08 Feb 2006 - 00:27

## mind the gap

This is starting to be funny.

I mentioned a couple of days ago that Christopher went into his Chapter 10 quiz knowing the material cold.

He could find:

• area of a square

• area of a rectangle

• area of a triangle

• area of a parallelogram

• area of a trapezoid

• area of a circle

He had learned all this stuff in about 3 days (YAY!), AND he could do it the KUMON way, with speed and accuracy (DOUBLE-YAY!)

source:
Bitter Single Guy

So what does the test look like?

This:

do we see the problem here?

Christopher has never, in his life, ever, figured the area of a complex figure.

He would never even have seen a complex figure if I hadn't shown him a few and bugged him about how he maybe ought to learn how to figure the area of a complex figure because "It might be on the test."

IT'S NOT GOING TO BE ON THE TEST! NO! IT'S NOT ON THE TEST! WE DIDN'T DO THAT IN CLASS! SHE DIDN'T TEACH US THAT! IT'S NOT ON THE TEST!

etc.

So now, the good news is: Christopher thinks I have Top Secret Mom Knowledge of WHAT'S GOING TO BE ON THE TEST.

That's a Good Thing.

update: Old Grouch says the drawing is wrong

I love this drawing Old Grouch left!

I love it so much I'm completely distracted from the question Old Grouch is raising — (in my next life I may have to be an artist who paints paintings of MATH)

Which dimensions on the drawing are Christopher's, and which were given as part of the test? If the (3cm+6cm) and the center 9cm lines are really parallel, the hypotenuse of the "one triangle" on the left CAN'T be 12cm... it has to be greater than 14cm.

Here's Anne:

If you use the marked numbers on the test of 4cm for the height and 13cm for the base, the hypotenuse of the triangle on the left is the square root of 185 which is between 13 and 14. But the base of the triangle on the left marked 10cm cannot be correct. If you draw a line parallel to this 10 cm base with its start at the right hand corner where the circle is and drop it to the bottom, you get a right triangle with a hypotenuse of 14 and sides 3 and 10. This is not possible since 32 + 102 does not equal 142.

Also, since the horizontal line marked 6 cm is parallel to the horzonal line marked 9 cm, the two vertical sides of the resulting parallogram have to be parallel and the same length. But, as I've pointed out, the length cannot be 14 cm because the sides are 3 and 10.

So the student who said he couldn't do this problem was absolutely right. You can't do this problem if you try to get all the number right because they don't come out right.

The funny thing is, when I put in my original post about this being a problem for high school geometry students, I believe it was because my intuitive brain recognized the problems, but didn't articulate the words to my verbal brain.

Thanks to Old Grouch for pointing out all the errors.

update: Ed and I just looked at this —

This drawing is wrong, no question.

Ed and I see that if the (3cm + 6cm) line is parallel to the 9cm line, then the left line labelled 12cm has to be 14cm.

We don't remember our high school geometry well enough to pick up on Anne's observation.

I do think this is a case of Anne's cognitive unconscious knowing something Anne's conscious mind didn't. To wit: this problem is way too hard for an 11 year old.

Ed just said this problem might be good for a sophomore, if the assignment is to explain what's wrong with the figure. (Yes?)

update: the test was 'easy'

Christopher just told me that when he went in for extra help on Thursday (he sees the math teacher once a week for extra help) she told him the test was easy.

I just told Ed, and he said, 'Then you have to bring this to her attention.'

I guess so!

I have this t-shirt.

The X-rated version is here. I figure if eduwonk can perseverate on big-girl panties, I can post x-rated tourist-wear.

-- CatherineJohnson - 08 Feb 2006

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Awesome news that Chris got 80% on this test. It shows his knowledge is becoming flexible.

And his mum is getting adept at reading Mrs Kahl's mind. Or are you breaking into her classroom at night and photographing the test papers?

-- TracyW - 08 Feb 2006

This is a totally and completely inappropriate problem for these kids.

This is an appropriate problem for a high school geometry class that has been working on these things.

I believe Carolyn and some others did a post about tests and testing knowledge. I believe the gist of the post was that you had to give enough questions so that the students who had the knowledge but it was not yet flexible were able to anwer some questions. Then you added in the extended questions to see whose knowledge was flexible.

I would love to know what Matt and the other teachers think about this on a PRE ALGEBRA test!!

-- AnneDwyer - 08 Feb 2006

Is it just me or does that triangle to the leftmost of the figure require some pretty advanced calculations to work out its height?

-- TracyW - 08 Feb 2006

This comment/question is only (barely) tangentially related to the post, but it's something that I have been thinking about for awhile (if there is a more appropriate place for this to go, it would be great if someone could move it there for me):

What exactly does a letter grade mean? At the end of each semester I am required to assign each student with an 'A', 'B', 'C', 'D', or 'F' (or 'I', but that's a different matter). As it stands, I base things off a cumulative percentage of total points (90% and above is an 'A', 80% and above is a 'B', etc.). I'm not entirely satisfied with this approach. Perhaps a couple of hypothetical examples might help explain some of my difficulties with the approach:

Suppose I have two students, each of whom received a 'C' in the class. One of them did every assignment I gave, took advantage of every opportunity for points, and barely squeaked by on each of the tests. The other student blew off homework, didn't take advantage of many opportunities for 'extra' points, made silly mistakes on tests (to get B's), but pretty clearly knew the material over all.

Although both students received C's, they manner in which they did so was very different. The first student probably has a weak grasp of the material but is exceptionally discplined about doing what's asked for the class. The second student probably has a pretty decent grasp of the material, but not a lot of discipline for doing what is asked (perhaps due to boredom with the material or just a general lack of discipline). All of that is lost with a letter grade of 'C'.

