action research pROJECT

action research pROJECT home

(A paper written for a class at the UofM)

Mary Tigner-Räsänen

I have recently begun home schooling my 7^th grade daughter, Nora, in math. My action research plan was to make observations about Nora’s learning and misunderstandings, and in the process of helping her, reflect on these observations. Then I want to apply what I learn to my own teaching. I’ll do this in the form of a ‘diary’ with reflections, and come up with some applications to my teaching at the end.

Nora:

Her complaint: too much working out of problems in too much detail over and over – what purpose does this serve? To force the students to look at the process? To show the teacher where the student misunderstands without having to use other means to do that? - Too many problems of the same type. Overall complaint: math is tedious, slogging through mud. It made her irritated. Her self-assessment: I’m dumb, I’m not good at math, I hate math. Method: work problems in book, correct problems, work more problems, test, repeat. The teacher is there to help with specific problems, but does not teach concepts or ideas.

My observation: Nora doesn’t stop to think about what is being asked before she answers the problems, doesn’t pay attention to details so she makes little mistakes. I tell her this. I say, “think, ‘what am I being asked to do’ before you start – wait for ‘a-ha’. think, 'what do I know, what do I need to know’.”

Nora doesn’t seem able to sustain her focus, especially when the problems are difficult for her.

My brother’s observation about math: (he’s an engineer). Repetition is important in math. I decide I need to think about ‘repetition’ – what legit purpose does it serve, how can it be misused?

Before we go to home schooling I ask her if she can ask her teacher about the problems she doesn’t understand. She says, yes, but I can understand it better when you explain it. She also says, I’m not doing well in math, but when people have questions they always come and ask me.

When I ask, “Did you ever ask your teacher for help when you didn’t understand something?' she said, “Yes.” I said, “Did it help?” She said, “He’d explain how to do the one problem but didn’t explain the idea.”

My observation: Nora works best when she understands the idea and only has to use the details to fill in. She flounders if she just has to work the problems over and over without understanding. It’s almost like she refuses to do it.

Nora and I have conversations about her trouble with math, and we have conversations about math concepts. Through the conversations about her trouble with math, I observe that her attitude is becoming more open toward it. She enjoys the conversations about math concepts, and often explains things to me that I don’t understand.

Repetition: repetition should lead to automaticity. (math facts, for instance). How much repetition is required for automaticity to occur, say in solving problems of absolute value? Is it desirable that each set of skills become automatic before moving on to the next level, or is it enough that the student be able to ‘think through’ the problem? Are there skills/operations in pre-algebra that absolutely need to be automatic, and those that don’t? Can this be made explicit to students, with the reasoning behind it? What if a student is confused about working a problem - does it make any sense for them to work multiple problems without first clearing up their misunderstanding? Doesn’t this lead to frustration rather than clarity? Is the hope that the concepts/operations will become clear through multiple problem-solving, or can this be done through teaching, with the problems seen as a way of putting understanding into practice rather than the means to understanding?

What bothers me is that there seems to be little useful feedback to students about what they are misunderstanding. They just know that a problem was wrong. They need assessments designed to reveal what their misunderstandings are (Wiggins)

Spreadsheets: The students learned spreadsheets by working out spreadsheet problems from the book on paper. They weren’t taken into the computer lab. Nora: “ I hate spreadsheets.” They were encouraged to look at spreadsheets on a computer on their own time. Now Nora has Excel on her laptop. She works spreadsheet problems there and says: “I love spreadsheets – they’re so easy to understand.” Why didn’t they just learn them in the lab?

When I said, “Look at it like a puzzle,” she said, “I don’t like puzzles.” I accepted this at the time, but then reflected on it. She can look at a game boy game and work out all of the levels and puzzles so quickly I’m left spinning. Her sisters and friends use her as a resource. I went back and told her “Nora, it’s not true you don’t like puzzles. Think of game boy.” she said, ‘That’s logical. I like logic.’ I said, ‘Math IS logic’. Then I realized that she doesn’t see the logic. How can I help her see that? One way is not to leave a concept or problem until it’s clear that she ‘gets it’. Her general attitude now is ‘Well, I can do it but I don’t get it. It must not make sense.’

As I reflect on this process several things are popping up:

It is possible, through conversation, for students to reflect on their understanding or lack of it. It takes time. Initial response may be ‘I don’t get it, I don’t like it, it’s boring,’ etc. Though it is impractical for a math teacher in a class to have ‘diagnostic’ conversations with each student, (and English teachers), such reflection could be done through journaling. Or students could be put in pairs and be asked to come up with a few questions about a certain concept or operation, and the teacher could address those questions in class, through mini-lessons.

Students may make assumptions about themselves as learners based on confusion that they are unable to clarify. This attitude can affect their whole approach to the subject. I know this is nothing new, but I see it so clearly in Nora’s ‘I hate math.’

