Nov. 17th, 2008
Student understanding of mathematical concepts is what teachers strive for, or at least, what teachers should strive for. Classrooms that see students interacting, communicating and helping each other solve problems, is what I would argue, a classroom where students actually understand the mathematical concepts being taught. Marian Small also agrees. Small (2008) notes that “it is through interactions with other students as well as with the teacher, and with the opportunity to articulate their own thoughts that students are able to construct new mathematical knowledge” (p.4). Small (2008) goes on to note that it is in these classrooms, one in which she calls constructivists classrooms, “that students are given the opportunity to develop richer and deeper cognitive structures related to mathematical ideas and students’ development of a level of mathematical autonomy”(p.4).
Often times, what you read in a book as theory, may seem really out to lunch. “yeah, it may be alright for them to say that, or do that in their classroom, but it would never work for me.” Well, I would like to demonstrate what this looks like as I take you, once again inside my classroom.
Please keep in mind we have only been engaged in learning mathematics through problem solving for a little less than two and a half months. The particular class I am about to describe was a concluding class after doing investigations with using patterns to explore division of 0.1, 0.01 and 0.001.
The question was posed at the beginning of class that went something like this : Frank divided 42.8 by 0.1 and got an answer between three and found hundred. How do you know his answer is incorrect? Show your thinking using pictures, numbers and words.
I talked a little about the problem, making sure everyone understood what the question was asking. I then let them go and begin solving the problem. What happened next, illustrates what Marian Small has just described.
Students began talking through the problem with their partners, they used things like Base Ten blocks, calculators (yes, I allow them to use calculators), graph paper and pictures to begin making sense of the problem. Most, if not all used Base Ten blocks to work out their thinking. I was hearing students correcting others thinking and explaining how they know in their own ways, talking about their own strategies in solving the problem. Once I seen that everyone had some type of answer written, I brought the class together as a whole and began discussing what they had found out.
I began by asking students to explain the problem, telling me what was required. I asked particular students to tell me how they thought about the problem. One student said she thought of it as money and you wanting to split $42.80 into dimes. Another student said that they thought of it by using Base Ten blocks and said that because they were dividing the number (42.8) into tenths, they would use a rod as a whole one and the unit block would represent a tenth. At this point I seen some students were a little lost, so I stopped. I asked another student, the one with a blank face, to explain what was just said in his own words. He began trying to make sense of what the other student was saying and as he was playing with the base ten blocks in his hands, things were beginning to make sense. Other students were helping this process of understanding evolve, understanding the concept of using a unit to represent one tenth and the rod would then equal one whole. With the support of the class and the use of the base ten blocks and conversations, the reluctant student understood why a unit was used as a tenth to represent this problem. He was then able to recognize that he would need 42 rods and 8 units to represent this number. I got him to come to the overhead (which is a part of every math class) and show the class what he had just discovered. But there was a problem. This student quickly realized there was not 42 overhead Base Ten blocks to put on the overhead. He looked at me so innocently and said, “but Miss, there’s not enough.” I laughed and said, you are right, so what are you going to do about it. He quickly recovered and said, well I know that a there are 10 rods in a flat, so I could use 4 flats. He then places 4 flats on the overhead. I asked him to look at the number he had on the overhead and say what it was. He said 400. Before I got a chance to respond, another student quickly corrected him and explained to him where he made the mistake. She pointed out that because he was using the unit as a tenth, the rod had to be a ‘one’ and the flat was now 10, so four flats would be 40. (Bingo, I now know that she understood the concept!) Then the reluctant student, who is now demonstrating a better understanding of this place value concept, placed two more rods and 8 units on the overhead and confidently said, “there you go miss, 42 and 8 tenths.” I looked at him as seriously as I could and said, prove it to me. After a little look, he went into detail about how he was able to come up with this arrangement. Once his explanation was complete, I looked at the class and asked if anyone agreed with him or if anyone would like to challenge him. I could see that everyone had the same base ten arrangement and they all agreed that this student’s explanation was correct.
So, from here, to ensure they really understood this, I posed the question: Why are we using the unit block as one tenth and why the rod would be equal to one? One of my weaker students, looked at me and simply told me because we are breaking the number into tenths miss, and a unit is one tenth of a rod, so that means all we have to do it count how many units or tenths there are and we have our answer. Well, I was a little blown away, to say the least. If my weakest student could think of it in that way, I knew something right was happening.
To keep the momentum going, I then asked the class, what would Frank’s answer be if he did divide 42.8 by 0.1. At once, they said 428. Eureka! One student stated,’ well miss we knew it was going to be that because dividing by tenths is just like multiplying by ten and 42.8 ‘times’ ten is 428. Horray!!
For the sake of time and space, I did not include all the conversations that was had during this class. More students were asked to come to the overhead in front of the class and explain how they went about solving the problem, or to prove that what was just said was indeed, true. The point I want to make is that through much interactions, conversations and student talk, students were given the opportunity to think about this problem, to talk about the problem and to talk through the problem. We used students’ lack of understanding as a springboard to help them, and the rest of the class, to have a deeper understanding of the problem and the concept being taught. I, as their teacher, did not intervene or disrupt this thinking, even if it was incorrect. They worked together to collaboratively ‘fix’ any misconceptions about this concept and used their own invented strategies and thinking to explain how they seen it to others. This type of classroom can be achieved, and this is proof that teaching through problem solving do work.