What I would like to do is have a fairly comprehensive and clear set of standards/tasks that students should be able to do. As an example, let's consider solving linear equations. In order from lowest to highest, these are some expectations I might have:

1. They recognize a linear equation

2. Can consistently solve (for x) linear equations of form a*x=b

3. Can consistently solve linear equations of form a*x + b = c

4. Can consistently solve linear equations of form a*x + b = c*x + d

5. Can consistently solve linear equations of form a*( b*x + c ) = d*( e*x + f )

6. Can consistently solve any linear equation with more complex forms than that given in 5.

In each of the above cases, a, b, c, d, e, and f are assumed to be decimals/fractions that are 'nice' (i.e., not involving 'unpleasant' fractions, square roots or other symbols representing irrational numbers). A higher level of expectation is that students can do each of these things when a, b, c, d, e, and f might be any fraction, square roots or other symbols/variables (I am not sure whether these would/should be interleaved or placed after all of the above). Perhaps if students were better prepared with arithmetic (understood that sqrt(2) is a number just like 1.4 is a number) it would not be such a hard step to take.

As an example: yesterday I gave my algebra students a quiz question (which I warned them was coming last week) that none of them could do. I asked them to solve for 'x' exactly (that is,without converting square roots to decimals). Here is the problem: (1/2) * (x - 3) = sqrt(2) * (4 + x)

When we discussed it after the quiz, I asked them if they could do this problem: 0.5 * (x - 3) = 1.4 * (4 + x)

Most of them said they thought they could do the second problem. Note that 1/2 = 0.5 and sqrt(2) is approximately 1.4.

Ideally, they would be able to solve the most complicated linear equations I can throw at them. But do they really need to know how to do it to be successful in the courses that follow? Not really. I would like to have a 'C' indicate that a student who receives this grade has demonstrated that he or she has the knowledge/competencies that are minimally necessary to be (minimally) successful in the next course. A 'B' should indicate the student has greater knowledge/competence in the material, and an 'A' should represent the highest level of competence. This still does not address the issue of gaps, however. How do you grade a student who has 'D' level knowledge/competence in one (limited) area of the course material and 'B' level knowledge/competence in other areas?

-- MattGoff - 08 Feb 2006

Regarding the triangle on the left: Note that the base and height have been labeled in green ink (the graders comments). Height is 4cm and appears to have been given. I didn't notice it at first, however and assumed that the 3cm dotted line was a mis-drawn height (for the base of 12cm).

-- MattGoff - 08 Feb 2006

MattGoff? - out of curiousity, how often do you get students who have "D" level knowledge in one area of the course and "B" level in other areas?

-- TracyW - 08 Feb 2006

In the most frequent case, they might have 'B' level knowledge on the early material (often because they have seen it before) and 'D' level knowledge on the later material (because it's new and they got caught by surprise by what they didn't know). However, even appart from this it's possible to have patchy knowledge (i.e., gaps).

Despite the general trend of accumulation in math, there are different strands that might run along more or less in parallel (with varying amounts of overlap and/or prerequisite knowledge in common). As a specific (fairly common) example, I have students who can solve equations just fine, but can't figure out story problems consistently. Another example would be a student who can solve simultaneous equations but does not know how to analyze rational functions (find domain, identify asymptotes, roots, and graph the functions).

Depending on how far the student goes in math, gaps here and there might not be a big deal later on. Should I keep a student from passing a algebra to meet a general education requirement because they have a few (relatively small) areas of 'D' or 'F' level knowledge? If you just take course averages, the answer is yes. If you demand mastery of every course objective, the answer would be no. I don't know what the best (if there is a best) solution is.

-- MattGoff - 08 Feb 2006

Anne Dwyer:

"This is a totally and completely inappropriate problem for these kids.

"This is an appropriate problem for a high school geometry class that has been working on these things."

I actually disagree with this, at least in part. While it may be too difficult for Christopher's class right now, it's far too easy for a geometry class. For that I'd expect the students to be required to rigorously prove the parallelism of the top of the righthand rectangular piece and the horizontal edge near the top of the figure (for instance).

I actually don't think the problem is inherently too difficult for the kids in the class; no piece of the problem is too difficult to figure out with a very few basic rules. But the concepts of how to figure the area of complex figures should have been explicitly studied before being tested. (Unless this is being used as a discriminator between, say, the A students and the B students.)

-- DougSundseth - 08 Feb 2006

oh wow!

and here I was planning to clean my desk this morning!

-- CatherineJohnson - 08 Feb 2006

I just take issue with the fact that he seems to not have seen any before the quiz. Why the surprise? I do think it's do-able, but since it really is new material coming at them fast and furious, why not have them practice on several examples before throwing a big one at them? Or maybe they did and he doesn't remember. Like Doug said, it might be used simply as a challenge problem to see who can put it all together.

Area of complex figures are in Saxon 8/7 (7th grade math course with some pre-algebra) are introduced, but explained pretty thoroughly and practiced before being tested on.

Hey, at least you come off as Super Math Mom for knowing that the problem would more than likely be on there.

-- SusanS - 08 Feb 2006

Catherine,

Another comment on this test:

Did Mrs. Kahl give any area problems that involved fractions or decimals?

The math wisdom collected at this site says the following: math is sequential. Every child needs to memorize the basics. Then each higher algorithim will give the student spaced practice in the basic skills.

So, what I'm saying is that once the students have learned the basics of area and have done a few calculations with whole numbers, the teacher should give them practice with fractions and decimals. I don't know about you, but I have very rarely in real life gotten a measurement that was a whole number.

Do you think Christopher could calculate an area and perimeter of a rectangle with the dimensions of, say, 33/8 in by 2.8 in?

-- AnneDwyer - 08 Feb 2006

first: YES! I was thrilled to see a score of 80 on this test. Christopher came home thinking he'd failed another test; I was expecting a grade in the upper 60s.

I'm COMPLETELY with Tracy on this; I interpret his score of '80' to mean Christopher is starting to get some degree of flexible knowledge.

I hope to heck I'm not jumping the gun here, BUT......suddenly Christopher seems to be 'getting' math — or, rather, getting the math being taught in this class.

We've seen 3 things:

1. Ed said the other night that Christopher's mental calculations are fantastic (I realize I'm saying that to a bunch of Math Brains, so pls correct for the hyperbole.)

