kp on the MathCounts course and SAT math

kp writes:

I took the AOPS Advanced MathCounts class this summer (I'm a coach and wanted to see what it would be like for my students and what new things I could learn from it.) I appreciated that the class taught the shortcuts but also focused on how the shortcuts worked and how you could adapt them when the problem was given a new twist. (For example, to find the number of factors a number has, first find its prime factorization, then add 1 to each of the exponents, then find the product of these numbers. They explained why this made sense, then assigned different variations of problems on this topic.)

I'm no expert on the SAT (I'm a middle school teacher), but there does seem to be a large overlap between hard middle school math and what is on the SAT. We sometimes use SAT practice problems in our MathCounts practices. Our district's merit scholars often participated in MathCounts in middle school. Perhaps that is because the type of kid who stays after school to do math is the type of kid who is also successful on the SAT, or perhaps it is because MathCounts helps to prepare them for the SAT.

The MathCounts course sounds like a blast.

Teaching the shortcuts is a great idea -- it's the shortcuts that help you see what's actually going on, I think.

I remember years ago reading an article -- it may have been a study -- about smart-works-hard type students versus the 'naturals.' The smart-works-hard types went on wild goose chases trying to solve problems, while the naturals produced short, elegant proofs and solutions. I laughed, reading that, having been on many a wild goose chase myself.

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.

PSAT Math

Since my son gets to take the PSAT test as a sophomore in October, I thought that it would be a good place to start. Taking into account the warnings of "imposter" questions, I decided to get some real tests. Unfortunately, they are not offered by the College Board anymore, so I had to get them through Amazon. I got them the other day, and I thought I would share my initial impressions.

First, the real questions are so enlightening compared to a 3rd party PSAT test book I bought. The questions on the test are simple, ... but speed and accuracy are very critical. In the first section, you have 20 questions to answer in 25 minutes. Even though some questions are trivial, it still takes you time to read the question, check to make sure you are not jumping to conclusions (that it really is that simple), and then fill in the bubble. It's one thing to do this informally, but if you try to simulate a real test situation, the seconds slip by very quickly, and the pressure to rush increases.

Second, some of the questions go out of their way to try and trick you into the wrong answer. In one question, you find x+y, but they ask for the average of the two numbers. One answer is x+y and another is (x+y)/2. Clearly, the test is not just about your knowledge of math. I think there were 3 out of 20 questions where they went out of their way to mislead you.

Third, there are certain things you need to know very quickly, like the ratios of the sides of a 45-45-90 triangle and a 30-60-90 triangle. It's not good enough to know the Pythagorean Formula or to find the answer with trig. The formula is too slow and trig gives you the result in decimal, when the answer is in radical form.

One test example was to find the height of a equilateral triangle with a side length of 8. Right now. Fast. With trig, you get 8*sin(60) = 6.928, but the answer is in radical form. With the formula, you get sqrt(64 - 16) = sqrt(48). Quick, what is the reduced radical form? Your brain might start to freeze before you find 4*sqrt(3).

You really need to be able to immediately write down any other side of these common triangles given any one of the sides. One would like to use simple concepts and mathematical understanding to solve many things, but that's not what the test is all about. The test is all about saving seconds. I only finished it with about 3 minutes to spare.

I know this is nothing new to SAT wonks, but I hope to share some of the math shortcuts I find.

How To Study, from A Pro

From the British Psychological Society's blog, BPS Research Digest:

Some advice from Nate Kornell:

Although as students we have all spent countless hours studying, we receive little guidance in how to study effectively. There are no shortcuts to effective studying, but in general, being actively involved in learning makes studying effective. Some specific points are obvious: pay attention in class, do the reading, don’t procrastinate, while others should be obvious but aren’t: study in a quiet place without distractions, don’t send text messages during class, ask questions if you are confused.

Here are three unintuitive but very effective ways of studying based on findings from psychological research:

Also check out Professor Kornell's webpage for more provocative information.

alert the vice president

I wonder if Lynne Cheney is aware that the DOE is telling parents not to subvert fuzzy math.

MATH HOMEWORK TIPS FOR PARENTS

• Encourage your child to use a daily math assignment book.

• Follow the progress your child is making in math. Check with your child daily about his homework.

• If you don't understand your child's math assignments, engage in frequent communication with his or her teacher. [ed.: good one!]
  

• If your child is experiencing problems in math, contact the teacher to learn whether he or she is working at grade level and what can be done at home to help improve academic progress. [ed.: If students need distributed practice, parents can find worksheets online.]

• Request that your child's teacher schedule after-school math tutoring sessions if your child really needs help. [ed.: been there. done that.]
  

