The
Center for Mathematics Achievement director, Anne M. Collins and staff Steve
Yurek, Katie Aspell, and Matt Rogers welcome you to our center, our blog and
our inaugural post. We are dedicated to transforming mathematics from rote
recitation, symbolic manipulation, and procedural prowess to engaging
student-centered experiences that build conceptual understanding, emphasize the
inclusion of multiple representations and knowing which representation best
represents a situation, and computations which can be justified, and through
which the reasonableness of the solutions is discussed.
So,
let’s look at an example. In my research, I have queried thousands of students
to solve the following equation:
3
+ 4 = ∆ + 5
The most common responses include 3 + 4 = 7 + 5; 3 + 4 = 7 + 5 = 12; and 3 + 4
= 7 + 5 = 17. One, thankfully the only one, actually neglected to honor the
equal sign altogether and illustrated their thinking with something along the
lines of the following:
3
+ 4 = 7 + 5
7
+ 11 +12 + 5
18
+ 12 + 5
30
+ 5
35
All of the solutions are incorrect but for different reasons. I contend that if, when equations were first introduced they were done so with balance beams, pan balances, counters, number lines, or algeblocks the misconceptions would not have occurred and intervention or remediation would be unnecessary.
Students who understand that an
equation is simply two equivalent expressions set equal to each other would
have little difficulty in solving such a simple equation. If taught, however,
as a procedure then difficulties and misconceptions arise.
Let’s
examine how parallel number lines might have prevented the expressed
misconceptions.
3 + 4 = ∆ + 5
So, as you can see, this blog intends to discuss
mathematical concepts, student misconceptions and ways in which to avoid them,
share interesting problems, and discuss any and all mathematics issues of
interest to our followers. We welcome you to comment on this post and/or to
begin a new thread. We are looking forward to our on-going discussions and
learning together as a community!
Anne
I was wondering what you thought about working backwards from the dashed line. After working the one side of the equation and seeing that it equals 7, could you work backwards on the lower number line. Move back 5, and land on two. Do you think one would be more effective than the other?
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