Three children have a foot race in which the leader after 1 minute is declared the winner. Each child runs at a constant speed. At the end of the minute, Alicia has run 720 feet. Ben, who started 40 feet in front of Alicia, has run 600 feet from his starting point. Cheryl, who started 40 feet in front of Ben, has run 400 feet from her starting point. For how many seconds during the race was Ben in the lead?

A) 8

B) 10

C) 12

D) 15

E) 20

How would you tackle this using rote knowledge or memorization? What conceptual understanding is enough to solve this problem? Draw a picture? Work backwards? If you could solve this problem in the allocated 3 minutes rather than 20, what would that mean? What do you do to get to the point where you can solve virtually any D=RT problem quickly? Is that done with some sort of better conceptual understanding? Is speed meaningless? At what point, if ever, does it become meaningless? What happens when you practice? Is that just about speed? Is experience just about speed? When you see a new problem for the first time, will you be as slow as the first time you solved a DRT problem?

I think it would be hard to argue that experience and practice just mean speed; that they are not transferable to new problem types. We are told, however, that some sort of vague conceptual understanding is all that you need, even if it takes a long time to solve a problem. So what happens with practice and experience? What is that called? Drill and kill? Either practice matters or it doesn't.

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## 21 comments:

There is a big difference between practicing problem solving and doing 500 almost identical distance-rate-time problems.

The AMC 10 questions generally rely on thinking about problems that are somewhat different from any the kids have seen before. *Some* kids try to drill old AMC tests, in the hope that very similar questions will be re-used, and they can do ok at that. But the top students usually have worked on many different problems mainly to develop problem-solving skills, rather than to be able to recognize a problem they have solved before.

The particular problem you gave looks like it comes from about position 5 out of 25 for an AMC 10 test (the problems are ordered in roughly increasing order of difficulty). The top students expect to solve the early problems in much less than 3 minutes, so that they have more than 3 minutes for the later, harder problems.

I have found that in Middle School, doing Mathcounts helps kids do problems like this. That means on those who can/want to spend the extra time benefit. Being able to solve problems like this comes from being able to see what kind of problem this is, and then being able to set up the approrpriate equations, etc. to solve it.

Having the conceptual understanding but not having the skill needed to set up and/or solve the equations it nearly useless. Likewise, being able to solve the equations but not being able to "see" them and set them up is equally useless.

The arguments that conceptual learning trumps all, or the skill trumps all, both miss the point. Both are needed.

So I believe - JE

I've observed that our local public schools refer to the completion of any word problems in Algebra as "extra credit" yet they go on and on about how Everyday Math in K-8 prepares students by teaching "concepts" over algorithms. I never see conceptual understanding demonstrated on worksheets full of equations.

That is why I'm still homeschooling.

"But the top students usually have worked on many different problems mainly to develop problem-solving skills, rather than to be able to recognize a problem they have solved before."

What is the difference between working on problems to develop problem solving skills and working on problems and only developing pattern matching skills?

In the question above, some students might figure out that it would be better to define position equations (relative to some fixed point) rather than use simple DRT equations. This is what they will remember to look for in other problems. How would you solve this problem and only derive pattern matching ability?

What if you solve the problem without thinking about position? What if you get the correct answer using some vague sort of problem solving technique? Will you be faster on the next problem that you run into? What have you "discovered" that can be transferred to another problem?

It seems to me that specific knowledge, skills, and concepts are very important for efficient problem solving. Should these be learned (discovered) by starting with just a Polya-like framework for problem solving? Volume I of "the Art of Problem Solving" is filled with understanding and mastery of basic skills as a prerequisite for problem solving.

This came up for me because I'm trying to figure out a good way to teach my son about combinations and permutations. The basic equations are easy to understand and apply. One might say that he has good conceptual knowledge. Or maybe not. This is an area where you can do one problem well and be completely lost on the next. Is the solution just to do a lot of problems with some sort of problem solving mindset? It doesn't work. The understanding you need is much more than conceptual. Most of the explanations I've seen work well for simple problems, but leave the student stranded later on.

