From The College Board: Reconciling AP Exams With Common Core

I'm very concerned about the information College Board is giving to AASA, the school superintendents association, regarding calculus. "Despite these measures, there are still difficulties in reconciling many AP courses with the Common Core. In particular, AP Calculus is in conflict with the Common Core, Packer said, and it lies outside the sequence of the Common Core because of the fear that it may unnecessarily rush students into advanced math classes for which they are not prepared. The College Board suggests a solution to the problem. of AP Calculus “If you’re worried about AP Calculus and fidelity to the Common Core, we recommend AP Statistics and AP Computer Science,” he told conference attendees. Moreover, the College Board may offer an AP Algebra course (although no plans are definite), which may supplant AP Calculus, particularly in schools rigidly adhering to the Common Core standards." I'm an apcalc and alg2 teacher with a b.s. in classical applied math. The statistics course that I took in college years ago was a calc-based course. Lisa Jones @proudmomom

Derek Owens - AP calculus

Crimson wife also points us to Derek Owens' course.

Courtney James

re: what to do about C. and calculus, Crimson Wife writes:

I have no personal experience with him, but I have heard numerous raves about Courtney James' tutoring from fellow moms in our local homeschool support group. He tutors AP Calculus BC according to his website. 

Exciting!

AP calculus scores & stereotype threat

As I understand it, stereotype threat is essentially a self-fulfilling prophecy that takes place below the level of consciousness -- although when I experienced stereotype threat while competing on a television game show my thoughts were conscious, intense, and pretty close to crippling.

I had won a spot on Sale of the Century, and when the big moment arrived I was to play against two men: one white and one black.

I was gripped by stage fright. Sitting in the studio with the group of aspiring contestants who had made the first cut, waiting to see whether I would be called, I felt so terrified I wanted to bolt from the room.

The only thing stopping me bolting from the room was the stage fright I felt at the prospect of people watching me bolt from the room. Which is worse? Having a roomful of strangers watch you play a game show? Or having a roomful of strangers watch you run screaming from the room instead of playing a game show?

I chose door number one and stayed put in my chair; and when I was called to play, two men were called to play, too. I was led to the seat between them.

Suddenly, entirely unbidden, my Betrayer Self began to think: "I can't win against men. I can't win against men. I can't win against men." Over and over again. "I can't win against men."

I was aghast.

I had an Ivy League degree and a Ph.D.; I was a committed feminist; I thought the idea that men were my betters was hooey. Plus I knew boatloads of random factoids and trivia, and Sale of the Century was a random factoids and trivia contest. Yet there I was thinking -- thinking very loudly -- "I can't win against men." Until that moment, I had had no idea I felt that way.

So when it comes to stereotype threat, I'm a believer.

I won that game, but the reason I won was that the black contestant, who was a better player than I, appeared to be suffering an even worse case of stereotype threat than the one threatening to derail me. I won't tell that part of the story here. Suffice it to say that he made an on-camera allusion to what he was feeling just before choosing the wrong square on the board and giving the wrong answer: and losing the game. It was a painful moment. More than painful; it was excruciating. I recall a murmur of what sounded like distress running through the audience.

I managed to defeat the white male contestant mostly because a) he appeared to be just as panicked as I was and b) I knew a lot more random factoids and trivia than he did. And heaven only knows what kind of stereotype threat might have been hammering his brain. Stereotype threat isn't just for for blacks and women; it's for everyone. In fact, you can lower the math performance of white male students attending Stanford University if you remind them, before they take the test, that they aren't Asian.

After I won the game, I calmed down and won two more games, then retired with cash instead of gambling my certain winnings to play on in hopes of making it to much bigger winnings at the top: a classic example of Kahneman and Tversky's concept of loss aversion.

Back to stereotype threat. Sian Beilock discusses stereotype threat at length in Choke. Turns out there is a (disputed) study of stereotype threat and AP calculus scores finding that when test-takers fill out the personal info form, which includes gender, after they've finished the test instead of before, girls score better. (The dispute has to do with statistic methodologies and significance.)

From the 'pro' authors:

Pragmatically speaking, the “trivial” differences carefully dismissed in Stricker and Ward (2004) can translate into very large practical effects, with real theoretical meaning. The inquiry manipulation reduces the gender difference to less than one third its original size. Instead of a ratio of about 6 girls receiving AP credit for every 9 boys who obtain credit, the new manipulation generates a ratio of about 8 girls receiving AP credit for every 9 boys.

