If the difference of 2 numbers is less than the sum of the 2 numbers, which of the following must be true?
(A) Exactly one of the numbers is positive
(B) At least one of the numbers is positive
(C) Both numbers are positive
(D) At least one of the numbers is negative
The answer given in the original source was (B). Do you agree? See notes below for further discussion of the wording of the question (before you react!).
This SAT-type question was posted about a year ago in my discussion group, MathShare (which is still extant but possibly being phased out). On that forum, I discussed how students struggled with the subtleties of logic in their analyses of the problem. That online discussion led to a meaningful debate (involving some exceptionally thoughtful educators) about how and when logical thinking needs to be developed in our students. All agreed that critical thinking and logical reasoning must begin when children enter school, long before the formalism of an axiomatic approach. Do you believe this is currently happening in most elementary schools? What materials are being used by those districts or teachers who are infusing critical thinking and logic? If we move toward a more standardized curriculum nationally, how important is this? I'm sure you know how I feel!
Other Notes about the problem above:
(1) Is the question ambiguous or flawed because the term difference fails to specify in what order the numbers are subtracted? Should the domain of numbers be specified (would integers be better?). On the SAT, it is understood that the domain is always real numbers.
(2) Why do you think so many students (these were strong SAT prep students) struggled with this and had great difficulty accepting that (B) was the correct choice? Do you think phrases like at least one and exactly one are problematic for many students?
(3) What methods do you think were used by students? There were several approaches as I recall.
(4) Do you think students should be encouraged to use an algebraic approach here rather than plugging in numbers and testing various cases?
(5) Would restating the question in its contrapositive form make it easier to grasp? (how many students remember this from geometry?!?)
(6) Would this question lead to a richer discussion if it were open-ended, i.e., no choices given?
(7) Could this question be given to middle-schoolers after they have learned the rules of integers or do you believe they do not have sufficient maturity to handle the logic?
Your thoughts....
Friday, July 6, 2007
If the difference of 2 numbers is less than the sum of the 2 numbers... Developing logical reasoning in our students
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10 comments:
Dave,
I posed this question to two of my sons, who are both gifted in math and participate in math competitions.
The 13yo who has just finished honors Algebra I got the correct answer immediately, and explained his reasoning to me.
The 11yo has finished most of pre-algebra did get hung up on the logic. He knew it was guaranteed when both were positive, impossible when both were negative, but only sometimes true when one was positive and one was negative. But since it wasn't guaranteed when one was negative (depending on order) he chose (C). (He was also thinking "positive difference" for a while.)
Even though he is just entering middle school, I'd guess that his reaction is typical of most middle schoolers. So, I would say that you could pose the question to middle schoolers, but you'd have to help them work through the logic carefully. I think this kind of reasoning is developed mainly by practice, though you might find something at Critical Thinking Co. that would provide exercises that would help with this.
I think the "must be true" part is more confusing in this case than the "at least one" or "exactly one" parts. Because the implication only works one way, it is confusing. If at least one of the numbers is positive, it's still possible for the difference to be greater than the sum, depending which order you subtract. That's what threw my 11yo off, and I think that's what would throw most students off.
As to the question about whether students should be encouraged to think about this problem algebraically, I think not. I think it's most sensible to think of problems like this in terms of cases, and perfectly reasonable to try out the cases with sample numbers. (That's certainly how I thought about it.)
mathmom--
Again you've articulated my thoughts exactly about challenging our middle schoolers (not just the gifted) far more than currently happens in most districts. Of course, there are wonderful exceptions and we should be celebrating those teachers and their administrators. I've had the opportunity to work with K-5 teachers in the classroom and it was so rewarding to see 'everything illuminated.' These children were wide-eyed (4th graders were the best!) and, as you know, soaked up my problems and challenges like a sponge. Some children were of course more self-confident, meaning they were not afraid to take risks, aka, 'not afraid to be wrong' (perhaps the most important quality in learning).
