*Recently, I began a MOOC (massive open online course) called How to Learn Math – For Teachers and Parents (2). More than 40,000 people took the last class – mainly teachers, parents and school administrators. 95% of people completing the end of course survey said that they would change their teaching or ways of helping as a result of the course. The course offers important new research ideas on learning, the brain, and math that can transform students’ experiences with math and is based on the work of and narrated by Jo Boaler. It’s divided into 8 sessions and each one ends with a prompt to write a one paragraph summary reflection on the ideas. My plan is to post those reflections here.*

The sixth session is titled, *Conceptual Learning, Part II: Connections, Representations, Questions*. The module focuses further on strategies to help students conceptualize mathematics learning. There are 4 ideas that spoke to me here.

Visual representations of math thinking help all students. When students or parents claim that they’re not visual thinkers, that’s really a misuse of Howard Gardner’s theories of multiple intelligences. His theories were not intended to pigeon-hole people into categories of “all or nothing learning”. In fact, brain research tells us that it helps everyone to engage in creating visual representations and those that indicate that they’re not visual learners, probably need visual representations more than anyone. Visual thinking and solutions help us to see mathematics as conceptual and meaningful and that has become more evident to me as I make my way through this course. Visual representation has never been my first choice when solving problems. I usually approach them with a well-remembered process because it’s usually quick. Visuals usually take more time which traditionally, has always been associated with less intelligence. But what good is a quick answer if I have no strategies for checking its validity? And what if I need to refine my answer? How do I do that without a mathematical thinking process?

Mathematics students need to use a mathematical thinking process. Many students, as I did, believe that they should instantly be able to solve a math problem or they’re doing it wrong. Part of the reason for this is that teachers start these problems with providing formulas to substitute numbers into. Many students don’t understand them so they’re not useful to them and as a result they won’t remember them. A mathematical thinking process is a necessary process for students and teachers to develop and participate in. Some components of this process should include the following:

- Process the problem without formal mathematics.
- Read the question.
- Think about it. (What’s it telling me? Do I have a related experience?)
- Create a visual. (drawn and/or manipulative)

- Talk to others. Consult resources.
- Estimate an answer or a range within it should fall.
- Convert thinking to more formal mathematics.
- Try something. Refine it. Revise it.
- Verify the answer. (Does it look reasonable?)

This list does not imply a certain order that should be adhered to but I think it is certainly important to not rush to an answer immediately and I think a visual is important to create early in the process. As well, students may want to verify reasonableness throughout the process. Also, I’m not suggesting that all work in mathematics requires all steps in the process but certainly verbally dense algebraic problems require a mathematical thinking process and an integral component of the process is collaboration.

Mathematics classrooms should spend considerable time collaborating. Collaboration is the act of sharing ideas with at least one other person. It’s through this process that we’re invited to make connections and think conceptually about mathematical processes. Active collaboration inherently shapes our thinking which is critical to conceptualizing our mathematics thinking. I’ve listened to many students question why they need to explain their thinking if they got the right answer. When we explain ideas in mathematics, we are reasoning and reasoning is an important part of mathematics – it’s really what mathematics is. A scientist proves scientific theory by finding cases for or against. Mathematicians prove ideas by reasoning and making logical steps. If students are not reasoning their way through problems, they’re not engaging in mathematics.

An inquiry relationship with mathematics helps students discover connections. An inquiry relationship with mathematics is about being curious and using intuition to solve problems. Students with an inquiry relationship with mathematics display courage and confidence in taking risks and are comfortable doing so. We struggle with nurturing this relationship that students should have with mathematics and part of the reason why is our educational policies focus on cramming knowledge into students’ heads. Math knowledge is certainly important and we shouldn’t have one without the other but it seems that, in many cases, our educational focus has killed that inquiry relationship students entered school with or entered a class with. Intuition is necessary for mathematics knowledge to grow. This curiosity with math is hard for students to maintain when they’re being asked to remember lists of methods and to crunch numbers. When students are curious, they use their own intuition to discover the connections in mathematics which in turn, sparks more curiosity.