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%

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.

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.

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.

ReplyDeleteGlad 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).

ReplyDeleteSo 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.

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.

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