We would like to be a community. Please start by introducing yourself to your group
members. What do you like to be called? (Consider including your preferred
pronouns!) What you are currently thinking you might like to teach? What is
something interesting about you?
It’s always helpful when you begin group work to make sure you’ll be ready to
discuss your work with the class when we are finished with work time. For today, who
in your group will take down notes? Will those notes be shared or sent to
the other group members? Who in your group will speak for the group?
(These roles should change with each class, even if you work with the same
group!)
While we will not discuss teaching methods in this course, we won’t ignore the fact
that as teachers, we are always learning from those around us. What teachers in your
past have inspired you, and why were they inspirational?
Everyone comes to this course with different experiences with mathematics. What
has been your own experience? How do you imagine your own experiences might
influence your future classroom?
Ultimately, we are here to learn together. What does it actually mean to learn
something? Discuss a time when you learned something. Your example could be
something you learned in school, learning to play an instrument or a sport, learning
to do a new job, or anything else! What was the process of learning like? How did
you know that you had learned?
Now, let’s do some math! How many squares are in the following image?
Spend a few minutes thinking about this problem either by yourself or by discussing
the problem with your group. After you have a guess as to what the answer might be,
discuss the following questions.
- (a)
- Did each member of your group come up with the same answer? Did each member come to their answer in the same way?
- (b)
- Pick one method and one answer from your group. Discuss this method in detail, as if you were presenting it to the class. What ideas are the most important? What steps are the most complicated?
Now imagine that you are a teacher, and you have given this problem to your
students. On the next page are four example explanations that you could get from
your students. With your group, discuss the following.
- Compare and contrast these four explanations. What do they have in common? What is different about them?
- What do you like about the explanations? What do you dislike? Why?
- From this work, what can you determine that each student understands about this problem?
- What feedback would you give to each student to improve their explanation?
Explanation 1:
I counted 9 squares in the picture, and labeled them on the figure.
Explanation 2:
Explanation 3:
First, I remember that to be a “square”, you have to have the same length on both sides. The easiest squares to see are the tiny ones, and I can see three rows with three tiny squares in each of those rows for a total of nine tiny squares. The next easiest square to see is the biggest one: the entire picture is a square! Lastly, there are some medium-sized squares. Look at my picture to see an example. I found four of these medium-sized squares: two with their tops on the top row, and two with their bottoms on the bottom row. So, there are 14 total squares.
I’m convinced these are all of the squares in the figure, because the tiny ones are , the medium ones are , and the large one is , which is the entire original figure.
Explanation 4:
| Step | Why |
| We are looking for squares. | The problem tells us so. |
| Squares have the same length on both sides. | That’s what it means to be a square. |
| There are 9 tiny squares. | We count the squares. |
| There are 4 medium square. | We count the squares. |
| There is 1 big square. | We count the entire square. |
| There are 14 squares total. | We add up the squares we found. |