This post doesn't fit this blog particularly well, but it's staying here for now....
Concepts and procedures: research by Bethany Rittle-Johnson
I have been thinking about the relationship between conceptual knowledge and procedural knowledge in maths. It is often argued that it is OK to learn and use a mathematical procedure before it is well understood - understanding can come later. I remember a group of maths undergraduates claiming, in a session I was giving on maths education, that this fitted their school experience. So OK, it might have worked for them, but might it perhaps have worked even better if the order had been reversed? And what about their school friends? Did it work for them too?
Finding research evidence on this has been frustrating. I have been looking at papers by Rittle-Johnson, whose work on productive failure I particularly admire. She seems to have come to the rather bland conclusion that each form of knowledge will enhance the other and that conceptual and procedural knowledge tend to develop iteratively.
It may be that a lot of maths learning does play out in this positive way, but think of all the adults who say they are no good at maths. And what about those who develop a deep anxiety about mathematics and a feeling of helplessness? In part, are such attitudes not often a consequence of having been taught procedures that don’t seem to make sense? Here I am thinking particularly of procedures such as column addition and subtraction, long multiplication, long division, the use of common denominators to compare or add fractions, and division by a fraction using the multiplicative inverse. In my childhood and again now, pupils in the UK are expected to become proficient in these quite early in their school lives.
These particular procedures are very powerful, in the sense that they allow us to perform a large number of standard calculations very efficiently - assuming a procedure has been learnt correctly and has become automatic. However, this can have negative as well as positive consequences. It can lead teachers and pupils and parents to think that the essence of school maths is to learn such procedures, rather than to learn about mathematical properties and relationships (as well as to engage in and to develop mathematical behaviour).
Also, while an automated procedure can free up working memory, it can also divert attention away from activities that could help pupils develop a better feel for mathematics. For example, while it might be empowering to be able to check the bill in Figure 1 using column addition, it can be enriching to think of other ways of performing the calculation, for example by spotting that the costs are equivalent to two lots £1.70 (which gives amounts that might be easier to sum in one’s head).
Similarly, while it might be empowering to be able to compare the fractions in Figure 2 using common denominators, pupils might develop a better feel for fractions if they were encouraged to find informal methods such as these:
½ is the same as 3/6; 3/6 is more than 3/7.
½ is the same as 3½/7; this is more than 3/7.
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Figure 1 |
Figure 2 |
Of course, pupils could use the freed-up working memory resulting from the use of a well-learnt procedure to examine the procedure itself, in order to understand it better, but this is less likely to happen if the focus is on getting answers, lots and lots of them.
It is also possible that pupils encounter a procedure in quite a gradual way, perhaps as a pattern emerging from meaningful mathematical activity. For example, pupils might be asked to consider the division 8 ÷ 3/7, by coming up with a story like this:
A 12 metre length of rope is to be cut into pieces,
each 3/7 of a metre long.
How many pieces will there be?
Then, after modifying the fraction in various ways (eg from 3/7 to 1/7, or from 3/7 to 6/7), pupils might notice that instead of dividing by the fraction they can get the same result by multiplying by its reciprocal. This might be an empirical observation, or pupils might (to some degree) appreciate how it arises out of the relations in the story. Depending on how such work continues (and what has preceded it!), it is possible that conceptual and procedural understanding of this rule could develop step by step, much in the way that Rittle-Johnson proposes.
The notion that conceptual and procedural knowledge develop iteratively is expressed in the very title of a paper by Rittle-Johnson: Developing conceptual understanding and procedural skill in mathematics: an iterative process. This paper, from 2001, is co-authored by Martha Alibali and by Robert Siegler, someone I particularly admire for his work on the variability of individual children’s thinking (see Emerging Minds, 1996). However, the paper is bewildering.
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The paper reports on two related experiments involving the representation of decimals on a number line. The first experiment involved 74 fifth-grade pupils (mean age 11 years 8 months) who were given a pre-test and post-test on conceptual knowledge of decimals and a pre- and post-test on procedural knowledge. The procedural knowledge tests were paper and pencil tests and consisted of problems about decimals on a number line. The intervention, which took about 40 minutes, involved similar problems but presented on a computer with each pupil working on their own. There were 12 problems in all, and after each answer, the pupil was informed of the correct answer and asked to explain why it was correct.
