In the last decade or so there has been quite a push back regarding discovery math. I have always found this to be very perplexing. How can it be denied that we are not born knowing mathematical concepts other than a very base form of quantity (Dehaene, 2011)? At some point, we all need to discover these mathematical ideas for ourselves. The question is really what is the best method for discovering these mathematical ideas. We could discover them through someone showing us (direct instruction), or it could be us figuring them out for ourselves through an experience, or it could be some combination of the two, but regardless at some point, we all need to discover these concepts and begin growing our own personal conceptual images for these ideas.
I am most interested in what types of activities contribute to the richest conceptual understanding and what are the most engaging and interesting. It is this combination that will cause students to excel and stick with mathematics. In my experience, this process of growing personal images and learning to think mathematically is developed when students experience a mathematical concept through spatial reasoning or pattern recognition. We want to foster a curious nature, and allow kids to enjoy those eureka moments. These ways of experiencing math help students to connect to their understanding, which is ultimately what we want to do – connect to understanding.
This activity will help your student discover what happens visually and symbolically when you multiply two fractions together.
Start by presenting a question such as 1/2 x 1/3. Ask them to fold the square in such a way as to represent one-half. Ask them to color in the section that would highlight the one-half. Then have them rotate the square and fold those halves into thirds. Have the student represent one-third by using a different color and color in the section one-third. At this point, take a look at your square and have a discussion about the meaning of the section that is colored in twice, which represents one-third of one-half.
Continue in this fashion with a variety of questions, where the student writes the answer in both the unsimplified and simplified form.
The goal is to never make math magic but to connect to understanding. Ask them why does this simplified form make sense with the paper folding images that we have created. Go over a number of questions where this simplifying equality can be proven to the student visually.
As much as possible I try to bring the mathematical concept back to a useful situation in everyday life, such as:
These types of questions will help your student connect more deeply with the mathematical idea. Feel free to have them build a paper model to represent the question but also have them express it numerically and reinforce the top x top, bottom x bottom discovery. Do a few of these type questions, they don’t have to all be so practical.
You then want to make a point of varying the questions to extend and push their understanding with such questions as:
Remember, mistakes are opportunities that allow for the most growth in understanding. Never make a student feel bad about mistakes, but use these as opportunities to understand what the student is thinking and extend their learning, by asking questions like “Interesting. How did you get that?” This way either the student will see their error or you can see why they made the error and help them understand where they went wrong by building a model for the question using the sticky note squares. Have fun discovering together!
P.S. When your student is writing fractions discourage writing them in slanted form such as 2/3 this will just need to be unlearned in high school. They should be written as one number directly over top of the other. Once questions get more complicated, such as in high school, it needs to be clear which number is on top and which is on the bottom.