It also raises an issue that I've confronted many times over the years. I will be studying new material and get the feeling that it's got to be easier than that. I will then look for another book or paper and find that a new presentation makes all of the difference. I've never had the feeling that "Oops, they told me too much and now I won't be able to understand it because I didn't struggle to discover it."

This is not a simple argument about conceptual understanding versus skill or practice. It has to do defining exactly what concepts, understanding, mastery, and problem solving skills mean. It's also about how much you teach and how much you let students struggle. If you had a great direct teaching technique where kids would learn the material with little effort, is that wrong or not possible by definition?

When teaching DRT problems, do you not teach kids about variations where positions are important? This is exactly what textbooks do with problem sets, so you could say that kids are forced to discover things at home by themselves. Would it be bad for the teacher to introduce these variations before they had to struggle at home? There might be a certain amount of satisfaction and confidence that is achieved with discovery, but it isn't necessary, sufficient, or even possible to for everything.

Conceptual understanding is not some vague conversational understanding. It's not recognition "oh, yeah, that's a XXX type problem" only, either.

Conceptual understanding means you understand the concepts involved so well that you automatically know why you're doing what you're doing. You know why the procedure works.

A good way to tell if your son has conceptual understanding is to ask him "why". Why are you solving the problem the way you are? Why does that equation work? Ask to a depth of 3 and see if the answers are still right.

Take permutations. The procedural fluency is easy--ask someone how many permutations there are of n distinct objects, and the answer should be immediate: n!

Ask why, and they should be able to explain: a permutation is a linear ordering of the n objects, each with its own position in that order. To count the total number of permutations, recognize that you can generate all permutations by thinking about all the places the first one can go, then all the place the second can go, etc. To understand conceptually, you need to understand that this covers the whole set of available permutations--this is the part you need to "see".

To do the counting, start by considering all the way the first object can go in any of the positions: it has n slots available, labelled 1 to n. Then, it picks one, and given that one, the second object can't go wherever the first one did, but can go anywhere else--that leaves n-1 locations, and so on down the line. OR you could think about it this way:

each of the n objects was labelled with a number on it, from 1 to n. Picking at random (say from a sack), you put the first one you pick in slot 1. How many ways were there of picking that first labelled object? n ways. on slot 2, you've got n-1 objects left to choose between.

You'd like your son to see how these two formulations are the same, and what's changed in each case, so he doesn't get confused between them. That's real conceptual understanding.

A good teacher knows that you need to move a student just a little bit past where they are comfortable, in small steps, for them to actually succeed at grasping the new material. Too big a leap between here and there and they'll struggle all right, but they won't learn anything from struggling, even if in the end they are "scaffolded" up to the right answer.Too little a leap and they aren't learning.

Of course, different students are different, so some kids will be able to handle a bigger leap right now than others can.

So in class, you present the basic constant rate problem. Then you do a few more where things are almost exactly the same. Then you do a few with some more variation from the original. If you can, you do one that's even farther away now, and you provide homework that covers all of the above, too.

Properly done in a classroom, you're not just "presenting" the material--you're working with them, asking for feedback, asking how this problem is similar to the one they've already done, asking how it's different. You're getting them thinking about the functional mapping between the inputs and the outputs, and the functional mapping between a problem they know how to solve and the new one they don't yet know.

Trying to present it from a variety of angles can help, too--can this problem be done by equating the distances? What about equating the times? Can you always equate both in a constant rate problem? Why or why not? What are you comparing, etc.?

Experience is more than speed. Experience is the ability to recognize how this problem is like problems you already know. Your brain needs to actually make the synapses, the connections, between the neurons. You can't do that without actually doing the problems. Mastery can't come without concrete experience.

Mastery takes practice, but once mastery is achieved further practice of almost identical material induces boredom. Gauging the right number and difficulty of problems to challenge students is difficult, as each student needs a different level and number of problems.

My son dropped his "trig" class this semester, because they were doing 40-50 problems a night of algebra 1 review, with no trig for at least a month. This was not what he needed. The only conceivable goal was that they were trying to get him to do problems in his sleep by first putting him to sleep. Other students in the class may well have needed the intensive drill---I don't know them, so I can't tell.

