So, what did I take away from the class? Simple...that I never want to hear the word "manipulatives" again. OK, I jest, but there is definitely a grain of truth to it. Allow me to explain.
For our class activity, we split up into groups and a different set of manipulatives was put on each of 5 tables. Along with the manipulatives there were a couple of sheets of paper. One was the instructions for what we were to learn and discover, the other was a question we could look at from a teacher's point of view.
I learned very quickly as I moved from table to table that using manipulatives to learn math makes my brain hurt. I just don't learn that way! True, there were a couple of tables where I was able to connect the manipulatives to the concept, but for the most part I just really didn't enjoy myself at all.
At one table, we were supposed to be working with fractions by creating shapes (like a parallelogram or triangle) that were, say 1/6 red, 1/3 green, and 2/3 blue using the different shaped blocks on the table. Sounds fun, right? Well I thought it was difficult and frustrating! In fact, at one point I said to my colleagues, "I'd rather be doing Calculus." And this was likely an activity that would have been done with students in elementary school! I would much rather have just worked with fractions the traditional way - doing drills and written problems with them.
At another activity, we were playing with algebra tiles. I'd only really seen pictures of those. They are essentially different coloured tiles that can be used to represent algebraic equations like 3x + 5x or (2x+1)(3x-2). I didn't know how they worked and my colleagues weren't sure either; our job was to work it out. Even when I began to grasp how they worked, I ended up looking at these tiles thinking that this is NOT the way I would want to learn. If I want to solve the two equations I put above, I'll just add the first one and FOIL (First, Outside, Inside, Last) the second! I don't need complicated algebra tiles to do that. In fact, I found myself just wanting to mess around with the tiles instead of trying to learn. I actually took the tiles and started literally spelling out the phrase 3x + 5x using the long skinny tiles. I was trying to amuse myself and get a laugh. You know, "I am representing the equation with tiles, like the instructions say!" Just not the way it was intended - I was being creative!
At the last table, we were given 11x11 pegboards and a bunch of rubber bands. We were asked a question about a farmer who had a certain length of fencing, and who wants to build a rectangular field. The question asked what the possible dimensions of the field could be and which dimensions would give the maximum area. So what did I do? Well, I took a piece of paper and began to draw rectangles with different side lengths to show the different possibilities. From there, I could have simply done some area calculations and, by trial and error, found the field with the maximum area. (What I actually did was use Calculus to solve it). I know I was supposed to be using the pegboards and rubber bands, but again, all I wanted to do was play with the things on the table. I actually picked up two pegboards and pretended to play Battleship! People....I am 36 years old - I am not a little kid who just wants to mess around - I'm usually very good at "just getting things done" while in class, but this was a challenge! It was so hard to focus!
The first thing this told me is that manipulatives do not help me to learn math. I would far rather be taught the formulas and procedures right off the bat, and then maybe be shown a visual to have a second way to look at it.
The second thing this told me is that not everyone learns like me. There will be students for whom manipulatives are the best thing ever to help them focus and understand math. Maybe those students feel the same lack of focus during traditional learning that I did while trying to use manipulatives. So, I know at some point when I'm teaching math, I'll have to "teacher up" (as opposed to woman up or man up) and bring out the manipulatives.
However, I'm sure that I represent a good chunk of students who just can't deal well with manipulatives. So, if I do have a particular lesson which lends well to using manipulatives, this is what I might do:
I would figure out which students in my class learn best with manipulatives and which students learn best the traditional way. I might have done this beforehand with observation over time, talking to the students, or getting them to experiment with the idea of manipulatives to see how they feel about it. Then, I would split the "manipulatives students" into groups and they would go to the few stations set up around the room with instructions on what to do.
Next, I would instruct the students who would like a traditional lesson to come close to the board (the "formula group"), where I would teach the concept the traditional way. After the lesson, I would assign those students practice questions to do while I walk around the room to see how the manipulatives groups are doing, and assist if necessary in understanding the connection to the concept.
Finally, once all students have some grasp of the concept, I would ask a representative from the formula group to come up and explain what they've learned about the formula and procedure to the manipulatives group. Likewise, I would have a manipulatives group representative come up to explain to the formula group the connection between the visual representation and the formula.
That way, all students leave with similar knowledge - the manipulative group will have learned the formula, and are now comfortable using it because they were able to discover it visually. And the formula group not only knows the formula, but now has a visual to connect it to, which furthers their thinking and understanding. It's the best of both worlds.
Now, that doesn't mean I'll be using manipulatives too often in my high school classes, but it is something I can think of adding to my repertoire when it fits well with a concept.
The class activity we did really, if nothing else, highlighted two very different ways of learning of which I'm very much in one camp. It emphasized to me that although there will be students who learn like me, not all students learn that way, and I need to try to keep that in mind when I'm teaching, whether that means using manipulatives, or maybe just enhancing my more traditional lessons with visuals. The possibilities are endless.
Rachel, it is not unusual for adults that have learned mathematics more procedural to find the use of manipulatives frustrating - I felt this way myself often when I first started working with them. A few things for you to consider: 1. you were trying to "relearn" mathematics that you already know with these tools - for students in your classroom they will be using these tools often to make sense of mathematics that they don't already know which is a different experience. 2. These models are difficult for us because it was not our experience. I have found the more I work with them the more natural it is for me and it has, at times, deepened my understanding of mathematics. I like your idea of having students that make sense of mathematics in different ways explore it in a way that is comfortable for them but them pushing them to make sense of it another way as well.
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