What Do You Mean, You've Forgotten!: An Investigation into Mathematics and MemoryAuthor(s): Gordon MooreSource: Mathematics in School, Vol. 31, No. 5 (Nov., 2002), pp. 42-44Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215568 .

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What Do You Mean, You've Forgotten! - An Investigation Into Mathematics And Memory

by Gordon Moore

We've surely all been there. Looked at the scheme of work for the next topic, prepared it, taught it and then at some time tested the pupils on it. When one sees the results ... well it's scary isn't it? Did I actually teach them anything? Surely they can't have forgotten it already. I only did this work with them last week!

In fact one teacher I worked with explained the situation by telling me of the 'mental magnet' he felt must be suspended above his classroom door. As soon as the pupils left his lesson and went through the door, the magnet sucked everything they had learnt that lesson out of them - and there was no mess.

This scenario occurred too frequently for me to ignore and so I took advantage of the Best Practice Research Scholarship run by the DfES to do some classroom research of my own into the topic of retention of mathematical knowledge by high school pupils.

The research that has been done into memory is vast and wide ranging, yet it seems that there is still no agreed understanding as to what is really going on in the mind. A typical example is that it is easy to remember the details, plot and characters of a novel you don't intend to memorize, but when it comes to learning the material needed to pass an examination - well we all know how difficult that is.

Serious memory research started off over 100 hundred years ago with Hermann Ebbinghaus (1850-1909). He did research into memory by memorizing lists of nonsense syllables and seeing how long he could remember them for and in finding out how many times he would need to practise to gain mastery of such a list. He discovered the characteristic forgetting curve or retention function (Fig. 1).

Retention Curve

65-

650 55

so

450

20 0 100 200

Tim q 0 4O 00O Tim

Fig. 1 Typical retention curve showing percentage of original learning against passing time - after Ebbinghaus

The graph shows the percentage of original learning against time. The characteristic rapid fall off and then slowing rate of forgetting is typical. John R Anderson, a noted researcher into memory, has shown (using log-log

scales) that this is actually a power law of forgetting (not exponential).

The first major theory of memory, and the one that perhaps most people are aware of is due to Atkinson and Shiffrin in 1968. They postulated the idea of a hierarchy of memory stores. They suggested that after objects are perceived, they end up in a short-term memory store. The contents of this store are constantly being updated by new input and if the previous contents are not transferred into long-term store - essentially by the method of rehearsal - then they are lost. However, there are problems with this hypothesis. Simple repetition does not always result in a better memory - if only it would!

In 1972 Craik and Lockhart argued that the memory trace (or engram) - the actual representation of the memory in the brain - was a result of the amount of processing that was carried out on the material to be learnt. The more processing that the material received, the deeper and stronger the memory. Much research has added to this basic idea and it is widely accepted and fits in with the neurological evidence on memory traces.

There are three basic ideas connected with memory. Firstly, there is the acquisition of the memory in the first place, then there is the maintenance of the memory in the mind and finally there is the retrieval of the specific memory. Failure in any one of these areas results in overall failure of memory - forgetting.

Interestingly it is very difficult to decide if forgetting is due to an inability to retrieve the memory or due to its decay and loss from the mind. How can one possibly tell? It is felt by many that we don't actually lose anything from the mind; forgetting is due to an inability to retrieve the memory. The reason for this is that many experience the 'oh, of course' response when given the answer to a question they cannot answer - they had the memory, but could not access it, perhaps not even realizing they knew it. It is also clear that it is easier to re-learn items that have been forgotten, than it is to memorize them in the first place.

One of the most important factors in remembering things is that of making some kind of elaboration of the material. For instance in trying to remember a list of words one might try using rhymes, e.g. clock - crock or tock, or one could use a synonym or other semantic processing, e.g. watch, timer. Apparently words are learnt better if the semantic representation is used. It also works better if the learner themselves generates the rhymes or synonyms - more processing is taking place. Even better is if one creates a whole picture about the word clock: what type, what age, what colour, the history of the clock, etc. Another advantage is that more ways of accessing the word are provided. For instance, a common memory technique is the 'method of loci'. In this the objects to be memorized are mentally pictured as being along a path, such as the journey to work that the learner is used to taking. The other common technique for memorizing words is that of visualizing some

42 Mathematics in School, November 2002 The MA web site www.m-a.org.uk

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strange and wonderful association for the word, e.g. for the clock, one might visualize a giant clock, with a face, hands and feet running quickly to get somewhere. Both these techniques are relying upon much processing of the material.

Making material distinctive also makes it memorable. We may have a favourite restaurant that we go to many times, but we only remember clearly the time when the waiter spilt the food on us. Well actually my wife seems to remember every occasion we go to one restaurant and everything that I said! It must have been distinctive! One interesting experiment had people learning words that were written upside down - they remembered the words better than when they were written the right way up.

Experiments have shown that the mind is well organized and that material that is better organized gets remembered better than disorganized or logically incoherent material. Surely in maths we have got that well sorted out - well until we remember that perhaps we haven't yet done directed number with a particular class and here we are trying to substitute negative numbers into a formula! Oh, please don't say it's only me that this happens to!

Memory can also be affected by the attitudes of the learner. For instance, we have already seen that intention to learn doesn't seem to matter - the novel gets learnt whether we want to learn it or not. Motivation does have an effect - probably because the learner will process the material in a more meaningful way. Anxiety may interfere with memory, but high arousal seems better than low arousal. Time of day has some effect, but this may be due to the fact that it is quieter at night and there is less distraction. Of course the learner won't make much effort if the material isn't relevant - Sir, what's the point of this?