Nora’s confusion stems from being unable to identify why she is doing something, not what she’s supposed to be doing. She’s unwilling/unable to ‘memorize’ what to do in a situation. So we are talking about problems until she can ‘see them’. She is totally willing and happy to do this, and it energizes her. Otherwise she says math ‘saps her’ of her energy.

My earliest assumptions about why Nora didn’t like math were not terribly accurate, nor was her self-reporting. She knew it was tedious, but part of that tediousness had to do with having to work problems about which she had no conceptual understanding. Some of the repetition and writing out of problems in detail (which she abhorred) have become learning tools FOR her rather than meaningless work imposed on her.

Conversation with Nora:

Me: Well, today we’re going to do transformations on matrices!!

Nora: Why in the world do I need to know how to do transformations on matrices?

Me: Well, if I knew the math better I could give you the real answer, because there is one. But since I don’t know it, I’ll give you the short answer: because you’re going to have to learn something soon that will build on doing them.

Nora: (silence)

Me: Think of it like the cello. Carolyn (her teacher) has you practice different skills in one piece, and she tells you that later on you’ll build on that skill in a more difficult concerto. It’s the same with math.

Nora: Yeah, but Carolyn always tells me where I’m going, and with math I never know. It’s like stumbling forward in the dark. And I hate it.

Today as we started algebraic expressions, Nora was totally confused. She got all of the problems wrong, so we sat down to talk about them.

First I told her that she was able to learn this, she just hadn’t yet, and that she was capable of being an ace math student, even thought she isn’t now. Then we simplified some expressions. First I substituted ‘apples’, ‘oranges’ and ‘zucchinis’ for x, y and x-squared, because she wasn’t seeing them as entities. She got the concept quickly. Then I had her do factoring, and when she did them wrong, I asked her questions that would lead her to see the implications of her wrong answers. Pretty soon she was saying, ‘wait a minute – let me look at this.’ And off she went. At the end of our session she said, ‘I really like this’. Hopeful, but thinking she might be talking about her coke, I said, ‘You like what?” She said, “This math.”

Reflection:

Students need to believe that they are capable of learning what they need to learn.
Automaticity: I’ve been thinking about this again. Today I wrote out some of my own expression-simplifying problems for Nora because there weren’t enough in the book for her to become automatic at working them. The problems in the book looked at the problems from a different conceptual vantage point too quickly. It’s a good concept, but she needed more practice at this one skill. Now she’s comfortable, and we can look at it from a new angle. I think sometimes the book is confusing, because before a concept is clear from one angle, it has the students look at it from another angle. They don’t have time to clarify their thinking.
When I explained the concepts first using concrete terms (the apples, oranges, and zucchinis) it was easier for her to apply the concepts to the math.
What seemed obvious to me (for instance that x + x + x =3x) wasn’t obvious to her. She also was confused when there was something like x – 3y + 4x + y. She wanted to know what she was subtracting 3y from. She was using the + and – more like a grade schooler, as operations, rather than movement in a direction. They’ve been doing number lines, so I assumed she would make the transfer of this concept, but she didn’t. When I made it explicit, she understood it. It occurred to me that often in math, assumptions are not made explicit; rather it is assumed that students will make correct inferences and transfer what they learn.

Implications for my teaching

Although I teach English rather than math, I’ve been able to see some general principles that I can apply in my teaching to be more effective.
This is confirmed in both Wiggins’, Educative Assessment and Bransford’s How People Learn, but it is important for students to understand, rather than just memorize, and often performance can mask misunderstanding. Student self-reporting may also mask misunderstanding. So I see the importance of taking the time to really know what the student’s understanding is before beginning.
It has become totally clear to me how much easier it is for Nora to learn when she understands the underlying concepts and knows where she’s headed. I think in my own teaching I sometimes haven’t made this clear enough to my students, in a way forcing their dependence on me while urging them to become independent learners. It has become clear to me the importance of clearly articulating goals and processes for students. This is also stressed in Wiggins and Bransford.
I believe that ‘conversation’ with students about their learning is essential. The conversation can take a number of forms. It may be a quick verbal check of understanding or it could be students’ reflection in a journal about what they know and don’t know. I also think this would be a good place to explore the use of small collaborative groups in which students could clarify with each other their understandings and misunderstandings.
While I don’t know the math well enough to create ‘authentic’ learning experiences for Nora, I can see how valuable that would be for her, and how much more engaging. While I can’t do it in math, I can do it in English for my students, and will focus on creating learning experiences for students that are meaningful, authentic, engaging, responsive, effective and feasible.
Agency seems to be very important. Now that Nora believes she can learn math and has had some success in mastering the concepts, and most importantly, I think, has begun to see her way through the process of approaching an unfamiliar concept, working some problems, and wrestling with the ideas (in our case, in conversation) until the concept becomes clear, she is taking ownership, slowly, of the process. As she takes ownership of the process, those things that just seemed like tedious tasks arbitrarily assigned her (work out every step of the problem, work many problems of the same type) have become ways for her to clarify her thinking and solidify her understanding, and she is doing them eagerly. This has important implications for my own teaching.