What he meant was that Christopher was quickly and naturally performing necessary calculations in his head — the same calculations he would have had to sit & mull over not so long ago.

At one point apparently Christopher needed to divide 63 by 2, and he quickly came up with the idea of dividing 62 by 2 and going from there (not sure what the problem was).

I know that sounds simple, but believe me: when you're learning math, it's not.

I attribute that to KUMON, probably. Or to the combination of KUMON with everything else. (btw, Ed has been seeing this when he's playing Poker or board games with Christopher, too. I'd say Christopher is reaching automaticity in mental math at that level.)

2. Suddenly, Christopher can do his homework on his own and get the problems right. I don't remember this happening even once until very, very recently. (Possibly an exaggeration, but not by much.)

Last night he had a huge homework assignment compared to what the teacher has been giving.

In the past, she's assigned 3 or 4 problems to do.

Typically he would miss every answer by a mile — either that, or have no idea even where to start to get his wrong answer.

Yesterday the teacher assigned a worksheet out of the worksheet book that accompanies the textbook. 26 problems on ratio. He did all of the problems independently, rapidly, and with reasonable accuracy.

26 problems!

btw, Wickelgren says kids need to do approximately 30 problems a night — or 30 problems per lesson. Last night's assignment was perfect.

Saxon Math alert: This weekend I taught Christopher the Saxon Math 8/7 lesson on ratio — before he did anything in class on ratio. Saxon is an incredible book.

Golly. I wonder if the strategy should be 'preteaching' not 'reteaching'?

I'm going to have to think this over. 'Priming' is a major concept in behavioral research.

That's what I did with Christopher. Priming.

Nevertheless, there've been several other assignments he's done on his own.

3. Last week Christopher was able to perceive that one version of a formula was the same as another, simpler version.

Prentice Hall teaches the formula for finding the area of a trapezoid as:

A = 1/2h(b1 + b2)

Saxon teaches the formula this way:

A = b1h/2 + b2h/2

I prefer Saxon's approach, because you can see, in the equation, exactly what you're doing and why.

Meanwhile, Christopher was busy memorizing all the area formulas in Prentice Hall, just as I did when I was a kid, without attaching much meaning or logic to any of them.

So I insisted he take a look at the Saxon version.

Normally this would have produced screaming, yelling, & eye-rolling, but not this time — or not so much. Christopher obviously didn't feel particularly 'threatened' by this heretofore unseen version of the formula, and he seemed actually to follow it.

I doubt he could do the calculations necessary to turn the Saxon version into the Prentice-Hall version, even though he 'knows' the rules (isolate the variable, do the same thing to both sides)....and he probably didn't completely follow what I was telling him, since in fact he didn't want to pay attention.

BUT his attitude was completely different.

The idea that the 'same number' or 'same formula' can be expressed in different forms is starting to seem normal to him. (It's SO exciting to see this. I can't tell you.)

I sure hope this is real.....but I guess it has to be 'real,' because he's doing it.

If his performance falls apart tomorrow, we'll just have to keep doing what we're doing, which is hanging in there, reteaching, preteaching if possible, and practicing until the clouds begin to part.

-- CatherineJohnson - 08 Feb 2006

Tracy

And his mum is getting adept at reading Mrs Kahl's mind. Or are you breaking into her classroom at night and photographing the test papers?

You have a wicked mind!

(btw, I LOVED Wicked. I was primed to think it was going to be dumb....and then I loved it. I cried at the end, then went out to the lobby and bought souvenirs.)

-- CatherineJohnson - 08 Feb 2006

I actually own this t-shirt.

I wore it to the middle school one day.

I think.

-- CatherineJohnson - 08 Feb 2006

The Koegels' manuals, including a manual on priming

PRIMING MANUAL: Increasing Success in School Through Priming
Intended for parents and educators to help a child with a disability who is included in a regular education classroom to acquire new academic material with more ease and thus also to help reduce disruptive behavior.

-- CatherineJohnson - 08 Feb 2006

Anne & Doug — I'd like to split the difference, if you don't mind.

This is an extremely interesting question (what's too hard, what's not too hard, etc....)

1. If you're following the rules of cognitive science, this is a bad test. The kids haven't seen or done any such problems; they've had no practice; etc.

Christopher's very bright friend M. got a 63 on the test, and the teacher hasn't put the class average on the test, which she normally does. Did a lot of kids flunk? I don't know.

I'm a teach-to-mastery 'radical' on these issues; I want to see every child earn a score of 90% before moving on. (That might be done through testing & then re-testing.....)

2. If the kids had been given practice I don't think it would be too hard. The fact that Christopher could manage an 80 is probably solid evidence.

Personally, I'm enjoying the 'integration' of geometry into pre-algebra....I studied geometry & algebra separately, and, as usual, never made the connection. Algebra was One Thing, geometry was Another Thing.

3. I have zero ability to answer Tracy's question about the triangle.

4. This test is another case of teachers (and everyone else) not realizing what the component parts of skills are.

The visual processing system makes it very difficult to SEE the separate parts of this figure AS SEPARATE PARTS. You must have practice. I say that from direct, personal experience. And any drawing teacher will tell you the same.

people who teach drawing always say that learning to draw is learning to see

What they mean by that is that it's unnatural for human beings to perceive the component parts and shapes of objects & landscapes.

That's the skill required here. The kids have to see parallelograms 'inside' a larger figure.

Furthermore, working memory makes it very difficult to keep track of where you are in a problem of this magnitude. VERY HARD. I'm having a terrible time with my KUMON problems now, which are very long, complicated order of operations problems. I can do every component calculation in the problems in my sleep, but putting them all together in one long calculation, and knowing where I am inside a problem, is shockingly difficult. (I've been meaning to write a post about this, & will soon.)

This is a case where the test is testing several skills at once:

1. how to find area of various figures
2. visual processing
3. working memory
4. flexibility of knowledge & ability to generalize

I don't think it's inherently wrong to test all four things in one exam, but I do think teachers & textbooks should know what they're testing & assign plenty of practice first.

