anonymous on the SAT math learning spike

anonymous writes:

I am like that with names. At the start of the school year, I need to learn 150 names. For the first week it is slow going and mostly by memory tricks. Then all of a sudden, I know almost every name automatically.

I wonder if your speed has increased because you have become better at instantly categorizing the math questions.

That's a good question!

I'm going to start keeping notes.

I continue to have the "implicit learning" experiences I've mentioned before, where I'll know that an answer is right without knowing why -- or, in some cases, I'll find myself on the path to solving a problem correctly while consciously thinking I'm doing it wrong.

I'll have to check my books on learning and memory to see if it's the case that implicit knowledge shows up before explicit knowledge. (Implicit knowledge is sometimes called the "cognitive unconscious," a term I'm keen on.)

If I had to guess, I'd say anonymous is right: I'm recognizing problems faster. I'm already as fast as I'm going to get at doing the actual calculations, and I don't think I've boosted my speed at setting up word problems (which I need to work on).

Also, my 'number sense,' for want of a better term, isn't especially good. That is, I don't read a problem and think 'the answer has to be in the neighborhood of thus and such because of thus and such.' My math knowledge continues to be fairly inflexible, so I don't take shortcuts doing the problems because the obvious shortcuts aren't obvious to me.

Not unless the problem is super-easy. Here is problem number 2 from yesterday's test:

A machine requires 4 gallons of fuel to operate for 1 day. At this rate, how many gallons of fuel would be required for 16 of these machines to operate for 1/2 day?

I started out setting up unit multipliers and quickly got stuck: I cannot for the life of me make unit multipliers work on an SAT math section. Why? Very frustrating.

So I was sitting there burning time on QUESTION NUMBER 2, the 2nd easiest question I was going to be doing, and finally I just bagged the dimensional analysis, looked at the problem again, and said to myself: "If it's 4 gallons for 1 machine for 1 day, then it's 2 gallons for 1 machine for 1/2 day, so if I've got 16 machines that's 16x2 and that's 32."

I was very happy to see 32 amongst the answer choices.


SAT genre

The other piece of evidence that anonymous is right -- our increase in speed is due to increased recognition (categorization) of what we're looking at -- is the fact that I can now tell a "genuine" SAT math problem apart from an ersatz SAT math problem. (I'm going to try to find out whether C. can tell the difference.)

Interestingly, I could tell the difference between a real SAT reading question and an imitation SAT reading question from the get-go, just about, and of course the reading section is what I'm good at.

young lives destroyed by failure to learn dimensional analysis

"I was a Wilton High School student who dozed off while Mr. Laptick taught us dimensional analysis in physical science. I never quite got the hang of it. It irritated me... all of those fractions. I never really liked fractions. Although my grades had been pretty high, I got a D in physical science and subsequently dropped out of chemistry in the first quarter of my junior year. It was not long before I started on drugs, and then crime to support my drug habit. I have recently learned dimensional analysis and realize how simply it could have solved all of my problems. Alas, it is too late. I won't get out of prison until 2008 and even then, my self image is permanently damaged. I attribute all of my problems to my unwillingness to learn dimensional analysis." Jane

"I thought I knew everything and that sports was the only thing that mattered in high school. When Mr. Hoogenboom taught our class dimensional analysis, I didn't care about it at all. I was making plans for the weekend with my girlfriend who loved me because I was a running back and not because of physical science. While other kids were home solving dimensional analysis problems, I was practicing making end sweeps. Then one day I was hit hard. Splat. My knee was gone. I was despondent. My girl friend deserted me. My parents, who used to brag about my football stats, started getting on my case about grades. I decided to throw myself into my school work. But I couldn't understand anything. I would get wrong answers all of the time. I now realize that my failure in school came from never having learned dimensional analysis. Alas, I thought everyone else was smarter. After the constant humiliation of failing I finally gave up. I am worthless. I have no friends, no skills, no interests. I have now learned dimensional analysis, but it is too late." Bill


I was at home, sick with the flu when Mr. Mycyk taught my class about Dimensional Analysis. Despite opportunities given to me to make up the assignments that I had missed, I chose to not do them. I thought that my mathematical abilities were already sufficient. How wrong I was! It’s been five years since I took that class--Now I spend my afternoons panhandling at traffic lights, hoping for passersby to give me spare change. If I ‘m lucky enough to scam a buck after a day’s work, I’m still not sure if my hourly rate makes cents. --Mario

source:
Dimensional Analysis

dimensional analysis problems from ktm-1

You may have to hit refresh a couple of times.


