Information binge

Image credit, Merwing: http://www.flickr.com/photos/merwing/7842171/ (Creative Commons)

So I sat down the other day (as I often do), and thought, “What about exponential and logarithmic functions might a high-school student find interesting?” Yes yes, exponential functions are essential in a number of financial math applications, but figuring out mortgage calculations is not the most engaging activity for the average teenager.  Surely I could find something more worthy of student attention.

Soon enough, I found my rabbit hole: “Malthusian catastrophe.”  Since stumbling on this half-familiar phrase nearly 24 hours ago, I have devoted an unusual proportion of my time to flesh it out through internet research.  This post is an attempt to organize this information and maybe even get some suggestions for new resources.  Well anyway, behold! :

“The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction, and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world”.

—Malthus T.R. 1798. An essay on the principle of population. Chapter VII, p61 [via Wikipedia]

Famine, pestilence, and plague … I’m in!

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Homework (v 5.0)

I have a split mind about homework. In my first years of teaching, I shared a commute with students attending one of the city’s elite high schools. I overheard complaints about how late they stayed up to do their homework; they occasionally whipped out their graphing calculator and beat up spiral to scrawl out some math homework. Not that this is a model of how things should be, but I thought: these students are already more proficient than my students, and the gap is increasing everyday. Thinking of my own high school days, I remember long sessions of homework which surely had some effect on my learning.

At the same time, I never developed a homework system that I liked. For the last school year, I essentially didn’t assign homework at all, which is certainly one solution to the issue. However, my students were expected to do DAILY homework in their previous class, and I’m not sure what message they’re getting by getting no homework in a class that’s supposed to be harder. So… homework is on the agenda for the next year (my fifth year of teaching!) I hope to use this blog post to share my thinking and propose a system that I could imagine working for me next year.

Guiding principle: Homework can let students use as much time as they need to try out some math questions. Therefore it can be an effective forum for practicing and receiving feedback. Feedback is the focus, not grades and the pressure they bring.

I highly recommend looking at Sam’s blog post HOMEWORK SURVEY RESULTS. It’s a thought provoking synthesis of what different teachers out there are doing. In the survey data I came across a great counterpoint to homework review in class, as follows:

Counterpoint: “Homework review, next to disciplinary conflicts, has been the most efficient way I know to waste classroom time and sink classroom morale.”

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a fun fractions problem!

Well, it’s been a million years since the last post, but I’ll spare any readers the doleful reasons.  I’m trying to figure out how to be self-disciplined with my writing and learning, so hopefully this summer sees many more posts.

But for now I thought I’d share a little problem that I saw today.  This summer I’m an adviser to a group of high school students who are taking college classes.   A student in a college algebra type course showed me this “bonus” question on the bottom of his quiz:

Find the sum, \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} + \cdots + \frac{1}{99\cdot 100} .

I won’t spoil your fun by posting a solution, but I encourage you to do so. I’m definitely interested in different solution methods on this one.

Enjoy!

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The Q*Bert Problem

Ah, new year, new math toys.  This time I finally fulfilled my math teacher imperative for cubes, figure 1.

Figure 1

I had some ideas about what I wanted to do: something simple involving triangular numbers.  But with the cubes, I knew that the temptation to build in 3-D was just too great to avoid.  And maybe I’m just playing too many video games these days, but I started building something and came up with this.

The Q*Bert Problem

How many cubes to change?Q*Bert is a video game that was popular in the 1980s.  To beat each level, the main character had to jump on all of the tops of the cubes so he can change their color.  Look at the picture above — how many cubes must Q*Bert change to beat level 7?

Before they made the final version of the game, one of the “designers” had an idea:

  1. Level 1 should have only one cube.
  2. Each next level should add one more layer of cubes.
  3. There aren’t any hidden cubes in the back.

Problem Statement:

A. How many cubes does Q*Bert have to change to beat level 99?

B.  What is the highest level that requires less than 1000 cubes to change?

The Goods

Pretty Posters

Notes for Future Use

  1. The order of questions 3 and 4 could be reversed — finding the rule is more abstract than finding the solution to the original problem.
  2. After working for a couple of days, some students had an answer but nobody could provide a rule and justify it.  So I used the “Rectangular Numbers Activity” to make the rule and a justification within their reach.  I’ve learned that I like the tension of students not knowing but searching, but at some point there needs to be resolution — an answer!
  3. I entered 4 different grades for this problem, one for each question.  So, I have learned something from the SBG approach.
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bad dream

I had a bad dream last night.