We'll be putting my son in the Art of Problem Solving Precalculus class, which uses fewer, but harder homework problems, so that he'll get to think about math, rather than doing mindless drill. It is a better match to his learning style

These are the kinds of problems found in the Singapore Math "Intensive Practice" books. I do think practicing word problems helps students learn to solve new ones faster.

It's a far cry, however, from the kind of "drill and kill" of doing page after page of just basic equations that I remember being assigned as homework growing up (and that I see in something like Saxon Math). Those kinds of problems did not require me to do much in the way of thinking- it was nothing but rote calculation.

"...they were doing 40-50 problems a night of algebra 1 review..."

That's a different issue.

"...kind of 'drill and kill' of doing page after page of just basic equations ..."

But what my son had to deal with in Everyday Math was the opposite; some sort of conceptual knowledge in place of practice. How do you know whether you have enough conceptual knowledge? Prove it by solving the problems.

All of the real(?) math textbooks my son had NEVER repeated the same kind of problem over and over. As you go down the list, you see variations that test your understanding. Often the teacher would give just the odd problems. When I gave out problem sets to my son, I would go through the list and pick out specific ones.

The problem is that my "drill and kill" is quite different than that of most educators. What do they tell kids who get wrong answers; that they need better conceptual understanding? Is the "kill" level determined by success or is it determined by something else? It seems to be replaced by the spiral and the assumption that success will happen. If it doesn't, then it must be the student's fault.

In my son's algebra II class, homework sets are graded on whether you have tried. Unless you completely blow it off, you get a 100. However, the homework adds up to only 5% of the grade.

"Drill and Kill" is used to justify all sorts of non-teaching because on a superficial level, it sounds great.

"Why are you solving the problem the way you are?"

The issue is that for some new problems, he doesn't know where to begin. Understanding with combinations and permutations has many levels. I've gone over the development of combinations and permutations with and without repetition, but many advanced problems don't seem to fit. Is it a conceptual problem or is it a different way to look at a problem? It's a little of both, it seems. Some of the problems can be broken down into parts, one of which could be a combination and one of which could be a permutation. How about taking groups of three letters from a word containing 10 letters, but keeping them in alphabetical order? I think there is a classic "MISSISSIPPI" problem about this. Deriving P(n,r) won't help if you can't figure out 'n'.

Many of the explanations I've seen do a good job on the simple problems, but offer little or no help when you get to advanced ones. It seems that once you get past the basics, you need to start over and tell students what they really need to know in a more general sense. My next job is to look at the AoPS approach starting with "Learning to Count" and comparing it with my Schaums outline of Probability and Statistics. This is one of those areas where I think there has to be a good explanation somewhere. It won't come from my mixed-up background in probability.

"... you're working with them, asking for feedback, asking how this problem is similar to the one they've already done, asking how it's different."

Unfortunately, the teacher is not allowed to do this in the modern classroom.

Singapore Primary Math I believe has the best balance of practice with basic calculation and higher order problem-solving skills. Unlike EDM and TERC Investigations, Singapore teaches the traditional algorithms. But unlike the kind of math I had growing up, it also teaches *WHY* those algorithms work. It's neither "fuzzy" nor "drill and kill" but superior to both.

I think one of the ideas in how to prepare students for this type of problem is the expert/novice research that been done. E.D. Hirsch talked about this a lot in the Schools We Need...

Experts have seen many different problem situations, and are familiar with the conceptual pieces that make up a particular type of problem.

A new problem then can be recognized as some type of permutation of pieces of the previous problems they've solved.

This problem certainly has aspects to it that we cover in Elementary Algebra rate, time and distance problems, but it then extends those ideas considerably - but not in an unreasonable way.

I would give this to my classes as a project so that they could take some time with it.

Steve: Piagetian research has found that students who are younger than a certain age (roughly 15-16 years old) don't have sufficient brain development to do complicated permutation/combination problems. They've found that if the answer is less than ~30, then the younger kids are OK, but if the answer is more than ~30, the younger kids just don't get it. Once they reach the necessary level of brain development, then they can figure it out. Until then, it just doesn't work.

They have also found that schools and curriculum committees don't read much educational psychology literature about brain development. :)

Pshaw. Nearly all of what Piaget said has been debunked. Zig Englemann has troves of such debunking examples.

