Although overlap exists between certain kinds of formative and summative assessments, and although you can design informal and formal means to assess your students, in this article, you'll learn about informal, formative assessment you can use in the mathematics classroom. The ideas presented will help teachers know more about their students' mathematical processing, including their reasoning, communicating, and problem solving.
Many of us would love to open up our students' heads so we could see the wheels turning as they solve problems. Since this is not possible, we have to find ways to get the students to show us what they are doing. With some students, this is fairly simple because they are extremely metacognitively aware (thinking about their thinking) and don't mind letting others in on their secrets. Other students, though, rarely "think about their thinking" and so, of course, rarely let their teachers in on how they process (because they aren't even aware of it). The following ideas work with both groups of students (and everyone in between); we just have to be more patient with the latter group. The good news is, from my experience, that once I begin asking adolescents to reflect on their thinking and problem solving, it starts happening more and more.
1. Interviewing students is a powerful way to tap into their processing. In the best of situations, teachers would have time to interview students on an individual basis - at length. In reality, we don't have the time to do this, so we need to make time to 1) briefly interview students on an individual basis and/or 2) interview groups of students - either briefly or at length. Long and Ben-Hur (1991) found that the following phrases or statements are helpful when interviewing and encouraging students who may not be fully forthcoming or expansive in their answers:
A = Always
M = Most of the time
O = Occasionally
S = Seldom
N = Not at all
The first time I use this sheet, I model what it is I want students to do - and the specificity of examples I am looking for. Then, I accept nothing less than what I know that they can do. Students are sometimes resistant to this because it involves a kind of thinking that some are not used to - but I persevere and within a short period of time, am able to get the kind of formative assessment data that is helpful to me in my teaching.
Regardless of whether we are teaching seven-year-olds, seventh graders, or seventeen-year-olds, part of our task is to teach them what our expectations are. If we expect students to reflect on their thinking and problem-solving processes, then we must teach them to do so. We need to:
1) model our own reflections,
2) ask them about their reflections, and
3) give them opportunities to reflect on their processes.
In other words, we can't give up just because they are resistant to giving us information about their mathematical processing.
Many of us would love to open up our students' heads so we could see the wheels turning as they solve problems. Since this is not possible, we have to find ways to get the students to show us what they are doing. With some students, this is fairly simple because they are extremely metacognitively aware (thinking about their thinking) and don't mind letting others in on their secrets. Other students, though, rarely "think about their thinking" and so, of course, rarely let their teachers in on how they process (because they aren't even aware of it). The following ideas work with both groups of students (and everyone in between); we just have to be more patient with the latter group. The good news is, from my experience, that once I begin asking adolescents to reflect on their thinking and problem solving, it starts happening more and more.
1. Interviewing students is a powerful way to tap into their processing. In the best of situations, teachers would have time to interview students on an individual basis - at length. In reality, we don't have the time to do this, so we need to make time to 1) briefly interview students on an individual basis and/or 2) interview groups of students - either briefly or at length. Long and Ben-Hur (1991) found that the following phrases or statements are helpful when interviewing and encouraging students who may not be fully forthcoming or expansive in their answers:
2. Another marvelous idea for assessing students' mathematical processes is shown below and is an adaptation from Greenwood (1993). Students are asked to use the letters at the top of the page to rate their own answers to each of the prompts - and to rate it both as THEY see themselves and as they believe THEIR TEACHERS perceive them. There is rarely a 1:1 match - and so this opens up some areas for discussion between teacher and student(s).
- I am interested in your thinking.
- Please help me understand. Suppose you are the teacher and I am your pupil.
- I don't think that this problem is easy. Sometimes I get confused...don't you?
- Sometimes when I have difficulties with a problem, I break it down into small steps. Let's do that here and find out.... (The problem is modified).
- I like it when you take the time to think.
- I understand now, but... (p. 107).
A = Always
M = Most of the time
O = Occasionally
S = Seldom
N = Not at all
- I can give clear and understandable explanations of the thought processes I go through when I am solving a problem.
- I can use the materials to show that the mathematics I do makes sense to me.
- Whenever I get stuck on a problem, I can use what I know to get unstuck.
- I am able to identify errors in answers, in the use of mathematics materials, and in my thinking.
- When I do a computation, I don't always need paper and pencil.
- When a strategy doesn't work, I try another one instead of giving up.
- I can extend, or change, a problem by asking extra questions or posing different conditions.
- I study and practice before tests and quizzes.
The first time I use this sheet, I model what it is I want students to do - and the specificity of examples I am looking for. Then, I accept nothing less than what I know that they can do. Students are sometimes resistant to this because it involves a kind of thinking that some are not used to - but I persevere and within a short period of time, am able to get the kind of formative assessment data that is helpful to me in my teaching.
Regardless of whether we are teaching seven-year-olds, seventh graders, or seventeen-year-olds, part of our task is to teach them what our expectations are. If we expect students to reflect on their thinking and problem-solving processes, then we must teach them to do so. We need to:
1) model our own reflections,
2) ask them about their reflections, and
3) give them opportunities to reflect on their processes.
In other words, we can't give up just because they are resistant to giving us information about their mathematical processing.
And to access scores of free resources that you can use to support the learners in your classroom, including PowerPoints, PDFs, and Word documents, just go to
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