-- CatherineJohnson - 08 Feb 2006

Susan

I just take issue with the fact that he seems to not have seen any before the quiz. Why the surprise? I do think it's do-able, but since it really is new material coming at them fast and furious, why not have them practice on several examples before throwing a big one at them?

EXACTLY

-- CatherineJohnson - 08 Feb 2006

But here's my other thing, and this is more.....slippery.

Psychologically speaking, it wasn't a bad thing for Christopher to have this problem thrown at him on a test.

He's proud he got an 80; he knows it's higher than it would have been just a couple of months ago.....

PLUS the fact that the score is an 80, not a 90, reminds him that he doesn't know what he doesn't know.

He went into the test thinking he knew area formulas cold.

He came back out realizing: gee.

There's more to know about area than just the formulas.

In this case, a score of 80 has been reinforcing & probably motivating.

This is a nebulous area for me.

Educators (using the word purposely) talk about 'challenge.'

The kids need to be challenged.

So far, I haven't thought about challenge. I've been consumed just by the need to get knowledge inside Christopher's head & my own.

My goal has been to break things down sufficiently so he could get it. Make math 'easier,' not 'harder,' then build up.

But obviously, somewhere around this age kids need to.....start seeking challenges.

What I'm saying is: this test violates the principles of cognitive science.

That's obvious.

But for Christopher, at least, it probably respects whatever the rules are of normal boy development.

-- CatherineJohnson - 08 Feb 2006

Anne

As to fractions & decimals, I'm beginning to think this may be a strength of the Prentice-Hall text.

The text constantly uses decimals, fractions, and integers in all chapters.

This particular test doesn't (and I don't think this test was created by Prentice Hall).

I've been trying to figure out why Christopher suddenly seems to be so much more on track in this class.

There are 3 variables, at least:

1. with the switch to the new English class, and the fantastic support the school has supplied, he's been de-traumatized. (MUST SAY IT, SPACED REPETITION: I love this school. It has the standard problems all U.S. middle schools have with curriculum, but the whole place acted when they realized they had a student falling apart.)

in a nutshell: C. is doing better in all his classes, not just math

2. extra help once a week - he's meeting with the math teacher once a week, and this has to be helping. C. has always said her explanations are clear; now he's getting these explanations repeated to him, and being assigned problems to do with her watching him do them

3. distributed practice in Prentice Hall - I'm wondering whether he's now had enough distributed practice to be 'getting' whatever it is he needs to get to do the course

for instance, the first or second chapter was on integers.

He had a terrible time. I was making up worksheets, buying workbooks, Ed was giving him constant practice.....we had a time of it

Ed kept telling me, 'He knows NOTHING about integers.'

Well, somewhere in there he started to know something about integers.

It was during the exponent chapter.

They had to do all kinds of negative exponents, which of course meant more practice on integer operations.

SO.....I think the Prentice Hall book probably embeds 'enough' or at least 'quite a lot' of practice throughout the book.

-- CatherineJohnson - 08 Feb 2006

Susan

Hey, at least you come off as Super Math Mom for knowing that the problem would more than likely be on there.

oh boy, that was fun

this age is SO obnoxious

the eye-rolling enough will do you in

-- CatherineJohnson - 08 Feb 2006

Matt

I'll get your post pulled up front later....

I would like to have a 'C' indicate that a student who receives this grade has demonstrated that he or she has the knowledge/competencies that are minimally necessary to be (minimally) successful in the next course.

To me, this is exactly what a C should mean.

I learned something interesting about one of our neighboring towns (I've forgotten which).

The report card will actually break down the component skills being taught in each course, and use the 1 - 4 scoring system to tell students & parents where the student is on each one.

For the uninitiated:

1 = far below proficient (don't know the wording)
2 = does not meet standards
3 = meets standards (don't know if this means 'proficient')
4 = above standards (again: I don't know if this means 'proficient')

I'm very interested in reform efforts like these.

In fact, I have no idea what an 'A' or a 'B' or a 'C' means (usually).

I'd like to know much more precisely what Christopher does & does not know, and can and cannot do.

I've mentioned this several times, with the TONYSS (Test of New York State Standards, a private test schools can purchase to administer in 'off-years')

We were given no information whatsoever about the nature of the items and the subscales.

I know Christopher failed 'Measurement.'

I have no idea what the 'Measurement' items tested, and neither did Christopher's teacher.

-- CatherineJohnson - 08 Feb 2006

btw, I sent Matt's post about teaching algebra to Carnival of Education; next week I'll send in Smartest Tractor's post.

This is a good one to send, as well, along with Anne's observations on her Math Boosters class.

-- CatherineJohnson - 08 Feb 2006

The way our district defines the "letter" grades is an A is " greatly exceeds district expectations", B is exceeds, C is meets expectations, D is needs support to meet expectations, and F is does not meet. The expectations are defined for each grade and each subject. Then there are also sub-categories below each subject (eg-performs multiplication and division computations, understands fractions, etc.) where they get a + if met, a checkmark/plus if progressing, or a checkmark if not met.

So while the letters are initially based on percentages, I think a teacher has some leeway depending on how the student performance matches up with the expectations.

-- KathyIggy - 08 Feb 2006

Kathy

How well do these categories work for parents?

And how is the report laid out (easy to read, easy to discuss with teacher, etc?)

-- CatherineJohnson - 08 Feb 2006

I think the categories work well once they are explained. They start using letter grades in 3rd grade (before that it's an E for exceeds, M for meets expectations, P for progressing, and N for needs improvement).