How would this manipulation affect females at the population level of all students taking AP Calculus AB? Stricker and Ward (2004) told us that 52,465 boys and 47,275 girls took the test in 1995 (p. 669). ... [C]hanging the way the tests are administered would increase the number of girls receiving AP Calculus credit from 15,081 to 17,870 in a year—an increase of 2,789 young women starting college each year with Calculus credit.

This size number should not be below the radar. The number of people taking the AP Calculus AB test is increasing. In 2004, there were 88,809 boys and 81,521 girls who took the exam (College Board, 2004), which represents an increase of 70.8% since Stricker and Ward collected data in 1995. All other things being equal, we estimate that 4,763 more women would receive AP Calculus AB credit if the timing were changed. We are convinced that stereotype threat in real-world testing situations can have a significant effect on test takers, and Stricker and Ward’s (2004) data support this conclusion.

Stereotype Threat, Inquiring About Test Takers' Ethnicity and Gender, and Standardized Test Performance
Lawrence J. Stricker, William C. Ward
Journal of Applied Social Psychology
Volume 34, Issue 4, pages 665–693, April 2004


Stereotype Threat in Applied Settings Re-Examined
Kelly Danaher and Christian S. Crandall
Journal of Applied Social Psychology
Volume 38, Issue 6, pages 1639–1655, June 2008
[figures drawn from Danaher and Crandall]


Stereotype Threat in Applied Settings Re-Examined: A Reply†
Lawrence J. Stricker, William C. Ward
Journal of Applied Social Psychology
Volume 38, Issue 6, pages 1656–1663, June 2008

Jaime Escalante, RIP

His wife and son later joined him in Pasadena, where his first job was mopping floors in a coffee shop across the street from Pasadena City College, where he enrolled in English classes. Within a few months, he was promoted to cook, slinging burgers by day and studying for an associate's degree in math and physics by night. That led to a better-paying job as a technician at a Pasadena electronics company, where he became a prized employee. But the classroom still beckoned to the teacher inside him. He earned a scholarship to Cal State Los Angeles to pursue a teaching credential. In the fall of 1974, when he was 43, he took a pay cut to begin teaching at Garfield High at a salary of $13,000.

"My friends said, 'Jaime, you're crazy.' But I wanted to work with young people," he told The Times. "That's more rewarding for me than the money."

When he arrived at the school, he was dismayed to learn he had been assigned to teach the lowest level of math. He grew unhappier still when he discovered how watered-down the math textbooks were -- on a par with fifth-grade work in Bolivia. Faced with unruly students, he began to wish for his old job back.

Motivating students

But Escalante stayed, soon developing a reputation for turning around hard-to-motivate students. By 1978, he had 14 students enrolled in his first AP calculus class. Of the five who survived his stiff homework and attendance demands, only two earned passing scores on the exam.

But in 1980, seven of nine students passed the exam; in 1981, 14 of 15 passed.

In 1982, he had 18 students to prepare for the academic challenge of their young lives.

At his insistence, they studied before school, after school and on Saturdays, with Escalante as coach and cheerleader. Some of them lacked supportive parents, who needed their teenagers to work to help pay bills. Other students had to be persuaded to spend less time on the school band or in athletics. Yet all gradually formed an attachment to calculus and to "Kimo," their nickname for Escalante, inspired by Tonto's nickname for the Lone Ranger, Kemo Sabe.

[snip]

The Advanced Placement program qualifies students for college credit if they pass the exam with a score of 3 or higher. For many years it was a tool of the elite; the calculus exam, for example, was taken by only about 3% of American high school math students when Escalante revived the program at Garfield in the late 1970s.

In 1982, a record 69 Garfield students were taking AP exams in various subjects, including Spanish and history. Escalante's calculus students took their exam in May under the watchful eye of the school's head counselor. The results, released over the summer, were stunning: All 18 of his students passed, with seven earning the highest score of 5.

But the good news quickly turned bad.

Testing controversy

The Educational Testing Service, which administers the exam, said it had found suspicious similarities in the solutions given on 14 exams. It invalidated those scores.

