Friday, February 26, 2016
Math problems of the week: Russian Math, Math Kangaroo, AMC, and Math Olympiad vs. Expii math
Compare these with the problem from Expii.com that Peg Tyre cites as representative of all of the above in her recent Atlantic article:
(Options for this last problems: bacteria, a ladybug, a dog, Einstein, a giraffe, or a space shuttle).
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math,
math buffs,
Reform Math,
Russian Math
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24 comments:
What is it that you don't like about the rope around the earth problem? If you know either the circumference or radius of the earth, then it is a trivial problem, but since those measurements are not given, I assume that the point is to do the problem without them. It's actually kind of an interesting exercise--it not only is a good introduction to what it means to solve a problem for the general case, it also yields nonintuitive result.
The mathematical challenge is indeed interesting. The problem is the excessive verbiage and the notion that this problem is representative of these other ones.
I would class many of these problems at different levels, but solving them requires prior mastery (and understanding) of more basic skills. You don't master basic skills using these sorts of top-down problems, and these problems, while interesting and perhaps useful, are neither necessary or sufficient to develop in math - to get a STEM degree in college.
These problems may test your mathematical flexibility, but it's a hit or miss proposition. Figuring out sequential integer problems doesn't necessarily prepare you for solving anything else. Likewise for problems that are sums of the digits. You get good at those problems by having seen them already. Once you've seen the trick, it's often easier to apply it to other slight variations. How many students figure these things out the first time? Few. Most of them figure it out by studying the solution and then applying it to other problems. That's how you win competitions.
So what does getting these problems right require; the general ability to do mathematical problem solving or having seen this sort of problem before? In competitions, the goal is to directly learn each class of problem. There might be some sort of vague flexibility skill developed, but those getting the best scores in competitions are the ones who have seen each one of these before. Test makers go out of their way to find interesting or unique variations, but success might be a matter of luck or inspiration (after studying the archetypical form) and not any sort of general problem solving process.
Dick Feynman talked in his books about studying all sorts of trick questions when he was young so that he could answer anything. He got a lot of mileage out of it and later in life it really annoyed many of his colleagues. Feynman did admit to the effect and also referred to more than a little bit of luck. I had some colleagues who loved to create that sort of aura which only amounted to having seen the problems before. In other cases, they would go out of their way to obfuscate the math just to impress the less knowledgeable.
Then there are the Fermi-type problems that many like to use as some sort of indicator of IQ or math capability. How many golf courses are in the US? How many molecules are there in a mountain? Ho Hum. Skill practice and memorize some key facts and numbers - along with bottom up mastery of basic math skills and understandings. Do you really need to know every single trick variation of logarithmic manipulation so that you can apply it on a timed test or competition?
The main problems are those who think that these questions should drive all learning in math and those who think that they somehow define good math students. No. The problems all require practice, memorization, and basic skills. Many are neither necessary or sufficient for learning math.
Isn't the Kangaroo question #1 wrong? Shouldn't the wording be: "The difference between a two-digit positive integer and the sum of its digits is always divisible by...?"
Since the difference between single-digit integers and the sum of their digits is always 0, which isn't an option: 1 - 1, 2 - 2, etc.
In addition, isn't it a stupid question? Since all a kid is going to do is go: 1-1=0, huh? And then maybe go up to 10-1=9, 11-2=9 and draw a conclusion from there. It doesn't take anything other than the most basic addition and subtraction.
#1 is pretty easy, but most lead off questions are warm ups, no? It requires that the student know his vocab - what's an integer- and to know that zero is divisible by all of the answer possibilities. Some of the students will fiddle with it after the test and prove it.
The excess verbage I notice is sentence #3, but its there to help the student visualize, not to trip the student up. The point of the problem is not successful computation, but may be revealing a cool factoid that the student may not have thought about plus developing visual and thinking skills. That can lead to piquimg intellectual interest. What do you see as excess, and why?
The last rope question is just stupid. "What is the largest thing that could fit underneath the rope?" "Largest thing?" Why not the tallest thing? Are they referring to mass? This isn't math. This is an educator or psych major's idea of what math is about. It's anti-math. This is not the sort of problem mathematicians or engineers would have to solve. I wondered how much of the 710 inches was used up by tieing the ends of the rope. When I got to "largest" I knew that it was stupid. Is this supposed to be a no-look-up Fermi-type problem?
How about "What is the difference in radius in feet between circumferences of 24,901.55 and 24,901.5612 miles?"