The authors suggest that the pupils in the study would have “some knowledge of decimal fractions” (ibid, p348), having been “exposed to relevant decimal fraction concepts such as place value, magnitude, equivalent values and the role of zero as a place holder”. However, number line problem like theirs would not have been “part of traditional curricula” and so they expected that the pupils (whose schools used traditional maths textbooks) would have “little or no procedural knowledge from placing decimal fractions on number lines”. But if that is the case, why are the number line problems deemed to be testing procedural knowledge? The authors’ reason is very simple, but does it make sense?
We distinguished between assessments of conceptual and procedural knowledge on the basis of the novelty of the tasks at posttest. Because children received repeated practice and feedback with number line problems during the intervention, this task became familiar and routine and thus tapped children's procedural knowledge.
I don’t understand this. Is novelty the sole criterion? It is possible that in the course of the 40 minute intervention, some pupils might have created some kind of pattern-based rule for locating decimals on a number line, which they then applied to the procedural post test, which came quite soon after the intervention. But is it not also possible that pupils would have used their understanding of decimals to try to solve the intervention problems and perhaps developed their understanding in the light of the feedback, and then applied this conceptual knowledge to the procedural posttest?
If I were asked to place a decimal like 0.63 on the number line, I could describe my actions like this:
Draw marks to divide the interval from 0 to 1 into 10 equal parts. From 0, count to the 6th mark and now divide the interval from the 6th to the 7th mark into 10 equal parts. Count to the 3rd of these new marks and identify it with an arrow.
Does this mean my thinking is procedural? I could perhaps instruct someone (or a machine) who knew nothing about decimals to execute these steps, but for me, each step is imbued with meaning and derives from my knowledge of decimals.
Perhaps a crucial issue here is the nature of the task. Imagine I were adding two 10 digit numbers using column arithmetic and at some stage this involved mentally adding 6 and 7, say, from the same column. I would know to ‘carry the 1’ from the result 13, but I wouldn’t, at that instant, give the 1 a great deal of meaning. I would not be thinking, “6 thousands (say) plus 7 thousands is 13 thousands, which is 3 thousands and ten thousands, which is 3 thousands and 1 ten-thousand”. I would though, I hope, be in touch with the more general idea that 10 entities in any column can be exchanged for one entity in the column immediately to the left.
Whatever sense one might make of the 2001 paper, it seems to me its general conclusions about the relation between conceptual and procedural knowledge have to be put to one side. More disappointing, perhaps, it doesn’t address the crucial question for me of what might happen to the development of relevant conceptual knowledge when pupils have spent considerable time on a procedure which, when it was introduced, they didn’t understand?
A few years earlier, Rittle-Johnson and Siegler produced a 35 page review of ‘The relation between conceptual and procedural knowledge in learning mathematics’ (in Donlan, 1998). They start with counting and consider aspects of single and multidigit arithmetic and some work with fractions and proportion. They pay some attention to the sources that might lead to the use of procedures but look closely at schooling in only one study (by Hiebert and Wearne, 1996), where some of the participants “received conventional instruction that emphasised practicing (sic) procedures” (ibid, p96). The study concerned multidigit addition and subtraction. Not surprisingly, perhaps, by the time children used correct addition and subtraction procedures, a greater proportion from the group who had received “alternative, conceptually oriented, instruction” were also classed as ‘understanders’, compared to the correct users from the conventional instruction group. What we don’t know is how children from the different groups felt about the work.
In a conference paper from 2002 (R-J & Koedinger), the previous year’s number line paper is used to support the claim that “Recent research indicates that conceptual and procedural knowledge often develop in an iterative process’ (ibid, p969). Thus does research evidence accumulate! The paper makes a reference to the ‘math wars’ in the United States but rather blandly asserts that “the maths education research community has begun to move beyond this dichotomy” (ibid). This would be all to the good if it didn’t mean that we now ignore the damage to pupils that can result from an overemphasis on procedures.
The 2002 paper involved 6 individualised, computer based ‘lessons’ with immediate feedback, that were given to 72 grade 6 pupils. The aim was to learn (something about how) to add and subtract decimals. Three of the lessons were classed as ‘conceptual’, with the other three classed as ‘procedural’. (I find these descriptors problematic, but let’s take them as read for now.) Half the pupils were given the three conceptual lessons first, followed by the three procedural lessons. The other half were given the lessons alternately, starting with the first conceptual lesson, followed by the first procedural lesson, and so on. The first lesson in both sets involved a money context, the rest were purely numerical (‘abstract’). The conceptual lessons focussed on ‘regrouping’, as in these assessment items:
Show 5 different ways that you can give Ben $4.07
List 5 different ways to show the amount 4.07.
