Building a Strong Mathematical Foundation

Before the days of GPS systems if you needed to go somewhere in a new town you would get out a map and take a look at what route you might take.  You might decide to take the freeway.  You would do that route for many days since that was the one route you had learned and each day it would get a bit more comfortable and familiar. 

After many days of following the same route, you would move to an abstract level.  You wouldn’t need to consult your map anymore to double check the name of the exit.  You would be sure to turn right at the T in the road rather than turning left.  If you wanted to stop and get a coffee on your way, you would know how to alter your route slightly to hit the Starbucks.   If you were on the freeway one day and there was a traffic jam, chances are you would have a good sense of how to exit and take some side streets because you had studied the map and had an idea of the surrounding area. 

Building a strong mathematical foundation with students proceeds in much the same way.  We want them to move from a concrete understanding to an abstract understanding but this process takes time.  We start with the students using tools and building models to develop their understanding of a concept just as you used a map to plan your route.  Once students develop some comfort and skill at this concrete level we can start moving them to more and more abstract levels of understanding just as you did when you were able to alter your route to grab a coffee on your way.

We want students to have a good sense of the surrounding area.  If they get stuck in one place, they should have ideas for how to go in another direction.  The new math practices call for students to “Use appropriate tools strategically.”  In order to chose a route and decide upon the best course of action, students need to have options available.  When we were in school most of us only learned one method for solving a problem.  Teachers have learned, however, that we do students a disservice when we only teach them one method of solving a problem.  If they get stuck or want to check their answer, they don’t have any other tools to choose from.

Additionally, teaching multiple strategies helps to meet the diverse needs of a class.  At any point in time students will be spread out along the continuum of understanding of a concept from concrete to abstract.  All students can be working on the same problem but they might not all be working on it in the same way, at the same level.  Students can learn from their classmates’ methods and ideas about solutions.  The whole group grows mathematically stronger as their consider the problem from different perspectives.

Recently I have been studying some of the math programs that are available for elementary schools.  Though Common Core standards are very clear about having students work at a concrete level before moving on to an abstract level, most of the math programs are not aligned with this thinking.

One program I reviewed had pictures of a few models but then when it came time for students to work independently, all of the work was abstract.  It takes time for students to move from a concrete to an abstract understanding.  Considering the example of the route in a new town again, you might have needed to consult your map each day for a few days before you could travel the route unassisted.  You traveled the route over and over and soon you became very comfortable with it.  We cannot just show students a few pictures of objects and assume that they are now ready for abstract thought about the topic.  Students need to build, model, and discuss their ideas.  They need to consider other students’ models of the concept and look at theirs in light of the new information.  The only way that students can build a strong foundation is to move through these steps from a concrete to an abstract understanding of the concept.

I see too many fifth graders who tell me they hate math.  It doesn’t make sense they say.  It isn’t interesting to them.  When I talk with them further I realize that most of these students never had an opportunity to explore concepts at the concrete level.  They were given algorithms and told to memorize them.  They were then able to apply those algorithms if the new problems looked just like the problems they had practiced.  If they hit a problem that looked slightly different or asked them to use their skill in a new way, they did not have enough of a mental image of the concept to be able to choose a new route.

Teaching students multiple ways to solve problems gives them a stronger mathematical foundation.  They can consider new problems from different perspectives and then choose the best method for solving them. They are flexible thinkers who can try new approaches when the one they are using doesn’t work out.  Teaching in this way also helps to meet the diverse needs of a group of students.  This is one of the strengths of the new Common Core Standards.