I suggested an algebraic approach, not initially as an option for the middle schooler, but, later on. There is something beautiful and essential in the algebraic solution that, IMO, needs to be demonstrated:
If a - b < a + b, then
-b < b. This leads to 0 < 2b and finally, b > 0.
The a's drop out, which implies mathematically that a could be positive or negative, but b MUST necessarily be positive! Thus, at least one of the numbers MUST be positive.
I completely agree that the logical necessity of 'MUST' is a developmental issue and quite challenging even for older students. Children need many experiences to develop the idea of logical implication so that they understand that when demonstrating 'A implies B' they are not falling into the 'Converse Error' trap!
Thank you for you thoughtful replies and testing my theories in real time with your sons. More importantly, thank them. I would like to hear their reactions to some of these problems (are they enjoying them?).
I'm beginning to realize that I'm not going to change people's views by writing this blog. I'm only inviting those who already share my beliefs!
Dave,
I'm not sure that your blog won't change the views of a few people. However, I think only people who are open to other views will be reading your blog (and others like it). Which isn't a bad thing.
I'm assuming you want to change the world and show everyone the wonder that is true mathematics! Sounds like the reason I went into teaching (well, one of them). Do I expect to change the world? Uhm, some moments. Do I keep trying anyway? Of course.
I can personally say that reading your thoughts & problems has helped me clarify my thinking about teaching mathematics. Will my thinking change? I hope so, as I don't want to remain rigidly entrenched to any one ideology.
Enough pontificating. I too worked the problem in a "by cases" method, then went to an algebraic method. In your experience, do all students see/understand the algebraic method - even after you've demonstrated it?
Dave,
could they have goofed with this question? Or was it at the very end of a section? I am not as familiar with SATs as I should be, but this seems on the hard end, no?
Because of the "at least" qualification, I'd think that algebra rather than checking values would be the easier way to go, as long as the student considered both cases. But even with plug in, there is a danger of not considering both cases.
jackie--
To wax poetic, teachers do change the world one child at a time (if you open the door for one child you are opening doors for this child's future generations in the exponential sense!)
The algebraic solution is subtle. Apart from the issue of following the rules of inequalities, the idea that a variable 'dropping out' of an algebraic sentence implies that this variable can assume an arbitrary value in its domain is fairly sophisticated. I usually precede this with a discussion of identical vs. conditional equations. E.g., 3x = 2x + x is an identity, which could lead to 0 = 0 if you wish. Current texts do not devote as much time to this distinction as the 60's New Math did so it's up to us to provide more details. The group that I worked with were strong in algebra so most picked up on the significance of the 'a' disappearing from the inequality. In a technical sense, variables don't disappear, they're still there with zero coefficients! Thus,
x = x appears to lead to 0 = 0, but one could also argue for 0x = 0 since 1x - 1x = 0x! 0x = 0 more strongly suggests that x can be any values in its domain! This is subtle but beautiful IMO!
jonathan--
this was not an actual SAT problem since I'm careful not to post these for copyright reasons. In fact, it was based on a question from an SAT-prep book which usually doesn't have the same quality control as the College Board. However, questions very similar to this have appeared on the SAT and, yes, they are at the end of the test and are considered difficult. I agree that, for myself, it was easier to 'see' the logic algebraically, however most students would test cases and a rich discussion can ensue from this.
Thank you for you thoughtful replies and testing my theories in real time with your sons. More importantly, thank them. I would like to hear their reactions to some of these problems (are they enjoying them?).
It's summer, so I bribed them with candy to try it. ;-)
Nature, nurture or both, I think all my boys (I also have a 6yo) were destined to enjoy this kind of thing, and they certainly do.
I volunteer to teach problem solving at their school, and each group (K-2, 3-5, 6-8, approximately) is excited when it's their turn to work with me. They certainly all enjoy it more than the more "routine" math.