Dan K on dimensional anlysis
dimensional dominoes from Dan K
dimensional analysis worksheets from Dan K

a way to teach unit conversion (Carolyn Johnston)
teaching Christopher unit multipliers

dimensional analysis word problems and answers
another cool dimensional analysis problem
solution
dimensional analysis problem from Math Forum
Dr. Ian talks about fractions and units
another triumph for dimensional analysis

dimensional analysis emergency

teach your children unit multipliers....

from Gary Carson:

I managed to pass both intro chemistry and intro physics in college without learning anything at all about either subject just by being able to manipulate dimensions to arrive at an answer for exam questions. In one semester of Chem I even made a B and had no clue about the actual subject matter.

I guffawed (inwardly - I do not guffaw out loud) when I read Gary's comment.

This is exactly the way I felt when I first laid eyes on unit multipliers and figured out what they were. (For the uninitiated, the single best place on the web to look at unit multipliers is Donna Young's homeschool website. Click on "Math" at the left. Then click on "Unit Multipliers" at the bottom right.)

Ed's cousin, a chemistry teacher at a high performing high school in IL (has a Ph.D. in chemstry) told us that a lot of his students come into his class not having the first clue about fractions or ratios. The best students do, he said, but no one else.

So the question is: is he going to teach remedial math, or is he going to teach chemistry?

He teaches them chemistry and unit multipliers.

dimensional analysis advice from Susan J

As a former chemistry teacher I would say you cannot overemphasize an understanding of dimensional analysis.

For kids who don't see the point, ask them "backwards" questions such as how many feet are in an inch?

Also, have them do long chains such as determining how many centimeters in a mile. It's good to have figured these out in advance yourself so your student is instantly rewarded if they get the right answer.

decline at the top

a comment left by Miller Smith in the Direct Instruction thread at joannejacobs:

70 to 80% of the time I do DI. My students don’t know enough about the math they should be perfect at in order to complete activities in the constructivist method.

For example: I wanted the students to find the mass and volume of five selected metals and then plot that data on a mass vs. volume graph, find the slopes of the lines (find density of the metals) and then compare the order of the densities to the metal’s position of on the Periodic Table to show the trend of density of the elements.

The students ALL have taken Algebra I and II and Geometry with very good grades in all (I have all honors classes and in 11th or 12th grade).

They could not find the mass of the samples using electronic scales (didn’t know about the tare), or the volume of the metals (didn’t know about finding volume by diffence), could [not] plot a graph on paper (they used graphing calculators), couldn’t find the % error from a provided equation, on and on. They couldn’t DO the constructivist method of ‘discovering’ the trend of the elements in the Periodic Table since they did not know how to do the math and science in the classes they already passed with wonderful grades!

When the University of Maryland science professors held a meeting with the science department heads in Prince George’s County this past fall, they told the science folks for my county that is was assumed that students from the county didn’t know anything about math or science.

I do DI almost all the time. I directly teach the lower level skill the students should have learned years before before I put the students in the lab. This is very slow - at first. Since we use math all the time in chemistry (ha! Who’d a thunk it!) the student start getting very good at these skills. They then start getting the point of the labs.

These students have been so abused academically by my county with constructivism. DI should be first and foremost. When basic skills are mastered then, and only then, can you put students in an environment to discover things using the tools they have.

A couple of years ago I talked to the Dean of Liberal Arts (I think it was) at a college out on Long Island. He was a math guy (I'm thinking a mathematician, but he may have been a scientist).

When I asked him about students' knowledge of math he told me, "We can't assume students know anything we would want them to know."

This included being able to solve a linear equation with one variable.

(The phrase "decline at the top" isn't mine, but I don't remember who coined it. Waiting for Utopia discusses decline at the top.)

update: Diane Ravitch used the term in 1997.

Over the years researchers have debated the meaning of the decline in SAT scores. Some have concluded that it is solely a reflection of the democratization of American higher education meaning a growing number of minority, low-income, and low-ability students in the test-taking pool. Certainly, changing demographics contributed to the decline, yet something more was happening. Declines occurred at the top of the ability distribution, especially on the verbal part of the test. For example, in 1972 (the first year for which comparable data were available), 116,585 students - 11.4 percent of test takers -scored higher than 600 on the verbal test. By 1983 that number had fallen to only 66,292, or 6.9 percent of the total. Since then the proportion of high-scoring students has remained around 7 percent. By contrast, in mathematics the decline at the top was only temporary. In 1972, 17.9 percent of test takers scored over 600. That proportion dipped as low as 14.4 percent in 1981, but by 1995 it reached 21 percent - the highest proportion of students ever to exceed 600 on the math test.

question for the instructivist

Instructivist left this comment:

When I was teaching DI to 8th graders some of the hurdles were understanding that conversion factors like 1 ft/12 in mean one and that we are taking advantage of the identity element for multiplication. Another hurdle was to figure out which unit of the CF should be in the numerator and vice-versa. I thought I had developed crystal-clear strategies for a foolproof approach, even though the approach didn't sink in easily.

