Problematised Situation

After sharing the story and using the cards as mental routine, pose a problem such as the following:

In what different ways could
you show 45 feet?

After reading the book, a Grade 2 class were asked to find combinations of feet that would make 45 feet altogether. As usual they could use concrete materials if they wanted to and could draw pictures or write numbers, whatever they chose to show their thinking. The following examples show the potential of this activity in terms of allowing the students to show what they do know and what they can do.

The range of strategies and results was as expected broad. There was one student who had difficulty putting 10 legs on her crab. She drew 11 knowing it was too much but could not fix it up. She continued and drew a dog with 4 legs and counted to 15. She was keen to talk about what she had done and to count the legs. The only strategy available to her was a count all strategy. She knew she had pushed herself past her comfort zone and was pleased with her effort.

As always there were surprises in store for the teachers. Clarissa's response shown below was one such surprise. Clarissa has missed large chunks of schooling and while the teachers suspect that she is 'bright' they find it hard to judge because of her lack of attendance. Her work sample is impressive and gives some glimpses into her thinking and mathematical ability even though it contains some errors. Notice the way in which she set out the first four rows of numbers in a triangular number pattern and then switched to a counting in fives pattern. Notice too the way in which she tried to keep count as she went by writing numbers beside each snail. What would you want to ask Clarissa about her thinking here?

Laura's work sample shown below shows that she has used a method that has worked for her before. Although she didn't quite get finished you can see that she has written all of the numbers from 1 to 45 correctly in sequence and that she has begun to circle legs to match the creatures in the book.

You will also notice that she uses the snails to finish off her rows leaving no extra legs to carry over to the next line. She would have arrived at a correct solution given a little more time. Was she discovering things about using loose 'ones' or about odd numbers etc as she went along? This is something to ask her in an interview perhaps.

Jessica's approach as shown below was based on a tens strategy. It took her no time at all to draw 4 crabs and count by tens to get to 40. It was also easy for her to count on from 40. She uses the same strategy in the second line. Notice how she pairs a boy with a spider to make 10 and then three spiders with three boys to make 30.

She has made 40 again and can count on. She realised that there were in fact a lot of possibilities and turned her paper over so that she could create a table to help her. This is quite a sophisticated idea and had she had more time she could have used it to find many other possible combinations.

Ben's work sample shown below again uses a tens counting pattern as a starting point. First he drew 4 crabs to make 40 and counted on. Look at the symmetry he has created with his layout. Next he turned his paper over and used 40 snails with the same count on as shown. He was beginning to focus on ways to make 40 knowing that he could pick up the same array for 5 each time. He is working smart.

Amber's strategy shown below was different again. Notice how each time she draws a creature she creates a running total of the feet shown. She systematically crosses of the last total. No way is she going to forget where she was up to.

The work samples shown and discussed here demonstrate quite clearly the range of entry points into a problem as well as the ways in which such problematised situations can be solved. Because the students had invented their own tools for solving the problem, they could explain clearly to others what they had done and why. They knew that they had been personally successful in their efforts even if their results were not quite correct and given more time most of them could have fixed up any errors that they did make. As observers we gained a great deal of anecdotal information about where each child was at and about where to move to next.