Re-engagement Ideas for Company Logo
Re-engagement:
Confronting misconceptions, providing feedback on thinking, going deeper
into the mathematics.
1. Start with a simple problem to bring all the students along. This allows students to
clarify and articulate the mathematical ideas.
2. Make sense of another person’s strategy. Try on a strategy. Compare strategies.
3. Have students analyze misconceptions and discuss why they don’t make sense. In the
process students can let go of misconceptions and clarify their thinking about the big
ideas.
4. Find out how a strategy could be modified to get the right answer. Find the seeds of
mathematical thinking in student work.
5. Develop ideas about what constitutes a convincing mathematical argument or
justification by analyzing work of others.
Re-engagement Overview: The re-engagement lesson is designed to promote as much student
discussion as possible. Students have already done the task on their own and now the
important ideas need to be brought out and examined. In the process, students need to have
the opportunity to confront and see the error in the logic of their misconceptions . Often, as
teachers, we try to prevent errors by giving frequent reminders, like “line up the decimal
points”. But actually errors can provide great opportunities for learning. Students don’t let go
of misconceptions until they understand why they don’t make sense. To the student there is
some underlying logic to their misconceptions. Instead of going back to re-teach a unit or lesson
from scratch, we want to take advantage of the time students have already spent familiarizing
themselves with a context. In this way we can maximize student learning from the work and
build on their previous thinking. By using student work, students become very engaged in the
process. They are genuinely interested in trying to figure out what someone else is thinking.
This also ups the cognitive demand of the task and helps students learn to be more reflective
about their thinking.
During the re-engagement lesson students do not have their work in front of them. This is so
they will be flexible in their thinking and not married to defending their previous ideas. We are
hoping that discussion will allow them to change their minds. Teachers often give blank papers
or “think sheets” to students to write on as the lesson progresses. The idea is to allow students
to structure their own thinking and processes, and not confine their solution strategies to the
teacher’s way of thinking. Teachers often begin the lesson by talking about wanting them to
have a discussion like mathematicians. A teacher might say, “We want you to discuss why
strategies make sense. We also want you to listen carefully to each other to see if you hear
something that makes you change your mind. “
Classroom Norms: To maximize the amount of discussion in the classroom the following norms
have proved helpful. Pose a question and then give everyone an opportunity for individual think
time, before starting the discussion. In this way everyone has a better chance of contributing
and develop a reason or curiosity for listening to the ideas of others, not just the quick
processors. Before going to a group discussion, do a pair/share to maximize the number of
students getting to share their ideas and to give students the opportunity to use and develop
mathematical language for a purpose of convincing others. Then open the discussion to the
whole class to allow divergent ideas to be shared and examined. Continue probing to allow
ideas to be “almost” over-clarified, so students really solidify their thinking. Try to avoid
feedback about the “rightness” or “wrongness” of their ideas. Let the group try to reach
consensus about what makes sense and why. Feedback stops the discussion.
So the flow of the lesson might look like this: Pose a question, allow think time, pair/share, and
then class discussion of the question, and then pose a new question. As a follow up, many
teachers give back the original tasks with red pens a few days later to allow students the
opportunity to edit their thinking. Other teachers may choose to give students new copies of
the original task and their original paper so they can write a final draft of their ideas
incorporating ideas they have learned in the course of the lesson.
The key to writing an engaging prompt is to use to small snippets of work devoid of the labels.
This forces students to actively think about where the numbers come from, what is happening
because of the computation, and why is this helpful. The cognitive demand of digging into the
work in this way is much higher than that required to just carrying out procedures.
Lesson Ideas for Company Logo:
Start with a simple problem to bring all students along:
Even though most students were able to identify the 3 squares in part 1 and give some kind of
description, I might start here just to get the conversation going and think of ways to push
student thinking. I might start with showing a picture of the problem and the prompt, using
either a poster or document reader. After giving students a chance to think, I might then show
them the following response:
“What do you think the student means?” Give students time to think, then share with partner
and discuss as a class.
“I overheard someone say that we could make this clear using letters. What do you think the
person meant? How do we use letters in geometry?” Again give students the chance to think,
then share with a partner and finally share with the whole group. Some students might suggest
lettering the squares, A, B, and C. Hopefully other students will come up with the idea of
labeling vertices. Give them time to discuss and try to give rationale on their own without
giving hints or leading questions. The purpose of re-engagement is to foster mathematical
thinking and discussion by students.
Now I would show a poster of part 2 of the task. I might start by having them brainstorm ways
of comparing triangles talking in pairs. Then I would show the work (in a poster so the student
remains anonymous):
I might then ask, “What does it mean to justify the answer? How could we justify this?”
After think, pair, share, I might continue with, “What other things might I be able to prove?”
Or, I might continue with student work. For example, I might take either of the pieces below:
Student A
Student B
“Visually they look congruent.”
Then I would ask, “How we might prove this?” I would have them work in pairs and then have
different pairs present their ideas. I would encourage students to ask questions (if they are not
convinced) and have the class work until they have a good solution. I might have them discuss
the idea of what it takes to convince a friend, what additional information or detail might it take
to convince someone who is skeptical or disagrees with a conjecture.
It might be fine to end the lesson here, because this will have moved many students from their
scores of 3’s to a higher level. However, if students seem engaged with the ideas, I might
continue with part 3. I might start with the work sample below:
“I overheard a group discussing this solution. One partner said I don’t think this can be correct
because it doesn’t show how the squares relate to each other. What do you think this person
meant?”
Another piece of interesting work is:
“They’re all similar but none of them are the same size.” “I am curious. What does it mean they
aren’t the same size? How could we find out the sizes of the squares when we don’t have
measurements?”
Again, I would give students individual think time--to give them a chance to have something
meaningful to say--then have them share ideas with a partner. As part of formative
assessment, I might walk around and check students ideas so I have some ideas about what I
want discussed during the whole group discussion. This will help build the ideas I want to come
out, rather than trying to deal with surprises that might take the lesson off course. This does
not mean that I might not allow some incorrect or incomplete ideas to come up first, to help
build to a complete solution.
In an advanced or engaged group, I might also pursue this idea:
“They’re all similar, but not of them are the same size.” I think it is an interesting idea because
squares are always similar, but triangles aren’t. I am hoping that ideas such as this can come up
in discussion. I might even press them to think if there are other shapes that might always be
similar. What are the properties of the shape that make this true?
What other questions or pieces of student work might be used to help students think more
deeply about part 3.