I have been an avowed dis-liker of Desmos. I never argued about the quality of the math, but I didn’t love the organization of it on the teacher side. However, having spent a few hours on Labor Day weekend turning card sorts and matching that Irene and I created into Desmos CardSorts for use in virtual/remote education, I have become more of a convert.

I have no formal training in assessment design–somehow, despite one Bachelor degree, two Masters, and additional training for my fourth license, I never had to take a course in how to create effect assessments. How do you avoid bias? How do you ensure that your questions assess all aspects of the content you are trying to assess? How do you avoid creating problems where students can have an incorrect process that generates a correct value? How do you choose values that create just-right challenge, not too easy but not so complex that the content is obscured by the mechanics of the computation? Anything I consider that I “know” about these topics comes from painful, painful trial and error from years of experience, not from a really valid training experience.

In creating these Desmos CardSorts I came face-to-face with some of these experiences again.

### Creating a Misconception

In building the LCM/GCF CardSort, I caught myself setting up a misconception: both “bogus” questions involved 5:

- 5 is the ____ (neither LCM nor GCF) of 10 and 11
- 5 is the ____ (neither LCM nor GCF) of 13 and 15

I can’t use 5 for both questions. Why? It’s like when we always orient triangles in the same direction or when we only solve with *x* on the right (or when we only use *x*, for that matter): if we want students to generalize, we have to avoid situations where we lead them to false generalizations. Based on the problems I had included, a student would be legitimately lead to believe that 5 is *never* the LCM nor the GCF of any pair of values, which is, of course, patently false! We know that as adults and as fluid practitioners of mathematics, but students do not always bridge that gap in understanding and our responsibility is to be caution in not creating false patterns.

### Bypassing Guessing

Students (adults too!) are excellent at finding the patterns that make life easier, especially if they are students for whom learning for learning’s sake is not enjoyable. These students tend to spend an immense amount of mental energy looking for patterns that allow them to bypass deep thinking…even though they might not be able to actually frame it in those words. In the LCM/GCF CardSort, that meant I needed to have more than just pairs, so that students couldn’t just pair up two cards and walk away. I also left some as pairs, so students couldn’t assume a pattern:

** Version One**: two Least Common Multiples and one Greatest Common Factor with same value of 12

- 12 is the ____ (GCF) of 24 and 36
- 12 is the ____ (LCM) of 4 and 6
- 12 is the ____ (LCM) of 3 and 4

* Version Two*: one Greatest Common Factor and one Least Common Multiple with same value

- 18 is the ____ (GCF) of 18 and 36
- 18 is the ____ (LCM) of 6 and 18

* Version Three*: pair of same values

- 2 is the ____ (GCF) for 2 and 4
- 4 is the ____ (LCM) for 2 and 4

AND

- 2 is the ____ (GCF) for 4 and 10 (second use of 2 as the Greatest Common Factor, but with a different factor pair)
- 20 is the ____ (LCM) for 4 and 10

* Version Four*: “Bogus” pairs (neither Least Common Multiple nor Greatest Common Factor; they would have 1 as Greatest Common Factor, which is valid, but outside of the scope of this activity).

When I was creating the CardSort for improper fractions and mixed numbers, I realized I needed to create more than one pair with a denominator of four, so that students couldn’t depend on guessing by just matching the denominators. I also put in some “distractors” of 1/3 (there are two pairs with a denominator of 3, but neither one related to 1/3) and one pair with a denominator of five, plus one whole number value of five, so students who are guessing and/or moving quickly will be “caught.” Finally, I did NOT make two pairs with fifths for denominators. I wanted to increase the chance that students would have to actually engage with the process of converting between the forms, rather than using “context clues” to make the matches.

The narrative and documentation of evolving consideration for Demos is a timely contribution for distance teachers.

Well done.

Robalee Chapin

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Thank you!

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