I have a handful of kids who are enthusiastic about the Math Olympiads and MathCounts. I have another chunk in elementary who do the Olympiads to get out of French that day. ;-) Many of them get enthusiastic as they get older and more practiced. We have all the middle school group do the Olympiads during a math period, and they all look forward to it. BUT... when I tried to get 4 kids (out of my middle-school group of 7) to go to MathCounts, I had two enthusiasts (one of them my son, the other a girl (yay!)) and it was like pulling teeth to get two more girls (as it turned out) to give up a Saturday to do math, no matter how cool the math, no matter how much free pizza and other bribes were offered. (In the end, they had a great time and those who are still eligible will go again next year!)
I don't think you and Jackie and I can change the world in a big way. I tried having a conversation related to this with parents on a gifted kids parenting list, and I might have changed one person's mind about what the ideal math program is for kids gifted in math. But most people have no interest in changing their way of thinking. Most people whose kids are gifted in math want their kids to be able to fly through the core curriculum as quickly as possibly, and not have to slow down for "enrichment" like problem solving using math skills they already know. They want their kid to learn "new" things every day, and if not, feel that they are being held back. I posted a link to The Calculus Trap (someone else recently linked to that here as well, I think) but that did not really convince anyone either. Some people are unwilling to entertain the notion that problem solving experience/expertise is worth spending math time on!
I do think I change the world in a small way, and that's important too. And I know that you do this too. I have definitely influenced the teaching principal (who teaches the middle schoolers math when I'm not with them) and she is now a believer in this way of doing things. And I change the world for individual students. And that's what I'm there for. Nothing wrong with changing the world one child at a time. Nothing at all.
(And yes, the algebra is slick. I think I shied away from it since I mostly work with middle schoolers, for whom that algebra would be a stretch. For SAT prep students more confident in the algebra, that's certainly a clean way of looking at it.)
Dave,
I love the exponential analogy! (can I borrow it?)
I wasn't implying not to show/discuss the algebraic method, just wondering about your experience with student understanding of such. Aside from math team, my student experience has been in the "lower" tracks, so sometimes my views get skewed (despite my efforts not to lower my expectations). Hence my asking about your experience.
Mathmom,
Personally, I don't think enrichment problems/problem solving should be limited to "gifted" students. I think if more students were exposed to these problems, we might have more students actually understanding the mathematics. Perhaps this wasn't the intent of your message, so sorry if I extrapolated!
Jackie, I also don't think problem solving should be limited to gifted students. I work with everyone at my kids' school (although those who are struggling badly with the basics sometimes get extra tutoring on the basics instead of working with me). My point was that on a list of parents of gifted kids, I could not even get those parents to see the benefits of their gifted kids working on problem solving rather than charging forward through the next bit of curriculum. Gifted kids tend to have more time available for such enrichment than others, since they can cover the base curriculum so quickly, so it's easier to add problem solving work without "shortchanging" the rest of the curriculum. But even then, most parents want their kids moving forward, and they don't think of working challenging problems with the math they already know as moving forward. sigh.
very heavy sigh (of empathy).
mathmom, jackie--
Perhaps they will listen to someone who has had experience with how MIT, Stanford, and the other top Math/Science/Technology schools view college admissions. These schools are not just interested in whether the student has completed multivariable calculus by 11th grade and has exhausted every AP course offered. MIT has traditionally asked for students' AMC scores because they expect all of the applicants to have SAT-I and SAT-II scores around 800. The American Math Competitions and AIME scores discriminate far better. Calculus is not needed on these contests. Students have to have advanced problem-solving skills and knowledge of particular strategies and facts that are not covered in any traditional curriculum. That's why the AoPS website is doing so well. If you'd like me to ever address the parents of these kids I would be happy to. Of course, they'd want to know how many titles I have and why they should take me seriously. I'd be happy to respond!
My blog would not impress them but I do have experience with admissions into the most competitive schools so they might listen for about 30 seconds...
OR have them email me!
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