Math Hombre

I've been a big fan of #slowmathchat on Twitter from Michael Fenton in general, and last week's joint effort with #probchat was especially good. (Cole Gailus has been doing great storify work archiving these; here's the slowprob mashup.) Some of the discussion was about NRICH problem 996.

I'm teaching College Algebra for the first time this summer, as apart of trying to revise the class. We're focusing the syllabus, and shifting some emphasis to practices from just content. I thought this problem was a good one. We're not doing proportional reasoning as an individual focus, but rate of change and difference quotient is a part of the class.

Then over the weekend, Marty & Burkard, the Australian math popularizers who collect math in the movies clips among other things, sent a link to this video from Burkard's new YouTube channel, Mathologer. It features a great math clip from Little Big League. If Joe takes 3 hours to paint a house and Sam takes 5 hours...



So much good in there, you wonder if the writer had some teaching experience.

• Number mashups - check.
• My uncle was a painter - check.
• Trick question trope - check.

I also liked that it was a typical messed up pseudo context (paint a house in three hours), and got at what math looks like without making sense. I found a clip that had the whole scene without interruptions, and we started our math problem solving at 2:40, pausing the clip. I asked the class: what makes this a dumb question? They said the obvious, got into why you'd want to know, and discussed if two people working together would be like that. One student who is a supervisor at work countered - he would like to have an idea of how long a job should take. I added that it would be important for a bid, too. I got to paraphrase von Neumann: People think math is complicated. Math is simple. Real life is complicated. (Actual: "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.")

I told the story Glenda Lappan tells, about the shepherd with 132 sheep, who delivers them evenly to four fields,  how old is the shepherd? They laughed, then one person gave the common student answer: 33 years old.

So then they tried to solve the problem. Some people decided 8 hours,  some 4. I asked what's the most it could be: we got to less than 3. That gave one person the courage to share their answer of 2.

We showed the rest of the movie clip, but stopped at 3:22 before the math explanation started. Were they satisfied with 3x5/(3+5)? No. Earlier talking about the course (doing the Piece of Me activity) someone had asked what my teaching style was. I didn't know how to describe it, but had said I would not be up front telling them stuff, they would be working and discussing. I used this part of the clip to support that, him telling the answer did nothing for us.

I did a bit of a demonstration (more like an early 'with' in gradual release terms): what do we know for sure? Set up a timeline: at 3 hours, 1 house plus a part. (Should be more than half, students say) At 5 hours, two houses plus more than half again. At 6 hours, 3 houses... what time would make sense to think about here? 30 hours, a student said. "It's like finding a common denominator." So at 30 hours, Joe's painted 10 houses and Sam has 6.  16 houses in 30 hours. Does that tell us anything about one house? A student suggested 30/16. Why? "Because that's their average. Per house." Awesome! They invented unit rate and in hours per house not houses per hour hour. Then someone pointed out the average was the same as the answer from the clip.

Jot down what you're thinking about after solving that. Share it with your table.

So onto our next problem. I warned them that it's not the same as the painter problem because it's solved the same way, but it is the same in that we want to make sense of it. I adapted the NRICH problem for less obvious units and less information.

Work in Progress

A job needs three people to work for two weeks (10 working days).

Andi works for all 10 days.
Burt works for the first week and Claire works for the second week.
Dave works for 6 days, but then is too sick to work.
Edie takes his place for 3 days, then Fred does the last day.

When the job is finished they are all paid the same amount. At first they could not work out how much each man should have, but then Fred says: “If Edie gives $150 to Dave, then at least Dave’s got the right amount.”

If we have enough information, how much was paid for the whole job, and how much does each person get? If we don’t, what’s a little bit of information that would let you figure it out?

They really gave it a go. A couple people got to a quick answer with the numbers involved, and all their tablemates couldn't dissuade them.  I recorded their strategies as I heard them:


What they got from making sense of what the problem asked.







Asked a student to record her chart on the wall.







After they worked for a while, most students felt like they did not have enough information. Those who felt they had a solution could not convince the class of it.

They consolidated the chart into the center information on days worked, and grouped them into three ideal workers, who worked all ten days.

In discussion I pointed out that they hadn't used the information that $150 made Dave's pay fair. They kept losing track of the idea that they were all paid the same, and wanted to know how much Edie had left.  Once they decided for sure they didn't have enough information I gave them more. (I considered just saying 'yes, you do' but they were stuck. ) So I said Fred had another idea. If he gave all his money to Andi, she would be set.

 Then they were off and running. Several solutions popped up. One was more algebraic, which killed the interest that many people had.











This was revived when someone presented a just logical idea. If Andi was set, so were Burt and Claire. Because they worked half of Andi's work, and had half of Andi's money. If they were fair, then Dave had pay for 5 days, so then the $150 was one day's pay.










Once they knew one day's pay, they backward engineered the pay per person and the total.

Someone asked about the original information, and I supplied that we knew how many days of pay (person-days, like man-hours, in my head) there were - 30 - and that was split among 6 people. This didn't have a lot of traction, as I think man-hours is a weird unit. Some people connected it with work and got it.

In general, use of symbols was a barrier, not a help, which means we have our work cut out for us. On the other hand, this lesson with these two problems was packed full of the values of the class culture I want to establish, and got them discussing math for the first or one of the few times in their life.