At the yearly "curriculum night" at the beginning of the year, the principal always explains a "C" is meeting the expectations for each grade level, and that sometimes (depending on the child, etc) a "C" is just fine. Our school's attendance area includes the "richest" subdivisions in Bloomington (along with more average areas like where I live), and in that group are a lot of parents who are real "pushy" beginning even in Kindergarten and get obsessed with grades even at that point. There is no tracking; I think they "cluster" the gifted kids and special needs kids (so they can decide which classes get an aide, usually)

I think the report is fairly easy to read. There is also an subsection talking about "Learner Characteristics" (does homework, pays attention, etc) where just plus signs or checkmarks are used, along with an area for teacher comments which are not limited to "canned" comments.

-- KathyIggy - 08 Feb 2006

I've got a couple of opinions:

If it is true that the kids had never seen a problem about such a complex figure, then this is really a cheap shot to put on a quiz.

This goes back to a point that I think is very important in learning math, but isn't really in itself math: use lots of paper. Paper is cheap; it is recyclable; show your work. I'll even go further to say that kids should be taught to do as little erasing as possible. It wastes time and makes a mess. Just cross out mistakes and move on elsewhere on the paper. I think it might be good practice to teach kids to use pens rather than pencils when doing math. The first step in doing a problem like this is to write an equation like:

Atotal = Atri + Atrap + Asqu + Acirc + Aparr + Arect

Then, you solve for each component area, and plug back into that big equation. If kids haven't been taught to write this out, then they will try to balance too much in their heads, leading to confusion or failure to include some piece that may have been correctly calculated.

I think it is hard enough to solve this problem that is really six area problems in one. I think that red herring 3cm dimension in the triangle is, therefore, over the top. I think it is already complex enough, which is compounded slightly by the fact that the drawing is not to scale (the 14 cm segment is much longer than the 15 cm segment). I think that's plenty to cope with.

Anne, I see your point about incorporating fractions and decimals. I think, though, that dealing with six different area formulae and properly organizing the solution is enough to do for these kids. In fact, I encounter whole numbers and very simple fractions in real life all the time. When you buy carpet, for example, you don't compute the square yardage to the nearest hundredth, you approximate. The same is true when buying, say, bulk fabric. It's true that once the carpet is installed or the dress is made, you would need more precision to accurately compute the area of material used. But who would care? Do you need to know the exact area of carpet in your room? In many projects, you get to choose your own dimensions. If I am going to hang a banner to advertise a garage sale or design the deck on the back of my house, I get to pick the size. I will always use nice, round numbers.

-- DanK - 08 Feb 2006

This goes back to a point that I think is very important in learning math, but isn't really in itself math: use lots of paper.

I'm having a HUGE problem with lack of paper in KUMON.

The problems are VERY complex (for 4th grade, that is) and there's no paper at all.

It's a mess.

-- CatherineJohnson - 08 Feb 2006

Kathy

That sounds fantastic — do you know how your schools started using it?

Do you like it, personally?

-- CatherineJohnson - 08 Feb 2006

Dan & Anne

I think it's a bad test to give....with a 'positive' side effect in this case.

I would prefer that he do the separate problems he's been taught — and do them with fractions & decimals.

I just scanned some KUMON pages to post; KUMON seems to, from the start, mix fractions & whole numbers in all of its linear equations.

Today I did the first page of equations with unknowns, and there were fractions on it.

KUMON seems to have a rule of never letting you forget fractions are everywhere.....

-- CatherineJohnson - 08 Feb 2006

I'm having to figure out all kinds of tricks to know where I am inside a problem.

These aren't hard problems at all.

They're just problems with multiple super-simple parts.

Actually, now that I think of it, it's a lot like knitting a fair isle or, even worse, XXXX pattern. (I've forgotten the term....)

-- CatherineJohnson - 08 Feb 2006

I really don't know when the format started; it's been the same since Megan was in Kdg the 2001-02 school year.

Though the district academic expectations are on their website, and are sent home at the beginning of the year, I sometimes think they should include these nearly verbatim on the report cards.

FYI, here are the 4th grade expectations: Shortened versions of these are listed on the report card. Mathematics • Demonstrate knowledge of gradeappropriate multiplication and division facts. • Collect, organize, record, display, and analyze mathematical data sets. • Perform more sophisticated operations using addition/subtraction with and without regrouping. • Read, write, compare, order, and round numbers up to 100,000. • Demonstrate knowledge of standard measurement instruments in practical application settings. • Determine equivalent fractions and reduce fractions to lowest terms. • Construct and analyze charts, tables, and graphs. • Identify and describe types of three dimensional figures, angles, triangles, lines, and quadrilaterals.

They also send home progress reports in the middle of every quarter as to the student's grade at this point. I think this is relatively new; they used to only send something home mid-quarter for D/F or "needs improvement" grades.

I don't know how typical our case is though, since our parental helicopter(s) are usually hovering somewhere over the 4th grade classrooms. We typically do not have many surprises at report card time, but of course do have surprises on a daily basis!

-- KathyIggy - 08 Feb 2006

ok, I've had a couple of life-extending glasses of red wine tonight (with Kris!)....so if this sounds wrong tomorrow I'll delete....

one of Christopher's friends actually wrote, on the top of the test, "I was never taught this, just flunk me" — something like that update: that's not what he wrote; that's what he told me he wrote. He said something like, 'I totally messed up, just flunk me.'

then he had to go see the guidance counselor who asked him, 'Why didn't you ask the teacher for help instead of writing something rude on the top of the paper' update: nope! didn't happen (though this is what he told me, so I'm assuming this is how he felt about it. The guidance counselor didn't tell him he was rude, didn't administer scoldings, etc. At this point, I'm not exactly sure how all this came to the attention of the counselor in the first place — the note at the top of the paper? I don't know. He did feel pretty bad about the whole thing, which I didn't like. I don't think a 6th grader should have to flunk a test when he went into it knowing the formulas at least fairly well. He certainly knew the material at a level above an 'F.')

apparently he actually had the presence of mind to say to her, 'I didn't ask her for help because I never saw a problem like this, so I didn't know to ask her' update: I think he did say this to the guidance counselor, who said she would talk to the teacher.

update: Now I'm wondering how Christopher managed to earn an '80,' and I'm wondering whether overlearning helps you survive a test like this.

