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.

References

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


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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.

References

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

 

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.