Specifically for combinatorics, I think it helps to get a kind of direct instruction template in place, and you repeat the template as possible, but yes, it will be hard to see how various puzzles connect up.

So the first questions are things like: which items *in this question* are considered "distinguishable" and which are "indistinguishable", and what's the state space for the question.

the state space question is really a big question: what do all of the answers in the universe of this question look like? How can we be sure we've counted them all individually?

Distinguishable vs indistinguishable is a sophisticated concept. In the physical world, only bosons and fermions are indistinguishable--everything else is actually distinguishable, so our intuition can get confused by problem where we're supposed to treat things as indistinguishable.

Several copies of the letter S are meant to be understood to be indistinguishable. This isn't obviously true, it's a convention of the problem that's often left unsaid. Similarly, when you're solving the order of flags of the same color on a pole, the flags of the same color are considered indistiguishable.

Labelled objects are always distinguished by their labels.

Just practicing the pieces of these problems--which objects are labelled, which aren't; which things are being overcounted, which things have we not yet counted, can build up mastery.

Complicated (?) permutation/combination problems aren't possible before 15-16? I don't think so. My 14 y/o son is quite capable of understanding these problems. The difficulty is not understanding, but teaching.

"... 'distinguishable' and which are 'indistinguishable', and what's the state space for the question."

Yes. I was reading my copy of AoPS and they take a more basic approach than starting with permutations and combinations. Work on the state space seems to be the only way to develop a feel for different problems. There is a lot of skill just understanding the problems and there are a huge number of variations. How about circular ordering and whether the direction matters or not, such as the ordering of keys on a key chain.

My point in bringing these things up is to go beyond simple arguments of conceptual understanding versus rote skill. There are different levels of understanding, some tied to basic concepts (like distinguishable and indistinguishable), and some based on mastery of basic skills. Some problem solving skills depend on having seen the problem before (common knowledge), and some skills can be applied in a more generic fashion. It seems that each type of problem (like work or DRT problems) requires a different emphasis.

My son was doing moderately difficult combinatorics problems at age 12. Not all kids are ready then, but not all need to wait until they are 15 either. Each kid develops mathematical maturity at a different rate, due to differences in teaching, differences in interest level, and differences in neuronal development.

The Art of Problem Solving books are excellent examples of practice a problem solving (as opposed to drill). The problems in Larson and Hostetler’s Precalculus are mostly drill.

The AOPS pre-algebra book is supposed to be published this fall. It's one of the possibilities I'm keeping in mind for when my DD finishes Singapore Primary Math 6B (she's about 90% of the way through 4A at the moment).

I've never liked competing in math, but I'm coming to the realization that those problems (many of them, not the really sneaky ones) define what I want my son to know, and that he can get all A's in math, but never get to that point. I should have realized this and started sooner.

I would suggest that being part of a team that prepares for these problems is even more beneficial than him just preparing them by himself.

Being able to explain your thinking and answers for these problems is truly important, and bares little resemblance to the "math communication" that educators want to see in Everyday Math. Here, you explain your reasoning not by analogy, but with more math, appealing to other math you know, with pictures that size up the constraints, etc.

To the extent that we can do this with our kids, it's better than not doing it, but it's better to do it with others.

How I tackled it.

I drew a diagram - which was not to scale and would have been even if I had a ruler to hand.

I then worked out the speed of each child in feet / second - the units did throw me a little but that is because I like in one of the 192 countries that uses the metric system. determined how far a child had travelled in a given time i.e. d= s * t - a formula drilled into me at school along with the other versions of s = d / t and t = d / s - the last formula is, to me, the least natural one.

I then thought - I am interested in when people pass each other and I thought - they have to be at the same distance when they pass.

I then made statements about the distance travelled by each using the some starting point and then had a set of linear equations and then I set these as equal dist(A) = dist (B) and dist (B) = dist (C) and worked out the two times as I figured that to start with C is in front and then B and finally A - therefore I found the two points in time - and took one from the other.

Crimson Wife:

The AOPS pre-algebra book is supposed to be published this fall.Where did you find out about this? I never know what they have in the pipeline until it is released. An AoPS pre-algebra book might be something I could use, too.

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