Try out different forms of the question with simpler words to see what effect the words have on how many kids get it right or wrong. This is just a simple word problem with some units conversion. What are they trying to test? Students start to struggle with these in 7th and 8th grades using much more useful examples, like:
Given supplies of 10% and 25% acid solutions, how many liters of each need to be combined to get 10 liters of a 15% solution?
I could change this to make it more wordy and vague and then pretend that it tests math ability more. While it's very important to have students learn how to solve word problems, the words matter. I've had too many tests in my life where the teacher/professor could not get the words right. They were trying to elicit a particular understanding and failed, as shown by the random results (and student reactions) when the tests were handed back. I've done it myself, but I could tell when I corrected the tests and would always drop that question.
A more mathematical approach to the question would be to ask students to create an equation that relates a change in radius to a change in circumference.
C1 = 2*pi*R1
C2 = 2*pi*R2
C2-C1 = 2*pi*R2 - 2*pi*R1
C2-C1 = 2*pi*(R2-R1)
(R2-R1) = (C2-C1)/2*pi
This shows more mathematical understanding, it's more useful, and it provides much more insight to what's going on. It's a linear relationship. Do students show that understanding with the "largest" question above?
It all depends on the level of the student. If the student is not at this level yet, then what, exactly, is the point of this question? To waste a lot of time on something that is simple when it's worded better or when they have more basic skills and understandings? What are they discovering - only that the result is not what they expected? What engagement is happening? Did it work? In a student-driven group learning process, how many students fail to make any discovery and end up being directly taught (poorly) by other students? Does this engagement work for them or are educators simply blinded by all of the "active learning?" Do educators use their critical thinking process to separate their beliefs (and rote training) from reality? How is this possible if they don't have content knowledge and skills?
SteveH--Did you solve the general case of the rope around the earth problem? I agree that parts of it are ambiguously worded--and that the "largest thing" verbiage probably should have been changed to "how much space would there be between the rope and the surface?--but the essence of the problem certainly isn't "anti-math."
X = change in radius
2piR + 60 = 2pi(R + X)
2piR + 60 = 2piR + 2piX
60 = 2piX
X = 60/2pi
X = 30/pi
X = 9.5 feet
So if you increase ANY circumference by 60 feet, the radius increases by 9.5 feet. So, I'm assuming that the "largest thing" would be Einstein, as giraffes are like 15+ feet tall.
Frankly, I'd love a source of problems like this one.
I was not talking about "the essence" of the problem when I said anti-math. The essence of this problem is trivial, and the extra words don't do anything more than add vagueness and misdirection. It's also neither necessary or sufficient and I don't know what a source of these problems would attempt to achieve in terms of mathematical understanding. All that I see are random problems rather than a set that covers a sequence or domain of knowledge and general problem solving skills. I've seen Diophantine-type problems that test makers expect students to solve without ever studying them beforehand. Some create problems where normal mathematical solving techniques are too tedious so you have to find the trick solution. (X-4)(X-5)(X-6)(X-7)=1680 is one of those problems. In my whole career of engineering and programming, I've never encountered one of those problems, and learning to solve this one provides no guidance to solving any other. The real world is made up of general problems, not special cases. It's not that working on these problems is a complete waste of time, but the time could be so much better used.
Explain how the essence of the problem is trivial for *beginning algebra students*. I agree that the problem is trivial if you know either the circumference or radius of the earth. But for *beginning algebra students*, solving this problem without either of those is absolutely *not* trivial. This problem is the perfect stepping stone to help students understand the idea of the general case.
*beginning algebra students*.
Who said it was for "beginning algebra students?"
Unit 10-7 on "Circles: Circumference and Area" is on page 551 of my son's old Glencoe Pre-Algebra book. Problem 40, which is referred to as "Standardized Test Practice" (which is not at a high level) is the following:
"The Blackwells have a circular pool with a radius of 10 feet. They plan on installing a 3-foot wide walkway around the pool. What will be the area of the walkway."
If kids are taught properly, then the first thing they do when they see a word problem is to identify the governing equation. For this problem, it's
A = pi*R^2
Any proper math textbook expects students to be able to solve for any variable, not just for the one on the left side of the "=" sign. The equator problem is really a standard pre-algebra problem. It's just being dressed up and confused with words that do not increase mathematical understanding.
"I agree that the problem is trivial if you know either the circumference or radius of the earth. But for *beginning algebra students*, solving this problem without either of those is absolutely *not* trivial."
It's trivial for any modern student to Google "circumference of the earth." If they haven't had a unit on circumference and don't know C=2*pi*R, then the problem is useless. What is your "general case"; to find information using Google?
