If THEY build it, Learning will come.

As the Indians are set to open their Spring Training schedule today against the Reds, I couldn’t resist the reference to the movie  Field of Dreams’ , “If you build it, he will come”, when thinking about the importance of having students build their own understanding of math (or any other subject for that matter) rather than being shown how to understand math.

This is only my second year as Curriculum Director of our district, but in that short time even I was able to see the power of allowing students to build on their understanding through examples taking place in our elementary math classes.  Here is an example of what I mean:  A fifth grader, early in the year, was given this problem 16/.25 = ?

-As an adult who has gone through a traditional way of learning math and taken classes up through Calculus, I was stuck on this myself when it was presented to me until I resisted the urge to use my calculator on my iPhone and reached back in my memory to figure out .25 is really ¼ and if I remember the “trick” to flip a fraction when it is being used to divide and then multiply the fraction,  I can then come up with 16x4 = 64 by pure luck of remembering the algorithm.  However, because the fifth grader had been given the opportunity earlier in his learning of math to look for the patterns and relationships of math rather than just the answers he had a completely different way of looking at the problem.  

-The fifth grader looked at the problem and recognized .25 is equal to ¼ and that it takes four ¼ to make a whole, so if he had to take four groups of the ¼ all he would need to do is do the same and take 4 groups (times) the 16 to get his answer that 64 groups of .25 goes into 16.  No tricks, calculator, or really need to struggle to remember a long buried algorithm.  Instead, he applied his understanding of the patterns and relationships to which he was previously exposed. By the way, it took me 10 minutes to figure out how he solved this because I couldn’t get my mind to go there at first (in fact, I am not sure I am even explaining it right) and this particular fifth grader wasn’t the “top math kid” in the class.

The point to all this is to emphasize the power of allowing students to build their own understanding of math (or anything else for that matter) upon what they already know rather than believing they need to be shown a method  or told a  “trick” (algorithm) before they can apply it to a new concept.  As adults, we need to resist the urge to “tell” and nurture the inclination to listen by providing the opportunities for the students to develop their conceptual understanding.  Mathematician Conrad Wolfram emphasizes this in this TED Talk Teaching Kids Real Math with Computers.  

Wolfram points out that about 80% of what kids do in math education is calculating, yet calculating is only one part of understanding math.  He states math education should be about:

1. Posing the right question
2. Connecting a real world problem to a mathematical formulation
3. Calculations
4. Reconnecting the formulation to the real world for verification

The “calculation” part, Wolfram argues, is the least necessary to learn because computers can do that for us (remember my urge to reach for the iPhone) and that math education should be about the other three parts.

I believe Stanford Professor and “Math Guru”, Jo Boaler (@joboaler), would agree with Wolfram because in her book Mathematical Mindsets, she emphasizes the importance of dispelling the widespread myth that math is about speed and answers and instead nurturing the students mathematical mindset by allowing them to make connections, think logically, and use space, data, and numbers creatively.  She talks of mathematics as a cultural phenomenon; a set of  ideas, connections, and relationships that we can use to make sense of the world and emphasizes that mathematics is about patterns (not answers).

In order for students to discover these patterns and relationships, we need to be sure to ask the right questions and put them in the right situations, but most of all we need to resist the urge to be too quick to “tell” them.  Our math coach, Mike Lipnos (@mlipnos) always says, “Every time I give a student something, I am taking an opportunity away.”, it is not that you don’t support the students, but rather how and when you support them. (I also mentioned this in an earlier post: They do not understand shallowness because they do not experience depth). I believe Boaler would argue that the best time for “telling” a student about a mathematical concept is after they have had time to explore the problem because their brains are then primed to learn and are more motivated to learn due to being allowed the opportunity to build on what they know.  So,  I encourage you to slow down the next time you find yourself running around giving answers like a chicken with his head cut off, and instead think about what is the next best question we can ask students and what situations we can put them in that allows them to build on their own thinking.