[2016 Blogging Initiative] Week Three: Understanding Questions


After making the transition to Standards Based Grading a few years ago (which is awesome!), over time I realized that my class had become too skill-oriented. To fix this I first tried to create standards that were “understanding standards”, but this overwhelmed my students with too many grades.

It took me an entire semester, but I realized that what I should be doing is asking “understanding questions” on assessments instead of only skill-oriented questions.

I try to limit my assessments to three questions (sometimes a question might have multiple parts), but I now always try to include an “understanding question” as one of those three questions. I never grew up answering these on math assessments, and they’re harder to grade because there’s usually not “one right answer”, but it has helped me get a better grasp of what my students understand (or don’t).

What do these look like? Here are some examples.


Exponents: Explain why b^x \cdot b^y = b^{x+y} is true.[1]

Polynomials: What does multiplying polynomials have to do with the distributive property?

Polynomials: Why can you combine some terms of a polynomial but not others? (3x^2 + 4x^2 can be added but 3x^2 + 4x^3 cannot)

Rational Expressions: Before factoring was the opposite of simplifying. What has changed and why do we factor first to simplify rational expressions?

Functions: Give an example of a function and a non-function outside of math class.

Transformations: Why does (x+3) move a graph left and (x-3) move a graph right? Isn’t that that the opposite of what you would expect?

Logarithms: Explain why \log_b{M} + \log_b{N} = \log_b{(M \cdot N)} is true.[1]

Reflection/My Own Questions

(1) Are these “understanding questions” enough to check for understanding? Probably not by themselves. So I need to get better at assessing repeatedly over time to check for retention of understanding.

(2) Should I give students the questions beforehand? Right now I do because if they want to figure out the answers on their own, great! As long as I have enough possible questions so they’re not simply memorizing and spitting back what I say, but really understanding it. (Or should they be able to get these questions even without me providing them ahead of time?)

(3) Is there a place to get these types of questions? I primarily look in the textbook or come up with my own questions, but surely there’s a bank of these somewhere online that I haven’t found yet.


These questions are ones that get at understanding, though harder to grade (at least they take longer), are worth it. When I started these questions, I was sorely disappointed how little my Precalculus students understood (even though they have seen some things, like exponents, in Algebra II and probably even in Algebra I!). I’m really curious what people think about the three reflection questions above.

Thanks for reading!


[1] I’ve flip-flopped between using the vocabulary “prove” and “explain”. The former suggests there’s one right answer to students whereas the latter allows for various explanations. “Explain” also is harder to grade, but I’m very excited if I see students start to write out examples in their explanations. No, it’s not as rigorous as professional mathematicians, but it shows me that they’re starting to understand.



Filed under Teaching

5 responses to “[2016 Blogging Initiative] Week Three: Understanding Questions

  1. Wow! You are truly assessing for understanding and preparing for transfer of knowledge. Though I am not aware of a bank of this type of questions, your style ties very closely to the work of McTighe and Wiggins (Understanding by Design). I wonder if you explored their work if you may find more support for your direction. Well done and keep it up! I am definitely bookmarking your blog!

    • Thanks! I’ve heard of “Understanding by Design” but it’s been a long time and I don’t recognize the names, so thank you for sharing those with me!

      Thank you for the kind words as well and thanks for reading!

  2. I think you’re right, that these type of questions are needed alongside the skills questions so students aren’t just regurgitating the processes but are truly thinking about the maths. Thanks for sharing 🙂

  3. I started using rubrics in two of my classes as kind of a transition into SBG that I hope to be doing soon, and I’ve started to realize the same thing…I never seem to assess anything besides just the skill. There’s no real *thinking* problems on my assessments. When you use SBG to grade these understanding questions, do you still have that “understanding standard” or do you lump them into the standard itself? I really want to use SBG next year but am trying to work out the kinks for myself now, so thanks so much for sharing!

    • I’ve taken away the “understanding standard” because that was too confusing to my students. I had to remember that most of these student’s aren’t excited to get a grade, or see what their grade means–they just want to pass. That meant that a grade which was even a little challenging to understand was too much for them (I tried this for a full semester). So I now just have one standard, but I try to make sure that there’s an “understanding question”.

      To go into it a little more: I grade on a 1,2,3 scale: 1 is “not there yet”, 2 is “proficient” or passing, and 3 is “mastery”. I put 3 questions on each assessment for a given standard. This means that students can pass without understanding, but to get the grade that most of them want (to get mastery), they have to answer the understanding question. How the 1,2,3 converts to A,B,C is enough for a whole other blog post, though I think I’ve blogged about it before.

      Thanks for the comment!

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