I always insisted on writing out each step and wrote the direction of the conversion on top of the problem with an arrow to minimize confusion, e.g. sec --> hours. Not all students were converts to my approach to conversion.

I don't quite get the arrow part.

I wish to heck I'd kept more notes on dimensional analysis.

Saxon teaches dimensional analysis throughout all his books, starting maybe in 7-6.

Several times I've thought I had it down cold, and then encountered a problem that stumped me.


notes:

• every single time we work on dimensional analysis I say to Christopher: "What does 1 ft/12" equal? Then I wait 'til he tells me it equals 1. Sometimes he doesn't tell me it equals 1, so I tell him. Then I say, "Why can we multiply the initial value by 1 ft/12"? That he always gets: we're multiplying by 1. Then I say, "Have we changed this initial amount? Is it a different amount after we've done all this multiplying by unit multipliers?" He gets that one, though he's slightly hesitant.... "...No..." Finally I say, "What has changed?" He may or may not say that the unit has changed, but that's only because he's not necessarily following my train of thought. As soon as I say it he gets it. I've become a huge fan of scripted instruction. I do this script every time we work on unit multipliers; when Christopher reaches the point where the script seems stupid and obvious to him I'll know he's got them conceptually as well as procedurally. (Or at least that he's got a far more solid conceptual understanding than he did when we started out.)
  

• Christopher has no trouble figuring out which unit has to go in the numerator and denominator, and neither did I including back when I first learned unit multipliers. I think there's something visual about it (and I believe visual memory is "stickier" though I have yet to review all that research).
  

• He sometimes gets confused about which number is which: for a particular problem he'll know he has to put yards in the numerator and feet in the denominator, but he'll write 3 yards/1 ft because 3-to-1 makes more sense or is more familiar (you probably know what I mean). So, although he has zero confusion about the canceling aspect of unit multipliers, the very fact that sometimes yards will be in the numerator and sometimes they'll be in the denominator can trip him up.

• I had a bit of trouble moving from "easy" unit multipliers (centimeters to yards) to rate unit multipliers (mph to meters per second), but Christopher has had no trouble at all. I'm sure that's because Christopher still writes the number 1 in the denominator of the rate: 60 miles/1 hour. I wish I'd thought of that. For quite awhile I kept thinking things like, "Wait! I have two units in the numerator! (60 mph - it's all one chunk) What do I do now!" This is one of those times where having a fresher brain is an advantage.

• I think the single hardest aspect of unit multipliers is knowing which number to put first. To this day I don't quite know whether it matters; I've gotten jumbled up in long problems before and had to unjumble myself by deciding there was one, and just one, number that could start the whole thing out. When Christopher reads a simple unit multiplier word problem I have him circle the value to be translated and underline the unit he's supposed to end up with. He's not particularly interested in doing that, but on the other hand the fact that we have done it seems to have made it fairly easy for him to figure out where to start.

• I have him do the cancellations as he goes along. I learned this the hard way. By the time you get to Saxon Algebra 2 you're doing some long-chain dimensional analysis; more than once I've lost my place and had to start over.

• having the student write two unit multipliers for each conversion (1 yd/3 feet versus 3 feet/1 yd) is a very good thing to do.

• dimensional analysis word problems are also a very good thing to do. For me it was a terrific exercise to use dimensional analysis to solve everyday word problems I had never used DI to solve before.
  

• Terrific DI problem from Saxon Math: The Adams' car has a 16-gallon gas tank. How many tanks of gas will the car use on a 2000-mile trip if the car averages 25 miles per gallon? 
(source: Saxon 8/7 Lesson 96 page 660 #3 - answer: 5 tanks)

  
  

Dimensional analysis is the simplest procedure on the planet, and yet it's strangely challenging to learn. I think this is entirely due to Wickelgren's observation about all math looking alike.

Dimensional analysis is the ultimate exemplar of the practice, practice, practice theory of knowledge.

It's easy, but it's confusing.

Practice solves that problem.

2nd exposure

Yesterday Ms. K taught a lesson on dimensional analysis.

The last time she taught a lesson on dimensional analysis was March 10, 2006.

Today is February 9, 2007.

According to the math department an 11-month gap between a first exposure and a second exposure is fine.

It's more than fine, actually.

If a student has been exposed to a topic for one week in 6th grade, and then again for another week in 7th grade, he should be ready and able to take a test.

And not just any test, either. He should be ready and able to take a complicated test filled with multi-step first-applications of the topic or skill.

I'm not surmising this, by the way.

Ed and I were directly told this by the math chair, who was defending Ms. K's latest test which half the kids had hosed, and which we had no interest in discussing in any event no matter how many kids hosed it. We've given up on Ms. K's tests. We've given up on Ms. K! We'd come to discuss curriculum and pedagogy; the math chair had come to defend the test. Under no cricumstances, she said, would she discuss curriculum and pedagogy with a parent. Any parent.

