06 May

# The Setup

One time I made my students measure the heights of buildings around our school with an engineer’s transit. They had to use trigonometry.  I was a pretty great teacher so I made them take all of their measurements multiple times.  Man, they loved it.  It was a great activity because I give out points for each successful thing they do, which really motivates them. Everything is on a scale of five to ten (I have a progressive attitude about grades), so when a team gives an answer I give them some points, and at the end of the week we average them to get final grades. (side note: this is extra great because I can give students feedback pretty quickly)

Anyway, when they were done we did some serious analysis on the board. One team measured the gym, and they took four separate measurements. They found that the gym building was 40, 39, 42, and 39 feet tall. To really make the significance of that stick I made a spark graph.

I gave that team three 9/10s, one for each incorrect measurement, and one 10/10, for the correct measurement of 40 feet.

The other team measured the science building, and I guess they were slow or something because they only measured it twice – 19 and 21 feet. In the end it worked out because we could still make a spark graph. Two 9/10s.

I was starting to think that the spark graphs weren’t that useful, but luckily this was on a smart board so I could have one of the students come up and drag them around.

We figured out that the science building was about 20 feet tall and the gym was about 40 feet tall.

Here’s what I couldn’t believe, though.  I asked them what the average height was, and here’s what they did:

$\frac {(19 + 21) + (40+39+42+39)} {6} = 33.33$

Whoa guys! I know the scale is supposed to start at 5 but I’ve got to give a zero for that.  Very disappointing.  Are you sure you’re in the right class?

# The Problem

The students couldn’t understand that the numbers they collected shouldn’t just be added up and averaged together.  I mean, you can add up numbers and average them, but you have to understand what you’re doing.  Averaging is an algorithm that really only works for equal, independent measurements of the same thing.  Obviously the average height of the buildings is 30 feet. We have to make sure that the numbers we’re sticking into our averaging algorithm are actually compatible.  Even though all six of our measurements were in feet… some are measurements of the science building and some are measurements of the gym.  If you average them you get the average measurement, not the average height!

# The Punchline

Overall, I think the activity was a success.  After I added up the points everyone got during the day and averaged them together, everyone had over 90%!  Then I added up the points they got on homework and tests, and averaged that in too.  Finally, I added up all the points that everyone had earned in the whole week and averaged those together, and I had a final grade of B+ for the class overall – pretty good!

Afterwards, someone asked me what they needed to work on to improve.  I looked up their grade and saw they had an 85%, so I suggested they try to get more points the next time I asked a question. I love that self-motivation that points systems provide.

But I was the proudest when the director of maintenance heard that our class had been measuring the buildings.  He actually came into the class to ask some advice! He needed to get new ladders so that he could easily repair the roofs of the building, and almost my entire class could easily answer that he should get – you guessed it – the thirty-foot model!

1. May 6, 2011 at 10:26 am

2. May 6, 2011 at 10:58 am

I love the metaphor. If you don’t get it, you don’t get it!

3. May 6, 2011 at 11:10 am

So what do you hope to accomplish with your 0 for the group that made a mistake (and I agree it is a mistake)? Teach them not to take risks?

• May 6, 2011 at 11:30 am

Well, as was pointed out by Mike at http://activegrade.com/blog/?p=78#comment-53, “This system rationalizes poor performance and encourages mediocrity” when you give a student with poor performance a 50%.

4. May 6, 2011 at 11:20 am

I think the kids did a valid estimate for the average.

“Averaging is an algorithm that really only works for equal, independent measurements of the same thing.’

Where did you get that idea?

Averaging is linear, and since you’re looking at the average height of of two buildings, each of whose heights was estimated by averaging, the students are, IMO, correct in using the overall average as an estimate of the average height of the two building. Their estimate, IMO, is more accurate than yours. Theirs is based on actual data. Yours is based on a belief you know the “right” answer.

• May 6, 2011 at 11:28 am

But if the gym team had taken 100 measurements, all around 40 feet, they would say the average of the two buildings’ heights would be much closer to 40 feet, right?

If they had found the estimated heights of each building (20 feet and 40 feet) and then averaged THOSE numbers, that makes sense. You get 30 feet. But they didn’t do that preprocessing – they just added up all of their measurements and got 33 feet. The individual measurements cannot be combined!

How can 33 feet be the average height, when one building is 20 feet and the other is 40? My estimate is based on their data too: [ (19+21)/2 + (40+39+42+39)/4 ] /2

5. May 6, 2011 at 11:23 am

Someone has suggested that the point of this piece is to be critical about the flaws of averaging grades, and that you are trying to be satirical. If this is true, then I apologize for my off-the-cuff remark.

• May 6, 2011 at 12:13 pm

Yeah, I caught the sarcasm, but just barely. It’s a bit buried!

• May 6, 2011 at 12:43 pm

Yeah, feeling pretty foolish. I read this the first time waaaay too literally.

• May 6, 2011 at 4:53 pm

Yeah, thank goodness I hesitated to script a reply about how out of character this was. It was only on a 2nd read through and in the last paragraph that I stopped and said…wait a minute.

Well played sir.

6. May 6, 2011 at 2:39 pm

I don’t get it. Are you suggesting teachers don’t weight their grading systems appropriately and get wrong answers? Which ones? Or this thus just a combo of bad teaching ideas? You got me too for a minute there…

7. May 6, 2011 at 4:38 pm

Great sarcasm. I especially love “I was starting to think that the spark graphs weren’t that useful, but luckily this was on a smart board so I could have one of the students come up and drag them around.”
Classic. I love it.
You make some great points about the priorities of our current system

8. May 6, 2011 at 7:34 pm

I read several math blogs a day. This is the first time I’ve had such a good laugh. What a great way to make a valid point. I know a few teachers I should forward this to but I don’t want to explain it yet again.

9. May 9, 2011 at 7:51 pm

Brilliant.

10. August 1, 2012 at 10:45 am

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