate over content.  Such emphasis is often represented by statements made by teachers, or school administrators such as “In the past students were taught by rote; we teach understanding.”

In order to ensure that students have understanding rather than performing a procedure by “rote”, students in lower grades may be required to provide written explanations for problems that often are so simple as to defy explanations. Also, they may be asked to solve a problem in more than one way, either by pictures as well as numerically, or by different methods. Failure to do so may result in the student judged to be operating via rote procedures and not possessing true mathematical understanding.  In lower grades, mental math and number sense are often emphasized before students are fluent with the procedures and number facts to allow such facility.

In lieu of the standard methods (algorithms) for adding, subtracting, multiplying and dividing, in some programs students are generally required to use inefficient procedures for several years before they are exposed to or even allowed to use the standard algorithm). This is done in the belief that the alternative approaches confer understanding to the standard algorithm.[iii]  In reformers’ minds, to teach the standard algorithm first would, once again, be considered “rote learning” thus eclipsing the conceptual underpinning of the procedure. 

Whole class and teacher-led explicit instruction (and even teacher-led discovery) has given way to what the education establishment believes is superior: students working in groups in a collaborative learning environment reducing the teacher to a mere facilitator of holistic “inquiry-based” or discovery learning experiences. Students teach themselves. Providing information directly is done sparingly or in combination with a group activity. 

The grouping of students by ability has almost entirely disappeared in the lower grades—full inclusion has become the norm. Reformers dismiss the possibility that understanding and discovery can be achieved by students working on sets of math problems individually and that procedural fluency is a prerequisite to understanding. Much of the education establishment now believes it is the other way around; if students have the understanding, then the need to work many problems (which they term “drill and kill”) can be avoided.

The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, Manitoba, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of some of the other panelists, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.” This sentiment was also echoed in an article written by Keith Devlin (2006). Such opposition has had limited success, however, in turning the tide away from reform approaches.

The Growth of Learning Disabilities

Students struggling in math may not have an actual learning disability but may be in the category termed “low achieving” (LA). Recent studies have begun to distinguish between students who are LA and those who have mathematical learning disabilities (MLD). Geary (2004) states that LA students don’t have any serious cognitive deficits that would prevent them from learning math with appropriate instruction. Students with MLD, however, (about 5-6 percent of students) do appear to have both general (working memory) and specific (fact retrieval) deficits that result in a real learning disability. Among other reasons, ineffective instruction may account for the subset of LA students struggling in mathematics.

A popular textbook on special education (Rosenberg, et. al, 2008), notes that up to 50 percent of students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction. What Works Clearinghouse finds strong evidence that explicit instruction is an effective intervention, stating: “Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review”.

Also, the final report of the President’s National Math Advisory Panel (2008) states: “Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities, as well as other students who perform in the lowest third of a typical class.” (p. xxiii). The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes math memorization and the other traditional methods for teaching the subject that have been decried by reformers as having failed millions of students.

The Stealth Growth of Effective Instruction

The Individuals with Disabilities Education Act (IDEA) initially established the criteria by which students are designated as “learning disabled”. IDEA was reauthorized in 2004 and renamed the Individuals with Disabilities Education Improvement Act (IDEIA). The reauthorized act changed the criteria by which learning disabilities are defined and removed the requirements of the “significant discrepancy” formula. That formula identified students as learning disabled if they performed significantly worse in school than indicated by their cognitive potential as measured by IQ. IDEIA required instead that states must permit districts to adopt alternative models including the “Response to Intervention” (RtI) model in which struggling students are pulled out of class and given alternative instruction.  If they improve under RtI, then the student is presumed to not have a learning disability and is returned to the normal class.

The RtI process provides a strong indication that LD diagnoses may in fact be caused by other factors such as poor instruction.  Although the number of students classified as learning disabled has grown since 1976, the number of students classified as LD since the passage of IDEIA has decreased. Why the decrease has occurred is not clear. A number of factors may be at play. One may be a provision of No Child Left Behind that allows schools with low numbers of special-education students to avoid reporting the academic progress of those students. Other factors include more charter schools, expanded access to preschools, improved technologies, and greater understanding of which students need specialized services. Of particular note, the decrease may also be due to targeted RtI programs. Since students who improve under RtI are presumed to not have a learning disability, the RtI program itself may have reduced the identification of struggling and/or low achieving students as learning disabled.

