I have been on Twitter long enough to notice an idea that is increasingly taking hold, especially in the U.K. education: memorization as the main tool for learning. Danniel T. Willingham, a cognitive scientist whose work I consistently shared for a few years now, has irrevocably (and most likely unintentionally) created a meme when he wrote in one of his widely known books (Why Don’t Students Like School?): “**Memory is the residue of thought**.” The truth of the sentence is obvious, almost trivial: to remember something you need to have thought about it quite hard. What happened though is that its meaning got completely twisted in some educational settings and turned into “Memorization is the only way to learn”.

This becomes more transparent when you read blog posts of “(neo-) traditional” teachers, posts that celebrate memorization, drilling, chanting of timetables, and severe disciplinary measures. Education programs such as Uncommon Schools (Doug Lemov) or Direct Instruction (Zig Engelmann) are trumpeted as being highly effective, evidence of test scores is given, and schools begin to adopt the philosophy and the structural dimensions of these programs as fast as they can.

“Why should you care? You don’t even teach in the U.K. or the U.S.” you might ask, and rightfully so. But as a teacher I can’t but wonder whether they are, in many ways, doing a disservice to their own students. And while the ethos of these schools is a different matter altogether and I won’t discuss it (after all, each school community has the right to their own philosophy and values), their means of achieving “effectiveness” is questionable in one of the areas I love: mathematics.

This subject is particularly prone to being the target of drilling and memorization for two reasons. First, mathematics is an exact science, therefore it is relatively easy to select a learning objective and assess it. An equation is easier to assess than a written composition: subjectivity has no place in calculating 234:2. Secondly, primary schools are expected to “deliver”: in other words, students are expected to have mastered procedural skills and fluency so that they can use them effectively in the following years. Nothing wrong so far – my 4^{th} graders, too, have memorized their timetables and can do up to four-five mental math operations using 3-digit numbers, including squaring numbers without any resort to writing.

What is the problem then? Well, the problem occurs when *only* procedural skills and fluency are emphasized as the new tendencies in these schools show. You won’t hear a word about “application”, let alone “conceptual understanding” – the latter being continually dismissed as “flimflam” and not something we should focus on, not in primary school anyway. Conceptual understanding is not a “skill”, neither quantifiable nor suited to assessment so let’s throw it in the bin, together with other “progressive” ideas.

For those who haven’t actually read the book (Why Students Like School?), nor all D.T. Willingham’s articles and relied solely on blog posts of other teachers, I have news: conceptual understanding is actually a matter of great importance, mentioned in his article I shared four years ago (Is It True That Some People Just Can’t Do Math?).

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**Mathematics, the three-legged stool**

Richard Askey, a professor of mathematics at the University of Wisconsin, is an experienced veteran of the math wars, having followed changes in mathematics education for many years. When referring to the reforms in mathematics in the U.S., he stated back in 1997:

* “Mathematics is like a stool: it sits on three legs. In the New Math period the only leg used was the structure of mathematics. The feeling was that if you understood the structure of mathematics, then you could compute and solve problems. That turned out to be false for all except a small group. Then we got Back to Basics which was founded on computation. However, the level was too low, and good problems and structure were both ignored. This failed badly. Then NCTM tried Agenda for Action and later the Standards. Both of these were built on the idea that if you could solve problems, then you could do mathematics. You can, but at too low a level. All three are needed – problems, technique, and structure. I fail to see why this obvious fact is not appreciated but it does not seem to be. Maybe then we can start to try to do this right.”*

It is no coincidence that D.T. Willingham mentions these three pillars of mathematics as well: **factual** knowledge, **procedural** knowledge, and **conceptual** knowledge. He, too, criticizes the unequal emphasis on one over others:

*“At the extremes, progressives claim that traditionalists would be happy for students to execute procedures without understanding what they are doing, and traditionalists claim that progressives care only that students understand concepts and are unconcerned about whether they can actually solve math problems. Most observers of math agree that knowledge of procedures AND of concepts is desirable.”*

* Factual knowledge* (knowing your number bonds or timetables, for instance) is critical in mathematics. Being able to retrieve basic facts such as 4 x 8 = 32 quickly from memory is essential when solving math problems of greater complexity because it frees up space in the working memory so the student can focus on the structure of the problem itself. I, too, insist on mental math (see here). However, even in this case, caution needs to be exercised, Willingham writes:

