Hi everyone,

My name is Bryan Penfound. Awhile back I was asked if I would be interested in helping out at MathMistakes and I said yes not knowing how challenging this term would be for me. Now that I have settled in a little bit, I thought I was a bit overdue for a post, so here goes!

Recently while volunteering at a local high school in a grade 9 classroom, I had to opportunity to observe students’ answers to the following question: “Create a trinomial in the variable t that has degree 3 and a constant term of -4.”

Here are five of my favourite responses:

I would love to get some discussion going. Choose one of the polynomials above and try to deconstruct what the student knows and what the student still has misconceptions about. What follow-up questions might you ask to learn more information about how the student is thinking? What follow-up questions might you ask to help with any current misconceptions?

• Howard Phillips

This has got to be a very dumb question. 10 times more sensible is the word “polynomial”, but American Math is full of this notational overkill.

• Donald Ey
• Donald Ey

I pick 3 and suggest that it is correct – 4 is a constant and the t is a 3rd order polynomial

• Bryan Penfound

Not quite. Recall that a constant term is one in which the power of t is equal to zero. This student perhaps was confusing the words “constant” and “coefficient”?

• Bryan Penfound

I’m not sure I understand what you mean. I use “trinomial” and “binomial” and “monomial” almost every day in Calculus. I am sure that this teacher was trying to determine whether or not the students understood this term.

• Bryan Penfound

(I also forgot to mention that this example was from Canada.)

• AnneMarie Steele

In #4, the student seems to understand the meaning of constant as unchanging.

They seem to not understand the role x plays in change. On a lower level, they don’t seem to understand the naming conventions for polynomial terms.

• Bryan Penfound

I agree that there might be something about the words ‘constant’ and ‘coefficient’ that the student has some misconceptions about. What I find particularly interesting here is that there are three copies of -4t, and the question says “degree 3” and “trinomial”. Is the student seeing degree 3 as three terms? Three copies? Or are there three copies because of the word “trinomial”? I might ask “Create a trinomial in the variable t that has degree 2 and a constant term of -4.” to see what kind of follow-up information I could gather.