Putting the Cart Before the Horse (Mathematically Speaking!)

In my experience as a teacher, I had many students that were learning English as a Second Language (EL). Maybe this is your experience as well, or you may have had the experience of trying to learn a second language yourself. When introducing new vocabulary around a concept, I would tap into to students background knowledge and experiences and try to connect it to something they already knew or were familiar. Now think about trying to explain or teach a language to a student that has no background knowledge or experience to draw from. This would be very difficult to do and to a certain extent, this is what we do during math time with K-2 students. Many times we put symbols in front of students and expect them to understand the concept connected with the symbol without enough (or any) experience to relate to it.

Let me explain further. I was listening to the radio on the way to work a couple of weeks ago, and two of the local D.J.s were talking about how they tried taking some of the standardized tests for elementary math, and they barely passed the second-grade test. They were describing one of the questions they answered as “Which addend should go into the blank so the equation has a sum of nine?” There is a lot of vocabulary there. Addend, equation, sum, and each of those has a mathematical concept (and symbol) attached to them as well as related understanding. As an adult, I had to stop and think about what each of those words meant mathematically. Now think about a six or seven-year-old trying to work and communicate with words and symbols that have no previous experiences or connection. Pretty crazy right?! But this is what we ask young students to do with numbers and symbols all the time. As adults, our experience of constructing our understanding of number is so far behind us that we forget that there is a substantial amount of conceptual understanding connected to them.

The research we have been studying at the AIMS Center this year has reminded me that we need to listen to and observe the “mathematics of children” so we understand where they are in their construction of number concepts. I remember a few years ago my niece, who was four at the time, referred to tomorrow or anything happening after today as “next day” and yesterday (or anything that happened before today) as “last day.” Could we understand what she was trying to communicate with us without using the specific word? Yes! Were we confident that she would eventually use the words tomorrow and yesterday? Of course, but only when those words make sense to her and not before. In other words, not putting the cart before the horse. We should guide students to use words and symbols to help them communicate their math thinking when they are ready.

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