## Friday, October 2, 2009

### Math Competitions

Does anyone have any comments about K-8 and high school math competitions? I was asked last year if I wanted to form a MathCounts team at our middle school. Has anyone done this competition? How about other competitions? I'm reconsidering doing this as a way to help kids and to focus attention on the needs of K-6 math. Are there other competitions that would be better? Actually, I've always hated the idea of math as a speed competition. However, it could be a way to help many more students than just the math brains. Are some competitions better for more kids or does it just matter how you set it up? For example, the Science Olympiad and the First Lego League are full-day events where all kids go and have a great time.

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## 42 comments:

SteveH,

A fellow parent and I started an after-school Continental Math League club for our school's 2nd and 3rd graders last year. (Continental Math League is the only major math competition that reaches down to 2nd grade). I blog about our experiences on Out in Left Field, starting here (with a couple of sample problems): http://oilf.blogspot.com/2008/05/math-problem-of-week-investigations-vs.html

For various reasons, some of which I discuss in my blog, it's not clear whether we'll be able to resume CML at our school this year. Stay tuned! And keep me posted on how it goes at your middle school. Math Competitions can be a great way to put pressure on the local math ed power brokers and nourish the many math-starved kids out there.

Katharine Beals

I've seen the Math Counts books--the stuff they suggest doing at weekly meetings--and they look pretty good.

Competitions are a great way to engage kids who either are under-challenged in school or just thrive on the game aspect (I've got one of each kind of kid!). There are lots of options, I've got a list going at http://stemology.wordpress.com/stem-competitions/.

Katherine, I didn't have CML on the list, so thanks for that tip!

These math competitions are the "farm teams" for the elite competitions -- like the International Math Olympiad and the China Girls Math Olympiad. Almost every U.S. student I know who has competed at an international level started out with either MathCounts or ARML.

Thanks for all of the feedback. I need to do my homework. I currently help out with the Science Olympiad, but with this, I would have to be more than a helper and make a long term commitment, even after my son gets to high school.

"MathCounts or ARML"

Is one or the other more popular in one region of the country than another?

Steve,

We did Mathcounts one year and the kids that were involved enjoyed it. I think it helps to have a real mathy coach, which we didn't have. The problems are tough and they only reward a small few at the actual even, but most kids don't seem to mind.

You can send kids as individuals or as a team, or both.

I thought ARML was high school. I hear the AMC now has an AMC 8 for middle schoolers.

Our high school has had all of the competitions posted in the past, but I haven't found out how they handle any of that. My son joined the math team, though, so I'll let you know if I find out anything.

I'm curious about the Olympiads. Has anyone done those? I don't mean the Science Olympiad, but some of the others. I know they added a Biology Olympiad, but I know no one who has done it.

SusanS

I helped coach a Mathcounts team the last two years and found it enjoyable. Last year we had a group of math-hungry 6th graders (doing pre-algebra in 6th grade) and almost all of them are eagerly back for another year. Some of the benefits for the kids:

- They get to do challenging math problems with others of similar ability. The problems are almost always harder than what they typically do in class.

- They are friends with each other outside of school, so it is as much a social event as a math event.

- They learn things in Mathcounts that give them an advantage in class when those concepts are introduced in class. Last year they learned Pythagorean Theorem months before it was taught in class.

- It provides an opportunity to show them higher levels of math, things that they will be learning in later years. We did some probability problems using a probability tree and then tackled two much more difficult problems. A few of the kids really enjoyed the challenge.

- While the actual competition is limited to 4 (or 8) students, those younger ones realize that in a year or two they will be the ones competing, and they learn from going to event.

If you have kids who are willing and able, it's a very enjoyable process.

JE in PA

JE in PA,

Thanks for the feedback. I suppose I won't get many students, so most of them will get to go.

There are some interesting contests for high schoolers at the sponsored by the University of Waterloo in Canada.

One of the issues I have with math contests is that you end up with problems like this: (from MathCounts)

"When its digits are reversed, a particular positive two-digit integer is increased by 20%. What is the original number?"

... when the effort might be better spent on moving to more advanced material. I can understand that for a 6th, 7th, and 8th grade contest, it really couldn't work to allow questions on trig, but my son enjoys and understands it when I cover some topics in trig and calculus. I would rather take the time to talk about vectors and parametric equations than to prepare him to answer questions like this.

