Brockton Week 2

  We had some intense discussions last night for sure and there exists confusion about the models we were using. This surprised me somewhat since in our first session when we modeled the problems everyone seemed to understand how the models worked. Then today I looked at samples of student work to discover a serious misconception about bar or tape diagrams (either term is acceptable but I will try to stick to TAPE diagram since that is what is used in the frameworks).

If we think about a ratio of 2 reds : 3 blues we are describing a multiplicative relationship that is represented in a comparative model as shown below.

Reds
Blues

Now if we are also told there are 32 reds and some blues we DO NOT need to add anything to the drawing BUT we need to show those 32 reds are evenly divided into the 2 cells/blocks we ALREADY drew.

Reds		16	16
Blues

And each cell must have the exact same scale factor…each cell represents a variable and the same variables must have the same value. So we know that there were 3 x 16 blues or 48 blues.  The problem then stated an additional 10 red skittles were added to the jar and the students had to find the new ratio of reds to blues.

The student misconceptions included adding 10 new cells for a ratio of 12 : 3 rather than understanding that 10 reds is less than one new cell…this indicates the students who underlined “key” words saw “added to” in the problem and simply added…illustrating a serious misconception that ratios are additive. Those students who thought about the problem correctly recorded 42 reds : 48 blues. So…what caused this widespread misconception???? A misguided belief that underlying KEY WORDS is a helpful math strategy to work with “word problems.” RATHER the students should be guided to restate what they think the problem is asking and to focus on the meaning of the problem…this will alleviate that serious issue of reading words outside of the context.

Next, I erred in handling the question about the Larry, Tom, and Georgia money and percents. What I should have done was to put our thoughts on the conjecture board and for us to rationally discuss what is going on in the problem, what we were representing and that we were talking around each other. MY FAULT! I apologize for not modeling what I preach in this regard. I have given this problem and our discussion lots and lots of consideration since Thursday…even losing sleep over it. Never very helpful.  It turns out we were representing and solving two different problems just as Helen and Rich pointed out. I will go over it at our next meeting and we can examine the model and connect it to the algebraic solution.

We also need to complete the problem we left hanging after our first class. Lots to do, so little time.

Anne

Brockton Week 1

            I thought Saturday’s class went very well. My greatest challenge is trying to move your thinking away from doing calculations to representing problem situations using models. How many times did you want to get the answer then try to fit the diagram to the computations? Think about the last problem we worked on. The amount of cards was 4/3 and we added 48 cards to one child which is a nice even number so we can divide that nicely into 2, 4, 6, 8, 12, 16, or 24 smaller sections…the drawing is easy. But!!!! The other child collected 13 so the model has to show 13 not 12 so will be a different length than the 12 most of you were playing with…and I admire those who persevered with the model. But, how many of you sat back and said the answer is 11 but I don’t know how to show it? Now fast forward to your own students. How many times do you show them something that makes sense to you but your students say “I don’t get it?” I would hazard to say, “Very often!” I know that phrase is frustrating to hear but most often our students do not know how to articulate what they don’t get. They do know that what you are saying makes no sense to them.

            Now, back to the model for the problem you are working on. How can you show one child adds 13 cards to what he already has even if the other child is adding a multiple of 12?

            The whole purpose behind both the mathematical practices and the new content standards is to engage students in sense-making, reasoning, and modeling mathematics. This translates into our, every one of us who teaches mathematics, to change how we approach the mathematics, and to recognize how uncomfortable it is going to be for us on this journey to help a generation of adults who embrace mathematics rather than dreading it or worse avoiding it at all costs. So, I will be pushing you past your comfort zone or as Vygotsky says, The Zone of Proximal  Development, to enable you to visualize and model some of the mathematics that we all can do algebraically or arithmetically.

            Think about the next few months as a journey of learning and understanding.

Anne