* “One would expect that interventions to improve automatic recall of math facts would also improve proficiency in more complex mathematics. Evidence on this point is positive but limited, perhaps because automatizing factual knowledge poses a more persistent problem than difficulties related to learning mathematics procedures.”*

** Procedural knowledge** (such as long division) is another layer that contributes to the development of problem-solving skills. Knowing the steps involved in solving a certain type of problem requires extensive practice but in a

**variety**of contexts. Two words of caution here.

Relying exclusively on ** modeling and examples** reduces students’ ability to see the underlying structure of a problem and usually direct instruction is exactly that: I show you the steps and you practice 10 similar problems. The gain is only apparent and short-lived: students practice 10 similar problems, solve them correctly (they embed the procedure in their current memory), and you notice 2 months later that they are stuck when given the same type of problems. Moreover, because of the

**effect, students considered they have mastered the topic and are confident they can solve the problems in the future. In other cases, when the problem has the same surface features but different**

*familiarity**, students tend to solve them incorrectly – this is also an effect of block practice and lack of variety of contexts where the procedure (and concept) had to be applied.*

**underlying, deep structure**Excessive modeling is just one reason for this failure to embed procedures in long-term memory; the other is mathematics taught in a* linear fashion*, that is, once a procedure is taught and practiced, teachers move students to the next one. This was a problem signaled by mathematicians in textbooks, too: lack of variety of topics being revisited simultaneously with new concepts. D. Rohrer and K. Taylor mention (The Shuffling of Mathematics Problems Improves Learning, 2007):

* “The arrangement of practice problems in most mathematics textbooks is one that most readers will recognize. Each set of practice problems consists almost entirely of problems corresponding to the immediately preceding lesson. For example, a lesson on the addition or subtraction of fractions (e.g., 5/6–4/5) is followed immediately by perhaps a few dozen problems, all of which require the addition or subtraction of fractions. In brief, each set of practice problems is devoted to the most recent lesson. Moreover, problems of the same type are usually in blocks (e.g., 12 fraction addition problems, followed by 12 fraction subtraction problems).”*

The authors of the paper mention many research findings over the years (R. Bjork, Bhrick et al, Bloom and Shuell, Carpenter and DeLosh, Reynolds and Glaser etc.) that show how * spaced* and

*is more beneficial to learning:*

**mixed****practice*** “If a practice set includes a randomly arranged variety of problem types, students learn to pair each kind of problem with the appropriate procedure. In other words, a mixed practice schedule requires that students learn not only HOW to perform each procedure but also WHICH procedure is appropriate for each kind of problem (e.g., Kester, Kirschner, & Van Merrienboer, 2004).” *

The results of their experiments involving spaced and massed practice are presented on pages 488, and 493 respectively (see charts below).

Why am I bringing **textbooks** in this discussion? Precisely because it is a venerated tool for teachers in direct instruction settings. There is nothing counterintuitive about them: they are sequenced in a logical manner, the topics follow one after the other, and involve a large amount of practice. Teachers might want to reconsider this approach in light of research – so, no, careful sequencing of topics followed by extensive practice does lead to better learning at all, and the effect of ** overlearning **is most evident in mathematics practice (Shuffling of Problems in Mathematics, p. 484).

Now, to the most “obscure” and denied aspect of knowledge by traditionalists: ** conceptual knowledge**. In his article, D.T. Willingham writes:

*“ More troubling is American students’ lack of conceptual understanding. Several studies have found that many students don’t fully understand the base-10 number system. A colleague recently brought this to my attention with a vivid anecdote. She mentioned that one of her students (a freshman at a competitive university) argued that 0.015 was a larger number than 0.5 because ’15 is more than 5’. The student could not be persuaded otherwise.”*

*“ The cost of poor conceptual understanding should be clear (…) Learning new concepts depends on what you already know, and as students advance, new concepts will increasingly depend on old conceptual knowledge. If students fail to gain conceptual understanding, it will become harder and harder to catch up. Students will become more and more likely to simply memorize algorithms and apply them without understanding.”*

Well, as you can see, **conceptual understanding does exist, and it is neither facts nor procedures**. How do you build it, though? It is quite elusive and it cannot be assessed as easily as the other two.

One way is to allow space for inquiry. In structured inquiry settings (I work in an IB school), it is a vehicle for building conceptual understanding, for spotting misconceptions, and for exploring mathematical concepts in depth. I will exemplify this in the second part of this post.

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