Comments

From what I understand that is pretty typical middle school teacher behavior. I've been protected from it due to my dealings with mostly Special Ed teachers who are not that way.

They could all take a lesson from the Special Ed teachers. Those teachers treat it much more as a partnership because they have to. Their success is completely dependent on their knowledge of the child, at home and at school.

I call it the "Run along, little parent, we can take over from here" syndrome because that's how they make you feel. And that would be just fantastic except for what you just mentioned. You have no idea your kid is tanking until way late in the day.

You also don't get tests, papers, graded things of any sort like you did in grade school, and generally, you're not going through the backpack everyday like before because you assume everything is going fine because, of course, you would have heard from someone.

-- SusanS - 22 Dec 2005

I don't see why you can't run question #1 backwards:

8 + 1 = 9

9 * 4 = 36

36 ÷ 3 = 12

12 - 5 = 7

And I kind of like the occasional problem with an answer of 15' 2 ¼".

I don't know whether it's appropriate for kids Chris's age, though.

These seem like the kind of problems for which the reported Japanese method* might work.

* Let the kids work for a while, then show them how to do it and why.

-- DougSundseth - 22 Dec 2005

I don't see why you can't run question #1 backwards

I couldn't do it.

-- CatherineJohnson - 22 Dec 2005

I got something incredibly wacky.

-- CatherineJohnson - 22 Dec 2005

oh!

I messed up on the last number!

I added 5 instead of subtracting 5!

THANKS!

I'll have Christopher do this tonight!

-- CatherineJohnson - 22 Dec 2005

The problem is actually fine (I think) for Christopher IF you have a teacher walking them through it.

After all the work I've done, I didn't get it (I thought I was screwing up order of operations when I worked backward, and had no way to tell whether I was or not....)

Ed couldn't do it, either, I don't think.

When you're maxing out the parents AND assigning a problem the kids have no idea how to do, you're not teaching.

Period.

-- CatherineJohnson - 22 Dec 2005

I call it the "Run along, little parent, we can take over from here" syndrome because that's how they make you feel. And that would be just fantastic except for what you just mentioned. You have no idea your kid is tanking until way late in the day.

good one

I've had it

I have SO had it

Of course, I go through all the papers, because I was never notified of anything in grade school, either.

-- CatherineJohnson - 22 Dec 2005

What's wrong with the word problems?

Ignoring the wrong answer in the text book to question 2, I think it's worthwhile showing kids that they can solve problems that require working backwards.

Sometimes I've had to recreate people's working to work out how they got to the particular answer - e.g. if someone says the NPV of installing energy-efficient lightbulbs to reduce peak demand is $500/kW, and given a set of assumptions about the cost, the energy savings per kW, replacement time, what interest rate did they use?

On the other hand, I've never had conceptual problems myself in working backwards, so I'm not the right person to assess the difficulty.

-- TracyW - 22 Dec 2005

These seem like the kind of problems for which the reported Japanese method* might work.

* Let the kids work for a while, then show them how to do it and why.

These problems are wildly over Christopher's head.

You can't do the Japanese way, either, when you're that far ahead.

Actually, that's wrong.

The first two could be done the 'Japanese way.'

It's the third problem that's ludicrous.

-- CatherineJohnson - 22 Dec 2005

Actually, ignore my comment above - I think I understand Chris's problems on re-reading your post.

The teacher probably should have started out with a few starting time questions, and then worked the way up to the tougher ones.

-- TracyW - 22 Dec 2005

Is Christopher's teacher the Ms. Kahl whose students make comments like:

I liked math until I had her. She hated my whole entire class just because of about three kids and it was so boring!

and

she might have good intentions and all, and want us to learn, but she needs more teaching experience. also, way too much hw!

and

She's looking for negative grades to give, but she's good

?

-- GoogleMaster - 22 Dec 2005

I think that one's doable with assistance, too. Oh, converting to feet and inches is annoying, but probably not necessary.

It ends up as:

Original Height = 2 * (3/2)^5 = 243/16 feet = 15 3/16 feet.

