In the Literacies paper Working with student resistance to math tools Kate Nonesuch talks about changing one aspect of her practice and the impact this had on her relationship with her learners.
I hoped that making manipulatives part of the assigned work in the class would mitigate the difficulty that always comes in a teacher-paced class, that is, that from the first day there is a gap between how much explanation, attention and practice is needed from one student to the next, and, as time goes on, the gap gets wider. In our program, we prefer to struggle with these difficulties rather than go to a self-paced delivery style. We want the advantages we see in group teaching and learning, and we notice that self-paced is often dead slow.
She then goes on to identify "fractions work" as the area she thought manipulatives would make the greatest difference.
This was my first "hmm..." moment. (I think I'm supposed to call it an "a-ha" moment, but I don't really get flashes of insight - just doubts I have to talk about for a month before there's some sort of gradually evolving understanding.) My "hmm..." was this: I've noticed that a lot of talk about math manipulatives centers on either fractions, decimals or percents - but mostly fractions. Is there a limit to how we make maths concrete? Are there manipulative types that correspond to types of math? Is there a chart out there featuring recommended hands-on tools for each... whatever-they're-called. you know... parts of math?
In her paper, Kate doesn't want to talk about fractions. Her interest is with the learner's reaction - resistance in many cases - and how she helped them shift their perceptions.
I had expected some resistance from students, but was not prepared for the strength of it. Students resisted using both my homemade manipulatives and the commercial manipulatives, their responses ranging from silent withdrawal to open refusal to use them. Over the years, I have tested different strategies of honouring student resistance and working with it rather than against it. I find that students need to be able to express their resistance in order to maintain their sense of self in the class, and that when they can do so with dignity, they are more likely to be able to stay present and attend to the work. When Arleen Pare did some research for her MA thesis in my classroom, she found a positive correlation between student expression of resistance and student retention. The more complex and open their resistance to me and my teaching, the more likely they were to continue to come regularly.I'm not going to follow that conversation any further. I had some other "hmm..." moments, and if you're at all interested I'd encourage you to read the full article.
These results suggest a positive association between conscious, active resistance and regular attendance. It also suggests that the more that conscious resistance is encouraged, the more likely it is that regular attendance will result (p. 115).
Students sometimes express their resistance by leaving the class, but over the years I have developed a teaching stance that recognizes, honours and encourages open expression of their resistance, and hence many students will question the use of manipulatives, although, as you will see from the examples that follow, their resistance may be indirect, and often comes in the form of a question that is not a real question.
What pulled me completely over to the side of the road came along later when she started talking about using the manipulatives to prove facts about fractions.
I do not require students to use the manipulatives. Their assignment is to prove various propositions: that 3/4 = 6/8, that 2 2/3 plus 1 1/2 = 4 1/6, 1/2 x 1/3 = 1/6 and even that 1/4 divided by 1/8 equals 2. If they want to use drawings or apples or anything else that will prove it, they can do so.
The first thing I thought was "Why?" Are we supposed to prove stuff in math? It never occurred to me.
I should confess that I'm at a bit of a disadvantage here. I only vaguely remember what math classes were like in secondary school, and I haven't taken any math courses since, so I don't really know what the etiquette is here. What I've been doing for the past decade is figuring out the hows and whys of math and sharing my figurings with my learners. Am I equipping them to do math on their own? Um... sometimes. Do you prove the math you do? If I don't prove it, should I worry that I'm making an error?
These were the questions and sorts of questions I came up against.
In working out answers for them, I recalled again the distinction between learning math and learning to present math. The first has to do with how I come to understand and trust the inner workings of mathematics. The other has to do with things I do, or could do, to scaffold others so that they can come to trust and understand maths. The distinction is important not least because I can't assume my learners and I learn in the same way.
Anyway, that's all I wanted to say. Kate wrote this paper which is full of reflection-provoking bits. The bit on the role of proofs in scaffolding learning - or in learning itself - surprised me and it's what I thought about last night and will probably think about tomorrow.
Well, that and enactive research and the adventures of a Canadian university student in Scotland.
You know how it is.