Contrary to all advice educators receive about timely and high quality feedback, I have an alternate hypothesis. Most mistakes children make are the result not of a misconception they have about mathematics but instead the student has a totally coherent conception that is the result of an incomplete set of knowledge about the world. Within the set of knowledge the child knows, the idea makes complete sense. In many of these cases, there is no need to give immediate feedback; instead you should create experiences that extend what that child knows.
My son once told me that one third equals three quarters. I could have said, “Well that’s not true. Here’s why.” Instead what I said was, “Well, why is that true?” He told me, “The 3 in the denominator tells you how many quarters you have.” I then considered, when is what he is describing true? If we look at one half and two quarters, the two in the denominator of one-half is in fact the number of quarters in two quarters. So given his very limited experience with fractions, at the time he was really only familiar with quarters and halves, he has a perfect logical connection between one third and three quarters.
Look at this typical representation of three quarters.
If students are not clear about the part whole relationship in fractions, they might interpret this fraction, which an adult would likely label as three quarters, as having three equal parts, just like a representation of one third has three equal parts. If one thinks the important part of a fraction is the number of parts it has and not the relationship between the parts and the whole, then one could also interpret one third as being the same as three quarters.
What’s my response? I need my son to know about more examples of fractions, particularly sets of equivalent fractions. Telling him right now that that’s not the right answer actually doesn’t help him. I need to interpret his response as a call for me to give him experiences around other kinds of fractions, other kinds of representations of fractions, and just telling him he’s wrong isn’t likely to lead to him growing as a mathematician. If, for example, my son learns that fractions are numbers and have a position on a number line, then he is less likely to think that one third and three quarters are the same, since they occupy different positions on a number line.
As it turns out with my son, I used an instructional routine called Choral Counting to connect fractions to a number line representation. With his classmates, we all counted together first by halves, then by quarters, and then by thirds, starting from zero each time. As we counted, I recorded our count on chart paper on top of a number line like the image below.
We also explored many other fractions as part-whole relationships, being sure to carefully define the part and whole for each fraction. A couple of weeks later, when I asked my son about whether one third is equal to three quarters, he explained to me why that doesn’t make sense. I did give him feedback about his idea, but gradually over time, by introducing new experiences to him. Not once did I tell him that he was wrong and that his idea didn’t make sense.
Every time we tell a child that some idea they have doesn’t work, we might be inadvertently teaching this child that mathematics is not supposed to make sense; that mathematics is not intended to be something that is true within what they know. The next time one of your students tells you something that you know isn’t true, try finding out, as carefully as you can, why they think it is true. You may just find out that what this student needs is more knowledge about the world rather than specific feedback on their idea.
