Educational scenario 3: Concrete materials in a problem-solving task Jake, Remy, Molly and Eliza, all in Year 4, were working on a problem from the Country Areas Program (CAP) website. The problem asked the students about making a square-based pyramid using 140 oranges. The questions were: how should the oranges be arranged; how many oranges would there be in each layer; and how many layers are there? Remy started by guessing the number of layers. He reasoned that adding a series of numbers together until the total of 140 was reached would answer the question, but was unsure where to start. The teacher asked the group how many oranges would be in the top layer of the pyramid. They looked blank until she brought out a square-based pyramid from a box of solid shapes. They then quickly realised that there would only be one orange at the top. The next question How many in the next layer? was harder to answer, until the students got out a set of blocks to make a pyramid, starting from the top layer of one. They realised that each layer of the pyramid would be a square. Jake pointed out that the side of the square increased by one each time. At this stage, making the squares was still necessary to avoid errors such as assuming the next square would be just one or two oranges bigger than the one before it. Molly was writing down the number of oranges (blocks) in each square. The teacher asked whether the numbers were starting to form a pattern. They then moved to paper, and wrote down the numbers they had so far: 1, 4, 9, 16 and 25. Eliza saw that the differences between the numbers increased by two each time. (The teacher recognised that the numbers were all square numbers, but chose to leave that observation until later.) They predicted that the next square would have 36 oranges, and made it with blocks to check their prediction. The teacher reminded the group of the questions they had to answer, and asked them to review which questions they now knew the answers to, and which still had to be determined. They realised that they only had to make layers up to the total of 140. Remy wanted to continue their number pattern until it reached 140, and was puzzled when he reached 121 and 144. The teacher asked them to read the question again, and think about how many oranges they had to work with. Molly realised that they needed to add together the oranges in each layer to reach the total of 140. They did this, and realised that they didnt have enough layers yet. They added more until the total reached 140, giving the answer to the problem. They referred to the pyramid they had made first with blocks, then on paper to answer the questions in the problem. The next step was to explain the strategy they had used to reach the solution, which proved difficult to remember. With the teachers prompting, they retraced their steps and were able to write down both the answer and the strategy they had used to reach it. Questions: 1. How did concrete materials help the students to solve this problem? 2. What role did the teacher play? 3. Identify and describe elements of Vygotskys and Piagets theories in the childrens thinking.
