|Posted on February 25, 2016 at 2:40 PM|
Written by mathmatician Emma Devitt
“It is a profoundly erroneous truism, reported by all copybooks and eminent people making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of operations we can perform without thinking about them. Operations of thought are like cavalry charges in battle – they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
Alfred North Whitehead, An Introduction to Mathematics
Math facts have become the bête noire of certain educational theorists in recent years. Thus, for example, Boaler, Williams and Confer (2015) argue that what is important to a mathematical education is developing number sense – the sort of sense, for example, that allows a child to infer that if 20+6 =26, then 19+6=25 – in addition to math facts. They argue, further, that one of the best methods of teaching number sense and math facts is a strategy called number talks developed by Parker (1993) and Richardson (2001).
It is difficult to assess the arguments of Boaler, Williams and Confer (henceforth, Boaler et al.) because they frequently resort to attacking what seem to be straw persons. They argue, for example, that a complete mathematical education “includes learning of math facts along with deep understanding of numbers and the ways they relate to each other” (2015). It is, however, difficult to imagine any educational theorist disagreeing with such an anodyne claim. Of course it is important – indeed crucial – to develop this sort of “deep” understanding. No serious educational theorist will deny this. (The Common Core Curriculum, for example, calls for both conceptual understanding and fluency with math facts). If there is a substantive disagreement on this issue, it must concern the relation between the learning of math facts and the learning of number sense.
The considered opinion of Boaler et al. seems to be that number sense is more important developmentally, and therefore should be made chronologically prior to the rote learning of math facts. Indeed, at various points, they call into question the utility of rote learning of math facts at all. It is very doubtful, however, that their supporting arguments establish either of these claims. Commenting on the public failure of a British Politician to successfully calculate the product of 7x8 – a sort of Dan Quayle ‘potatoe’ moment – Boaler et al. comment: “When asked to solve 7x8 someone with number sense may have memorized 56 but they would also be able to work out that 7 x 7 is 49 and then add 7 to make 56”. But this assumes that the product of 7x7 is the sort of fact that will be learned through number sense rather than rote learning. And this is something that the advocate of rote learning will deny. Boaler et al.’s argument is, therefore, question begging (in the technical, rather than colloquial, sense, of assuming the truth of the conclusion they desire rather than arguing for it).
A similar charge of begging the question can be leveled at their advocacy of Parker and Richardson’s number talks. I agree that this is a powerful tool, one that should be used in classrooms and ideally at home too. However – and here is the rub – it is very difficult to do use this tool without prior knowledge of math facts. Thus, they employ the example of posing the problem of 18x5, and then describe students’ different strategies for solving it. Every single strategy they describe presupposes the knowledge of math facts and the laws of mathematics .
None of this, I should emphasize, is to deny the importance of conceptual or “deep” understanding of numbers in a mathematical education. On the contrary, Bezuk & Cegelka (1995) may be correct when they write: “Prior to teaching for automaticity… it is best to develop the conceptual understanding of these math facts as procedural knowledge.” (1995, p. 365). Moreover, Bruner wrote extensively about the CPA (Concrete, Pictorial, Abstract) approach that is now used by the Singapore Math Curriculum. Thus, I agree with Boaler et al. that understanding is very important. Such understanding can be reached through work with manipulatives and through modeling. But none of this is incompatible with according a similar degree of importance to the learning of math facts. When moving to the abstract phase one will have deep conceptual understanding of number sense. However, to progress to higher order abstract math you will still require automaticity with math facts.
Boaler et al. make much of the anxiety involved in the timed testing of math facts: such anxiety is “a debilitating, often life-long, condition.” If we wish to avoid such anxiety then, they suggest, we should jettison such testing. As someone prone to vomiting before major exams (and even some not so major ones) I can sympathize. However, there are few psychologists or psychiatrists who now argue that the way to deal with stress is to avoid the conditions that engender it. Rather, stress-inducing conditions need to be faced, and successful strategies developed to accommodate them.
This is another point at which Boaler et al.’s arguments are question begging. The guiding assumption is that since timed testing is stressful, the correct strategy is to eliminate such testing. But the best strategy for dealing with this stress might, in fact, be the rigorous and, therefore, successful rote learning of math facts. To quote a wise old karate sensei I once knew: “You need to do the kata until the kata does you”. Performing a kata – a ritualized but meaningful series of movements – in front of a crowded room is very stressful. One way to avoid this stress would be to give up karate. The other is to perform the kata so many times that it becomes burned into your brain and body, and you can perform it no matter what the environment. The cure for stress inducing timed testing of math facts may be more, not less, rote learning of math facts. Thus, as Cavanagh (2007) has argued, regular timed testing might, in fact, benefit students by allowing them to become accustomed to working under pressure.
