# Second Thinking Problem

Yesterday, I said:

I'll start by having groups try to describe the characteristics of each type of cube (for example, the cubes with 3 faces covered are in the corners).  Depending on how well that goes, I might challenge them with with an nxnxn cube, or give them another problem to work on.

Well it didn't turn out how I had hoped (understatement 🙂 ). Students had difficulty putting into words a description of which cubes would have 0 faces covered, 1 face covered, 2, etc.  Since this was required to go further, I didn't see any alternative to directly teaching this and giving them sample descriptions.

After I did that I asked them to use those descriptions to determine the number of cubes of each category for a 4x4x4 cube--yes we did this yesterday, but I was hoping they could use these descriptions rather than brute-forcing the solutions. Well that was very optimistic of me! Not only did they have lots of difficulty, but they were disengaging;  I think probably because they had had enough of this problem. (I didn't even get to trying to generalize--way too optimistic!)

So I switched to another problem:

On a table there are 1001 pennies lined up in a row. I then come along and replace every second coin with a nickel. After this, I replace every third coin with a dime. Finally, I replace every fourth coin with a quarter. After all this, how much money is on the table?

As with yesterday, initially there was a lot of standing around with blank faces. But within 5 minutes everyone was at least looking at a whiteboard, and greater than 50% were actively engaged with their groups.

Here's one of the most advanced group's work:

Four things stand out from my two classes:

• Each class had at least one group that was really struggling, but refused -- multiple times -- to try approaches that other groups were using.  Even when I said, "I guarantee you'll have more success if you try those approaches."
• Each class had a few keen students who wanted to work on paper, then put a final answer on the whiteboard. I tried to handle this by telling them the whiteboard was for their thinking, not final answers and that it would allow other groups to see what they were doing and get ideas. At least one student refused to do her work on the whiteboard.
• There was a lot of problem with mental math and assessing validity of results.  For example, a few times I pointed out that an absolute upper bound on the amount would correspond to 1000 quarters.  Students instinctively said, "25000 dollars?".  Even when I prompted them to reconsider, they still insisted it was a reasonable answer.  Others also had difficulty determining the dollar amount using a calculator.
• I mentioned a few times that it would be helpful to look for patterns that repeat.  Most groups didn't know how to deal with that, or gave up when they couldn't see a pattern after 10 coins.

Conclusions

• I think 80 minutes is too long to be working on one problem.  A majority of students were disengaging around the 60 minute mark, presumably due to frustration.
• Given the apparent lack of mental math ability and general lack of problem solving skills, I'll have to directly teach some strategies relevant to the problems being solved more than I had initially anticipated.
• Student engagement is actually much higher than I had expected after only 2 days. I think I can increase engagement by giving easier problems that result in more success and thus less frustration.
• Having students work on vertical surfaces is great for allowing them to see what other groups are doing, and for me to immediately get a feel for what's happening in the class

#### Second Thinking Problem — 2 Comments

1. I'm also wondering if that question is too hard. I'm surprised you didn't encounter students just trying to google the answer!

Are you starting the semester off with these types of questions to get the students thinking about and enjoying math? I'm really wishing I didn't start off with buying/leasing a car!

Are you eventually going to use the apprenticeship and workplace text or just give the students notes?

• Initially I thought it would be easy, but then I quickly realized it was not! However with sufficient prompting from me, there was engagement for the majority of the block. My prompting consisted of things like:
- "Just try writing out a list of coins and see what happens when you do the replacements."
- "Try the approach that other group is using."
- "It looks like the majority of groups are on the right track."

Then later on,
- "Joey, tell us what you've noticed." (He had noticed there was a repeating pattern.)
- "So the question for all groups to consider is, Can you find a pattern that repeats?"

A number of groups were successful in finding the repeating pattern, but got stumped at that point.

As far as googling the answer, it may have happened, but whenever students said they had an answer, I always asked them to show their reasoning on the board. Interestingly, the first result on google shows an incorrect answer :-).

I'm starting off the semester with these questions for the reasons I described in my "Creating a Culture of Thinking" post. At this point, I'm quite happy with engagement, and being able to see class understanding/difficulties at a glance. More about that in my next post later today :-).

Next week I'll start with actual 30-3 content, but I'm going to use "upside down lessons" as described in my Creating a Culture of Thinking post. I'll be using the MathWorks 12 workbook as a source for the problems, and as a reference I can point students to for things like definitions or extra practice.