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.

Hosting the Play Date
I am hosting the Math Teacher's at Play this month.  Please submit your own blog if you have something to share.  But also consider submitting something you've found valuable at someone else's blog.  It's the same form for either, and it just takes a second.  The link can always be found at the bottom of the right side column here.

Complex Instruction
I've shared before how NRICH is probably my favorite source for rich problems.  They also share bits of scholarly writing, host discussions and contribute in other ways to the math ed community.  This month, they've posted a wealth of resources on problem selection for group environments, leaning heavily on Jo Boaler's work with several of her articles or chapters; Dr. Boaler calls it complex instruction.  Take a look!


Concentration
Last and most oddly, I wanted to share some quotes from a Japanese gamer.  I play this game called Magic the Gathering, which is really the best strategy game ever invented.  (With the possible exception of global thermonuclear war, but you know the only way to win that...)  Players construct their own (or copy someone else's) deck of cards, and then face off against one or more other players.  It has a pro tour, and a small number of players who actually make their living playing.  The best analogy I've heard for it is a chess game with modifiable board, rules and pieces.  Each year the game expands and changes, yet remains simple enough for interested people to pick it up.  It has probability situations that will break your brain.  Frankly, it's amazing that I've gone this long without geeking out about it in this blog.

The author, Tomaharu Saito, is one of the best players ever.  He is known for his concentration, and wrote an article on how to learn to concentrate.  I love when people share their understanding on how to learn, and really feel like that the best reason to teach math (for most students) is that it is such a good context to learn how to learn.  The full article is here; selected quotes below.

I decided to write an article that would always be helpful to players, one whose ideas would not fade with time or format changes.

One of the best reasons to write!  That's one of the reasons I like so much for my students to write.  But then we need to give them ways and opportunities to share.  I'm still working on that.

In order to show 100% of your ability, concentration is crucial.  Frankly, as far as concentration is concerned, each person is different. And I feel like there are some people, although it may be only a handful, who can concentrate extremely well without training.
However, this does not mean that most players are naturally sufficiently focused. I think that time spent training concentration can affect one’s ability to refocus when their concentration is broken. This will boost your deck’s power, and is itself a way of taking countermeasures against weaker decks.

One of the things I am currently struggling with as a teacher is how to develop and support my students in becoming long thinkers, better able to apply themselves to significant problems, and to persevere through being stuck.  This feels relevant.

When practicing concentration, I find it particularly effective to pretend I am playing in a real event and focus accordingly. If my only goal were to boost my concentration, it would be good to always play as though I were in a tournament, but I also like to watch deck development, tune my build, learn match-up odds, and work on other goals which can capture my attention and cause a distraction. Because this can also lower my efficiency, I have found it is not a good idea to always split my focus. It is for this reason that I make time to challenge myself to work on concentration.

Remembering overall purpose, creating authentic conditions, assessing barriers to improvement.  Choosing to challenge yourself.  My teaching question is how to create purpose like that within the classroom.  You wouldn't do this for an exercise.  It needs to be relevant to your life.

Also after reading a book or watching movies or television, make sure you are able to explain the subject to a third party. This is always effective in improving awareness. When it comes to explaining the subject to someone else after reading or watching, you need to have watched it carefully and tried to concentrate in order to pin down the main point.

This is transfer!  He's talking about applying concentration in another circumstance.  The point about awareness in relation to concentration is new to me, but makes sense.

There are various methods for training your concentration, but there are none that allow you to master the skill in a short period of time. There is a feeling that steadily bit by bit the skill increases, so persist in your attempts. 

Also, one characteristic of concentration ability is that it is greatly affected by your every day life. In particular, people who stay up late should be attentive to this. If an individual does not spend enough time in sunlight various problems can ensue, and concentration ability is no exception.

Clearly true.  He's talking about life long learning, and how lifestyle impinges on performance.

Indeed, circumstances where concentration is broken and misplays are made are frequent, and naturally they make winning more difficult. I recommend self-confidence, but you can also learn my own method for recovering concentration.

Have you heard of the Saito Slap?  If there is bothersome noise around me, I will slap my cheek hard to recover my concentration.

He's not kidding.  He's not recommending everyone slap themselves, but instead talks about the usefulness of a routine that you can rely on, that by training triggers a response.  I think about that in my prayer life, but haven't thought about it for teaching.

What are the times when it is easy to break concentration? When you recognize the times when it is easy for problems to arise, it becomes easier to cope with them.

• The moment you think you’ve won
• When you’ve lost a previous game due to a play error
• When you’ve lost a previous round due to a play error
• When you’ve lost a previous game due to a poor draw, mana troubles, etc.
• When you’ve lost a previous round due to a poor draw, mana troubles, etc.
• After some kind of trouble occurs
• When something causes you to become irritated
• When you think about things other than the match

Perfect for an anchor chart!  I love the idea of thinking about ways you get stuck or when you give up on problems or...  What would this list look like in math class?

Magic is a wonderful game. Right now, I am betting my livelihood on it, but even misplays and losses do not cost me my life. I don’t let other people get to me.

If you relax a little, you can move forward.

That deserves an amen for brother Tomaharu.

Now to get more people to reflect metacognitively on their passions!

Where do you get good problems for your students?

One source is that problem-of-the-day widget at the bottom of the blog. A couple times a week, I'm copying those, put them into a Word document, and then save them for a good opportunity.

But my all time favrite source is from the English (or British?) parallel to the NCTM: Nrich. Problems are sorted by content, tagged, by grade band (stage) and challenge level (number of stars). Some are unsolved, but accessible. Almost all are clever and/or interesting. Soooo nice! Give them a try. Here's an account of a teacher and how they use Nrich.

Here's one that I gave on a math for middle school final this semester:
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Take, for example, four consecutive negative numbers, say
Now place + and/or − signs between them. e.g.
There are other possibilities. Try to list all of them. Now work out the solutions to the various calculations. e.g.
Choose a different set of four consecutive negative numbers and repeat the process. Take a look at both sets of solutions. Notice anything? Can you explain any similarities? Can you predict some of the solutions you will get when you start with a different set of four consecutive negative numbers? Test out any conjectures you may have. Try to explain and justify your findings.