kitchen table math, the sequel: 4/17/11 - 4/24/11

Friday, April 22, 2011

Teachers don't want to be blamed

The Blame Game -- from the School Institution Point of View

Throwing Curves reviews an article in the current issue of The Nation: Teachers are not the Enemy.


If you have time to answer, that would be great!

SAT prep and merit aid survey

sage on the stage

“The sage on the stage” versus “the guide on the side” is how the debate is often framed. Proponents of the former ruled the education roost throughout the 19th century, but in the 20th century a child-centered doctrine, developed by John Dewey in the gardens surrounding the University of Chicago’s Laboratory School, then refined at Columbia University’s Teachers College, gained the high ground, as “inquiry-based” and “problem-solving” became the pedagogies of choice, certainly as propounded by education-school professors. In recent years, the earlier view has staged something of a comeback, as KIPP and other “No Excuses” charter schools have insisted on devoting hours of class time to direct instruction, even to drill and memorization.

Eighth-Grade Students Learn More Through Direct Instruction
by Paul Peterson
Education Next
Summer 2011 / Vol. 11, No. 3
On the question of sages on stages versus guides on sides, we critically need parent choice.

You shouldn't have to pay private or parochial school tuition to get a teacher-centered classroom for your kids.

anonymous and Allison on choosing 2 out of 6 vs choosing 4 out of 6

I mentioned in a comments thread the fact that I'm having a very difficult time grokking the idea that the number of ways you can choose 2 items out of 6 is the same as the number of ways you can choose  4 items out of 6.

Why does 6C2 = 6C4?

Here is Allison's explanation of the specific problem I was asking about.

And here is Anonymous explaining why 6C2 = 6C4:
Image you have six balls (red, orange, yellow, green, blue, and purple, for example). You also have a box.

6C2: I have six balls, and I am choosing which two balls to put in the box.

6C4: I have six balls, and I am choosing which four to leave out of the box.

Thus, the combination "I put red and orange in the box," is the same way of splitting up the balls as "I leave yellow, green, blue, and purple out of the box." Counting the ways to put two in the box is the same as counting the ways to leave four out of the box. 

chemprof on parents and choice

re: the proliferation of global-themed charter schools, chemprof writes:
I don't have a problem with parents choosing a charter that I wouldn't choose for my own kid, anymore than I have a problem with parents choosing progressive private schools. I do have issues with these kinds of content-free methods taking over neighborhood schools when I don't have the option of sending my own child elsewhere. I think most of us are clear that many parents have different goals for their kids, but with the current neighborhood school system, there is no room for multiple approaches, and when charters are limited, then that's all you have.
That is exactly the way I feel.

I have no desire to force other parents to embrace the kind of school I want for my child. None!

But, by the same token, I feel strongly that other parents should respect and support my family's values and goals for our child.

After writing and moderating the Irvington Parents Forum for several years now, I believe that parents can't all be happy with the same schools.

Since we all have to pay the same taxes, I believe parents should support each other's choices, and for me supporting each other's choices means supporting charter schools, tax credits for private and parochial school tuition, and funding for homeschooling.

I realize there are different ways to configure a system based on parent choice, but my essential point remains the same: parents in my group need to be able to send their children to traditional, teacher-centered, content-rich schools without having to pay private tuition.

not puzzling, and not a paradox, either

Barry spotted this Thursday session (pdf file) at the NCTM's upcoming 2011 Annual Meeting:
The Chinese Paradox: How Traditional China Trumps U.S. Reform Attempts
(General Interest) Session

Using video of urban Chinese math classes and professional development, the speaker will explain the puzzling paradox for how seemingly traditional Chinese educational methods—large, teacher-centered, lecture-based classes; exam-driven curricula; and so on—produce students who excel over their U.S. peers, despite the United States’s recent reform attempts initiated by NCTM.

Thomas Ricks
Louisiana State University, Baton Rouge
The success of large, teacher-centered, lecture-based classes where students learn exam-driven curricula is not a puzzling paradox to me.


I'm wondering about that word seemingly.

seemingly traditional Chinese educational methods

I'm wondering whether Thomas Ricks is going to resolve the paradox by arguing that Chinese educational methods are only seemingly traditional.

I hope not.

Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series)


Thursday, April 21, 2011

help desk - balls in boxes

from Art of Problem Solving's Introduction to Counting and Probability:
5.16 How many ways are there to put 4 balls in 3 boxes if:
(b) the balls are distinguishable but the boxes are not.
I don't understand the part of the solution that explains how many ways you could put 2 balls in one box, 2 balls in another box, and 0 balls in a third box:
(2,0,0): There are 4C2 = 6 ways to choose the balls for the first box, and the remaining go in the second box. However, the two pairs of balls are interchangeable, so we must divide by 2 to get 6/2 = 3 arrangements.
To me, it seems like there would have to be 6 ways to choose 2 balls from balls 1, 2, 3, and 4 for the first box:


What am I missing?

For the option of (2,1,1), the solution is 6, not 3:
There are 4C2 = 6 options for picking the two balls to go in one box, and each of the other two balls goes into its own box.
I don't see how (2,0,0) is different from (2,1,1) when the boxes are indistinguishable.

Introduction to Counting & Probability (The Art of Problem Solving)

Clifford Stoll at TED

In Defense of Clifford Stoll

wrong turn

at the Reason Foundation:

reason: What is at the very top of your education reform list?

Here is my prediction: online learning is not going to save us.

I'm pretty sure.

I would like online learning to work. (I think.) I just don't expect that it's going to work, partly because it hasn't worked (well) for me or for anyone close to me, and partly because distance learning was an expensive flop when colleges tried it in the 1990s (see pages 98-99).

This brings me to my beef with education reformers who are advocating "technology": if they don't have an argument as to why technology is going to be "transformative" this time around, then they don't have an argument.

That's not to say they're wrong; maybe this time will be different. But I have yet to hear a reformer explain why this time will be different, and that's a problem.

They are not reckoning with the evidence against their position.

High-Tech Heretic: Reflections of a Computer Contrarian

High-Tech Heretic: Reflections of a Computer Contrarian 

Oversold and Underused: Computers in the Classroom

Oversold and Underused: Computers in the Classroom



The Computer Delusion by Todd Oppenheimer 

In defense of Clifford Stoll

Virtual Schools at Education Next

Chisenbop tutorial

just posted to the precision teaching list: Chisenbop tutorial

help desk - sine and arcs and circular functions

From Dolciani's Algebra and Trigonometry, Chapter 13-2 Circular Functions, p. 555:
sin s = y
cos s = x
tan s = sin s/cos s if cos s ≠ 0
cot s = cos s/sin s if sin s ≠ 0
sec s =1/cos s if cos s ≠ 0
csc s = 1/sin s if sin s ≠ 0
s is the length of an arc of a circle.

This may be asking too much, but I need help.

I have never seen sine, cosine, tangent, etc. applied to an arc. I've only learned sine and cosine in relation to angles in a right triangle.

I've looked back through chapter 12, but I don't see a section that explains this. I'm sure it's there, but it's not obviously there, and I'm in a hurry, sad to say.

Is there a short way anyone can explain to me how we get from sine, cosine, angles, and SOHCAHTOA to sine, cosine, and arcs?

Is there a website that has a succinct and lucid explanation? 

And is there a book you like for self-teaching trigonometry and algebra 2? (Do we know what book(s) homeschoolers use?)

I need a royal road to circular functions.

Algebra and Trigonometry: Structure and Method, Book 2

Digital Education at Throwing Curves

An attempt to take a serious look at the growth of technology-based instruction in the classroom:

Technology, digital natives, and the future of education

Wednesday, April 20, 2011

practice, practice, practice

C. and I just took Section 3 of Test 3 in the CollegeBoard online course.

20 questions, 25 minutes

C. missed one and skipped two; I skipped one (which C. got right!)

If we made the same score on the other 2 sections, C. would have a score of 670, and I would be at 750.

In January, C. was scoring in the upper 500s, and I was scoring in the low 600s. (Forget January; I'm pretty sure we were both bombing the test as recently as March. Must find my notes...)

Ed's goal for C. is a 2100, but I'd like him to break 700 on all 3 tests.

James Tanton on quadratics

I just interviewed Philip Keller, and he pointed me to James Tanton's website and books.

His Guide to Everything Quadratic has finally answered my burning question about the term "quadratic":
The prefix quad- means “four” and quadratic expressions are ones that involve
powers of x up to the second power (not the fourth power). So why are quadratic
equations associated with the number four?

Answer: These equations are intimately connected with problems about squares and quadrangles. (In fact, the word quadratic is derived from the Latin word quadratus for square.) Questions about quadrangles often lead to quadratic equations. For example, consider the problem:

A quadrangle has one side four units longer than the other. Its area is 60 square units. What are the dimensions of the quadrangle?

