In Math 1, we have been working on several different tasks, each of which is related to combinatorics, the first unit that we'll be officially starting next week. In each task, students start with an introductory question and then each group creates an extension to pursue next. The three tasks we've done so far are below. I'm still tweaking the fourth one and will post it when I'm done (hint: this is one of the things I need help deciding).

Task 1: How many paths from A to B if you can only travel down and to the right?

Extensions created by students: generalize for a grid of any size, allow travel up and to the left (without crossing over), allow traveling diagonally

Task 2: Consider a game in which you flip a coin four times. At the beginning of the game, your score is 0. Each time you get heads, you get a point. Each time you get tails, you lose a point. What are the different scores that are possible and how likely is each of these scores?

Extensions created by students: generalize for n flips, what about dice that have 4, 5, 6, etc sides?

Task 3: How many different monetary values can you make from these bills?

No extensions created yet, will have more time on this next week

Scroll down for the presentations from class for each of the investigations, which include slides about group/class norms.

Two big questions with which I'm wrestling in doing these tasks are:

- How much, if any, content teaching should there be? Students are practically begging for more efficient methods than just listing out all of the options, but should this unit really be about helping students get better at exploring their own thinking or is it better to teach some content while they're hooked and eager rather than coming back to it when it actually comes up in the unit? For those who incorporate student-driven investigations along with teacher-led instruction, when do you do the latter?
- Relatedly, how much should I be pushing students to make the connections between these problems more explicit? I feel like I've been dropping some (subtle) hints and revisiting student work from previous problems in the hopes that some students will point out the underlying connections, but no such luck. Again, is it better to show these connections now, even if it means they will mostly be teacher-driven, or better to wait until later and let these problems simmer for a while longer?

My current thinking on these two questions is that I will require each student to work on generalizing one of the tasks and then have students present their generalizations and ask more explicitly about connections between them at that time. I have to now choose a fourth task that I hope will make the connection more obvious... suggestions? What are some tasks/problems you've liked for hooking students on combinations?

P.S. I am super happy with how group norms and vertical whiteboarding is going so far this year. Using the same routine with a new math task each day so far has created a really nice flow and students are interacting well and starting to independently leave their groups to find out what other groups are doing to bring those ideas back. It was definitely worth taking a few days out of the content rush to set things up.

**Presentations from class**

Thanks for sharing your adventures. I am (noticing and) wondering:

ReplyDelete1. What answers did you come to on how much curriculum and connections?

2. What did you do for your forth day?

3. How has adding this unit imacted your class climate and students interactions as you move into your first unit?

Hi Leeanne! I decided to postpone explicit content instruction until the official start of the unit, but will encourage students to go back and revisit these earlier problems once we've looked at more efficient methods. Here is the task we used on Day 4: https://drive.google.com/file/d/0B_LfA0vBZWpCNlZIZ3htSnZVMGc/view. In at least one class, this did prompt some more efficient methods that will connect nicely to working on combinations and permutations this week. For the class that did not come to these methods independently, I think it will still create motivation and interest in learning more formal methods. In terms of impact on class climate and student interactions, it seems like it has done more to get students more comfortable with each other rather than with me. Most of our class time has been spent with students investigating and discussing together and I've been explicit about how I want them to work together and why I'm taking a backseat and encouraging them to explore and present. Students have also read and reflected on several readings excerpted from "Mathematical Mindsets" and "Make It Stick," which has supported my goals for this unit tremendously. They seem more comfortable jumping into unfamiliar problems, sharing ideas they are not certain are correct, and trying to figure things out even when they are stuck. I need to make sure to come back and refresh these aspects of the class periodically, but we are off to a good start!

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