The research of Miller, Hall & Heward actually shows that time trials improve accuracy and fluency, and students enjoy being timed (1995). Students can be motivated when they see their scores rising. Timed math fact tests are one of the occasions in a student’s education when there is an unequivocal, objective correct answer to each question, and if he or she learns it they are able to get 100%. After all, as Dewck (2006) has argued, dedicated students who put in the hard work and have a growth mindset can succeed at anything. There is no obvious reason why this cannot be applied to the learning of math facts.
I began this paper with a quote from the great Alfred North Whitehead, the co-author (along with Bertrand Russell) of the monumental work, Principia Mathematica. In this work, they try to deduce the laws of arithmetic armed only with the apparatus of set theory: essentially, relations between numbers are reduced to set-theoretical functions. Using this apparatus, it took Whitehead and Russell eighty-six pages to prove what they called the “occasionally useful” claim that 1+1=2. So, if there is anyone who has a “deep understanding of numbers and the ways they relate to each other” it is Whitehead. And, yet, far from eschewing the role of rote memorization – or automaticity – Whitehead emphasizes it.
The reason is relatively obvious. Holding things in working memory is hard work. There is much erroneous talk in Boaler et al.’s (2015) article about mathematical facts being stored in working memory – based, it seems to me, on a misguided interpretation of empirical work of Beilock (2011) and Ramirez et al. (2013). In fact, nothing is stored in working memory. Working memory is not a store: it is a (functionally defined) place where information is temporarily placed in order to be manipulated. This is hard work! There are severe limits on how much can be brought into working memory and how much can be done with it once there. Rote learning of mathematical facts acts much as an external store of information whose function is to reduce the complexity of operations performed in working memory.
Imagine how difficult it would be to do long division – say 1,000,000,870,050,705 divided by 353 – in the head. Instead we avail ourselves of an external representation. The function of this external representation is to break down the overall process into a series of smaller processes. Each stage in the process can be achieved by way of a pattern mapping operation – learned by rote memorization. We fill in the result of this pattern mapping-operation, write it down, and then use this result to direct us to the next stage of the operation. In this way, the complexity of task is offloaded – or distributed – onto the environment. The external representation, written on paper or screen, is an external resource we can use to reduce the complexity of the task we would otherwise have to perform in the head. This idea forms the basis of the embodied turn in cognitive science that has become prominent in the last fifteen years or so (Rowlands 2010), and which is, I think, deeply consonant with the work of Bruner. My point is that with respect to working memory, the results of rote memorization are akin to an external resource. The more we can avail ourselves of this resource, the less we have to do in working memory. Rote memorization is not the opponent of deep understanding: rather, it is its best friend.
Pace the claims of Boaler et al., this idea is, I think, now widely accepted by educators and cognitive psychologists. For example, Whitehurst (2003): the “ability to recall basic math facts fluently is necessary for students to attain higher-order math skills”. Delazer et al. (2003) argue: “Recent research in cognitive science, using MRI’s, has revealed the actual shift in brain activation patterns as untrained math facts are learned”. And, crucially, as Dehaene (2003) has shown: “Instruction and practice cause math fact processing to move from a quantitative area of the brain to one related to automatic retrieval” (Dehaene, 2003). These are all processes whose function, or result, is clear. As Geary (1994) argues: when you have automaticity of your math facts you free up your working memory to work on higher order tasks, problem solving and learning new concepts and skills.
Nowhere in any of these processes is rote learning the enemy of conceptual understanding. On the contrary, they complement each other very nicely. The reason for this stems from certain features of working memory and, in particular, its relation to both short- and long-term memory. As Willingham (2004) has demonstrated, transferring a fact from short- to long-term memory requires “over learning”—this involves not merely being correct in fact recall on one occasion, but doing so repeatedly. Moreover, as Rohrer, Taylor, Pashler, Wixted, & Cepeda (2005) have shown, retaining the memory for a significant period of time requires additional, periodic, practice subsequent to initial mastery—emphasizing the importance of regular review of learned material. Hill and Schneider (2006) have shown that repetition of this sort produces significant changes in the brain, the thickening of neuronal myelin sheath resulting in faster retrieval.
Elementary mathematics deals with many different concepts like money, telling time, using the four basic operations on large numbers, measurement, estimation and rounding. Working with all these disparate concepts requires the use of math fact fluency. Fluency with math facts means that students will be confused less, find calculations easier, and will be faster and are more accurate in performing such calculations. With fast automatic recall of math facts students have time to concentrate on the higher order math problems they are being asked to compute or navigate. They will also complete tests within the allotted time. In short, rote memorization of math facts is not the enemy of deep understanding. It is its best friend. Bryan and Harper (1899 - quoted in Bloom 1986) put it better: “Automaticity is not genius, but it is the hands and feet of genius”.
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