Having seen the results of ineffective math curricula and pedagogy as well as having worked with the casualties of such educational experiments, I have no difficulty assuming that RtI plays a significant role in reducing the identification of students with learning disabilities. The problem remains, however, that after a student shows improvement under RtI, he/she is then returned to the type of teaching that caused the student to be referred to RtI in the first place. In my opinion it is only a matter of time before high-quality research and the best professional judgment and experience of accomplished classroom teachers identify effective and non-effective teaching methods. Such research should include:

1) The effect of collaborative/group work compared to individual work, including the effect of grouping on students who may have difficulty socially;

2) The degree to which students on the autistic spectrum (as well as those with other learning disabilities) may depend on direct, structured, systematic instruction;

3) The effect of explicit and systematic instruction of procedures, skills and problem solving, compared with inquiry and other reform-based approaches;

4) The effect of sequential and logical presentation of topics that require mastery of specific skills, compared with a spiral approaches to topics that do not lead to closure, 

5) How students who improved under RtI fared when returned to a reform-based classroom and

6) The extent to which students who are doing well in a reform-based classroom are receiving outside help via parents, tutors or learning centers. 

Would such research show that the use of RtI is higher in schools that rely on programs that are low on skills and content but high on reform-based techniques that purport to build critical thinking and higher order thinking skills? If so, shouldn’t we be doing more of the RtI style of teaching in the first place instead of waiting to heal the casualties of reform math?

Until any such research has been conducted, the educational establishment will likely continue to resist recognizing the merits of traditional math teaching, based on conversations I’ve had with education professors. A statement made by James McLesky (2009), a professor at University of Florida’s College of Education, is typical of what I’ve been told:

If we provide only (or mostly) skills and drills for students with disabilities, or those who are at risk for having disabilities, this is certainly not sufficient. Students need to also have access to a rich curriculum which motivates them to learn reading, math, or whatever the content may be, in all of its complexity. Thus, a blend of systematic, direct instruction and high quality core instruction in the general education classroom seems to be what most students need and benefit from.

While Dr. McLesky recognizes the value of direct and explicit instruction, his statement carries with it the underlying mistrust and mischaracterization of traditional math teaching—a mistrust that defines such teaching as 1) consisting solely of explicit instruction with no engaging questions or challenging problems, and 2) failing to teach math in any complexity. Statements such as these imply that students who respond to more explicit instruction constitute a group who may simply learn better on a superficial level. Based on these views, I fear that RtI will incorporate the pedagogical features of reform math that has resulted in the use of RtI in the first place.

The criticism of traditional methods may have merit for those occasions when it has been taught poorly. But the fact that traditional math has been taught badly doesn’t mean we should give up on teaching it properly. Without sufficient skills, critical thinking doesn’t amount to much more than a sound bite. If in fact there is an increasing trend toward effective math instruction, it will have to be stealth enough to fly underneath the radar of the dominant edu-reformers. Unless and until this happens, the group-think of the well-intentioned educational establishment will prevail. Parents and professionals who benefitted from traditional teaching techniques and environments will remain on the outside — and the methods that can do the most good will continue to hide in plain sight.


Devlin, Keith. (2006). Math back in forefront, but debate lingers on how to teach it.  San Jose Mercury. Feb. 19.

Geary, David. (2004).  Mathematics and learning disabilities. J Learn Disabil 2004; 37; 4

Lyon, Reid (2001), in “Rethinking special education for a new century” (Chapter 12) by Chester Finn, et al., Thomas B. Fordham Foundation; Progressive Policy Inst., Washington, DC. Available via  http://eric.ed.gov/PDFS/ED454636.pdf

McLesky, James (2009). Personal communication via email; October 20.

