I’ve been thinking about our school district’s mathematics pacing guide lately. This past week we gave the required mid-year math assessment to make sure the kids are on track, and I’m seeing the predictable result: Some are; too many are not. So, what now? The pacing guide doesn’t say. It isn’t a teaching tool, you see. It’s a prod to keep us all moving like cattle toward a theoretical vanishing point. There’s a difference between teaching and program administration, and “covering the material” isn’t good teaching, even if more kids test OK and proficiency averages increase.
There’s a better way. Bill Kerr’s post about Jerome Bruner stirred my thinking about this, since I read Bruner for some of my M.Ed. project research. I looked back at that project this week, and I want to say something about it.
I entered our local university’s K-12 Reading Endorsement program after teaching for 17 years. It was a transformational experience for me, causing me to re-examine everything I believed about teaching. The philosophical underpinnings of the program were sociocultural, assuming education to be a social practice rather than a technical enterprise. The program was open only to practicing classroom teachers because we tested education theory by designing participant-observer inquiry projects. Unsurprisingly, I found that no approach or program works as advertised straight out of the box. There are always contingencies and anomalies, and these are things to pay attention to.
Early on, when we were discussing vocabulary development, I began thinking about math. I didn’t see any reason why comprehension instruction should be confined to reading, especially since the kids seem to have a terrible time remembering math jargon. I set up a small case study with a group of kids who read pretty well, but had trouble in math. I taught math vocabulary intensively to the whole class, hoping to see improvement in the math scores of the kids in this group. But there was no real difference for them.
I looked at the results of my first intervention as a new puzzle. Then I found the NCTM’s Communication Standard (available now by subscription), and dug through the bibliography for what was known about language and mathematics. I found a gold mine there, and read extensively from it. Magdalene Lampert’s “When the problem is not the question and the solution is not the answer” was hugely influential. She wrote about a teacher research project in which she taught in a fifth grade classroom, working to establish an authentic mathematics discourse community. In the abstract for her paper, she said, “…To change the meaning of knowing and learning in school, the teacher initiated and supported social interactions appropriate to making mathematical arguments in response to students’ conjectures.” The article is worth reading, as it maps out exactly how she went about her project, and her bibliography contains many of the same sources as the NCTM Standards document.
At the same time that I was off on this tangent with math, I continued to explore issues around reading comprehension, trying Literature Circles and Book Clubs – small book discussion groups – with my sixth graders. I wondered if the problem with my math vocabulary instruction project, and math instruction in general, was that authentic discussion is not emphasized in elementary level math classes. Nor, I would add, is analytical thinking much taught. So I set up a structure for small group discussion and problem solving that I called Math Chats. It was great fun. The kids loved it, and they got very involved in arguing and justifying their thinking about math problems.
I kept my case study group together and recorded some of their discussions. While I was transcribing and coding the things they said, I noticed that some of them – the kids who had the most trouble with math – were completely uninterested in any question about why a particular solution path worked, or didn’t. Invariably, the kids who didn’t “get” math only wanted to get to the answer, nevermind the reasons why it was correct, or whether there might be alternate approaches. Inquiry was lost on them. They had no patience for it, and continually drove discussions to the bottom line, or were altogether off task.
Bruner, recognized that social class influences how people use language, which might explain part of the trouble some kids have learning math. He found that the more prevalent use of analytic language among the middle class encourages those children toward the habitual use of formal categories and strategies such as featural analysis of tasks, consideration of alternative possibilities, questioning, hypothesizing, and elaborating. In contrast, he noted that lower class students “tended to have little categorizing ability except in affective terms; they were highly concrete and immediate in their approaches to objects and situations” (Poverty and childhood, The relevance of education, p. 145).
Bill Kerr’s post moved me to look up Bruner again, and I found a copy of Acts of Meaning online. It’s in two places, actually. The first chapter, The Proper Study of Man, a critique of the information-processing model of human cognition, turned up as a Google book search result. The remaining three chapters are part of a pdf file, which I am still reading.
From The Proper Study of Man:
…information processing cannot deal with anything beyond well-defined and arbitrary entries that can enter into specific relationships that are strictly governed by a program of elementary operations. Such a system cannot cope with vagueness, with polysemy, with metaphoric or connotative connections. When it seems to be doing so, it is a monkey in the British Museum, beating out the problem by a bone-crushing algorithm or taking a flyer on a risky heuristic….It precludes such ill-formed questions as “How is the world organized in the mind of a Muslim fundamentalist?” or “How does the concept of Self differ in Homeric Greece and in the postindustrial world?” And it favors questions like “What is the optimum strategy for providing control information to an operator to ensure that a vehicle will be kept in a predetermined orbit”?
