Thinking Tools (Manipulatives!)

Today we started by brainstorming a definition of manipulatives.  Our group discussed how manipulatives can help abstract ideas become more concrete but can also be used to take concrete ideas and extend them to become more abstract.  Other groups included lots of examples of manipulatives such as measuring cups, math cubes, algebra tiles, rulers or really any tangible objects.  We also discussed using the term “Thinking Tools” when talking to students about manipulatives.

Even before my experience with Scientists in School, I was a big fan of hands-on activities, but that experience reinforced my understanding of the importance of manipulatives.  I believe that students learn through doing and using manipulatives is a great way to encourage students to become involved and enjoy learning.  We did talk about some negatives which may occur while using manipulatives with students, like flying elastic bands and blocks being thrown.  For those who want to test the limits, there are computer simulations (eg. Gizmo) which allow students to do similar activities using a computer (eg. Geoboards with elastics).  What a concept that would be to have to use an ipad as the consequence of misbehaving with the manipulative!   In all seriousness, I think that a computer simulation of an activity should only be used when the real hands-on activity can not be performed in the classroom due to safety or equipment issues.

Some students don’t want to use manipulatives because there is a stigma in using them or they think that they are “toys for younger kids”.  Often, teachers bring out the manipulatives to help those students who are still struggling with a concept, as if an afterthought.  Instead, the activity should be structured such that all students need to use the manipulative in order to solve the problem.  This will allow all students to become involved and learn by using the manipulatives.  Then, these same manipulatives can be used in the future to build future concepts on past understanding.  For example, using algebra tiles to understand integers with number lines, collecting like terms and then extending that into using the “clothes line” area model of multiplying in grade 9 or quadratics or completing the square or the difference between sin and cos in grade 10.  The best manipulatives are useful across many strands and grades so that students become comfortable using them and can build on previous conceptual understanding. 

I also think that it can be useful for students to develop their own manipulatives.  I had groups in 12U biology make manipulatives in order to explain the concept of DNA replication.  During their presentations to the other groups, it became very obvious if the students had grasped the concepts.  After using the algebra tiles, it became apparent that if the teacher has the students make the algebra tiles, the number of “1’s” can not perfectly divide into an “x” tile but that the “x²” tile needs a length of “x” and a width of “x”.  Thus, some thought is needed if the students make their own math manipulatives.

As I was using the algebra tiles, I became more proficient at being able to visualize the math concepts.  The more I practiced, the faster I was able to represent the math expressions using the tiles.  In our group, we discussed the importance of teachers making themselves comfortable by using the manipulatives before using them in the classroom.  My daughter said that in grade 7, she did not understand algebra tiles at all but in grade 8, the way that the teacher explained the math using the tiles made much more sense and she found them very helpful.   I was surprised at the number of people in our class who have never seen algebra tiles.  I don’t think that I ever used them as a student, but my children have and so they must be gaining popularity with teachers.  I wonder if they are used more in elementary than in secondary?!?  Has anyone used them in secondary math and for which concepts??  Did you meet with success or what would you do differently next time?