I bet it does.

C.'s friend did know the formulas, but he was a bit shaky on them, and he seems to have forgotten part of the trapezoid formula while doing the test. Around here, we're so traumatized by tests & Ds that we're getting Christopher to some kind of 'overlearning' point going into the test.

I don't know that Willingham would call it 'overlearning'.....I'd be surprised if Christopher remembers the formula for finding the area of a trapezoid this time next year.

BUT, he had speed-and-accuracy going into the test.

So I'm thinking that when you're given a test much too hard for what you've learned, any degree of automaticity on the content you did learn must help free up resources to tackle the test question.

-- CatherineJohnson - 09 Feb 2006

boy, I hate that language....

(Does it work for you?)

I'm going to have to look at the CA standards again...

-- CatherineJohnson - 09 Feb 2006

I'm trying to think who it was, recently, who was saying a standard shouldn't be a verb.....

-- CatherineJohnson - 09 Feb 2006

I mentioned some time back the Bering Straight School District and the Re-inventing Schools Coalition (that was started in Alaska). There are a number of interesting things about their model that I find appealing.

Execerpts From the Re-inventing Schools Coalition website: (Standards Based Design)

******************

.....

When teachers are using effective instruction, anyone should be able to walk off the street and ask any student what they are learning, how they are learning it and why they are learning it. Students understand how the lessons taught fit into their education plan and what level they are at any time.

.....

Once the standards are in place, it is necessary to make sure students are really learning the lessons. ....students can demonstrate proficiency of what they have learned in a way that is meaningful to them. Multiple assessments that align directly to the standards allow students of all learning styles to be successful.

.....

It is absolutely critical to report to students, parents, staff; ALL stakeholders what is being learned, what progress is being made and how it is making a difference. We need to know that students are really learning. Showing this progress in a way that is understandable helps all stakeholders to embrace the educational process and create the excitement of learning for each student.... At RISC, there are no “traditional report cards” ....each student is assessed with a “Student Assessment Binder (SAB)” that shows what level they are in any given content area, where they need to be to graduate, and what comments they and the teacher have at any given time...

********************

There's a fair amount of language that seems a little high falutin' (for lack of a better term) to me, but the interactions I have had with people implementing the model lead me to think they are very serious about getting kids educated effectively. The Bering Straight School District even has put up a Wiki where people with an interest (their stakeholders) can go and edit the standards. The collaborative work done on the Wiki will be taken into account as the standards are revised (I think they consider revisions each year).

One of the things that is allowed in this system is for students to progress at different paces in different areas. There are certain minimum requirements in each area for graduation, but the standards/levels go beyond the minimum. Students are not tied to a specific level just because they are in a certain grade.

-- MattGoff - 09 Feb 2006

"one of Christopher's friends actually wrote, on the top of the test, "I was never taught this, just flunk me" — something like that"

And the teacher sent him to the guidance counselor?? I would have thought that might have been a clue of some sort for the teacher. . . .

I would be very curious to know what the teacher's purpose is with this quiz. Does she think that the kids should be able to make this leap without being taught how to do so? Or, is she trying to weed out the kids--to be able to differentiate for grading purposes? Or, is she testing to see which kids have flexible knowledge?

Of course, that probably "begs the question" as to what the goal or purpose should be (TEACH TO MASTERY), but it would be insightful to understand her teaching philosophy.

My hunch is that she probably hasn't even thought about it at any length. What's interesting is that the collective brain trust of KTM has probably spent more time analyzing and thinking through this than she may have.

I showed this problem to my 11th grader and here was her analysis:

"When young kids are asked to make these kinds of leaps it becomes frustrating and disheartening. Just when you are starting to feel like you have a good handle on the base knowledge, you take a quiz or test and discover that this base knowledge has been twisted on you in a very complex way."

I think what then starts to happen is that some kids (and their parents, if they are fortunate to have parents who can help them) then start trying to anticipate every possible angle that the teacher might throw at them.

Other kids may just throw in the towel out of frustration and start to think that they aren't good at math. It just seems to me that moving from inflexible to flexible knowledge should be done with incremental steps, not in big leaps.

-- KarenA - 09 Feb 2006

/*
"one of Christopher's friends actually wrote, on the top of the test, "I was never taught this, just flunk me" — something like that"

And the teacher sent him to the guidance counselor?? I would have thought that might have been a clue of some sort for the teacher. . . .
*/

-- if you're part of the solution,
you're part of the proplem. never forget:
the main lesson of public school
is submission to authority.

-- VlorbikDotCom - 09 Feb 2006

the main lesson of public school is submission to authority.

I have to say.....that's pretty much where I am.

I got more details last night.

He DIDN'T write 'I was never taught this.'

He said 'I totally messed up, just flunk me' — something like that.

The guidance counselor wasn't a punishment; apparently they feel like this child's grades are slipping.....

What I don't like at all is the......

It's the assumption that something is 'going on' with the child.

As far as I can tell, the focus will be on the child.

Has something 'happened at home'? (That's a HUGE one; EVERY TIME a child does ANYTHING anyone's concerned about, the question is: 'HAS SOMETHING HAPPENED AT HOME.')

I DO want to be fair.....all (most all) of the folks at school are concerned about the kids, caring, responsible people.

They've turned Christopher around completely.

It's not 'personal,' in other words.

It's structural.

If a child is showing signs of stress, there is no public analysis of how the school might be contributing to the stress.

Now, in private, I hope they're asking themselves: are we doing something wrong?

Or: can we improve our work with this child?

But if they do this, it's not part of the public discourse — neither of the school's, nor of the parents'.

This core assumption — the problem is in the child, not the school — gave us a great deal of power, paradoxically.

We could say: when C. went to school he was in great shape, nothing's changed at home, SO the only 'variable' is the school.

That 'worked,' because in fact everyone there could see that we had a school problem.

But in more nebulous cases, where a bright, happy, hard-working child is 'slipping'.....as far as I can tell, the first thought is 'what's wrong with the child.'