Also, problems like the one above have always bothered me, so I will ask a philosophical question.

Is this math?

That's such a general question that even I would have to answer yes, but what mathematical process do you go through that can be applied to other problems. (other than common sense) You know that the second digit must be greater than the first, and that for 20%, the numbers have to be close together. I can write an equation for this relationship and find a ratio for the two digits, but I still have to guess and check. Perhaps it's my over-reaction to many who want to use problems like this to define math. They look for questions that require more than math skills, as if that is the definition of mathematical competence. It's not as if this thought process is built on top of mathematical skills. It's something else. It's a philosophical attempt to break the linkage between skills and understanding.

Let the original number be (x,y).

10y + x 120 6

--------- = ---- = ---

10x + y 100 5

Let the original number be (x,y).

10y + x . 120 . 6

--------- = ---- = ---

10x + y . 100 . 5

50y + 5x = 60 x + 6y

44y = 55x

4y = 5x

So, for integer solutions, x = 4 and y = 5,

and the original number is 45.

check:

54

---- = 6/5

45

--Is this math?

Are they supposed to solve it using two equations and two unknowns? Or are they supposed to be clever?

My main problem with math contests was always that I wasn't very good at them. :)

Truthfully, lots of kids were very good at being clever. I wasn't. Over time, I began to resent "clever", and I began to think that I couldn't be a mathematician (particularly a mathematician in discrete math) because I wasn't clever. (I wish I could say I was wrong here, but I still don't know. There were cofounding variables.) I also began to really denigrate people who asked "clever" puzzles because they were really asking "have you heard this puzzle before and did you remember the answer?" rather than anything that helped people think mathematically.

So, to your question about what's math, the competitions are less important than the preparation. If you prepare your students for those questions by teaching them more math, by teaching them how to conver that word problem into the appropriate equation, about enough number sense to do sane guess-n-check, then you've taught math, even if the questions themselves don't seem to be.

And helping demystify the cleverness is good, too. A good prep course for those competitions can do that, and demystifying the cleverness can increase everyone's confidence, even those a bit less "natural" at math.

I have trouble being clever too, but I try not to admit it too often. :)

Once I got down to 4x = 5y and I knew x and y were both in {0, 1, ... 9}, then I could use "guess and check" to figure that if x was 5 and y was 4 then 4x = 5y = 20.

Normally I hate guess and check because it suggests that math is a hopeless mystery and only intuition and not logic can help you. I wonder it that is what makes it so attractive to folks who are a little scared of math in the first place: it sort of levels the playing field.

To me a good proof is as pretty as a poem.

Once you get it down to 4x=5y just express it as a ratio

4/5 = y/x and since x and y have to be single digit integers they have to be 4 and 5.

I know that I can count on someone here to correct me if I'm wrong, but I think this is a problem with 2 variables and only one equation; i.e., you can't solve it with simultaneous equations. But once you get it down to the ratio, the answer is evident.

I like this problem. It involves percentages, place value, and ratios. You can solve it by setting it up correctly with no guess and check. I think it's pretty mathy, frankly.

I was involved in FLL in Toronto for 4 years and I have to say that the kids who really got anything out of it were those who were on teams with enough resources to make it meaningful: by which I mean, enough after-school time for team meetings, involved and supportive parents, and a knowledgeable coach who taught them real mechanics and programming skills.

I only saw those variables come together in a few teams, and they ran the gamut from private to public to homeschool. The knowledge and ability of the coach was what made the difference, and in the vast majority of teams the coaching was not strong enough so the teams weren't doing much more than playing with legos.

I don't have experience with after-school math clubs, but I know that many students in Toronto (homeschooled and schooled) participate in the online Abacus math challenge at http://www.gcschool.org/program/abacus/index.aspx

Nikita

See, told you I wasn't clever or good at math competitions!

It doesn't have two equations. It has one equation and two inequalities.

10x + y = 1.2(10y + x)

simplifies to

y = 5/4 x

but there are inequalities, too,

that y is between 0 and 9

and that x is between 0 and 9.