I’m teaching a class in my old classroom in Bushwick, but all of the students are from this year.  The classroom is dark and loud.  The desks are disorganized, and students enter and leave randomly.  There is no lesson going on.  I go in the hallway, chasing after a group of students to find that some of them are disturbing another class next door.

This teacher (current school) leaves her class and effectively chews me out to get my act together.  This sends me into a rage of yelling at my class, which is actually somewhat effective in that they sit down and are quiet.  Stress levels are high, and one of my students curses at me, but then starts crying because she didn’t mean it and that’s not the type of student she is.

Then I wake up.

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November slump

I’m in a slump.  Need evidence? Unusual amount of prodding needed to start work — check.  Straggling late students — check.  A student asked me if I’m so good at math, why didn’t I become a engineer or a mathematician, don’t they make more money?  Maybe he was just curious, or maybe I seem out of sorts.  That’s how I feel.

Oh boy, well there’s a lot on the brain these days.  The team meeting today turned into a Joe meeting — the subject of another post.  (Do not worry… I still love my team.)  It has to do with this and this.  And of course my SBG report card is overdue [self reminder: just like my library books].  For now, let’s use “top hit logic” to figure a way out of my November slump.

7 Ways to Crush a Hitting Teaching Slump

[My modifications are pretty obvious; original article by Jon Doyle here]

Baseball Teaching is a very humbling game, even for the world’s greatest players. At any given time, there’s countless Major Leaguers going through a slump.

So, what do you do? How do you help someone in a slump?

The Causes of a Slump

Most people actually think slumps are created by the [teacher] doing something wrong with their mechanics management. However, what usually causes an ongoing slump is how the [teacher] is affected mentally. Physical problems are easy to identify and fix. Start with the following basic checklist and then move into the mental aspect.

Physical Solutions to a Slump

Videotape the [teacher] in a live game situation. If you can only get batting practice footage, that is ok, but a real game is best.

  1. Pick out any flaws you may see.
  2. Show the hitter so he or she recognizes the problem and then pick out one or two drills to help with this mechanical flaw.
  3. Get their eyes checked — you’d be surprised how often this fixes slumps!

The good news about getting out of a slump is all it takes is one good swing. A [solid lesson] and a ball that finds a hole is good enough for a two-hit game and the slump is history.

The quickest, yet most difficult, task is getting the [teacher] to believe the above is true — A [teaching] streak is just ONE swing away…

Mental Solutions to a Slump

Do whatever it takes to get the hitter some confidence. Some suggestions:

  1. Let them get a feeling for solid contact. Get them back in their comfort zone and let them get their feel back.
  2. Have someone they respect compliment them. Whether it’s a good player or opposing coach, have that individual tell them their swing looks good and to keep up the good work. More often than not, a “slump” is much bigger in the [teacher]’s eyes than it is to everyone else. Sometimes all it takes is a compliment for the [teacher] to see this.
  3. Practice visualization. Studies show that visualization is an effective form of “practice” and does carry over to the real world. All great hitters use visualization of some form. It doesn’t have to be weird and kooky, simply learn to “see and feel” yourself taking a good swing and making solid [lesson plans].

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Some temporary solutions: Start off everyday with a can’t-lose hook (sounds obvious and easier said than done).  Keep free-for-all work time to a minimum, and maximize the amount of student academic talk time.  And frame the lesson more clearly!

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keeping up with the habit

While I feel good about my blog-reading habit, my posting habits have been in decline.  In the meantime, I’ve had a number of thoughts that I need to work out.  So let’s make a little list, and maybe we’ll expand later, ok?

  1. I must improve my differentiation.  No more hand-waiving on “multiple entry points” — yes, there is a time and place for this — but it’s time for some specific problem types to address the different levels of understanding in my class.
  2. Similarly, I need to integrate more language development into my class.  (I teach ELL students.)  And definitely more practice on discussing math.
  3. Time to develop the next unit!  This is the big one (leading to a “portfolio project”) so hopefully the six weeks in class is enough.  I’m going to become a fake physics teacher and teach about objects in motion and free-fall.
  4. I need to be better at “making” kids keep their concept checklist updated.
  5. I’m really digging this post.

That’ll do.  More to follow.

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