National Center for Educational Statistics, (2015) Digest of Educational Statistics: 2013. Table 204.30Available via http://nces.ed.gov/programs/digest/d14/tables/dt14_204.30.asp

National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC, 2008.

Rosenberg, Michael, D. L. Westling, J. McLesky (2008). Special Education for Today’s Teachers: An Introduction; Pearson. New York.

Zimba, Jason (2015) When the standard algorithm is the only algorithm taught. Common Core Watch; January. http://edexcellence.net/articles/when-the-standard-algorithm-is-the-only-algorithm-taught


Worked Examples and Scaffolding, Dept.

NOTE: I’m talking about understanding in math at the researchED conference in Vancouver on Feb. 9. See here.


In teaching procedures for solving both word problems and numeric-only problems, an effective practice is one in which students imitate the techniques illustrated in a worked example. (Sweller, 2006). Subsequent problems given in class or in homework assignments progress to variants of the original problem that require students to stretch beyond the temporary support provided by the initial worked example; i.e., by “scaffolding”. Scaffolding is a process in which students are given problems that become increasingly more challenging, and for which temporary supports are removed.  In so doing, students gain proficiency at one level of problem-solving which serves to both build confidence and prepare them for a subsequent leap in difficulty.  For example, an initial worked example may be “John has 13 marbles and gives away 8. How many does he have left?”  The process is simple subtraction.  A variant of the original problem may be: “John has 13 marbles.  He lost 3 but a friend gave him 4 new ones.  How many marbles does he now have?”  Subsequent variants may include problems like “John has 14 marbles and Tom has 5.  After John gives 3 of his marbles to Tom, how many do each of them now have?”

Continuing with the example of adding and subtracting, in early grades some students, particularly those with learning disabilities, have difficulty in memorizing the addition and subtraction facts.  On top of the memorization difficulties, students then face the additional challenge of applying this knowledge to solving problems. One approach to overcome this difficulty has been used for years in elementary math texts, in which students are provided with a minimum of facts to memorize and  then given word problems using only those facts the student has mastered. Such procedure minimizes situations in which working memory encounters interference and becomes overloaded as described in Geary (in press). For example, a student may be tasked with memorizing the fact families for 3 through 5.  After initial mastery of these facts, the student is then given word problems that use only those facts.  For example, “John has 2 apples and gets 3 more, what is the total?” and “John has some apples and receives 3 more; he now has 5 apples. How many did he have to start with?”  Additional fact families can then be added, along with the various types of problems.  Applying the new facts (along with the ones mastered previously) then provides a constant reinforcement of memorization of the facts and applications of the problem solving procedures. The word problems themselves should also be scaffolded in increasing difficulty as the student commits more addition and subtraction facts to memory.

Once the foundational skills of addition and subtraction are in place, alternative strategies such as those suggested in Common Core in the earlier grades may now be introduced.  One such strategy is known as “making tens” which involves breaking up a sum such as 8 + 6 into smaller sums to “make tens” within it. For example 8 + 6 may be expressed as 8 +2 + 4. To do this, students need to know what numbers may be added to others to make ten. In the above example, they must know that 8 and 2 make ten.  The two in this case is obtained by taking (i.e., subtracting) two from the six.  Thus 8 + 2 + 4 becomes 10 + 4, creating a short-cut that may be useful to some students.  It also reinforces conceptual understandings of how subtraction and addition work .

The strategy itself is not new and has appeared in textbooks for decades. (Figure 1 shows an explanation of this procedure in a third grade arithmetic book by Buswell et. al. (1955).

The difference is that in many schools, Common Core has been interpreted and implemented so that students are being given the strategy prior to learning and mastering the foundational procedures.  Insisting on calculations based on the “making tens” and other approaches before mastery of the foundational skills are likely to prove a hindrance, generally for first graders and particularly for students with learning disabilities.

Figure 1: Adding by “making tens” from Buswell, et. al. (1955)

Figure 1: Adding by “making tens” from Buswell, et. al. (1955)

Students who have mastered the basic procedures are now in a better position to try new techniques — and even explore on their own.  Teachers should therefore differentiate instruction with care so that those students who are able to use these strategies can do so, but not burden those who have not yet achieved proficiency with the fundamental procedures.

