kitchen table math, the sequel: up the down staircase

Tuesday, April 19, 2011

up the down staircase

Sara (not sure who Sara is!) sends a link to this Everyday Math problem posted on the Well-Trained Mind forum.

Up is negative; down is positive.

That strikes me as a bad mnemonic to teach kids who are going to be encountering coordinate planes just a few years from now.

She got the answer wrong, too.

Up the Down Staircase

Up the Down Staircase


Catherine Johnson said...

I also find it a little 'off' to be talking to 2nd grade kids about how scary it is to have surgery----

Most 7-year olds don't expect to have surgery, and they don't think about their parents having surgery....

I guess I just find it a little inappropriate to be 'introducing' children to a fear of hospitals and surgery as a way of teaching math.

When I was that age, I didn't know that hospitals were scary. I thought they were safe places that made people well. A hospital was like a church or a school: these were all places that made my world stable and safe.

Allison said...

The whole thing is insane.

Most children that young don't have experience in hospitals; even the ones that did would still not be able to follow her translation of motion into emotion; bringing emotions into a math problem is a sure-fire way to interrupt the thinking process and instead connect math to emotions--in this case, FEAR; no child has a working memory long enough to keep track of the numerous steps of the analogy to relate it back to math at all.

Not to mention that up is down and down is up is also insane.

Wiping the slate clean and looking at the big picture here, no teacher should use *analogies* to teach arithmetic. Analogies are not math. Analogies are not precise. Analogies are open to interpretation. Analogies are pre loaded with ideas we can't control, and especially for younger ages, may simply not mean to students what they mean to our peer group. Worse, analogies only mean anything to adults who have already chunked them. For children, they take up working memory.

Arithmetic is best taught precisely. Using manipulatives, pictures, symbols is not the same as using analogies. Manipulatives and pictures are the elements needed to facilitate doing arithmetic. There is a huge difference between teaching place value and teaching "number houses". Teaching an analogy adds layers of complexity without elucidating what's contextually important in math.

rocky said...

Sometimes I use a similar picture. I imagine the numbers as little tug-of-war fighters. If they are both positive or both negative, then they are pulling the same way and I add them up to see how positive or negative is their combined result. If they are opposite signs, then they are fighting each other and I subtract. Of course the larger team wins, so I know whether the answer is positive or negative.

Ok, it's not for everyone, but it could help.

Allison said...

with 2nd graders? or 7th?

Catherine Johnson said...

The whole thing is insane.

MUCH more succinct!

Michael Weiss said...

What if you scrap the whole "elevator ride"/"hospital" nonsense (it is utter nonsense) but preserve the core of the method? It might go something like this:

Now let's subtract with the problem 807 - 294. Write the problem down with the 294 under the 807. Start with any place value you want -- you can either go left-to-right or right-to-left. For illustration, we'll go left-to-right.

We have 8 hundreds minus 2 hundreds; that's 6 hundreds, so write 600.

Next we have 0 tens, minus 9 tens. That's negative 9 tens, so we write negative 90.

Next we have 7 ones, minus 4 ones. That's 3 ones, so we write down a 3.

Now we have written down "600 - 90 +3". 600 - 90 is 510, 510 + 3 is 513. That's the answer.

Honestly, is there anything wrong with this as an algorithm? It is universally applicable, rooted in the properties of the base 10 system, and cultivates an understanding of positive and negative numbers.

I think where the teacher in the "story" went dangerously, dangerously off-track was in the "subtract up/subtract down" idea. She seems to think that the way to handle "0 - 9" is to "subtract up" and attach a negative sign to the result because "it's like riding an elevator up in a hospital". Ridiculous. But we should be careful to place blame where it belongs. Is that "explanation" found in Everyday Math, or did the teacher invent it herself?

Allison said...

2nd graders have typically only been told numbers are the natural numbers. They use manipulatives and visuals reinforcing this."teaching" or even introducing negative numbers to most 2nd graders at this point is unfair. No one has told them what a negative number is, and the time to learn it isn't when trying to learn subraction algorithms. One thing at a time, well motivated. Adding more elements for familiarity later is a way to lose most students.

Michael Weiss said...

Oh, I wouldn't propose teaching subtraction this way to 2nd graders. Not at all.

However, I have seen second graders do this (write a negative digit in one of the columns when subtracting a larger digit from a smaller one) spontaneously. Many 2nd graders already do know about negative numbers, whether or not it has been formally introduced. They do not come to us as blank slates.

The question for the teacher then becomes, What do you do when that happens -- when a student comes up with a creative and mathematically valid idea? Tell them "No, that isn't the right way to do it"? Or would you rather try to prevent it from happening in the first place, on the grounds that not all students will understand it?