Without the additional constraints, then there would be infinite solutions. Because they picked well, those constraints are met with simple guess and check, so most people didn't express them.

Understanding when a generalized problem of that sort gives you integer solutions is actually sophisticated college level linear programming. So it's all in how you teach it, and how you teach them to recognize it.

Clever. That's a good word for it. There is a cleverness factor.

Rocky -Did you happen upon that 5 : 4 ratio the first time you did it? When I created my equation, I got a ratio of 11 : 8.8.

In a speed contest, and knowing that the second number must be larger than the first, and that the two numbers are close, it wouldn't be too helpful to be mathematical and come up with a ratio of 11 : 8.8. You would still have to guess and check. It would have been faster to start checking close numbers in sequence, like 12, 23, 34, 45, 56. I would see how the percentages change when the digits are flipped.

Searching for a solution is a perfectly fine mathematical technique. There is Newton's Method if you know the derivative, or you can use the Secant Method if you just have a function evaluation. Even though this is an integer function, it could help. If you have a good starting point for the search, you might get lucky. Might. I might be more impressed if the guess and check taught in K-8 had any connection with the mathematical searches taught in college, but they use guess and check to avoid math, not enhance it.

I guess that's my point. This reminds me of those Mensa-like problems where cleverness and not mathematical skills win out. In fact, some problems seem designed to send you off in the wrong (slow) direction if you apply standard mathematical skills. To get the solution, you don't apply other mathematical techniques, you apply cleverness, as if that is a key component of being good at math. That's my point about trying to define math at "something else".

But, as Allison says, it's the preperation and demysifying cleverness that's important. I remember reading about how Dick Feynman loved to study all sorts of trick problems and calculation shortcuts. He could do cube roots in his head. He didn't have to impress anyone, but he worked hard at it, probably to the annoyance of many, because those people knew where it came from.

In fact, I have a colleague who seems quite intent on playing that game too. It surprised me when others exclaimed how incredibly smart he is. They didn't understand a word he was saying, but whatever it was, it was quite impressive.

In fact, a lot of what I do in my work is about demystifying mathematics, but many of the technical papers I read seem intent on the opposite.

Steve: I don't know how you got 11:8.8, but if we get rid of the decimal point...

11/8.8 = 110/88 = 10/8 = 5/6

There we go.

When I think of "increased by 20 percent", I just think the ratio of the new over the old is 120/100.

Yes, the modern way would be to start punching numbers into the calculator and dividing until you get "1.2" (120%). It could even be faster than what I did. I just don't like it.

... Dazzling them with simplicity. :)

Rats. I mean

11/8.8 = 110/88 = 10/8 = 5/4

"11/8.8 = 110/88 = 10/8 = 5/4"

I understand how to do it, but is that really what they expect you to do?

OK. How about this problem:

"Each letter represents a non-zero digit. What is the value of t?

c + o = u

u + n = t

t + c = s

o + n + s = 12"

I suppose you have to assume that if they gave you this problem that there must be a solution. I happened to substitute correctly the first time and got the solution, but what if you don't guess right? You could set up a matrix and do row reduction in the hope that something will show up, but I can't imagine that kids in 6th - 8th grades are expected to do that.

SteveH,

This one is a substitution problem, and you don't have to guess at all.

Start with

o + n + s = 12

s = t + c so o + n + t + c = 12

u = c + o so

o + n + t + c = (c + o) + n + t = 12 = u + n + t =12

and u + n = t so 2t = 12 and t = 6.

I just started substituting and looked if things started to simplify, which they did. Often the trick with these kinds of problems is NOT to jump into getting numbers but to work with the symbols. I don't know if that is math, but it is certainly a useful skill in "mathy" disciplines.

Happily, we can substitute:

o + n + s = 12

o + n + (t + c) = 12

(c + o) + n + t = 12

u + n + t = 12

t + t = 12

2t = 12

t = 6

and u + n = 6

and c + o + n = 6

The problem says none of the letters are zero.

c + o = u

u must be at least 3 (if c and o were somehow 1 and 2), but u + n = 6 and u and n can't both be 3. So u must be greater than 3, but less than 6.

So c and o must be either 1 and 3 (with u = 4) or 1 and 4 (with u = 5). They couldn't both be 2. That means that either c or o is 1.