• Advocate with the principal for the use of research-based peer tutoring programs for math. These tutoring programs have proven results, and students really enjoy them. [ed.: aarghhhy]

• Use household chores as opportunities for reinforcing math learning such as cooking and repair activities. [ed.: aarghhh]

• Try to be aware of how your child is being taught math, and don't teach strategies and shortcuts that conflict with the approach the teacher is using. Check in with the teacher and ask what you can do to help. Ask the teacher about online resources that you can use with your child at home. [ed.: aarghhh, aarghhh]

• At the beginning of the year, ask your child's teacher for a list of suggestions that will enable you to help your child with math homework. [ed.: how about no]

file under: the right hand doesn't know what the left hand is doing

from the same source:

Find out whether your child's teacher is highly qualified and whether the school follows state standards for mathematics instruction. Ask the school principal for a school handbook or math curriculum guide. If your school doesn't have a handbook, ask the principal and teachers questions such as the following:

• What math teaching methods and materials are used? Are the methods used to teach math based on scientific evidence about what works best? Are materials up to date?

• How much time is spent on math instruction?

• How does the school measure student progress in math? What tests does it use? How do the students at the school score on state assessments of math?

• Does the school follow state math standards and guidelines?

• Are the math teachers highly qualified? Do they meet state certification and subject-area knowledge requirements?

If you have not seen it, ask to look at the No Child Left Behind report card for your school. These report cards show how your school compares to others in the district and indicate how well it is succeeding.

so.... Don't interfere with the math instruction on offer at your kids' school, but do grill your principal to within an inch of his life about the curriculum, the research base, and teacher qualifications.

I'm sure that'll work.

The Feds must resolve disputes amongst the warring parties by lopping white papers in two & assigning half the pages to one camp, the other half to the other camp.

Party number one gets to write Homework Tips on page 5; Party number two gets to write the Homework Tips on page 11.

[pause]

I wish my school had a handbook.

Or a math curriculum guide.

Or anything in writing at all.

[pause]

Of course, we do have a handsome set of form letters.

Hard to say which one's my favorite.

if you listen

Ed always says that if you listen people will tell you what they think.

After he read the list of "tips" he pointed out that the list assumes the school isn't going to be teaching your kid any math.

Three of the tips are about peer tutoring and "extra help."

Note that peer tutoring is research based while extra help is not.

This has been our experience.

Extra help doesn't help.

[pause]

This weekend I discovered that one of C's friends, who is in the regular track and shouldn't be is being tutored by a high school student.

This is my middle school.

Very bright kids are a) in the slow track and b) not learning math there, either.

Hack the SAT on the writing section

Hack the SAT on tutoring the writing section:

The grammar portions [of the SAT writing test are far more important [than the essay]. And they're very easy to prepare for. In fact, there are only a handful of grammar rules that the SAT loves to test, and they repeat them over and over. So if you can just memorize the rules that we'll lay out in this section, you'll find that the actual test won't be that much of a challenge. During test prep, scores on the writing section tend to increase most reliably, and by the most number of points.
Hack the SAT by Eliot Schrefer
p. 159

I believe it.

Last week I skimmed an SAT roster for our high school from a couple of years back. I saw no 800s on math, one 800 on reading, and four 800s on writing. (Not sure how many students in that group -- perhaps 125?)

I think that's consistent with Schrefer's observation that the writing section is easiest to coach.

why is SAT reading so hard?

I've just recently focused on SAT reading ---

It's hard.

Until two weeks ago, I took SAT reading for granted. I had high verbal scores in high school, and when I took a sample SAT test a couple of years ago, I missed two questions on the reading, only one of which was a real miss in the sense that my answer was clearly wrong. On the other miss, I disagreed with the CollegeBoard's answer.

So I hadn't thought about the verbal section until I started working with C. and a couple of other juniors on SAT prep.

Now I find that working through the verbal section is a wretched, miserable experience, and this is true in spite of the fact that I get the answers right.

Here's Hack the SAT:

You know how some people have a gift?

[snip]

No one has a gift for Critical Reading.

Sure, plenty of people have a gift for reading.... [but] no one is particularly natural at critical reading, this seemingly endless stuff that jams the SAT full of tiny blue print and passages about the subtle revelations of medieval scientist nuns.

[snip]

[E]ven those students who get high scores on Critical Reading find it more draining than the Math or Grammar sections. 

True!

Working through a Critical Reading passage - especially a 'compare two passages' section (those are death) - I feel the brain equivalent of eye strain.

Why is that?