The tricky bit in all of these is that you are using the output of the first (internal) problem to become the input of the second and so on. And you're running it backward at the same time.

If I were trying to teach this, I'd start by explicitly running that problem forwards, though I'd probably use simpler numbers in the example. (1/2 previous height rather than 2/3, 16 feet as the original height, 4 bounces)

After running it forwards, I'd take may sample problem and run it backwards, step by step, showing that I'm now dividing by 1/2 rather than multiplying by 1/2. I'd explicitly mention that dividing is multiplying by the reciprocal, and talk about the reciprocal of a few numbers.

Only then would I let the students try again with the more difficult problem.

Then, after a few minutes, I'd show (or have a student that had figured it out show) how to solve the problem. Then reinforce with a few more problems as homework.

The problem here seems to be choosing which skills to use, not figuring out how to use the individual skills. I suspect that most of the kids in the class would get the idea, since (by your description) they have the individual skills necessary to solve the problems. The teaching goal should be to show how to choose the right skills.

Of course that wouldn't be as "challenging".

-- DougSundseth - 22 Dec 2005

oh my gosh!

that's her!

-- CatherineJohnson - 22 Dec 2005

This is my favorite so far:

Ms.kahl is really nice... she's also helpful when i have questions but i wish i was in a different class..... we can never move on because no on gets anything!!

-- CatherineJohnson - 22 Dec 2005

The teacher probably should have started out with a few starting time questions, and then worked the way up to the tougher ones.

Exactly, and I should be clearer. (WAY clearer.)

If he were doing Singapore Math, he'd be doing problems substantially harder than this now (don't know about the ball-bounce one...)

But he'd be DOING them.

He wouldn't be sitting around guessing and checking.

-- CatherineJohnson - 22 Dec 2005

The tricky bit in all of these is that you are using the output of the first (internal) problem to become the input of the second and so on. And you're running it backward at the same time.

I think this may be an 'enrichment' problem suitable for TAG kids.

(I think.)

As I become better at teaching, I may feel differently.

But at this point, there is nothing Christopher would gain from observing even a skilled teacher doing it.

For one thing, there's way too much burden on working memory (although I was able to get around that by drawing bar models.)

Here's a factoid to remember: Ed told me it would have taken him 'hours' to do this problem.

This is a guy who got a C in the advanced calculus for engineering students at Princeton AND taught high school math successfully to GED students.

It's just too hard for non-math-brains.

-- CatherineJohnson - 22 Dec 2005

And time is precious.

When you're pouring time into a problem you absolutely can't do, a problem your mom is going to do for you, start to finish, more of your childhood has been wasted.

-- CatherineJohnson - 22 Dec 2005

If I were trying to teach this, I'd start by explicitly running that problem forwards, though I'd probably use simpler numbers in the example. (1/2 previous height rather than 2/3, 16 feet as the original height, 4 bounces)

That's an interesting idea.

If I were going to attempt this, I'd cut it WAY DOWN, too.

No more than a couple of bounces, max.

-- CatherineJohnson - 22 Dec 2005

They don't know how to set up an equation and isolate variables.

That's what they should be working on.

Not 'enrichment' problems they can't do.

I'm tearing my hair out, I'll tell you.

-- CatherineJohnson - 22 Dec 2005

I suppose that 15 3/16 would be a bit difficult to guess. 8-/

I wasn't trying to suggest that he or the rest of his class should be able to do these problems without further instruction. Rather, I think that they could benefit by actual instruction in how to do these problems, because they have the prerequisites.

That would not be true of students without the basic skills knowledge that they have. I misunderstood your comments to be saying that these problems were inherently too difficult to be addressed at this time, because there was some skill or set of skills that they were lacking. Sorry I misunderstood.

-- DougSundseth - 22 Dec 2005

oh wow

good news

I showed Ed the NY state assessment problem:

he said Christopher could do it

I said there was no way he could do it

Christopher did it in 2 seconds

-- CatherineJohnson - 22 Dec 2005

I feel better

-- CatherineJohnson - 22 Dec 2005

For #3, a gentle introduction might consist of saying, look, after 1 bounce the height was 2', and that was 2/3 of what it used to be.

