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Can we do away with minus?


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When I was a child learning basic arithmetic, a problem like this was not uncommon:

5-3=2 (five minus three equals two)


For some reason, I started thinking about this and negative numbers. I was never taught the concept of negative numbers until years later. With my basic arithmetic skills, I would probably be baffled by something like: -2+3=1 even though I understood 3-2=1 perfectly well.


Are we doing a disservice to math students by teaching arithmetic this way?

What if negative numbers were taught from the beginning and a problem like 5-4=1 was no longer seen as "five minus four equals one", but rather as "the sum of negative four and five equals one"?


It might seem trivial, but having the concept of negative numbers earlier on would seem to help with gearing up a mathematical mind earlier on. We could do away with minus.


Of course, maybe others here were taught this way early on. If so, I'd like to hear about it. I'd also be interested in experience people may have with teaching one or both of these methods and their reactions to the outcomes.

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Minus numbers are important to math, but they imply something less than nothing, which sort of defies common sense. The minus numbers sort of imply debt, and are therefore not something tangible, in the present tense and often need paperwork to make them tangible. It is more based on the past and future. Positive numbers use the present tense in terms of tangible reality.


For example, if I have 5 apples and take away 3, I have 2. There is no smoke and mirrors and it is quite clear. But if I start with negative three apples, who knows what political party might be in power down the line, such that I may be able to negotiate paying back 1 of 3, thereby making -3 equal to -1 in the future. Theoretically, I could begin with negative 3 apples, give one back in the future, and end up with zero.


Math assumes the ideal world, where negative implies a fixed deficit, that has to be paid back in kind. But that is not the real world. If Joe steals three apples from John, John will have negative three apples. Joe might eat the apples (take away three) and have none. John will go to the police and have Joe arrested. Joe may not be required to pay back any apples, but may magically turn minus three apples into a week in the slammer. It does get confusing so it is taught later.

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I had the, IMHO, good fortune to attend US public elementary school 1966-1972, at the height of the New Math pedagogy, so had little difficulty recognizing in any grade that 5-3 = -3 +5 = 5 + (-3) = 2 etc, or even -5 +2 = -3.


New Math emphasized geometric and set theoretical approaches over rote arithmetic, and the teaching of presumably more advanced abstract ideas before practical arithmetic.


Although my memory of the 1st grade is distant, I recall that our classroom had a large paper number line attached to the top edge of its chalkboard, and that addition and subtraction was taught as addition and subtraction of 1-dimensional vectors, a very clear and intuitive approach.


How well a pedagogic approach works depends not only on the expert-dictated approach, but on the comprehension and acceptance of it by each teacher. I had the additional good fortune to have a 1st grade teacher who was a recent college graduate, and was unhampered by habits from previous pedagogies and enthusiastic about and capable of using the new teaching methods.


In later grades, my classmates and I encountered more senior teachers who rejected New Math, and to some extent tried to “reeducate” us away from it. The reaction of me and many of my classmates was the opposite of what these teacher intended, however, as we quickly discovered that, in many ways, we already knew more math – and knew it more deeply – than our teachers, leapt to the conclusion that we were all much smarter than they, and made a game of attempting to trip them up and humiliate them in from of the class, our New Math mindsets stronger than ever.


While evidenced based pedagogy strongly supports approaches like the 1960s' New Math, many parents and other students’ family members, however, reject it in favor of “the good old-fashioned three Rs”. While extreme versions of either approaches are non-optimal, and best approaches need to be somewhat tailored to individual students and teachers, the optimal approach is IMHO much closer to the New Math end of the continuum than the 3 Rs.

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