Which is pretty much another way of saying 'submission to authority.'

-- CatherineJohnson - 10 Feb 2006

I told my friend last night, 'I'm a radical on this issue' and I am.

I think she ended up signing the Grade Contract, which we refuse to sign, now or ever. (We've said so.)

She said, 'So how's that going to work. You want Christopher to spend his whole life blaming other people? It's the school's fault I got an F on the test. It's my boss's fault, it's society's fault?'

I need a better answer to that......

In fact, I should probably write a post.

I said, 'We are customers. The school needs to teach our kids, and, when the school fails to teach our kids, the school needs to analyze what they need to do differently.'

I've also said, in other conversations, 'Christopher is a highly responsible child,' which is true. He's super-responsible; he has the kind of responsible nature they warn you about, because he has handicapped siblings.

One thing I don't need in life is a middle school teaching him responsibility. HE'S GOT THAT.

My other friend, J., actually is great on that point.

She said, 'If a kid doesn't get that at home, he's not getting it at school.'

(J. knows some of the parent goings-on better than I do. Apparently there's a Soccer War happening.....so she's got the In's and Out's.)

-- CatherineJohnson - 10 Feb 2006

Karen

Sorry, I had the story wrong. (That was what this child told me, but that wasn't what he'd written.)

In fact, he DID tell the guidance counselor he'd never been taught complex figures before, and she said she'd talk to the teacher.

So it's not just that the school assumes the child has the problem, etc......and, as I've said about a zillion times now, we love this principal. With him leading the school, we're happy — happy meaning we'll carry on lobbying for changes in curriculum and paradigm, but we like and respect the people we're lobbying.

Paradigm is the right word here.

It's universal (as V says).

EVERYWHERE, in any public school I've ever seen, the child is the one who is analyzed when things go wrong.

If a child fails to learn, it is the child's 'issue.'

-- CatherineJohnson - 10 Feb 2006

"When young kids are asked to make these kinds of leaps it becomes frustrating and disheartening. Just when you are starting to feel like you have a good handle on the base knowledge, you take a quiz or test and discover that this base knowledge has been twisted on you in a very complex way."

oh that's beautiful

KEY WORDS POST UP FRONT

The only reason we're not in there screaming and yelling about the test — we will raise the issue — is that Christopher got an 80 AND that he seemed to be emotionally good with it.

He thought he'd failed the test, and to see that he'd been able to get as many things right on a way-too-hard test as he did valuable in his case.

It sure raised my confidence in his abilities. (It's funny that I say 'abilities.' That's how these things work. You keep attributing grades to 'abilities' instead of to 'teaching' or 'practice.' I know better, and I still do it.)

-- CatherineJohnson - 10 Feb 2006

I just stumbled onto a series of essays and editorials written by Terrence Moore, who is a Principal at Ridgeview Classical Schools in Ft. Collins, Colorado. They are terrific. Here is the link:

I wasn't sure where to post this . . . .

-- KarenA - 10 Feb 2006

Which dimensions on the drawing are Christopher's, and which were given as part of the test? If the (3cm+6cm) and the center 9cm lines are really parallel, the hypotenuse of the "one triangle" on the left CAN'T be 12cm... it has to be greater than 14cm.

-- OldGrouch - 10 Feb 2006

If you use the marked numbers on the test of 4cm for the height and 13cm for the base, the hypotenuse of the triangle on the left is the square root of 185 which is between 13 and 14. But the base of the triangle on the left marked 10cm cannot be correct. If you draw a line parallel to this 10 cm base with its start at the right hand corner where the circle is and drop it to the bottom, you get a right triangle with a hypotenuse of 14 and sides 3 and 10. This is not possible since 32 + 102 does not equal 142.

Also, since the horizontal line marked 6 cm is parallel to the horzonal line marked 9 cm, the two vertical sides of the resulting parallogram have to be parallel and the same length. But, as I've pointed out, the length cannot be 14 cm because the sides are 3 and 10.

So the student who said he couldn't do this problem was absolutely right. You can't do this problem if you try to get all the number right because they don't come out right.

Does this mean that Mrs. Kahl doesn't understand her geometry? The funny thing is, when I put in my original post about this being a problem for high school geometry students, I believe it was because my intuitive brain recognized the problems, but didn't articulate the words to my verbal brain.

Thanks to Old Grouch for pointing out all the errors.

What do you do now, Catherine, send it back with corrections?

-- AnneDwyer - 11 Feb 2006

Lessee if I can make this work... redrawn problem:

-- OldGrouch - 11 Feb 2006

Here:

-- OldGrouch - 11 Feb 2006

KEY WORDS TO POST OLD GROUCH

-- CatherineJohnson - 11 Feb 2006

Was there a circle tacked on, too?

-- SusanS - 11 Feb 2006

YES!

YES!

YOU FORGOT THE CIRCLE!!

-- CatherineJohnson - 11 Feb 2006

Remember, the 10 cm vertical line segment, CE, is bogus too based on a hypotenuse of 14 (DF) and a height of 3 cm.

Also, based on the parallelogram BDEF, that 3 cm (BC)has to be an extention of line CE. In order for BC to be the height of triangle ACE, it would have to be perpendicular to AE.

-- AnneDwyer - 11 Feb 2006

OldGrouch: Good catch!

-- CarolynJohnston - 11 Feb 2006

My posted comment (just before AnneDwyer?) has vanished!

-- CharlesH - 11 Feb 2006

(From Carolyn: Charles, it looks like it got deleted by accident. Here it is again:)

"YOU FORGOT THE CIRCLE!!"

Tagging on a circle like a balloon dancing on the nose of a sea lion is pointless. There is no reasoning involved. Either you know or don't know how to calculate the circumference or area of a circle. A discrete task will do as well.

Now, using semicircles requires some reasoning. A complex figure I like is to surround a square with four semicircles (find area of concoction) or sticking two circles into a rectangle (abuting) and giving only the length of the rectangle. The task is to calculate the area of the rectangle not covered by the circles. Solving this requires some exciting reasoning. Of course, first you need to know the basics, i.e. how to calculate the area of these figures.