So she discussed the test and we discussed curriculum and pedagogy.

That's how we know the chair of the math department thinks 11 months between exposures is fine and the kids should be ready to take a test.



This is the kind of thing that gets me even more revved than I already am.

I'm handing this one off to Mr. Engelmann:

Typically about 60 school days pass before any topic is revisited. Stated differently, the spiral curriculum is exposure, not teaching. You don't "teach" something and put it back on the shelf for 60 days. It doesn't have a shelf-life of more than a few days. It would be outrageous enough to do that with one topic-- let alone all of them.

...Don't they know that if something is just taught, it will atrophy the fast way if it is not reinforced, kindled, and used? Don't they know that the suggested "revisiting of topics" requires putting stuff that has been recently taught on the shelf where it will shrivel up? Don't they know that the constant "reteaching" and "relearning" of topics that have gone stale from three months of disuse is so inefficient and impratical that it will lead not to "teaching" but to mere exposure? And don't they know that when the "teaching" becomes mere exposure, kids will understandably figure out that they are not expected to learn and that they'll develop adaptive attitudes like, "We're doing this ugly geometry again, but don't worry. It'll soon go away and we won't see it for a long time"?

The Underachieving Curriculum judged the problem with the spiral curriculum is that is lacks both intensity and focus. "Perhaps the greatest irony is that a curricular construct conceived to prevent the postponing of teaching many important subjects on the grounds that they are too difficult has resulted in a treatment of mathematics that has postponed, often indefinitely, the attainment of much substantive content at all."

War Against the Schools' Academic Child Abuse, pp. 108-9

The good news is, I spent the past week having Christopher do dimensional analysis problems.

hah!

I've been teaching Christopher how to use unit multipliers off and on since January 24, 2006. It's been more off than on, extremely sloppy teaching.

But it's been "on" enough that he always has some residual memory when I wake up one day and remember I haven't given him any practice on unit multipliers in a great long while.

I desperately need an afterschooling drill-and-kill book.

I need a book that has pages and pages of dimensional analysis problems of all kinds, along with pages and pages of various multi-step complex problems using algebra, geometry, baby statistics & probability, stem and leaf charts, etc. - I need it all.

In one book.

Anyway, about a week ago I started having Christopher do dimensional analysis problems every day. Five or six of them. My goal this time, and I'm sticking with it until it happens, is for Christopher not only to be able to do dimensional analysis problems, but to do them fast.

We're going to carry on doing dimensional analysis problems until the state test in March; then I'm going to write on my calendar the next date he should do some more of them.

question:

when is that date?

how long am I supposed to wait?

how long can I wait?

I bet Engelmann knows.

One of these days I'll get around to reading his book.

After Christopher gains speed and accuracy I am going to carry on having him do dimensional analysis for the next 2 years so he'll remember unit multipliers for the rest of his life.

[pause]

Oh, fine.

I can no longer find the Dan Willingham article I was positive said that if you study the same thing 3 years in a row you remember it forever.

sigh

The good news is that because I just spent a week having Christopher do dimensional analysis problems he was able to do Ms. K's overly-complicated homework (multi-step dimensional analysis word problems) with ease.

In fact, multi-step dimensional analysis word problems were exactly what he needed.

I hope this is evidence I'm beginning to channel the mind of Ms. K.

Life would be a lot easier around here if that were the case.

rules for installing a new curricula

dimensional analysis word problem

I love this guy.

Because you never learned dimensional analysis, you have been working at a fast food restaurant for the past 35 years wrapping hamburgers. Each hour you wrap 184 hamburgers. You work 8 hours per day. You work 5 days a week. You get paid every 2 weeks with a salary of $840.34. How many hamburgers will you have to wrap to make your first one million dollars? [You are in a closed loop again. If you can solve the problem, you will have learned dimensional analysis and you can get a better job. But, since you won't be working there any longer, your solution will be wrong. If you can't solve the problem, you can continue working which means the problem is solvable, but you can't solve it. We have decided to overlook this impasse and allow you to solve the problem as if you had continued to wrap hamburgers.]

He's got one about college applications, too:

A Wilton High School senior was applying to college and wondered how many applications she needed to send. Her counselor explained that with the excellent grade she received in chemistry she would probably be accepted to one school out of every three to which she applied. [3 applications = 1 acceptance] She immediately realized that for each application she would have to write 3 essays, [1 application = 3 essays] and each essay would require 2 hours work [1 essay = 2 hours]. Of course writing essays is no simple matter. For each hour of serious essay writing, she would need to expend 500 calories [1 hour = 500 calories] which she could derive from her mother's apple pies [1 pie = 1000 calories]. How many times would she have to clean her room in order to gain acceptance to 10 colleges? Hopefully you didn't skip problem No 1. I'll help you get started.... 10 acceptances [ ] [ ] etc.