If we want to change the meaning of knowing and learning in school, we can start by asking better questions.
4 Comments
Hi Doug,
I live in northern Australia. Part of my job involves helping to train teachers to teach literacy in the bush, and getting them to raise their expectations of low achieving kids. I lurk on your blog occasionally, and have very much enjoyed reading it.
What you say about Bruner and his contribution to our thinking about classroom discourse resonates very deeply with me and has prompted me to stop just lurking and actually write something. I was particularly interested in what you say about some kids not tuning in to the ‘why’ of school activities.
For Aboriginal kids in the bush (and the whole demographic is currently failing dismally at school), the why of schooling often totally escapes them – why do we read, why do we answer the teacher’s questions, why do we even bother coming. And because teachers aren’t always very reflective on their own culture and its attendant assumptions, it often doesn’t occur to them to even explain why we do the things we do in school. But even when they do think about explaining, the explanation and the activity are often at odds – and as you say, the kids bring things back to the bottom line and to their comfort zone, which often results in very ritualised work.
Over the past few years I’ve been fortunate to be able to watch lots of teachers teaching, and to talk with them about what they do. What you say about kids who don’t ‘get’ school is very true of kids here too. Which brings me to a comment on your reading of Bruner – we have found that the difference between middle class kids and kids from low-literate backgrounds is not so much a question of language, as a question of ‘orientation’ to the text or the activity. So, for example, middle class kids from literate families know that when the teacher asks a question about the story they have just read, they are expected to find the answer in the text of the story. Kids who don’t come from literate backgrounds often don’t know this, and so they look for answers in their own experience. Similarly, kids who understand the purpose of schooling will look for answers to math questions from within the logic of the mathematics; kids who don’t really know where it’s all going invent their own logic, which is often ‘please the teacher so she leaves me alone’ or ‘get through this lesson and get out to lunch’. Or if that fails, ‘disrupt the lesson and get kicked out, or at least buy some time not to have to do the math, or the writing, or whatever’.
One strategy we use as a way of giving kids the orientation that we want them to have is a strategy called ‘preformulation’ (the term comes from Courtney Cazden, but she might have got it from someone else), where you basically tell the kids what you want them to know before you ask any questions about it. Then you only ask questions that you know the kids can answer. No more ‘guess what’s in the teacher’s head’. It sounds a bit daft, but it provides a strong scaffold for kids who may not otherwise know where you are coming from. Lots of teachers use this strategy intuitively anyway – but it’s interesting to become conscious of, and plan it into each lesson. The follow up to preformulation is ‘reconceptualisation’, where you take up from the answer to your question and elaborate on it, and talk about its significance. The idea is that as you work through a cycle of lessons, you have less and less need to preformulate, and you will be able to ask more open questions about your topic, because the kids share your orientation to it and know where you’re coming from. The craft of course is to know when to stop preformulating. Teachers who are good at the technique get the kids to take on the reconceptualising too.
I don’t know if I’ve made any sense here; I got a little carried away as your study struck so many chords for me. Anyway, if you’re interested, you can read something about our project here: http://www.nalp.edu.au/whatisal.html
Hi Helen. You sound like a person who I could spend some time talking with.
One of the things that I began doing as a result of this project was something like you suggest. Rather than simply explaining, demonstrating, and so forth, I also began commenting on the classroom discussions. Asking for explanations and justifications, and also offering my own for why I was saying and doing the things I did, making the ‘talk’ of the classroom part of the lesson.
You say, “Then you only ask questions that you know the kids can answer,” which I wish I could say that I consistently do because I understand the wisdom of it. But this is not easy with the curriculum materials we’re using. I’m thinking about how to work around the textbook, and the best idea I have is to create more of my own math problems.
I’ll follow the link you left in your comment. I am very interested. Thank you.
Wonderful links (from both of you) and comment from Helen – still reading but thanks in the interim
Thanks for your comments Doug and Bill. I’m looking forward to more conversations with you soon. In the meantime, I shall leave my air conditioned office and venture out into the sweltering Darwin afternoon, imagining the contours and shadows of an Alaskan mid-winter.
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