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]]>I will reiterate a point others have made: there’s no “Jack.” The scenario is almost certainly hypothetical. And I’m guessing (with the little we have to go on) the following: the method “Jack” uses isn’t intended to be either instructional or exemplary. It’s just a method. If this is early in the instructional process with primary (K-2) grade kids, I have little doubt that there are many who are going to do arithmetic of any kind with some amount (possibly quite a lot) of counting – whether on fingers, other concrete objects, items in a picture, dots on some sort of number line, etc. And that is quite normal.

Adults who already know and have mastered the “standard algorithm” for subtraction are aghast at the very idea that any time might be spent on ANY other approach, regardless of developmental psych., math education research, or anything else. There’s one right way to do math: the way they learned. Period. End of conversation.

But what if this were something I’ve witnessed both live and in videos, including from Japanese elementary classrooms, in which kids are asked to add fractions and at least one student proposes that 1/2 + 1/3 = 2/5? Would people object to the teacher, who is introducting the subject for the first time, just putting that answer on the board along with other proposed answers, and without comment, and then asking students to speak about the suggested sums and/or methods for arriving at them? Or is it incumbent on all teachers to: a) immediately point out mistakes; and b) immediately give the standard method?

I think the answer to both of those questions is a resounding “NO!” Kids benefit in many ways from hearing from peers and learning to think critically about the answers others come up with, along with the answers they come up with themselves. That’s how one develops as a mathematical thinker. So this exercise creates a (probably) artificial scenario and merely asks the student to look at the work and critique it. It’s not asking for a judgment of “Jack” but rather to find one (or perhaps more) error(s) in the proposed solution to a question. Period.

The adult who wrote the note to “Jack” couldn’t even manage to do that, likely because it was SO important for that person to make snide comments about a method of which s/he didn’t approve. What a glowing accomplishment on his/her part! Something to be truly proud of. Of course, I have to wonder if this person has some reading difficulties. But more likely, it’s a matter of being blinded by bile. And that’s commonplace in conversations about math teaching and learning in this country. Someone citing allegedly high-end math credentials weighs in snidely and almost assuredly misses the point.

If some child who already had learned the standard algorithm weighed in with the argument that it’s “wrong” because it’s not the standard algorithm, that would be very interesting. But most kids just learning multidigit subtraction would probably try to puzzle out what “Jack” was thinking. Some (many, I hope) would see the trouble and explain the error. Some might also comment about “efficiency,” but I think that’s an adult concern. Kids become aware (if allowed to do so by adults) that there are advantages and disadvantages to various strategies for lots of things (including sports & games), and make choices based on what suits them best, if not pressured to instantly conform. Seems like far too many adults are only concerned with conforming. As if any modestly bright child would be incapable of switching horses if s/he found a good reason to do so later on down the road. So many of us seem to have so little respect for or confidence in kids, even though we were kids at some point and at least some of us did learn arithmetic.

Give kids a little breathing room and they’ll make good choices. Stifle their opportunity to think and they won’t think. And therein lies the problem with so many American kids when it comes to mathematics. They are taught to believe that it’s beyond their capacity to handle via their own thinking and ingenuity. By 3rd or 4th grade, they’re passive, waiting to be spoon-fed the next magic method to be memorized but (likely) never digested.

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]]>Studies show us that advance problem solvers always have a number of different ways to solve a problem and tend to automatically gravitate to the easiest one. I think Jack’s problem here is he is using a sledgehammer to crack walnuts.

Is this a reasonable way to solve this problem? It certainly is one way, but it reduces math to counting on the tips of one’s fingers, so I wouldn’t count it as being subtle or efficient or imaginative or …

There is a classic story (probably apocryphal) about a student taking a physics test which asked the question: “state a procedure to determine the height of a very tall building using a mercury barometer.” The student responded with “hold the barometer up until its length appears to be as tall as the building and use “similar triangles” to calculate the height of the building (diagrams included). The professor, expecting an answer that used the barometer as a barometer gave him a score of zero on the question. The student complained that the question did not require the barometer to be so used. This lead to a rather large dispute ending up with the combatents in the Dean’s office. The Dean was not amused, and put the student at a table with paper and pencil and told him to answer the question (again). What followed was a list of ten different ways to determine the height of the building, correct ways, none of which involved using the barometer as a barometer. #10 was “Go to the Building Supervisor and say “I will give you this barometer if you will tell me how tall the building is.”

The problem with the question is it provides an inefficient solution to a problem with a mistake built in and asks the student to troubleshoot. Specifically what ability is being tested here? Should the test taker “fix” a basically poor choice of solutions or should he suggest that Jack’s mistake was to not select a more straightforward approach?

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]]>Now there’s an insightful, well-argued comment if ever I’ve read one. Guaranteed to enlighten us all: not about mathematics or the Common Core or this problem, but about people who think simplistic epithets are the same as facts and reasoning.

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]]>It’s Michael, Mary. And you quote one sentence out of dozens I wrote here, do so out of context, and suggest my comment speaks to all or most teachers, rather than to specific ones to whom those adjectives do in fact apply. If you need to see it otherwise, I can’t help you. If you want to read what was actually written, interpret it reasonably, and talk about what can be done to help those teachers who really don’t get it and particuarly those who don’t WANT to get it, then there’s a basis for discussion here.

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]]>what happened to 116?….five dots from 117 is 112

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