We had a fun class in the elementary math course today. Introducing SMP1 - problem solving, we got to an interesting question: what's a problem?

Here's the story as told by the residue on my whiteboards.

Schema Activation: jot down about a time you solved a real problem in your life. You won't have to share with anyone if you don't want to - this can be private.

After a few minutes to write, I shared how one of the big justifications for teaching and prioritizing math in school, other than the jobs to which it gives access, is that it teaches problem solving.


 Actually more yes than I expected!

People asked if I meant did math class help with the problem that they journaled about or in general. I said we want to know about problem solving in general, but they could use their instance as a specific.







Next question: is it possible that math class could help teach problem solving? Short time to discuss at table.








These were definite yesses, with a lot of confidence.




One of the reasons I like teaching teaching math is teaching is so much like math to begin with. So rephrasing: our problem is how to go from the current situation to get what we want.




So the next prompt was to quickly brainstorm. Ideas for making this happen.

















I shared that I liked how many of these were things that were up to the teacher. And I paraphrased Marzano, about how there is bad news: a small percentage of factors affecting student learning are under the teachers control; good news: that percent still makes a big difference.

What did they like?

• The emphasis on real life. This brought out uniform hatred for unlikely impractical story problems.
• Logic problems: one of the students shared how engaging and powerful these were for here. I asked about the contrast with real world, but people were comfortable with the crazy logic problems because the emphasis was on how did you do it.
• Teachers can ask for multiple ways and have students compare them.

Okay. So if we're going to teach problem solving, we need problems. I asked a question, then asked them to think about if that was a problem or not. There was a tub of square tiles on each set of two tables.



After they all had answers, I asked them to think about the second part. After a short time to discuss, we went to +cheesemonkeysf 's talking points structure for the statement.

We're still working on the structure, so some of my feedback was about that. The no comments idea is hard.






Pretty strong agreement. People were willing to share their reasoning with the whole class ...  even the small minority. Maybe especially the small minority?







I was really happy to see the "depends on what the teacher does" idea come up. And I added the "depends on students" too. This is not a problem for me. For first graders, a serious problem - there aren't enough tiles in the tub to cover! For them... well, they talked about methods, chose how to do it, discussed results; these are problem indicators to me.

Then we went back to the question. What answers did they find?




 There was shock at the diversity.

One question I ask a lot is: is this a question where different answers could all be right?

They discussed and...

There was concern about the largest answer being too big, but that table and an adjacent table figured out the first group's table is actually larger.

I shared how measurement is one of my favorite content areas, exactly for this reason that a diversity of answers is to be expected. It can be culture setting.

So, with all this, definitely time for a problem. One of my favorites. How many pentominoes are there?


We played with domino patterns last class, so that's a natural starting point. We agreed that there was only one way to make a domino with two squares. If they touch, they have to share an edge.

With three squares, the issue came up about putting them "in the middle." That's solved by the edge rule. What about the elbows in a different direction? No, the class agrees, if you can turn it to match, it's the same shape.

So what I want to know: is how many pentominoes?



People got to work with tiles and graph paper. No group found all the tetrominoes as a step, which I was trying to suggest. A couple of times I had tables report on how many they had and were they expecting more. 8 more, 9 more, 12 probably, etc. Next time, 12, 14, 15 and a mix of have them all and think there are more. When our time was up, they sent emissaries to the board to draw what they had:

Now the best part: math fight! Are flips the same or not?

They divided up into groups based on their answer: 14 flips matter, 9 flips are the same. They shared reasoning. Flips matter because these are flat things and a flip requires another dimension. And if you try to match them up they do not line up. Flips don't matter because you can get them to line up, and what's the difference between a flip and a turn, really?

I refuse to settle the issue, and ask them to make a complete set before Monday.

If you care to comment or tweet a response, I'll share your answers with them:

• How do you recognize what is a problem for your students?
• Are flipped pentominoes the same or different? Why?

Thought this little video had some excellent problem solving advice. It's an ad for a scholarship competition for a design program. Reminds me of a series of images called Advice to Sink in Slowly. Great graphic design and some pretty solid advice.

Problem Solving is a big deal in any math class I teach, and I, like most math teachers, use Georg Polya's problem solving phases as a framework. Though I used to teach it as a four step process, I now recognize it as four phases, which problem solvers can progress through in many different ways, back tracking and skipping. The ultimate reference on this is Polya's book How to Solve It. My handout (adapted from Dave Coffey's) is here; it focuses on Polya's questions. (Questioning being another important comprehension strategy.) The modern day successor to Polya as a researcher and teacher of problem solving is Alan Schoenfeld. The link leads to his site where he generously shares a lot of his research and work. I recommend at least skimming Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making In Mathematics (pdf link), a novella of a paper. Around pp 60-67 there's an amazing section on novice and expert problem solvers and teaching interventions.



One of my calc students was taking his final early, on his way to the month duty for army reservists, and commented on how he remembers Polya's steps by a connection with the troop leadership procedure. He sees:

• understanding the problem - receive the mission, issue the warning
• make a plan
• do the plan - start movement, recoineter, complete plan
• revise and check - issue plan and supervise

How cool is that! I just had to share. Thanks, James!