So what was the height before the bounce?

After that it's just chaining backwards.

But IMHO this is a tough question for 6th-grade unless the teacher's already walked them through an example.

-- VerghisKoshi - 22 Dec 2005

I misunderstood your comments to be saying that these problems were inherently too difficult to be addressed at this time, because there was some skill or set of skills that they were lacking.

If you misunderstood, then I miswrote.

(Seriously.)

-- CatherineJohnson - 22 Dec 2005

now I'm wondering why I had no idea he has mastery of that fraction problem

(apparently has mastery, at any rate)

-- CatherineJohnson - 22 Dec 2005

For #3, a gentle introduction might consist of saying, look, after 1 bounce the height was 2', and that was 2/3 of what it used to be.

So what was the height before the bounce?

I did that, and he sort of got it.

I think he would have completely gotten it if there weren't a vast long backwards chain

'chaining backwards' is a good term

I'll use that with him when we look at the problem again

-- CatherineJohnson - 22 Dec 2005

"The teachers and administrators listed below are eligible for tenure effective September 1, 2006. The Board of Education welcomes your feedback. Please send your comments to board@irvingtonschools.org."

Helllooooooo? (evil grin)

Be sure to share your feedback before (or after) you submit it!

-- GoogleMaster - 22 Dec 2005

But IMHO this is a tough question for 6th-grade unless the teacher's already walked them through an example.

This is The Way.

Give kids problems they can't do, then say you're challenging them.

If it didn't eat up entire evenings of time, I'd shrug it off.

But at this point we are badly pressed for time in which he can do real learning, not just sit around guessing numbers

-- CatherineJohnson - 22 Dec 2005

It would have taken a bloke with a 'C' in advanced calculus hours to solve the bouncing ball problem?

Here's how I did it:

Draw a diagram of the heights of the bouncing ball and number the bounces

           
           |    
           |  |  
           |  |  | 
           |  |  |  |
           |  |  |  |  |
           |  |  |  |  |  |
No. bounce 0  1  2  3  4  5

Please note the lines are not to scale.

Now we know the height at bounce 5 is 2 ft.

            
           |  
           |  |  
           |  |  | 
           |  |  |  |
           |  |  |  |  |  2 ft
           |  |  |  |  |  |
No. bounce 0  1  2  3  4  5

And we know that the fifth bounce is 2/3 of the height of the fourth bounce. We can reverse that, and determine that the fourth bounce is 3/2 of the fifth bounce, or 3 ft.

           
           |  
           |  |  
           |  |  | 
           |  |  |  |  3 ft
           |  |  |  |  |  2 ft
           |  |  |  |  |  |
No. bounce 0  1  2  3  4  5

And from here we can work out that the third bounce is 3/2*the fourth bounce, and so forth backwards.

         
          15 3/16 ft 
           |  10 1/8 ft 
           |  |  6 3/4 ft
           |  |  | 4 1/2 ft 
           |  |  |  |  3ft
           |  |  |  |  |  2 ft
           |  |  |  |  |  |
No. bounce 0  1  2  3  4  5

No calculus involved.

How was Ed trying to solve this problem?

-- TracyW - 22 Dec 2005

If you misunderstood, then I miswrote.

You're turning into Engelmann.

-- KDeRosa - 22 Dec 2005

I think Ed might have been thinking of the following problem:

A ball is dropped from a height of 12 feet and bounces. The first time the ball bounces, it reaches a maximum height of 8 feet. For each bounce beyond the first, the maximum height of the ball is 2/3 that reached on the previous bounce. If the ball is allowed to bounce forever, how far does it travel?

That problem is generally presented as a calculus problem (though it is possible to solve without it).

-- PaulMiller - 22 Dec 2005

Yeah, the old infinite series problem.

-- VerghisKoshi - 23 Dec 2005

Actually in some ways problem 3 seems a better one for pre-algebra students than the first two.

