4×4=1?

Sometimes, early mathematics learning is necessary even in college math courses.

A week or so ago, a community college math instructor told me, “You all at AIMS should visit my classroom to study a couple of my students.” He knows that we work with children between the ages of 3 and 8, but thinks there is a niche for us at the community college level as well. Here’s a recent example of how one of the conversations between he and one of his students went:

(Instructor and Student, working on a word problem together)

Instructor: Okay, so now that you have that, what is four times four?

Student (pausing to think deeply): Eight? (looks at Instructor for affirmation — none given) One?

This adult student has no identified special learning needs (EL, SLD, etc.), and thinks that 8 and 1 might be options as solutions to 4 times 4.

How did the student get to this point? By the way, I would love to hear your responses before I give you my opinions, so feel free to scroll to the bottom of this article and share before continuing to read.

My own take on this is that, to understand multiplicative concepts, a student must be able to: 1) know that a number word indicates a quantity of items that could be counted and a number word sequence that could be counted -without having to actually count (in additive situations, we say that students that know this are “counting-on with understanding”); and 2) understand that composites can be found in any part of a number sequence; and 3) be able to keep track of the composites as they are counted.

In my follow-up discussion with the instructor, it seemed as if this student did not have a firm grasp on any of those three factors. Is it any wonder that she is grasping at straws to answer 4 times 4?

Sure, it was probably part of the timed math facts she “learned” in elementary school. She might have even been fast enough to pass the 4s test and get a gold star on her chart.

But the key here is that she never really understood the concept of number.

A longitudinal study that analyzed multiple sources with large data samples showed that children who enter kindergarten behind in math achievement, on average, never catch up. In fact, they fall further behind. The chart below shows that the students in the lowest quartile do lower-quality work in Grade 8 than the students in the highest quartile do in Grade 3 (Schoenfeld & Stipek, 2011).

So, it should come as no surprise that there are college students that have never built understanding around even the most basic number concepts. This community college student was fortunate enough to have an instructor that was able to recognize that she wasn’t mathematically ready for the problems being posed in the course. The questions the instructor had for me were about how to help her move forward from where she was.

To me, equity in mathematics education is about giving each student the chance to construct mathematics based upon what they already know. No exceptions.