How did the use of manipulative helped the students build a preparation or a readiness to study calculus? The study of speed through time -velocity, was totally related to what exactly the students learned through their elementary, middle, and high school years: observe, discuss, analyze, organize, solve, and present the work with strong justifications.
The study of the cat's movements was completely related to four story building, pizza, game series ... projects.
"Mathematical knowledge grows from actual experience of moving around and living and thinking about it. How we are moving around?," said Prof. Robert Speiser. I do said the same in one of my video reaction paper, that the primary and most important skills of math and how to observe things and make differences between them and draw conclusions are built during our early age, Exactly at the moment we started be aware of the world around us. "Get in there and move!" said Prof. Speiser. However the students weren't exposed to calculus yet, but they were able to present their work in different format: graphs and calculation. The comparison between the cat's speed (acceleration and constant speed) allowed the student to make a prediction of the cat's next movements or velocity.
Overall, the students were able to study the cat's movement mathematically and present their findings in a very convincing way. Thus, good teaching can help students build powerful and lasting understanding to important ideas, as prof. Speiser said. Moreover, fro us teachers we must understand that students will not learn and their learning will not last unless what they are learning or the information which they are getting from us is making sense to them. Otherwise, the teaching will be like filling water in a punctured container.
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