If we denote the length of one side of the quadrangle as x units, then the other must be x+4 units in length. We must solve the equation: x(x+4) = 6, which is equivalent to solving the quadratic equation x^2 + 4x - 60 = 0.

Solving quadratic equations, even if not derived from a quadrangle problem, still
involves the geometry of four-sided shapes. As we shall see, all such equations can
be solved by a process of “completing the square.” 
The New Math SAT Game Plan

chemprof on merit aid, part 2

chemprof writes:
[M]erit aid is all part of "enrollment management." The goal is to attract strong students, partly because that helps with rankings but largely because of cohort effects. Schools with a core of strong students are more attractive to other strong students, who want to be in classes with other students at their own level. For less competitive schools, managing this is a tricky problem.

...I'd agree that knowing the [specific] criteria [for merit aid] would be interesting, but I don't know that you'd be able to make much use of it. ... [W]e are typically told we can offer 8 scholarships and there might be 30-40 students who admissions has targeted as possibilities (based on stated interest and their general ranking). Picking the 8 is a matter of reading the files and talking to admissions, and nothing about criteria or the decision making process is ever written down!

Oh, and our admissions department ranks every file on a 1-5 basis where 1 is highest. However, the faculty have often thought they overlooked strong students (as 2 or 3 ranked), because those students had fewer extra-curriculars or were otherwise less interesting to admissions. 

chemprof on merit aid and an upper cut-off

re: merit aid, chemprof writes?
The big thing to know as a parent is that the major opportunity for aid happens when students enter college. There is less merit aid available for transfers at most places, even if they admit a lot of transfers, and if you aren't awarded merit aid on entrance, there is usually no mechanism for getting an award later. 


For our science scholarships, we look at math SATs, and typically give them to students whose math SAT is 650-700. Why an upper cutoff? Because we've found that students above that are using us as a safety school, and might come if they get a major merit award but not if they only get a smaller one. But if someone was at 710 and admissions was really excited about them, we might push things around a bit.

help desk - even and odd functions

5. Determine whether each function is even, odd, or neither.
a. f(x) = x^4 = 3x^2 + 5
b. g(x) = 2x^2 + 6x + 4

Algebra and Trigonometry Structure and Method

Algebra and Trigonometry: Structure and Method, Book 2

First draft of New York teacher evaluation regs available for comment by the public

The first draft of the New York teacher and principal  evaluation regulations has been posted online for comments from the public.  It looks like it might be a slog to go through the 40-page document, but the six-page summary is here.  The deadline to submit comments is Friday, April 29.

In my first look at the proposal, it hit me that I probably oppose any evaluations based on New York state tests.  "Something fishy" about New York State Regents scoring
Last year, legislation was enacted requiring an annual performance evaluation of all teachers and principals. These evaluations will play a significant role in a wide array of employment decisions, including promotion, retention, tenure determinations, termination, and supplemental compensation, and will be a significant factor in teacher and principal professional development. The Regents Advisory Task Force on Teacher and Principal Effectiveness -- composed of teachers, principals, superintendents of schools, school board representatives, school district and BOCES officials, and other interested parties -- has been meeting regularly since September 2010. And the Board of Regents has discussed various topics related to the evaluation system at its meetings in January, February and March 2011.
Earlier this month, at the Regents April meeting, the Task Force submitted a comprehensive report containing recommendations for implementing the evaluation system in New York. . . . . The draft regulations will be on the Regents agenda at their meeting in May.
Cross-posted at Education Quick Takes

Tuesday, April 19, 2011

sage on the stage bests guide on the side

new study

from the press release:
[P]rominent organizations such as the National Research Council and the National Council of Teachers of Mathematics, for at least the last three decades, have “called for teachers to engage students in constructing their own new knowledge through more hands-on learning and group work.”

Harvard Study Shows that Lecture-Style Presentations Lead to Higher Student Achievement
Thirty years.

merit aid at the University of South Carolina

anonymous writes:
The University of South Carolina has an extensive Honors College, which offers upper-division classes in a wide variety of fields, in addition to the more usual freshman-sophomore ones. All Honors College courses are taught by professors (no TAs) and the classes are small - my son and his wife had classes as small as 4 students (2 grad, 2 undergrad) and few above 20-25. An Honors College degree requires a senior thesis. The last I heard, the Honors College required at least a 1400 SAT, but most Honors College students receive merit scholarships - for out-of-state students, such scholarships qualify them for reduced tuition (essentially the in-state rate). The campus is right in Columbia, and the housing on the historic Horseshoe is reserved for Honors College students. It's not a big brand name, but the opportunity for a great education is right there and the Honors College faculty want to help their students with special programs, internships etc.

up the down staircase

Sara (not sure who Sara is!) sends a link to this Everyday Math problem posted on the Well-Trained Mind forum.

