Throughout my career, I have passionately believed that using an investigative, problem-solving approach brings about a deeper understanding and longer-lasting learning.

I also believe that finding out what pupils already know before teaching them anything and using group work and self-assessment are key factors in successful learning. Interestingly, these approaches are now being revalidated under the rubric of formative assessment.

The National Numeracy Strategy is now working in key stage 3. What will this mean for teaching approaches, lesson structure and differentiation?

Recently, I had the chance to test my beliefs in a school. Focusing on a two to three-week unit of work on Pythagoras’ theorem (two double periods and one single period a week), we took two parallel, mixed-ability, classes in Year 9, roughly similar in attainment.

One class was taught Pythagoras’ theorem in the normal way by a teacher uninvolved in the experiment, while the other class was taught by the head of department. I merely acted as an observer and adviser.

We decided to:

* establish what the pupils already knew, understood and could do

* identify what they needed to learn

* write the learning objectives

* select investigative, problem-solving type activities that best matched the objectives

* assess informally whether the objectives had been achieved and then allocate remedial or extension work accordingly

* carry out a final formal assessment to find out what the pupils now knew

* carry out a test on both classes two months or so after completion of the work

* consider outcomes.

First, we gave all the pupils a response sheet with questions such as “I can draw an acute-angled triangle and know why it is so called” to establish what they already knew about Pythagoras and related matters such as squares, square roots and areas of triangles.

None of the pupils seemed to know anything about Pythagoras’ theorem or its application to right-angled triangles and it was clear there were some weaknesses in finding squares, square roots and areas of triangles.

For the first lesson, we put the pupils into groups of four or five and asked them to discuss each other’s responses to the original statements, to confront any differences and come up with solutions. They could then alter their own answers, but in a different coloured ink. The class, as a whole, then discussed any outstanding issues.

For homework, pupils were given a worksheet on slanting squares and had to use three different methods for finding the area of a slanting square drawn on dotted paper. Using the information gathered from this homework, the teacher developed a truefalse sheet, with 13 statements on the nature of squares and square roots which the class had to work through individually and in silence for 15 minutes at the beginning of the next lesson. Again, they got together in groups and discussed responses, making any changes on their own sheets in a different coloured ink.

Now the teacher was ready to produce learning objectives and the initial open-ended activity. The class had to investigate what types of triangles could be enclosed by using combinations of three squares from a range of different-sized squares (with areas written on each) and say if anything was special. Waiting for the right time, the teacher chose and developed one particular comment: was there, one of the pupils asked, a relationship between the areas of the squares and the type of triangle enclosed?

In the next lesson a right-angled triangle was drawn on the board with squares on each of the three sides. The teacher explained that he was willing to supply two pieces of information about the diagram, but the pupils had to ask the key questions.

Once these had been established, pupils then had to use the two pieces of information to work out other areas and lengths of sides shown on the diagram. In the third lesson, a statement was written on the board: “The area of the largest square is four times the area of the right-angled triangle. True or false?”

This statement had been made by one pupil, who for some reason had always worked on isosceles right-angled triangles. In the initial whole-class discussion the fact emerged that the largest square could be dissected to form four right-angled triangles (congruent to the original right-angled triangle) surrounding a smaller square. The rest of the lesson was spent trying to work out how the dimensions of the inner smaller square could be established from the dimensions of the original right-angled triangle.

For their summative assessment pupils produced a “revision booklet” which recorded useful things they had learned about Pythagoras.

The other class had followed a more traditional approach, with no pre-test, no open-ended activities, no independent group work. The teaching was excellent, but the approach was totally different, much of the class work done from a textbook with formal exercises and traditional-type questions.

Four or five weeks later, including two weeks over the Christmas break, and without any advance notice, we gave both classes a written test in the form of another response sheet.

The results showed quite significant differences in performance between the project group and the peer group. The average test score for the project group was 80.2 per cent (boys 76.2 per cent, girls 84.6); the peer group’s average was 56.6 per cent (boys 50.4 per cent, girls 73.6).

It would be too easy to jump to conclusions from such a small-scale experiment. Nevertheless, it seems that the project group remembered far more and had a deeper understanding of the topic than the other group.

The head of department said: “Establishing what the pupils already knew, understood and could do before starting on the topic had significant advantages because I could identify initial weaknesses. Otherwise these would probably have been left unresolved and would have hindered progress.

“The importance of group discussions and summative pupil self-assessment cannot be over-emphasised. The level of understanding shown by the pupils in their booklets was very high indeed and probably the main reason why their test marks were so good. If the department had a setting system, about half of those particular pupils would not have met ‘Pythagoras’. I believe this to be unacceptable.”

Our problem-solving, investigative approach may or may not fit easily into the KS3 National Numeracy Strategy way of doing things. It is not for me to judge. All I do know is that we have to own up to the fact that there was no three-part lesson structure, the learning objectives were not shared with the pupils and there was hardly any direct teaching from the front, apart from the frequent questioning, probing and pushing of the pupils’

thinking. And the learning wasn’t rushed, but it was still effective. Very effective in fact.

Peter Critchley is a former advisory teacher and numeracy manager in Suffolk