Since u + n = 6, if u = 5 then n = 1, but either c or o is 1. So u = 4 and n = 2.

Now we have:

n = 2

u = 4

t = 6

c, o = 1, 3

Now there are two answers:

1) c = 1, o = 3, u = 4, n = 2, t = 6, s = 7

2) c = 3, o = 1, u = 4, n = 2, t = 6, s = 9

If you really want kids to be good at these (though I don't know why you would), there are magazines in the supermarket and Wal-Mart called "logic puzzles". Each problem gives you a grid and you have to eliminate answers and cross check to find out what goes where with whom.

For example "Mr. White in the kitchen with the butcher knife", "Mr Green in the garden with the pruning shears", etc.

I see now that ChemProf beat me to it.

Yikes! I said:

So c and o must be either 1 and 3 (with u = 4) or 1 and 4 (with u = 5). They couldn't both be 2. That means that either c or o is 1.But I left out the cases where c and o could be 2 and 3 with u = 5.

Here are the four answers:

1) c = 1, o = 3, u = 4, n = 2, t = 6, s = 7

2) c = 3, o = 1, u = 4, n = 2, t = 6, s = 9

3) c = 2, o = 3, u = 5, n = 1, t = 6, s = 8

4) c = 3, o = 2, u = 5, n = 1, t = 6, s = 9

This is a very unsatisfying problem. It makes me wonder if it is designed to discourage anyone from thinking logically at all, and just guess and check.

Once you get the digit switching problem down to 4x = 5y you don't need ratios, you just need to realize that the number 4x (aka 5y) has to have a factor of 4 (from the left side) and a factor of 5 (from the right), and the easiest number for that is 4x5. One good things about these kinds of problems is that they train kids to recognize what's in front of them.

Also regarding cleverness, the saying is that if you use it once it's a trick and if you use it twice it's a technique.

I think this is a good competition problem since you can either use intuition, whereby some students will just sort of conjure up the solution and then just need to demonstrate that it works, but you can also directly translate it into equations and work with those.

And regarding intuition, it is like luck with the proviso that to a large extent you make your own luck.

I am not a mathy sort and I wouldn't have been able to easily construct an equation to solve this problem.

However, I did recall that transposition errors in numbers are always divisible by 9.

So it seemed logical to me to try dividing the factors of 9 by .2.

9/.2 =45

So I hit 45 & 54 on my first guess.

Sometimes the "clever" problems just need to reasoned through with logic.

So is this math? I don't know. Does logic fall under the venue of math?

Anonymous said "So is this math? I don't know. Does logic fall under the venue of math?"

As one anonymous to another, yes.

Yes, "clever" is really a disparaging word to me. Just say "how clever" in the flattest voice you can imagine. You can probably hear my disdain.

--In fact, I have a colleague who seems quite intent on playing that game too. It surprised me when others exclaimed how incredibly smart he is. They didn't understand a word he was saying, but whatever it was, it was quite impressive.

I despised this behavior as an undergrad at MIT. At the time, I couldn't tell when people were deliberately obfuscating because they knew they didn't know things, versus when people were unaware they were talking through their hat. All I knew was they demoralized me, whether they were right or wrong.

Now I know which is which, and the obfuscators are afraid of being found out, and afraid that they don't know anything, so they need to play the con man, while the clever are simply weak and don't know how weak their knowledge is, since they've relied on being clever instead of learning. I also have met brilliant people who make the truly clever look foolish, and to a one, the brilliant never showed anyone up, while the clever did, many of them did so for fun.

A related phenomenon is that of a professor giving a lecture that feels Light-From-Heaven-illuminating, but the feeling (and all of the knowledge with it) fades within 2 minutes of leaving the classroom.

I've come to realize that the professors who pulled those stunts were not the good teachers, but the poor ones. It took DECADES for me to realize that. They had us entranced, and so we were hooked, and since were were in love with that feeling and the experience, we thought we were learning. But we weren't. We were being dazzled. The best teachers didn't dazzle. They might on occasion give a final lecture that dazzled, but they understood that the role of teaching was for us to gain mastery, not to bask in their own glow. The dazzlers didn't demystify things. They spent lecture time connecting the dots in interesting ways, or made major leaps and let us ride on their backs, but they didn't ever lead us to enough mastery to recreate even a tiny piece of that understanding.