On the issue of false rigor, I just posted something that strikes me as a ludicrous example of false rigor. http://instructivist.blogspot.com/

-- CharlesH? - 11 Feb 2006

Many thanks for the resuscitation act, Carolyn.

-- CharlesH - 11 Feb 2006

I didn't include the circle out of laziness. So, assuming an invisible circle... ;-)

The contradictions led to my curiousity about the sources for the figures.

AnneDwyer? nailed it about either the 10cm height or the 14cm DF being wrong. (I hadn't noticed that!) Actually, 14cm COULD work, but only if BD isn't parallel to EF.

I started by assuming that BDEF was a parallelogram, which would make BE=DF=14cm so AE has to be greater than 14.

Then I looked at ACE. If all the lengths (except the 12cm) are "true," we have a 3-4-5 right triangle with hypotenuse AC and the third vertex the unlabeled point to the right of A. (Okay, call it "G"). If we assume (dangerous) that the 3cm side CG is an extension of EC AND is exactly perpendicular to EF, then that establishes where point A lies on the plane.

Which leads to an answer to the question in red: Either: (1) Does right triangle ECB have a height of 3cm, given that line EB must continue to point A. If true, BC is an extension of CD (if CD is parallel to EF).

Or: (2) Does triangle ACE have a height of 3cm? If true, BC is NOT an extension of CD (if CD is parallel to EF).

Then we have the case where CD is NOT parallel to EF...

At which point I threw up my hands.

-- OldGrouch - 11 Feb 2006

But BD is parellel to EF by the definition of the problem: if two lines are perpendicular to the same line, then they are parallel.

So by putting the right angle symbols in there, the teacher made BD parallel to EF, Thus, if CE 10, then DF cannot be 14. If DF is 14, CE cannot be 10. But whatever you do, because BDEF is a parallelogram (again by definition of the right angles), BE cannot be 12 nor can AE not be twelve. It is then impossible to get the area of anything above EF.

This problem cannot be done and certainly cannot be done by 6th graders.

-- AnneDwyer - 12 Feb 2006

Anne and Old Grouch,

I believe you have proven that the distances labeled on the problem are bogus. It can't work.

I think, though, that what the teacher had in mind was to call CDFE a trapezoid. The problem there is that she apparently labeled that little dotted line BC = 3 cm. She should not have included that. Then, it would be possible for angle BEF to have a different measure than angle BDF, and BDFE would not be a parallelogram. It is not entirely clear if the 12 cm dimension applies to segment AE or segment BE. Either way, it should have been left off, because it is another bogus value. If you know the lengths of segments EG and AG, you can find the area of triangle ACE. If only she would have left those two values unlabeled, it would have worked...I think.

-- DanK - 12 Feb 2006

A complex figure I like is to surround a square with four semicircles (find area of concoction) or sticking two circles into a rectangle (abuting) and giving only the length of the rectangle. The task is to calculate the area of the rectangle not covered by the circles. Solving this requires some exciting reasoning

Those are WONDERFUL problems.

That is exactly the kind of problem the kids needed to move on to next — and this is the kind of problem RUSSIAN MATH & SAXON move on to.

Those problems really do move you on to the 'next level,' make you think, make you generlize & transfer knowledge, etc.

I remember being SHOCKED the first time I saw a problem like that!

(Which leads me to wonder what on earth I was doing in high school geometry.)

-- CatherineJohnson - 12 Feb 2006

I haven't studied this problem, but just glancing at it, it looks interesting & potentially do-able:

One altitude of an equilateral triangle is a side of one square, and one side of the same equilateral triangle is a side of a second square. The area of the larger of these squares is 56. What is the area of the smaller of the two squares?

Geometry Problem of the Week

Apparently this was a problem given to 8th graders in Australia. They've got their solutions posted, too.

(Christopher certainly couldn't do this without some practice.)

-- CatherineJohnson - 12 Feb 2006

I wouldn't have the area of the square be 56, however (not for 6th graders who've never found square roots).

I've have it be 49 or 64....

-- CatherineJohnson - 12 Feb 2006

oh, wait!

'Senior High School' & Grade 8 — they must do their grades differently — what grade are these kids in?

-- CatherineJohnson - 12 Feb 2006

Here's another problem:

Find the area of this basketball key. The end is a semi-circle. Use = 3.14

-- CatherineJohnson - 12 Feb 2006

The Geometry Problem of the Week is fun, because you don't actually need to take any square roots.

-- TracyW - 12 Feb 2006

you don't?

I thought you did.....

-- CatherineJohnson - 12 Feb 2006

oh goodness.....of course you don't

56 ÷ 4 = 14

sheesh

-- CatherineJohnson - 12 Feb 2006

it IS fun, isn't it?

-- CatherineJohnson - 12 Feb 2006

We're turning you into a maths brain.

-- TracyW - 13 Feb 2006

LET'S HOPE SO!

-- CatherineJohnson - 13 Feb 2006

/* the main lesson of public school is submission to authority.

no ... but my main man j. t. gatto sure has.
for instance, the six-lesson schoolteacher (1991).

-- VlorbikDotCom - 14 Feb 2006

oh fantastic!

thanks!

(I've read...A DIFFERENT KIND OF TEACHER)

-- CatherineJohnson - 14 Feb 2006

Slightly off topic, for Catherine to tuck away for future reference:

If you want Christopher to be a successful calculus student down the road, when his math class covers slope (probably next year?) (probably in the context of graphing lines), make sure he understands every detail that is included.

Calculus is (roughly speaking) the study of finding areas, finding slopes, looking at tiny pieces of things, and how all these ideas are related.

-- RudbeckiaHirta - 15 Feb 2006

Rudbeckia THANK YOU

I know nothing about calculus, and plan to take it myself.

I've been writing down EVERY PIECE OF ADVICE anyone's offered.

-- CatherineJohnson - 15 Feb 2006