I had students (plural mind you) on my last placement that
had failed grade 10 math twice and were taking it for the third time. These
students were also likely to fail again. As expected, the big issue was their
poor arithmetic and number sense. One student told me that 1 + 0.25 = 0.7.
Another had no idea how to do addition and subtraction when negative numbers
were involved, and generally wouldn’t bother to include negatives. For this
post, I decided to do some reading on how other people explain negative numbers
and do a little review of the ones that stuck out.
Start – Direction –
Distance
This approach uses a vertical number line. They start with
addition and subtraction of positive numbers. Students use the first number as
where they start on the number line;
whether the problem uses addition or subtraction gives the direction they will move; the second number is the distance that they will move. Once that
is done, they try addition and subtraction with a negative start. Once that is comfortable, they try addition with negative
numbers, and eventually subtraction with negative numbers. I like the use of a vertical number line
because I think “up –down” is easier to conceptualize that “right – left”. The author of this approach specifically says
that he avoids analogies. I appreciate that, though I think stories are
sometimes easier to remember than procedures. Of course, analogies only go so
far and should be used judiciously. He also makes a point of separating
addition/subtraction from multiplication. As such, he never says something like
“two negatives make a positive”. Personally, I think this approach sounds
procedurally very useful since there is nothing arbitrary or ad hoc about it.
Perhaps the deeper meaning of why it works will appear spontaneously to the
students, but at the very least, they will know how to do the approach
correctly.
Hot Air Balloon
I like this analogy less, the more I think about it, but I
think it does have its uses. Imagine a hot air balloon. Puffs of hot air are
positive numbers. Sandbags are negative numbers. Adding hot air moves you up.
Subtracting hot air moves you down. Adding sandbags moves you down while
removing sandbags moves you up. I won’t elaborate on how the analogy works. I
think it is pretty obvious. It is certainly a cute picture and it gives a clear
idea of how removing something can cause an increase. However, I’m not sure
that the analogy is terrible useful for learning how to do actual calculations.
I would use it as a picture for a confused student, but I would be hesitant to
lead with it. One nice thing about this analogy is that there is a game that
goes with it, which may be a fun activity.
White and Black Stones
This analogy is from Marian Small and I think greatly
overcomplicates the matter. As a matter of honesty, I vehemently dislike almost
everything I’ve seen from her, so I may have been a bit biased reading this. In
her procedure, students are encouraged to represent positive integers as empty
circles and negative integers and shaded circles. One positive circle will
cancel with one negative circle (what she call the “zero principle”). Once all
the possible cancellations are done, the student counts what is left over. The
procedure works smoothly for some combinations of numbers but becomes very
complicated for others. I generally don’t like Small’s reliance on cumbersome
manipulatives and complicated visual metaphors. The metaphor should never be
the procedure, in my opinion. It breaks down a bit when we get to
multiplication where she admits that there is no easy way to model the product
of two negative numbers. I will concede that her “zero principle” did grab my
attention. It was an interesting way of demonstrating that zero can always be
rewritten as the addition of a number and its negative, which pops up later
down the road in completing the square.
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