If they have had this unit, then the equator problem is just one of many that could be encountered in that unit's problem set. It's not a stepping stone or discovery sort of problem. It's a simple circumference unit homework problem - one of many.
The difficulty I have is that this is some sort of special problem. I disagree with the idea that students learn best when they are given problems without being properly introduced to the material or slowed down with words or having to look up extraneous information. This is the old discovery claim, which is neither necessary or sufficient. Math competition problems also love to find trick or special case solutions where the general methods won't work because you don't have enough time. This is not the same as the math understanding that comes from putting together general unit skills and understandings into the solution of larger, more complex problems. However, these unifying skills can be directly taught and not obtained by wasting a lot of guess-and-check discovery time on a random series of competition math problems. The best competition students are those who directly study all of the past questions. They save time when their advisors break the tricks and techniques into separate topics and directly teach them. Discovery and struggle are neither necessary or sufficient. Students have plenty of that with each homework set and don't need to have educational pedagogues create more struggle as if that is a path to understanding.
Perhaps I'm not being clear enough. You have to analyze each of the problems above differently. They do not all fall into one class or type of problem. Let me start with the first one:
Prove that 5^101 + 5^99 is a multiple of 13
Do you learn or understand this more if you have to struggle without a clue? I don't remember if I figured out the trick or if I saw it somewhere long ago. It's actually based on an idea that I think is very important - backwards identities.
When students learn about exponents, they usually learn something like:
A^m * A^n = A^(m+n)
This is what I call the typical forward definition. However, it works just as well backwards:
A^(m+n) = A^m * A^n
This also relates to what's driven into students' brains: "simplify!" When they see an expression or equation, they think there is only one direction to go: simplify, such as no factors with negative exponents. They think that's illegal.
I made a point to teach my algebra students all of these backwards identities and directly teach them that "simplify" is not a mathematical requirement, just a common thing to do - and that many problems can be solved by doing what's mathematically legal, but in the opposite direction of simplify.
Another thing related to this is "order of operation." This matters in specific situations, but an expression or equation is really a two-dimensional object and not a string of characters on your TI-89 with a specific order of operations. There are things you can legally do and things you cannot. How you manipulate the equation is not defined by some strict order of operations or "simplify." What I call forward identities tends to limit students.
However, is the solution to this inflexible understanding a matter of discovery using problems like the one above? If you are told the solution, do you understand it less? What about my more general idea of backwards identities? How many of these problems do you have to do to really discover or understand the generalities of what I'm getting at? Competition problems are loaded with these sorts of backward or reverse-simplify algebraic techniques. Is discovery the only way to achieve this understanding? No, because even strict discovery classrooms do not have enough time to do this for all units. They just go through the motions for a few problems as if discovery is a transferrable skill. It isn't. There may be some sort of vague general benefit, but it does not compare with any direct approach, and my direct learning of these tricks have made me more able to discover new things on my own.
Discovery skills do not only come from practicing discovery. I have had grand discoveries when I was directly taught. I can still see myself sitting in my high school calculus class when the teacher directly taught us how integrating between limits gives you the area under the curve. I can still remember how I felt. And we students were not just passive learners. Our direct teacher got us all involved. We were all up at the blackboard at times. That pressure forced students to come in better prepared. Compare that to the modern approach of wasted time and poor direction with student-led guess and check group discovery. It's as if teachers think that students can do a better job of teaching themselves.
Once you get beyond classroom to competition-type problems, there are many different or new classes of problems to learn to solve, such as sequential integers and sums of digits. Does success on these problems indicate some sort of discovery ability or just plain hard work? Are these problems indicative of potential in math versus whatever other non-competition math students might opt to study? is this for engagement and excitement purposes? Does that really translate into the ability to individually complete homework sets to master the low level skills and understandings?
I have different issues with each problem above. There are vague wordiness distraction issues and special case solution issues. Some of the problems I like, but not if they are used in some sort of discovery process. I'm not a fan of the AMC test and math competitions in general, but they offer much better use of time than many other silly discovery math problems I've seen. Also, people have to understand that success on the AMC series of tests requires a lot of hard work and preparation, not some magic discovery skill. However, the AMC does not define (as Peg Tyre and others think) the only path to success in mathematics or engineering.
Part of the point of the rope problem is that you *don't* need to know anything about the circumference or diameter of the Earth.