Up is negative; down is positive.

That strikes me as a bad mnemonic to teach kids who are going to be encountering coordinate planes just a few years from now.

She got the answer wrong, too.

Up the Down Staircase

Up the Down Staircase

Debbie Stier on online learning

Here’s What’s Wrong With Online Learning: Kaplan, A Case Study

Monday, April 18, 2011

Test Professors

looks interesting

not a teenager I know

Chaeran Chung reviews John Chung

Dr. John Chung's SAT Math

Dr. John Chung's SAT Math

getting ahead of ourselves

from purplemath:
  • Given that f(x) = 3 x 2 + 2x, find [f(x + h) – f(x)] / h.
This is actually something you will see again in calculus. I guess they're trying to "prep" you for upcoming courses when they give you exercises like this, but it's not like anybody remembers these by the time they get to calculus, so it's really a lot of work for no real purpose. However, this type of problem is quite popular, so you should expect to need to know how to do it.
I'm a big fan of purplemath.

Branching Out

In a completely throwaway gag line, football writer (and high school math teacher) Mike Tanier makes a mocking reference to Everyday Math in his offseason column.

I could never be a documentary cameraman. At one point in Sunday's episode of Human Planet (a BBC/Discovery Channel production in the Planet Earth vein) a father takes his two children on a five-day hike along the frozen Zanskar River in Northern India so they can attend a boarding school. The river slowly melts during their journey. At one point, the 11-year-old daughter must crawl along a tiny, cracking ice ledge over the rushing, freezing waters.

...With the poor girl's luck, the school she risked her life to attend just adopted the Everyday Math curriculum, making the whole trip worthless.

Football Outsiders is largely a sabrmetrics site, so the readership is more numerate than average. Still, it warms my heart to see Everyday Math mocked so casually outside of the usual context. I had been planning a post linking Bill Walsh to practicing to mastery, so I guess this serves as a nice segue.

Sunday, April 17, 2011

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.

the learning curve

re: C's and my apparent learning "spike," here are Frank E. Ritter and Lael J. Schooler on the learning curve:
Most tasks get faster with practice. This is not surprising because we have all seen this and perhaps know it in some intuitive sense. What is surprising is that the rate and shape of improvement is fairly common across tasks. Figure 1 shows this for a simple task plotted both on linear and log-log coordinates. The pattern is a rapid improvement followed by ever lesser improvements with further practice. Such negatively accelerated learning curves are typically described well by power functions, thus, learning is often said to follow the "power law of practice". Not shown on the graph, but occurring concurrently, is a decrease in variance in performance as the behavior reaches an apparent plateau on a linear plot. This plateau masks continuous small improvements with extensive practice that may only be visible on a log-log plot where months or years of practice can be seen. The longest measurements suggests that for some tasks improvement continues for over 100,000 trials.


The power law of practice is ubiquitous. From short perceptual tasks to team-based longer term tasks of building ships, the breadth and length of human behavior, the rate that people improve with practice appears to follow a similar pattern. It has been seen in pressing buttons, reading inverted text, rolling cigars, generating geometry proofs and manufacturing machine tools (cited in Newell and Rosenbloom, 1981), performing mental arithmetic on both large and small tasks (Delaney, Reder, Staszewski, & Ritter, 1998), performing a scheduling task (Nerb, Ritter, & Krems, 1999), and writing books (Ohlsson, 1992).


Averaging can mask important aspects of learning. If the tasks vary in difficulty, the resulting line will not appear as a smooth curve, but bounce around. Careful analysis can show that different amounts of transfer and learning are occurring on each task. For example, solving the problem 22x43 will be helped more by previously solving 22x44 than by solving 17x38 because there are more multiplications shared between them. Where sub-tasks are related but different, such as sending and receiving Morse code, the curves can be related but visibly different (Bryan & Harter, 1897).


The learning curve has implications for learning in education and everyday life. It suggests that practice always helps improve performance, but that the most dramatic improvements happen first. Another implication is that with sufficient practice people can achieve comparable levels of performance. For example, extensive practice on mental arithmetic (Staszewski reported in Delaney et al., 1998) and on digit
memorization have turned average individuals into world class performers.

Draft version of:
Ritter, F. E., & Schooler, L. J. (2002). The learning curve. In International encyclopedia of the social and behavioral sciences. 8602-8605. Amsterdam: Pergamon.