Maybe those profs thought we'd just have to trust the spiral of undergrad and grad courses too. If we became profs, we'd have figured it out, surely.

btw,

there are four solutions to that substitution problem, but you're not asked to find them. Just to find t.

So, I find this a "clever" problem. You're supposed to realize that you don't NEED to solve for the others, you just need to solve for t.

I call this clever because it's something that if you've been taught to break the code, you know how to do, and if you haven't, you'll miss it, and the point along with it. I don't consider such problems instructive. But proper instruction into recognizing the trickiness is useful. And yes, then it becomes "not a trick but a technique".

I guess I'm trying to make a distinction here. Perhaps what bothers me most is that a lot of K-8 math is filled with guess and check and problems that try really hard to be something other than math.

"..if you use it once it's a trick and if you use it twice it's a technique."

I would then ask when you would use it a second time. That seems to be the whole point; to give problems that do not use known techniques, as if that defines mathematical ability.

For the substitution problem, one's first reaction is that you don't have enough equations. All you can do is substitute and hope that the other terms drop out. There are definitely faster ways to substitute, but is that some sort of skill to be developed?

I suppose I should just label this competition math - a world unto itself.

I think these competitions often mis-encourage mathematical cleverness for mathematical ability, rather than encouraging skills to mastery.

For one, many of these competitions are time-based--being faster is the important thing.

My favorite prof at MIT was Mike Artin. Mike Artin told us in his algebra class that we would *never* see a problem on a test that we hadn't been given before.

Because, he said, math was about GETTING THE ANSWER RIGHT, not how long it took you to get there. Math is about truth, and there's no time limit on truth.

If solving these questions requires finding the high falutin guess-n-check because that's the quickest way to the answer, and quickest is best, then absolutely, this isn't testing math.

I think it's wise to see them as just something other than math, too, like a clever person's version of guess and check. The guess and checks in the dumb fake-pre-algebra questions on state tests can be solved by knowing "try simple numbers". These guess and checks require knowing something you wouldn't otherwise know except you know some trick, but it still doesn't require thinking or mastery of some body of knowledge, really, more that you have these tricks in your back pocket, able to be pulled out at any time.

So the answer to when you'd use such a trick a second time is: on another math competition question!

That means these competitions self select for people who are good at these competitions. It does not mean they have gained flexible knowledge.

I don't think it's healthy for such stuff to be mistaken for math, or for youngsters interested in math to think they have to be good at these kind of stuff to do well. But then again, that could be sour grapes on my part. But rather than opt out of the competitions, I come back to trying to create a preparation environment that does explore the math being hinted at here.

Terry Tao, the young Fields medalist at UCLA, who is widely considered one of the brightest mathematicians in his generation, and who did participate in math competitions, has had this to say:

>I greatly enjoyed my experiences with high school mathematics competitions

>But mathematical competitions are very different activities from mathematical learning or mathematical research…

(http://terrytao.wordpress.com/career-advice/advice-on-mathematics-competitions/ )

and

>Mathematics competitions, when used in moderation, are indeed a good way to show high school students that maths has more aspects than the often rather dry material covered in classes.

(in the comments at http://terrytao.wordpress.com/2007/03/13/article-in-the-new-york-times-and-maths-education/ )

While Terry’s experience and judgment on all this are far more valid than my own, I’d generally concur.

People who are good at math (not necessarily mathematicians) do sometimes enjoy rather silly contests: I once got into an informal contest with a friend of my wife’s, a female Greek physicist now teaching engineering in Britain, over who could most rapidly convert from British home prices (pounds sterling per hectare) to US home prices (dollars per acre). Rather silly; we were just playing around – and there was no real winner: we were just challenging each other.

If kids take math competitions in the same spirit (as Terry said, “when used in moderation”), I could see how they could be fun.

If the math competitions become important to a kid’s ego (or a parent’s ego), if he or she thinks they show his real ability or promise in math, or if he or she thinks they are real math, I think they could be damaging.

I think I (and Terry) are agreeing with Allison.