In a school we have the problem where the need to memorize material in one lesson is followed by the need to memorize material from a quite different subject in the next lesson. It has been shown that having to learn some new material can interfere with the memorization of previous material.

The context for retrieving the memory should be as close to the context in which it was learnt. An intriguing experiment had divers learn some material on land and some under the water. The best retrieval occurred when the contexts matched. This of course has implications in the type of learning environment provided and the type of examination taken.

A most important phenomenon is that known as the 'Spacing Effect'. This is concerned with whether it is better to have one major episode of learning (massed practice) or many smaller ones, spaced out over time (distributed practice). Some experimental evidence suggest that distributed practice leads to better learning and retention, but most evidence for this is only related to the usual kinds of learning found in the psychological laboratory.

Indeed, that last point is very important and profound. By far the vast amount of research carried out into memory has taken place under strictly controlled conditions as psychologists attempt to identify the key factors involved in memory - of which there seem to be an ever increasing number. But what about in the real world where things are not so straightforward and where indeed in mathematics the material is cognitively challenging and complex.

My research looked into this. I actually used the ideas from the spacing effect and from making the learning

distinctive by using ICT as key factors in attempting to reduce the expected fall off in retention. As it happened what I discovered was unexpected.

The research method was to teach four classes of year 8 pupils (12-year-olds) two different topics - area of simple plane shapes and the calculation and use of mean, median, mode and range. The four classes comprised of two parallel middle to top ability pupils and two parallel classes of low to middle ability pupils. Each class was given a pre-test, were then taught the material, with one class at each level acting as a control and the other having some intervention applied to it. Each class was then given a post-test. At intervals of time after that, each class was given a retention-test to see how much they still retained.

The results of my studies are shown in the two graphs below (Figs 2 and 3). The graphs plot the week number along the bottom and on the vertical axis is plotted the percentage score in the tests. I haven't the space here to discuss the details, but the interested reader is directed to my full report on the DfES BPRS web site.

When one looks at the graphs one is struck immediately by the facts that pupils learn - which was most gratifying, but also they don't seem to be forgetting! There is none of the expected quick fall-off in retention even over quite a few weeks (20-25). This of course was most unexpected and made the use of interventions somewhat redundant, though of course since I was already committed to the research I did carry on! Of course there is a little fall-off, but it is quite slight really. In other words what the pupils do learn they seem to remember quite well. Some individuals even increased their test scores over time.

What is clear is that they do not initially learn to the expected degree. The big question really is why didn't they learn well enough to score top marks. Certainly for the mensuration experiment, the pupils were only being assessed on how well they could work at a surface or instrumental level, i.e. could they remember and use a few formulae. Even for the work on mean, mode, median and range the level of deeper or relational understanding being tested was low.

Retention of Mensuration Knowledge

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Fig. 2 Chart showing retention of mensuration knowledge over time

Mathematics in School, November 2002 The MA web site www.m-a.org.uk 43

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Retention of Mean, Median, Mode. Range Knowledge

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Fig. 3 Chart showing retention of mean, medium, mode, range knowledge over time

Since this does not fit in with the expected outcome it is useful to hypothesize on what might be happening. It could be that after instruction further processing of the material takes place, perhaps even subconsciously. More likely is that the learning of new mathematical knowledge in later lessons is causing the links to the older knowledge to become reactivated thus strengthening the memory. For instance, if a pupil has learnt about areas of rectangles and then moves onto another topic such as multiplication, their mind might well activate the schema containing the material for areas as it seeks to assimilate and accommodate the new information. It might well make a connection between multiplication and area, thus strengthening the memory trace. At times this 'spreading activation' might become quite strong and the pupil experiences insight - the 'Eureka!' effect. I am hoping to test some pupils to see if retention is lost over the summer holidays when this process shouldn't be able to happen. The DfES of course believes quite strongly that pupils do forget material over the long holiday - although despite requests they have not provided me with the references to research to support this belief.

Another possibility is that the effects of doing the tests may well reactivate and allow further processing of the memory. This is a bit of a paradox, isn't it! You can't test how well something is remembered, because the test will interfere with the memory - Heisenberg anyone! It is also clear that pupils will also perform their own post-mortems on a test and this is an unlooked for additional intervention in the experiment.

It could also be that the loss in memory is occurring over very short periods of time - minutes and hours, not days.

Finally, it could be that the research itself is flawed. As a first attempt at doing research I certainly made lots of errors and mistakes - documented in my report if you really want to see how bad I was! Of course I am also dealing with a very small sample of pupils - it could just be a local peculiarity.

However, my belief is that this effect is a real feature of memory for cognitively challenging topics. Contrary to expectation, pupils don't forget - they seem to retain what they do learn very well.

Additional Reading My full report: Retention of Mathematical Knowledge by High School Pupils, can be found on the DfES Best Practice

Research Scholarship web site. The full report is found by clicking on the option at the bottom of the summary report. http://www.dfee.gov.uk/bprs/reports.cfm

An in depth coverage of memory research is found in Learning and Memory - An Integrated Approach (2nd Edition) by John R. Anderson, Publisher John Wiley and Sons ISBN 0 471 24925 4. [

Reference

Anderson, J. R. An Integrated Approach, ISBN 0 471 24925 4, John Wiley and Sons.

Keywords: Memory; Retention; Mathematical knowledge.

Author Gordon Moore, Endon High School, Leek Road, Endon, Stoke-on-Trent ST9 9EE. e-mail: gordonmoore(i gordonmoore.co.uk

Crossnumber Puzzles

Solution to Key Stage 3 Number 1 Puzzle

(from page 41)

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44 Mathematics in School, November 2002 The MA web site www.m-a.org.uk

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