Don't know how this relates to the experience of having a sudden jump in learning....

the learning spike?

As I mentioned the other day, C. and I have suddenly jumped a good 50 points in our math scores on Blue Book tests.

On April 6, C. missed 7 and skipped 1 on a 20-item math section.

Four days later, on April 10, he finished all 8 questions in an 8-question multiple choice section and missed just 1. He got 7 of the 10 grid-ins right, missed 2, and skipped 1.

And: all three of the questions he missed were dumb mistakes. He knew how to do the problems, and did them quickly enough to finish the test.

I've had the same experience. Last summer I couldn't hope to finish a math section; yesterday I finished early enough to go back and check my bubbles.

At first I thought the higher scores were a fluke. But C. has now turned in the same performance on 5 math sections in a row, and for me that number is probably 6 or even 7.

the power spike of learning?

I'm surprised. I don't remember ever experiencing a sudden jump in learning like this, and my understanding of the "learning curve" is that it's a power curve (if that's the right term), not a right angle. You make more gains early on than you do later.

C. and I made practically no gains early on. While C and I weren't doing a lot of SAT practice fall semester, we have been working with some regularity since January, and in that time we've gotten nowhere. He's been stuck in the high 500s, and I've been stuck in the low 600s. (Very low.)

In fact, I've been stuck in the low 600s for a good two years now. Not that I was practicing SAT math per se -- I wasn't -- but I have been studying high school math off and on during that period, and I've seen no transfer to SAT math at all.

Yesterday, my score on all 3 sections of Test 2 in the College Board online course was 690. C.'s score was 640.

C. said, "It's like I jumped over a wall."

If you graphed our scores on an xy plane, it would be more like we leaped a tall building in a single bound. 

Now we have to leap another one.

arguing in French

The other night at dinner we were talking with our friends about whether they'd had this experience. One friend, an attorney, said tax law was her version of SAT math. She didn't get tax law at all until one day she did. 

Then Ed remembered learning French in France. He was doing what C. and I have been doing: grinding away, putting in the time, having nothing much to show for it. 

Then one day he was sitting around with some friends, and one of them made a provocative statement about something or other. Ed disagreed, an argument ensued, and at some point Ed realized he was arguing in French.

Arguing in a foreign language is the equivalent of an 800, I think.

10 tests

What does this mean, if anything?

Well, first of all, I have to see whether C. and I really are stable at this new level. I suspect we are, but we'll see.

Second: start early. I have no idea why it's taken us so long to experience this leap, but no one becomes an expert - or even a proficient novice (which is probably what we are now) - in a day.

Third: The Blue Book has 10 real SAT tests. C. and I have taken all of the math sections in all 10 tests, and I have taken all 3 math sections in Debbie Stier's January 2011 test.* Our scores jumped somewhere in the neighborhood of 30 math sections taken over several months' time.

My current thinking on SAT prep is that students should do all 10 sample tests in the Blue Book at a minimum and should spread that work out over at least 4 months.

I'm also thinking it would be a good idea to do the 9 tests College Board offers online for $70.

reading and writing

I'm going to start paying attention to the reading and writing sections. We've done far fewer of those because C. is a very good reader and has been since he was little. He was one of those kids who taught himself to read. That's a funny story, which I know I've told before. C's Kindergarten teacher called us in for a parent-teacher meeting and told us C's handwriting indicated that he was at risk for a reading disability, which was true. Very bad handwriting is a flag.

Naturally, I figured: we've got two autistic kids so now we're going to have a dyslexic kid, too. Just our luck. 

Two weeks later, C. could read. All of a sudden. He went from not reading to reading.

(Another case of a power right angle?)

Back on topic: because C's SAT reading scores are routinely in the low 700s, there's not a lot to learn about SAT prep from observing him, I don't think. He misses or skips questions when he absolutely does not know a vocabulary word and can't figure it out from context. I told him yesterday he has to get back to memorizing his SAT vocabulary words, so the challenge will be remembering to nag him to do it.

("Have you studied your SAT words?" "No." "Do you know where your SAT flash cards are?" "I have an iPod app I use." "Fine, but do you know where the cards are?" "They're in the family room." etc.)

The writing multiple choice questions are more interesting; he misses more of them and presumably will benefit from more practice. (We've done very few writing sections.)

I'm going to start paying attention and will report back.

* We had also re-taken all 3 sections of the first test in the Blue Book as well as 1 section of the first test in John Chung's book.
The Official SAT Study Guide, 2nd edition

Dr. John Chung's SAT Math