Saturday, February 9, 2013

The Five Blackboards Strategy: A Cooperative Learning Exercise in Mathematics

a proof from Euclid's Elements (Book I, Propos...
Euclid's Elements (Book I, Prop I) (Wikipedia)
by Richard D Boyce

In Year Ten, I spend as much time as I can with the more talented classes teaching logical thinking and deductive reasoning in their Euclidean Geometry Unit.

Additionally, I spend time teaching my class a formal approach to geometric proof.

One of the activities I always do is to give my students a series of five written questions such as the one below on Circle Geometry.

In ABD, DA is extended through A to a point E such that AE @AB. If the circle drawn through A, B and D cuts BE at F, prove that FE @FD.

This strategy has a number of steps and it is used after all the basic teaching and practice with circle geometry has been completed. It is the final step in my teaching of this unit.

Here is my strategy.

Step 1 The students in the class are given 5 problems similar to the one above for homework. They must draw diagrams for each one and write a proof for for each problem.

Step 2 At the beginning of the next lesson (we have 70 minute lessons), I divide the class into groups of five, giving each student a number 1 to 5.

Step 3 I have already organised five to six blackboard/whiteboard areas in my room or a nearby space (I use mobile boards). I allocate each group a board area.

Step 4 Each student in a group has been given a number which is the number of the problem for which they will write a full proof with diagram on their board. They will explain their solution to their group as they go.

Step 5 The students in their groups interact with the 'student' teacher where necessary to gain more explanation or to debate the 'correctness' of the proof.

Step 6 Once a student completes their 'proof', he/she rejoins the group to allow a new 'teacher' to give their proof to their problem.

Step 7 The teacher moves from group to group getting involved only as a last resort.

Step 8 Once the process is finished, the class gathers in their normal positions where the teacher asks the students to share any problems encountered or any innovative proofs.

Step 9 The teacher reviews the process, highlighting any important issues not raised in Step 8.

I feel my students always gained much from this process. Here are some of the comments they made during our evaluation of the activity.

1. I found I understood better because the student was explaining in 'our' language.
2. We all helped each other.
3. Others gave us good ideas.
4. It was more fun than a normal lesson.

For me, as the teacher, the positive outcomes were:

  • Communication skills were developing.
  • The students were on task willingly.
  • They had ownership of what they did.
  • Mentoring among students was beginning to develop.
  • This strategy could be used successfully in many topics, e.g. trigonometry and simultaneous equations.

Cooperative learning in Mathematics is not the norm in most class rooms. Our author, Rick Boyce, used it many times in his teaching career especially when he became a Head of Mathematics in an effort to show his staff its value to students' learning. He gave professional development workshops on how he used it in his classroom and has published an eBook on the subject on the website

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