In an article in the September 2001 issue of Mathematics Teaching (published by the Association of Teachers of Mathematics) David Fielker describes a couple of "demonstration" lessons he had given in front of inspectors as part of some numeracy training. This prominent and much respected maths educator bravely describes how the inspectors criticised his lessons on the basis that he had not made clear what the learning objectives were, he had not summarised the lesson, he had been too open-ended, he had left some questions unanswered, and so on.
A lesson on fractions that I saw given by a maths co-ordinator in a middle school several years ago came to mind. This was in the period post-Cockroft (Mathematics Counts, 1982) and pre-National Numeracy Strategy, and the lesson was very much in the "Fielker" mode.
The teacher settled the class and then wrote on the board: "Which is the larger of these two fractions 2/3 or 3/4?". This was all she contributed. Her class all knew what was expected of them. In their groups they had first to discuss the problem and ways of solving it, agree on a strategy, find some way of proving it and prepare a convincing argument to persuade others. They also knew they could use any equipment that was available. Of course, the teacher had "trained" the pupils to work like this. Try that approach in classrooms where the pupils have not been so well prepared and you would have a riot on your hands.
Some groups tried to use a calculator to solve the problem, others used pieces of string. Some used cut-out circles, one group used Multi-link. It took this particular group several attempts before they stumbled across a rectangle made from Multi-link which enabled them to easily find 2/3 and 3/4 of the same rectangle. They then tried to find a different-sized rectangle which would enable them to do the same thing and show them again that 3/4 was still larger than 2/3. One group thought they had solved the problem right away. Three-quarters must be larger - because the numbers are larger. The teacher posed them a question: which is the larger, 2/4 or 1/2? This caused them to pause and reflect again.
With about 20 minutes to go, each group was asked to share their thoughts. It was pretty obvious to all that the various strategies had all concluded that 3/4 was indeed larger than 2/3. Just before the end of the lesson, the teacher drew two circles on the board, one much larger than the other. She divided the larger one into three equal parts, shaded in two of the parts and asked the class what fraction she had shaded in. They replied correctly. She then divided the smaller circle into four equal parts, shaded in three of them and asked the class what fraction she had shaded in. They again replied correctly. She then said: "According to my diagrams, 2/3 is larger than 3/4 but all of you have just said that 3/4 is larger than 2/3. For homework, think this through and write down some thoughts". I wish I had been there when they returned with their thoughts.
In my role as an advisory teacher I was keen to try the same problem but with a group of non-specialist teachers, and using a slightly different approach. At the front of the room I put out three chairs. On the first chair I put one Mars bar, on the second chair two Mars bars, and on the third chair three Mars bars.
I selected one teacher and said: "I want you to go and stand behind the chair which will give you the most Mars bars." (She stood behind chair three.) I then selected a second teacher and said: "I want you to go and stand behind the chair that gives you the most Mars bars remembering that if you stand behind the same chair as someone else you will have to share the Mars bars out equally between you." (He stood behind chair two.) After the sixth teacher had made a choice there happened to be one person behind chair one, two people behind chair two and three people behind chair three. Consequently, the seventh teacher to be called had to make a decision between which was the larger amount of chocolate bar: one bar shared between two, two bars shared between three or three bars shared between four? In other words, he had to decide which was the largest: 1/2 a bar, 2/3 of a bar, or 3/4 of a bar. At that point we had a lengthy discussion, not about the answer, but about the strategies they would use to decide which was the largest of the three fractions. Because they all knew that 2/3 and 3/4 were both larger than 1/2, the discussion centred on the other two fractions.
The first time I did this, I was fascinated with the range of strategies used. Some used time, that is, used the fact that 3/4 of an hour (45 minutes) was longer then 2/3 of an hour (40 minutes); others changed the fractions to decimals; some found the common denominator and went from there; one or two compared 1/4 with 1/3 and used that; some imagined a "cake" cut into thirds and fourths; others imagined a bar of chocolate with 12 sections in it and so on.
Each of the strategies was discussed and evaluated. I then presented them with two more. One was "cross multiplying". Using the two fractions 2/3 and 3/4 you multiply 2 x 4 and 3 x 3 to give you 8 and 9. As 9 is greater than 8, 3/4 is therefore greater than 2/3. The second strategy was to use a "fraction" calculator to subtract them. If 2/3 minus 3/4 is positive, then 2/3 is greater than 3/4. If 2/3 minus 3/4 is negative, then 3/4 is larger than 2/3. (Note: if the answer had been zero, 2/3 would have been equal to 3/4.) After discussing all these strategies further, we then returned to the original task with the Mars bars. We noted there were 23 teachers in the room. I asked them to imagine the process being continued and to work out which chair the 23rd person would go and stand behind and what fraction of a bar that person would get. They worked on this task in their groups for some time before we all agreed on the "answer".
However, they did not manage to produce a strategy which would have enabled them to shortcut the calculations. I left them with that problem to think over for the next time we were due to meet.
Neither of these two lessons fit into the three-part lesson structure that is currently dominating teaching practice and I wonder what inspectors would make of either of them. Even though teachers are continually being told that the numeracy framework is just a guideline and that they can alter the structure of the lesson, most have neither the confidence nor the desire to do so. They play safe and stick to the guidelines. And the end result? A very monotonous diet.
Pointers for a successful maths lesson:
* You cannot rush learning. You cannot expect pupils to achieve every one of your learning objectives every lesson. Neither do you want them to think everything can be solved and completed in 45 minutes.
* There is also much to be said for frequently not telling pupils what the learning objectives are, so that they can approach the problem free from constraints.
* The "Fielker" approach still has a very important role to play in the teaching (and understanding) of maths. To hear teachers say: "We will do lots of investigations after the SATs" is very worrying. This either suggests that they are concentrating too much on teaching to the tests or that they do not think that the investigative approach has any role to play in the teaching (and learning) of maths.
Peter Critchley is a former advisory teacher and numeracy manager in Suffolk