Procedure versus “Rote Understanding”

It has long been held that for students with learning disabilities, explicit, teacher-directed instruction is the most effective method of teaching.  A popular textbook on special education (Rosenberg, et. al, 2008) notes that up to 50% of students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction. The final report of the President’s National Math Advisory Panel (2008) states: “Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities, as well as other students who perform in the lowest third of a typical class.” (p. xxiii). These statements have been recently confirmed by Morgan, et. al. (2014). The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes memorization and other explicit instructional  methods.

Currently, with the adoption and implementation of the Common Core math standards, there has been increased emphasis and focus on students showing “understanding” of the conceptual underpinnings of algorithms and problem-solving procedures. Instead of adding multi-digit numbers using the standard algorithm and learning alternative strategies after mastery of that algorithm is achieved (as we earlier recommended be done), students must do the opposite. That is, they are required to use inefficient strategies that purport to provide the “deep understanding” when they are finally taught to use the more efficient standard algorithm. The prevailing belief is that to do otherwise is to teach by rote without understanding.  Students are also being taught to reproduce explanations that make it appear they possess understanding — and more importantly, to make such demonstrations on the standardized tests that require them to do so.

Such an approach is tantamount to saying, “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.” Forcing students to think of multiple ways to solve a problem, for example, or to write an explanation for how they solved a problem or why something works does not in and of itself cause understanding. It is investment in the wrong thing at the wrong time.

The “explanations” most often will have little mathematical value and are naïve because students don’t know the subject matter well enough. The result is at best a demonstration of “rote understanding” — it is a student engaging in the exercise of guessing (or learning) what the teacher wants to hear and repeating it.   At worst, it undermines the procedural fluency that students need.

Understanding, critical thinking, and problem solving come when students can draw on a strong foundation of domain content relevant to the topic being learned. As students (non-LD as well as LD) establish a larger repertoire of mastered knowledge and methods, the more articulate they become in explanations.

While some educators argue that procedures and standard algorithms are “rote”, they fail to see that exercising procedures to solve problems requires reasoning with such procedures — which in itself is a form of understanding.  This form of understanding is particularly significant for students with LD, and definitely more useful than requiring explanations that students do not understand for procedures they cannot perform.


Ansari, D. (2011). Disorders of the mathematical brain : Developmental dyscalculia and mathematics anxiety. Presented at The Art and Science of Math Education, University of Winnipeg, November 19th 2011. http://mathstats.uwinnipeg.ca/mathedconference/talks/Daniel-Ansari.pdf

Buswell, G.T., Brownell, W. A., & Sauble, I. (1955). Arithmetic we need; Grade 3.  Ginn and Company. New York. p. 68.

Geary, D. C., & Menon, V. (in press). Fact retrieval deficits in mathematical learning disability: Potential contributions of prefrontal-hippocampal functional organization. In M. Vasserman, & W. S. MacAllister (Eds.), The Neuropsychology of Learning Disorders: A Handbook for the Multi-disciplinary Team, New York: Springer

Morgan, P., Farkas, G., MacZuga, S. (2014). Which instructional practices most help first-grade students with and without mathematics difficulties?Educational Evaluation and Policy Analysis Monthly 201X, Vol. XX, No. X, pp. 1–22. doi: 10.3102/0162373714536608

National Mathematics Advisory Panel. (2008). Foundations of success: Final report. U.S. Department of Education. https://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf

Rittle-Johnson, B., Siegler, R.S., Alibali, M.W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, Vol. 93, No. 2, 346-362. doi: 10.1037//0022-0063.93.2.346

Rosenberg, M.S., Westling, D.L., & McLeskey, J. (2008). Special education for today’s teachers. Pearson; Merrill, Prentice-Hall. Upper Saddle River, NJ.

Sweller, P. (1994) Cognitive load theory, learning difficulty, and instructional design.  Leaming and Instruction, Vol. 4, pp. 293-312

Sweller, P. (2006). The worked example effect and human cognition.Learning and Instruction, 16(2) 165–169