The procedural lessons involved addition and subtraction, where pupils entered a number or quantity into a ‘chart’ divided into columns, with a second quantity entered underneath and which was to be added to or subtracted from the first (Figure 3). The lessons involved items similar to these assessment items:
You have $8.72. Your grandmother gave you $25 for your birthday. How much money do you have now?
Add: 8.72 + 25.
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Figure 3 (from R-J and K, 2002) |
The study found that “Students in the iterative condition made greater improvements in procedural knowledge and comparable improvements in conceptual knowledge compared to the concepts-first condition”. In particular, pupils in the iterative condition did relatively well on items where the ‘align numbers on the right’ pre-conception had to be overcome.
It seems to me, the way the authors use the terms conceptual and procedural, here and throughout the paper, needs careful scrutiny. However, before doing so, let us consider the findings themselves. They don’t seem surprising when one considers that the conceptual lessons focussed only on regrouping - a rather strange activity designed to help pupils add or subtract quantities, but whose purpose only became clear in the procedural lessons. Further, the conceptual lessons did not engage with the alignment issue.
It is understandable that for the purposes of research the two kinds of lessons were designed to offer quite narrow experiences to the pupils. A teacher would surely have enriched them and made them more purposeful by, for example, letting students explore how to add or subtract quantities in the conceptual lessons. However, even in their limited form, a key feature of both sets of research lessons was that they were based on meaning, that is to say on pupils’ existing knowledge of decimal fractions and, in particular, their knowledge of money, such as the value of different notes and coins and how they combine. It seems to me that the procedural lessons, as well as the conceptual lessons, were conceptual, at least to some degree, even if they were looking at a particular way of performing arithmetic that could, if one wanted to, be inculcated in a purely mechanistic way. We are told very little about the computer-based system, Cognitive Tutor, that was used for the lessons, beyond the fact that it provides “on demand step-specific help at any point in the problem-solving process” (p972). But even if this help was procedural (perhaps along the lines of ‘don’t do that, do this’), it would not prevent pupils, if they were so inclined, from trying to interpret such ‘help’ in a meaningful way. As for the benefit of presenting the lesson-types iteratively, it should come as no surprise that if experiences are given that it is thought might help pupils attain a particular goal, then it is useful to give pupils some inkling, early on, of what that goal might be.
The findings from this study (as I’ve interpreted them!) are fine as far as they go: pupils benefitted from the iterative approach as it provided a more meaningful way of teaching procedures! If only procedures were always taught in that way! But would it not also be important to compare such an approach with a hard-core procedural approach and look at the long term effects? After a delay of, say, 6 months, how well would pupils be able to remember and execute the column method procedure and how would they explain it?
In a later account of the same research (R-J and k, 2009), the authors acknowledge “the potential downsides of practising procedures early in the learning cycle” (p485) but they don’t seem to have any interest in investigating this. The paper includes a task analysis of what they now call the ‘place-value’ and ‘arithmetic’ tasks. Interestingly, for the steps in ‘arithmetic’ tasks, they provide an occasional ‘conceptual alternative’ such as “retrieve fact relating current place value to place value of column to the left” (p488). Thus they allow for the possibility that tasks in the ‘procedural’ lessons are tackled in a conceptual way.
Conclusion
In sum, these studies suggest that there is a happy medium where the teaching of concepts and procedures can go hand in hand, with each form of knowledge informing the other. However, teaching doesn't always conform to this happy medium.
There are two, more extreme, scenarios that I think are of interest, namely those where teachers strongly emphasise concepts first and those where teachers focus primarily on procedures. An example of the former is Connected Mathematics, which R-J and K describe (in 2009) as "the most widely used reform-oriented middle-school math curriculum in the US" (p485). It seems to me, it is quite possible that there are advantages in the Connected Mathematics approach of giving pupils more time to explore a topic and get a feel for the relations involved before wrapping things up with procedures that might curtail such activity.
As for the other extreme, a glance at the textbooks currently being endorsed by the government in the UK, would suggest that the focus on procedures is commonplace. Consider, for example, how simplifying fractions is introduced in book 6A of Maths - No Problem! (MNP) and book 6A of Power Maths.
In the MNP book, the rule of dividing the numerator and denominator of a fraction by a common factor is introduced alongside a representation showing both the original and the simplified fraction as part of a whole (see below).
The book contains three such examples, followed by three items where pupils are asked to apply the rule, supported by a diagram of the given fraction but not of the simplified version.
The chapter concludes with six purely numerical items involving the rule (below). Pupils are then asked to complete a worksheet where 18 more factions are to be simplified.