Dave

Thanks for letting me know that my views are shared by others.

But I don't want to pick on math competitions. I'm just trying to figure out whether I want to commit to starting a MathCounts team at my son's school. I don't doubt that it would be a good thing overall, and I can try to emphasize the important skills. ..but...

But it reminds me of sports. When my son was young, he played soccer and baseball. He loved them. However, things change when you get to 10 - 12 years old. Kids get serious. Coaches get serious. Parents get serious. Competitive sports is not bad, but there is no in-between. So, my son doesn't play any organized sports. I know that many high schools offer some no-cut sports, but that must be difficult to pull off and make everyone happy. Does anyone have any comments (good or bad) about that?

When the school asked me to start a MathCounts team, I know that part of the reason was to show parents that the school is not ignoring the top end of the spectrum. This annoys me somewhat because it seems like an excuse to not do anything about Everyday Math. The kids in the middle lose out, but the school will get great PR.

Allison: thank you for showing me my errors. When I reread the problem, it just asked for t, as you said. Now I see that it is a fine problem, and I fell into the trap.

When I read the word "clever" I imagine your voice sounding like the Snipes character in Harry Potter.

I have two things I try to live by regarding cleverness. The first is that the better you know something, the better and more simply you can explain it without oversimplifying it. The second is that the art of the teacher is to make something so clear and sensible that it seems easy. The teacher uses his expertise to make the implementation obvious so that the student can concentrate on the concepts.

The downside of this is that the student must take notes (in college), or the teacher must make the students write down examples (in grade school), lest the student go home and the insight vanishes before he can do his homework.

Steve,

We had the same problem sports-wise. I have no idea how you play high school sports and participate with an academic team, plus keep your grades up. Schools often have intramurals, but I'm not even sure ours does.

My son loved sports, too, but when it got very serious, he seemed out of place. Many parents seem to be working for a sports scholarship, even in middle school. Nothing wrong with that, but like you, we noticed there was no other outlet for him.

My son has had some difficulty already in the grade department since he spends hours weekly participating in his one sport at the high school and being on the math team. He had to pass on Science Olympiad because there just wasn't enough time. We're trying to make it work, but once again, I've become his personal secretary. My guess is that he will just give up on the sports thing altogether.

SusanS

--The second is that the art of the teacher is to make something so clear and sensible that it seems easy. The teacher uses his expertise to make the implementation obvious so that the student can concentrate on the concepts.

I don't agree. I think the art of the teacher is to make something so clear and sensible that it seems clear and sensible. Making hard things seem easy is not helpful in my book. It is good to learn hard things are clear and sensible: that hard things can be broken down into smaller things, manageable things. This does not make things easy.

It is not good to learn that hard things are easy. It is not true.

I always took notes in college. My notes never could make up for the dramatic steps that the teacher elided over. The teacher may think that by glossing over various implementation details they are elucidating concepts. Done very well, of course, they are right. But the details are necessary to learn whether or not you've learned anything--we can all think we understand, but we're mistaken. So the problem sets should force you to work through those details, until, again, you've got mastery.

The best courses were ones where the profs gave out copious exercises to improve basic skills. Again, Mike Artin was the king of this. He explicitly stated in his textbook that the exercises in the book were to show you had sufficient mastery to attack the problems. The exercises should have been doable without much thought, without getting stuck. If you could not, then you needed to go back and learn more before going forward, he wrote in his textbook. The problems required discipline and thought and a lot of work.

His lectures were of proofs of theorems. Proofs in his course didn't show all of the dead ends he'd tried; they were clear and sensible (usually..on occasion, he'd forget how to do something and get stuck...a kind of surreal experience for us students, heart warming and terrifying simultaneously.)

But they weren't all easy. Maybe we're just arguing words, and we mean the same thing behind them, but maybe not. I don't want the illusion of learning, and I don't need the illusion that something is easy when it isn't. Clear and sensible is better.

I've thought about this more, and wanted to add that my take on this is deeply influenced by my personality, and it may be that I'm quite in the minority here, but:

when my teachers presented material to me that was difficult, but made it seem easy, they undermined my ability to work hard.

