Thursday, December 8, 2016

Day 71: Grades and Mathemagic

I screwed up.

I'm a numbers guy, and I screwed up the numbers!

One of the more difficult aspects of doing Standards-Based Grading inside of a traditional grading system is the conversation.  How do you equitably convert a 4 point scale to a 100 point scale?  In SBG, a 2 isn't failing, but means basic understanding of the concepts.  In a traditional system, this usually means a C (70-80%).  A straight conversion would put the 2 at a 50%, which doesn't accurately represent the level of understanding in the course.

When I set up my scale, I set the conversions as follows:

4 ⟹ 100%
3 ⟹ 85%
2 ⟹ 70%
1 ⟹ 50%
0 ⟹ 0%  (This is reserved for students who missed a test or didn't answer a question)

With this scale, my classes were averaging 73% overall.  This is a little lower than I would like, but well within reasonable parameters.

Except for the mistake that I made.

My new district doesn't use 70-79% as a C, 80-89% as a B and 90-100% as an A.

The scale we use is:
A ⟹ 92-100%
B ⟹ 84-91%
C ⟹ 76-83%
D ⟹ 70-75%
F ⟹ Below 70%

This isn't a major difference, but it meant that my points didn't correspond to the appropriate grade.

So I changed them!  I changed them for every student, in every class, for the whole year.

4 ⟹ 100%
3 ⟹ 88%
2 ⟹ 80%
1 ⟹ 50%

This change brings the class averages back into line with the appropriate letter grades.  This was entirely my mistake and it didn't occur to me to do this until today.

I accepted responsibility for it and asked my students for forgiveness of the oversight.

In 8th period, something incredible happened.

The warm up today was for students to come up with a strategy to solve 14*16 mentally.  Through most of the day, students had ideas to break the numbers down in multiple ways, the most common being breaking 14 into 7*2, 16 into 8*2, then 7*8 is 56 and 56*4 is 224.

At the end of the day, a kid came up with an odd strategy:

"I took the 4 from the 14 and added it to the 16, making it 20, so I had 10*20 which was 200.  Then I multiplied the 4 and the 6 to get 24, added it to the 200 and got 224."

I looked at him skeptically.

When students develop their own strategies and algorithms, they may stumble on the right answer without having a valid mathematical reason.  This happened with a few other strategies that students offered.  When I tried the previous ideas using different numbers (24*26) it fell through.

With this strategy, it worked!  So I tried a different pair of numbers. And it worked!  With 28*35, we added the 8 to 35, multiplied 20*43 to get 860.  When we multiplied the ones digits (8*5) it didn't work, but it DID if we multiplied 8*15 (15 being the difference between 35 and 20.)

What was that about?? Does it work with any pair of 2 digit numbers?  Like a good mathematician, I realized that I needed to do a general proof.

I started by setting up general terms for 2 two-digit numbers (10x1+x2 and 10y1+y2) and multiplied them.  Then I went through the process that the student described with the general terms and, lo and behold, it works in general!

My favorite part of this was that I was doing this proof and exploration while students watched, letting them see my excitement, my methods.  The pre-algebra students saw the algebra I put on the board and, while they found it a bit intimidating, many of them were interested and wanted to me to explain what I was doing.

Math isn't magic.  It's the language with which we describe reality.


  1. Just FYI, I think an area model works pretty well for showing why this works, too. You're moving 10 x (last digit of 2nd #) chunks to a different part, and the leftover x (last digit of 2nd #) is what you have to add on in the end.

  2. Glad you clarified the percentages and all that; it's the sort of thing that reminds me how, despite being so geographically close to the US, our scales are completely different. I go based on: A (80-100), B (70-79), C (60-69), D (50-59), F (under 50).

    So it's a little easier to map levels 4, 3, 2, 1 and 0. I'm curious, your level 1 maps to 50%, but that's an F? Not a pass? For our rubrics (in Ontario), Level 1 is a bare pass showing limited understanding.

    We also have a range in the level zero at my school, where 0%-30% means you basically didn't do anything and fail, versus 30%-49% where you did some stuff but didn't even show limited understanding. (In the latter case, you can go to summer school.)

    Weird how there's so many systems out there. So I feel like that sort of "mistake" happens more often among educators than we might thing. Anyway, good on your for being transparent about it.

    1. For our grade scale, 70% is passing. I have no clue why. In the first scale I listed, 60% was passing and that's pretty much the norm around the country.

      For mine, the level 1 is the "did stuff but didn't show even basic understanding" and 2 is the bare pass.


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