About a month ago a colleague of mine Kyle Pearce wrote a post “Does memorizing multiplication facts hurt more than help.” It was a very interesting read and I happen to agree with Kyle’s point of view. As many of my frequent readers of this blog know, I prescribe to the constructivist approach to learning mathematics. I believe that students through discovery and proper guidance will be able to understand a wide load of big ideas and theories. Not only do I believe this but I have witnessed this first hand with my students in every grade that I have taught.
However, this is not so for many people. In fact it was a discussion on Kyle’s blog post (feel free to read the thread) that has me thinking more and more about this topic. And not only thinking about it but trying to fix and insight thoughtful discussion around the ways in which we are teaching math.
Maybe a little background first. Math has been a hot topic for the past year, if not for the last century. For many countries, provinces, and states, math curriculum has undergone a significant change from what we grew up with as children. Some. like myself, believe that these changes are for the better, some have not. I was recently at the OAME and listening to Brent Davis a professor at University of Calgary. In his lecture he shared that the reason math curriculum was introduced was so we could have a work force to crunch numbers, nothing more and nothing less. As we have evolved beyond that (not saying fact crunching is not important) our skills have also changed and I think this is what we need to remember; we have evolved.
For this reason I and many others are proposing a more balanced approach to mathematics. Lets stop this war and needless debates and get to teaching good mathematical practises. One in which our students will push their thinking and really think about the numbers.
If it was up to me this is what I would include:
1) Math should be linked to Big Mathematical Ideas:
I think this is the first step to thinking about our students as mathematicians. Catherine Fosnot (2002) has some very interesting work around making our students mathematicians. One of the most interesting facts is there was a study done with so many mathematicians and they were asked to solve a problem. Not one of those mathematicians solved it the same way. I found this interesting because that is what I see math. Math is about the mathematics and there is not one way of doing things. We have to teach our students the understanding, the flexibility and the patience to be mathematicians.
2) Math is about real numbers:
Students need authentic experiences to learn. Let’s think about ourselves and how we learn. Now some do learn through reading and replication but if you honestly think about how you learn a concept the best; it is through trail and error and than guidance from a mentor. This is the same for our students. They need real experiences so that they can play and discover the mathematical concepts. In my personal experience both in tutoring high school students and teaching mathematics in the primary and junior divisions it’s the contexts that allow students to really understand what they are doing. It’s the context that helps them build models of representation. In my classroom, these are often done through social justice problems and real life contexts.
3) Students need time to explore:
This goes hand in hand with the above comment. As much as we need instruction, we also need exploration. Students need time to make mistakes, reflect, debate and discuss. These experiences allow students to make connections between concrete and abstract thinking. I was reminded at the OAME that every mistake makes a new synapse in the brain. We need these mistakes I order to solidify our learning.
4) Students need Mentors:
My most recent research in understanding teachers questions has shown me the importance of teachers, not that I didn’t believe that before. With a shift towards discovery learning, we as educators have forgotten the importance of our role, or what even our role is. I truly believe that we should not be the dispenser of knowledge but the mentor of that knowledge. Though through exploration students will learn (many studies to show this) they may not have tools to reflect or pull together the big ideas. I think this is where much of the back lash has come from discovery math, reform math or whatever you want to call it. As students explore and discover there needs to be some sort of guidance. This is where a teacher can shine and help students with the mathematics. However, my research shows that for this to happen, a teacher needs to 1) have a good understanding of mathematics, 2) have a good understanding of how children learn mathematics and 3) plan. I know that all teachers plan but this planning involves thinking of big ideas, landscapes and possible questions. It is these questions carefully placed that can allow students to figure out and make mathematical connections.
Take a look at this video of three of my students thinking about fractions.
They have never been taught fractions from me before this and in fact as a class we haven’t even started the unit. However, that being said think about their learning and the role they play and the role that I play as a teacher. Where do my questions come from? Why did I ask them at the time I did?
5) Time for debating, conjecturing, discussing and proving:
For me this is the time for a teacher to shine; however not in the traditional sense of standing in front of the class and lecture or tell students how it should be done. Just like students need time to explore they also need time to debate as a community. It is through this debate that students defend their thinking, conjecture, question and solidify their learning. Moreover, it goes beyond just showing and telling. As a teacher the types of strategies that you show matter. How are you building the learning up, what questions are you modeling? How are you focusing the talk? How are you fostering talk? These are all questions that a teacher needs to be asking.
Also take a look at my most recent grade two conversation of multiplication.
6) Repeated Practise:
Yes I said repeated practice! Students need it, but it’s not just doing procedural learning over and over again. When I say repeated practise I mean a similar problem for students to continue their exploration. Students, well most students, cannot solidify their learning through one experience. In a typical unit my students will solve about 7-8 problems that could take three to four weeks to learn. These problems build their learning and knowledge from day to day.
Yes skills are important. They are needed but they are not needed like we use to think about it. For me it’s how are we introducing facts. Do we make students think through facts? Are they taught in isolation or allong with concepts? In my classroom, students practise facts at home, they play math games in the classroom (here is a file of my math games: https://drive.google.com/folderview?id=0B4245QONE7HaaHl3M3ZKNWd3SUk&usp=drive_web
). In addition before my problems start I often use string lessons which builds on mental math strategies and learning how to be flexible thinkers and playing with numbers. This to me is more important then struck memorization. It teaches students that numbers are not confined facts but that you can pull apart numbers and use known facts to solve other facts. Through this process students often learn all of their math facts, can recall them and use them in problems, which to me is way more important.
These are just a few of my thoughts on what I am calling a balanced math approach. I have a few more to hash out around integrating and implementing a center approach within my problem solving approach.
What are your thoughts? Don’t you think it is better to discuss and fix our problems rather than lay blame about which is better? Shouldn’t we think about our students first and their needs in their 21st century world? Love to here your thoughts.