That's because I would get stuck. I would be stuck, and think "why isn't this easy, it's supposed to be easy, therefore I'm not doing something right, therefore I'm not good at this." Being shown something was easy didn't help me to practice, and often, it didn't even help to show me what I was supposed to practice. It was the opposite of explicit instruction. I was supposed to surmise which details had been left out and how they were relevant. Instead, I would collapse into confusion and insecurity.

Kids fall off the academic train at every level. Practically speaking, no one ever gets on once they've fallen off. It might seem that making difficult things seem easy is a way to keep them on the train. But it seems to me that it's a way for them to get derailed before they even know it, because it shapes their expectations of how it should feel and seem for them to do real work.

"...art of the teacher is to make something so clear and sensible that it seems easy."

I think the context matters. In K-12, there is a belief that kids should struggle, so they withhold information. There is also a problem with what I call top-down or thematic learning where teachers try to motivate kids into success with the basics. Unfortunately, the basics never quite get defined well and students are left to figure it out themselves.

Perhaps easy is not the right word. Teachers should make all of the details clear. There seems to be a belief among educators that you shouldn't do this; that it makes the process more rote. If you just write and write and write (and not quit first), you will become a good writer. (That point could be argued.) However, if you teach writing in a formulaic way, will the student get stuck there?

I'm clearly on the side that says that people are not stupid and that if you have any ounce of desire, the bottom-up direct approach is always better - with the lots of practice too.

Our schools are big on reading. Read. Read. Read. It solves all problems. It means that they don't have to be ery good about grammar, spelling, and vocabulary. I think the real reason is that it fits better with full-inclusion. It's easier to have kids read at their own level than it is to have everyone on the same page in grammar. This reading thing seems to be based on some simplistic studies that show how much reading can do for you. Therefore, you can skip the direct approach.

This is probably not what

Allison was talking about. In college, I wanted everything spelled out clearly. I don't remember any case where the professor got me to believe that something was easy and then I found out it wasn't. Usually, they talked as if it was all easy and then I wondered why I was so stupid. At least we students could form stupid homework support groups.

>>I know that many high schools offer some no-cut sports, but that must be difficult to pull off and make everyone happy. Does anyone have any comments (good or bad) about that?

With sports, cross-country is usually no cut. The team usually gets the students cut from the other sports who want to compete in something, anything in order to have a sport on the resume. They're happy it's inclusive,and generally accept that it's going to take work to come up to the competitive level & of course some never will. Some meets are only for the top runners, some are for everyone. A math counts program could be inclusive, as long as club members had a bottom floor in prior knowledge and the sponsor wasn't expected to teach the material taught in school.

>>"Each letter represents a non-zero digit. What is the value of t?

c + o = u

u + n = t

t + c = s

o + n + s = 12"

Thought this was a funny (ha-ha) problem for that age group...expected that it was going to say c+o+u+n+t+s=something at some point. Kids this age would think that's funny, as they are catching on to puns and jokes like spelling names backwards to make words in books.

So..using the add'n property of equality (also taught around this age) and combining equations

c+o+u+n = u +t (first two)

c+o+u+n+s=u+t+t+c (third equality)

since o+n+s=12, u and c are on both sides, one can simplify to

12=2t or t is 6.

I think the key is to see a pattern..to me the pattern was the letters spelling a word, but the puzzle didn't quite work out that way as I followed the facts down in order.

"expected that it was going to say c+o+u+n+t+s=something at some point."

Good one! I didn't see that.

Some of my kids attended a large suburban high school in a very affluent, highly-educated area. Soccer was a VERY big sport in the area, to the point that it was almost impossible for kids to get significant playing time at the JV level (if they made the team at all) unless they had been playing top-level travel soccer for several years prior to high school. A new principal arrived and instituted a no-cut policy for all sports. Since state or county rules prohibited seniors on JV teams, the soccer varsity ended up with several seniors who had never been able to make the JV. The gap in fitness, skills and tactics was HUGE. The kids pretty much never played and they really didn't even know the other team members, all of whom had played with/against each other for years on club travel teams. I was amazed the kids didn't quit, because the whole situation was very awkward. The policy was changed by the following year, at least for varsity level. I'd agree about the cross-country comment above and it may also be true of track. Some kids did it just for fitness.

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