(Ce = Circumference of Earth, De = Diameter of Earth, Dr = Unknown additional diameter added by the rope)
Ce = De * pi
Ce + 60' = (De + Dr) * pi = (De * pi) + (Dr * pi)
60' = Dr * pi
Dr = 60'/3.14 ~ 20'
-----
I used something similar to the walkway problem a while back when trying to calculate how far it is to walk around our block on the sidewalk. I couldn't measure the sidewalk directly, but I could drive the car several times around the block to make an estimation. Thinking through it, I realized that a walker would experience the block as a rectangle, and the car would almost do the same rectangle except for the rounded corners.
Walker = 2(W + L), Car = 2(W + L) + 2R*pi, where R is the distance from the center of the sidewalk to the center of the car, and with the 4 corners adding together into a complete circle. Running the calculations, it came to 0.44 miles to walk around the block.
Apparently you're not understanding what I'm saying, or you haven't read my comments fully.
I already said way upthread that if you google the circumference or radius of the earth then this problem is totally trivial. What makes it thought provoking for a beginning algebra student is figuring out how to solve it without those quantities. There is no need for guess and check, and there is no need for discovery learning. And I'm not talking about having kids solve these sorts of problems instead of receiving direct instruction in math or that these sorts of problems should be attempted by every student. Where did you get that idea?
All I said was that the essence of this problem, when it is solved without using actual quantities for either radius or circumference, is an interesting exercise for students at a certain level in their training. And I'll say it again: I would love to find a source of problems like these. The problem is special because it gives a non-intuitive result and can be solved for the general case fairly easily, so it is accessible and possibly even somewhat interesting to beginning algebra students. Believe me, problems like that are difficult to find.
Just to clarify, my comment at 10:59 was in response to SteveH's 8:24 comment.
SteveH, the rope problem is a special problem these days because all but the most basic word problems have been removed from the curriculum- full inclusion is basic only. The students that could solve it would be using their time talking about the choice of knot and what that meant to the amount used to encircle the Earth, as you noted.
"Part of the point of the rope problem is that you *don't* need to know anything about the circumference or diameter of the Earth."
So how much time do you spend on this concept? If students come up with the answer, does that skill apply to any other of the problems above? If they do not come up with this but have to be told, is the result less effective? The problem does not tell students that they cannot look up the circumference of the earth.
The context of these problems matter. I find that the special case nature of many of them are not worth the time invested. That time could be better invested in more general mathematical concepts and ideas as I mentioned.
"...the rope problem is a special problem these days because all but the most basic word problems have been removed from the curriculum- full inclusion is basic only."
K-6 math is a wasteland. It's over for most kids by the math track split in 7th grade assuming that the schools have not given in to curricula like CMP in 7th and 8th. My son's Pre-Algebra textbook was all that he needed and it was filled with all sorts of interesting homework problems for each unit. My position is that for K-6, there are fundamental structural problems that after-school math programs cannot solve. After that, proper textbooks give all of the mathematical learning necessary. At that age, any after-school Math League sort of program can focus on these types of problems as extras, but I've stated my position on their usefulness above.
SteveH, could you walk me through the 5^101 + 5^99 problem? I don't get it. Thanks --
@FedUpMom
I'm not SteveH but I think I got it:
5^101 + 5^99 =
(5^100)(5^1) + (5^100)(5^-1) =
(5^100) (5 + 1/5) =
(5^100) (26/5) =
(5^99) * 26 =
(5^99)/2 * 13
@Anonymous 1:56 PM, there's a less complicated solution:
5^101 + 5^99 =
5^99 * (5^2 + 1) =
5^99 * 26 =
5^99 * 2 * 13
Thanks, Anonymous & GoogleMaster!
"So how much time do you spend on this concept?"
Which concept are you referring to? I homeschool, and when I presented the problem to my son, we spent maybe 15 minutes on it. But there is more than one concept that this problem illustrates, and I've spent substantial time on all of them.
"If students come up with the answer, does that skill apply to any other of the problems above? If they do not come up with this but have to be told, is the result less effective? The problem does not tell students that they cannot look up the circumference of the earth."
As I've said several times, if students look up the radius or circumference, then the problem is trivial. When I did this problem with my son, his first inclination was to look these quantities up. I let him, and he did the calculation easily. The result was interesting to him because his intuition told him it would be much smaller. But then I told him that the problem could be done without knowing the radius or circumference and had him do it. That took some scaffolding from me. I could have outright told him how to do it, but my experience in teaching him, his brother, and my tutoring students has been that having some investment in solving a problem results in better attention and retention.
"The context of these problems matter. I find that the special case nature of many of them are not worth the time invested."
What a student learns in solving this problem without knowing the radius or circumference can be applied to many types of word problems. When solved using only the numbers given in the problem eliminates the special case aspect that you seem to be objecting to.
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