It bears repeating

Thanks to Vlorbik, I've revisted one of my favorite Rafe Esquith quotes. I just don't get tired of reading it. I hope you won't either.

"Many people in the public schools want certain things done for economic or political reasons. Frequently, teachers attending staff development meetings are taught not by master teachers but by publishing-company employees. Why do fifth-grade teachers need hours of instruction before they can use a fifth-grade math book? Book publishers don't go to bed at night worrying that Johnny can't read; they worry about sales and profits. If our teachers can't teach, they're unlikely to be reformed by textbook publishers.

My school currently has two literacy coaches-- former teachers-- who constantly beg me to use their materials with my children. When these coaches try to convince me about their approaches, they rarely rely on arguments about improving reading. At the end of the last school year, my fifth-graders scored in the ninety-first percentile in national tests while the rest of the school scored in the forties. The coaches know this, so why do they want me to change my methods? They want to satisfy their boss, who wants to please his boss, who wants to impress her boss-- administrators all. It's insulting (and boring) to listen to them drone on about how to use their books.

The same song-and-dance routine accompanies the promotion of the latest new-and-improved math textbook. Seriously, how many ways can you teach a child his multiplication tables? Arithmetic hasn't changed. If a math text is so complicated that the average fifth-grade teacher can't understand it without hours of instruction, then there's something wrong with the book, the teacher, or both. These companies interfere with creative and effective teachers. If we want Johnny to calculate better, we need to hire better teachers, not buy newer textbooks."

--Rafe Esquith, An actual classroom teacher
excerpted from There Are No Shortcuts

Vlorbik also suggested that I watch the movie: The Hobart Shakespeareans. It's now at the top of the queue.

help desk

Ed just spent 3 hours helping Christopher with his ELA assignment.

We are on "vacation."

The assignment is huge.

Select a poem (any poem) from a book or anthology; identify three elements of good poetry; interpret the poem in a "well-developed response"; cite sources according to MLA; type the poem and the assignment (school doesn't teach typing, so I guess that's my job); OPTIONAL: research poet, find out about poet, etc.

This is starting at the top.

So....recent assignments have been way over C's head, but the few books he's been required to read are below grade level.

aargh


Anyway, here's the question.

I am not an English major.

I have no idea how to analyze a poem.

I have no idea, even, how to read a work of 19th century literature intelligently.

I know nothing!

So I need books. Books with shortcuts. Books that will give me a superficial but useable map-of-the-world where poetry analysis is concerned.

I thank you in advance.


being your child's frontal lobes

We've just today managed to figure out that C., although terrifically responsible when it comes to homework he can do, is avoidant when it comes to homework he can't do.

Avoidant means:

• didn't select a poem to analyze from the books at school & copy it down (good thing I spent hours researching and purchasing poetry anthologies last school year!)
  

• avoids the assignment altogether; and fails to mention assignment to parents

I guess we can say he's not proactive when it comes to seeking extra help.


Ed awaits his B-.

.

smart people need more practice

This is kind of cool ---

Highly accomplished and talented people often choke under pressure because the distraction caused by stress consumes their big supply of short-term memory, says Sian Beilock, assistant professor of psychology at the University of Chicago, who conducted the research. In the business world, for instance, short-term memory is what lets you listen to what people are saying and yet maintain a point you want to bring up.

Because people in this group heavily rely on short-term memory to tackle challenges, they're at a particular disadvantage. When put under pressure, they resort to using less accurate short cuts to solve problems, such as guessing and estimation, much like those with lesser abilities.

[snip]

.... when talented people begin to feel the heat....

....they started worrying about screwing up. That consumed the working memory capacity of those in the study, which had allowed them to use complex strategies to solve problems in other circumstances.

"Once they had all this capacity to devote to a problem," says Beilock, "and now they're thinking about their worries, trying to suppress them and possibly causing themselves to freak out more."

[snip]

One of Gray's studies has shown that people with higher intelligence also are more taxed by having to control their emotions.

[snip]

If choking under pressure is a concern for you, Beilock's advice is practice, practice, practice--and not just problem solving, but problem solving in high-pressure situations. Memorizing methods of handling problems means you won't have to rely on your short-term memory.
source:
Why Pros and CEOs Choke

There's a strong association between high working memory capacity and high IQ. For awhile there, I think, people were thinking that working memory might actually be IQ.

I take these findings to mean that high-working memory types have "coasted" on WM; they haven't learned work-arounds or shortcuts -- like all these brainy little math kids who refuse to write out the steps because they can do them faster in their heads. (This may be completely wrong... I'm free associating.)

Because they're naturally fast, they haven't learned how to be efficient.