The first one is hard to keep track of unless you have enough algebra to set up the problem and solve it line by line.

The second one is hard to understand properly in the first place unless you write the algebraic equation (or draw a bar model).

The third, if students are lead into it properly, doesn't require any working knowledge. E.g. 3. a. Draw a picture of the ball bouncing five times, getting lower each time. Number each bounce. 3.b. How high does the ball bounce on time five? Write the answer above the fifth bounce on your picture. 3.c. How high does the ball bounce on the fourth time? Write the answer above the fourth bounce on your picture. 3.d How high does the ball bounce the first time? Write the answer above the first bounce on your picture. Feel free to write any working on your picture.

-- TracyW - 23 Dec 2005

Here's what gets me about problem 5-6 that you posted, with the huge splash sheet about 'working backward': all algebra is about working backward. So what's the big deal?

If you have a word problem, you start by labeling the unknown 'x'. Then you write an expression that tells you what happened to x, and equate it to a known quantity.

To solve the problem, you 'work backward', undoing everything that's been done to x in order to reveal it. That's exactly what the process is.

Why not just teach the kids algebra -- set the departure time to be 'x', add the 5 hours travel time and the 1-1/2 hour rest time, and set it to 3:30 and then solve for 'x'?

This business about 'working backward' effectively replaces the systematic equation-solving process with guess-and-check, wasting time that could be spent learning a much more powerful process for problem-solving.

-- CarolynJohnston - 23 Dec 2005

I hated textbooks like this when I was in school.

"Read, Plan, Solve, Look Back." I'm busy learning algorithms, and the stupid book wants me to memorize this sequence logo thing? Gee, let's make everything easier by adding the number of things you need to remember.

"Read." Well, DUH!

"Plan." It's just like Solve, but you use words instead of symbols, and you don't actually do anything. Did they really need this? They could have made a triangle logo thingy instead of a circle.

"Solve." Double DUH!

"Look Back." In the real world, we call this Check Your Work, Doofus.

-- BrendaM - 23 Dec 2005

clarity update: there's nothing wrong with these problems

Actually, there is something wrong with the "working backwards" technique to solving these problems: the technique doesn't generalize.

Not only that, but the technique will become obsolete the first week of algebra class.

In Singapore math, bar models would be used to solve these problems. Bar models generalize.

In DI, there is an explicit rule not to teach anything that doesn't generalize, otherwise it's a waste of teaching time.

It sounds like Christopher's entire sixth grade math class consists of merely learning a series of specific rules that don't generalize, i.e., a waste of time.

-- KDeRosa - 23 Dec 2005

At the end of every week my son's school sends back all the tests and HW for that week in an envelope. We sign the envelope and send it back.

All the HW and tests scheduled for the week is posted on the web, a separate page for each teacher. All we have to do is check the page and know what's coming up in the week.

Woould something like this help you to keep track of Christopher's HW?

-- VerghisKoshi - 23 Dec 2005

If you misunderstood, then I miswrote.

You're turning into Engelmann.

No!

This has been my rule for a long time (in fact, I realize I should write a post about it, because I think this is probably a simple 'rule' you could teach kids.)

For years my Rule has been: Anything a reader says about my writing is true.

This sounds crazy, but it works.

The fact is, unless a reader is crazy himself, anything he/she says about something I've written is true for him/her.....and there's something in my prose that's sparked that response.

Obviously, if I followed this rule in its strong form (EVERYONE IS RIGHT EXCEPT FOR ME!) I wouldn't get a lot written.

But what you find, when you publish, is that people's 'misunderstandings' cluster.

In this case, both Doug & Tracy had the exact same 'misunderstanding' of my post.

That means I miswrote, period.

It's true that if you combed through my post, looking at each and every word (which I haven't done yet), you'd probably find that I said what I thought I said.

BUT the fact that this post is glaringly open to another interpretation—even invites another interpretation—means I miswrote.

It's an incredibly useful principle.

I'm trying to think how I would put it for kids......

hmm

I'll have to give it some thought.

-- CatherineJohnson - 23 Dec 2005