The action angered the students, who thought the service would not have questioned their scores if they were white. But this was Garfield, a school made up primarily of lower-income Mexican Americans that only a few years earlier had nearly lost its accreditation. "There's a tremendous amount of feeling that the Hispanic is incapable of handling higher math and science," Escalante reflected later in an interview with Newsday.

He, like many in the Garfield community, feared the students were victims of a racist attack, a charge that Educational Testing Service strongly denied. Two of the students told Mathews of the Washington Post that some cheating had occurred, but they later recanted their confessions.

Vindication came in a retest. Of the 14 accused of wrongdoing, 12 took the exam again and passed.

After that, the numbers of Garfield students taking calculus and other Advanced Placement classes soared. By 1987, only four high schools in the country had more students taking and passing the AP calculus exam than Garfield.

Jaime Escalante Dies at 79; math teacher who challenged East L.A. students to 'Stand and Deliver'

Stand and Deliver Revisited

C-a-l-cu-lus AP, hardest class in his-to-ry

New words for a song stuck in my head.

l squared on college prep

From the university perspective, it is not particularly helpful for them to know a little engineering at the expense of the math/physics/chemistry they might otherwise take. I feel quite certain that engineering schools are good at taking students with a solid foundation of math and basic physics and getting them to where they need to be in engineering. They are not set up to take students who think they know engineering and remediating the math, physics, etc. that they should have as prerequisites.

I don't teach engineering, but I do teach calculus, and I have a similar problem. I would rather have students who have never heard of a limit or derivative, but can do algebra and trigonometry. The 40% of the class that come in with the backward set of skills (a little calculus, but not enough algebra) really struggle.

This is extremely good to know. I believe it. I'm pretty sure rudbeckia hirta has said the same thing over the years.

oh gosh. Speaking of rudbeckia, I found this terrific post from 2006 while Googling her blog. I should probably print that one out and scotch tape it to my collection of as-yet unused calculus textbooks.

David Klein's AP/IB grades changed by Fordham

This is distressing news.

Based on my own n of 1, I'm sure David's original grades were correct. C's camp counselor last summer, who is attending Rensselaer Polytechnic, told me the IB math sequence he'd taken in high school was a joke. He didn't learn much calculus, but he did write a "research" paper. Looked some stuff up and slapped it together.

I wish to heck I knew what I did with my notes on the conversation. He gave me a rundown of the many topics IB didn't mention while students were busy slapping together low-quality papers.

If I run across them, I'll post.

A few years ago I checked out the IB web site. The lower grades IB curriculum was entirely constructivist, which made me think the upper-grades curriculum couldn't be far behind.

.............................

It was a lot of fun having C. in camp with a math-brain counselor. The counselor would tell him things like, "I don't like to read and write papers. I just like to do math." This was an utterly novel concept to C, who quoted it to me verbatim on a number of occasions.

remembering and forgetting

from a comment left by Le Galoisien:

But really. If they for example, require you to know logarithms, the general attitude is like, "You probably forgot how to do these. Here's a refresher." and you learn the concept over again ... because it is true even the top students in the school forget them because the syllabus is structured in such a way that it is hard to exercise them all year.

I remember the seniors would cry, "awww, you mean we had to remember those?" when there was a rare AP problem that required us to know a trig identity we had learned a year before. And the teachers would respond, "of course. You didn't learn them for nothing," implying it was our fault (but begrudgingly teaching them to us again). But somehow, even though it was partially our fault for not revising the concepts we had learned over the years (even when we had been assigned no work that dealt with them after we finished the unit) I often wonder if it is someone else's fault as well.

I mean, imagine all the time that has to be used reteaching concepts, and generally just in time for the examinations, before we put them in the closet again.

If we reinforced them all along, I wonder if students would save so much time with progress so much quicker that doing linear algebra in your senior year would be no big deal.

That's the story around here, only worse.

Learn percent, forget percent.

Learn percent again, forget percent again.

Repeat, repeat.

Meet with math chair; math chair says class had no business flunking latest test because "they saw that material last year."


inputs, not outputs

I'm realizing, again, how deadly the inputs model is. When school quality is defined by class size, per pupil spending, and number of Masters degrees held by teaching staff (pdf file), there isn't much incentive to design curriculum & instruction that ensures students will actually remember what they've "learned."

In my next life I plan to live on a planet where schools and curriculum designers focus on:

a) how to get content and conceptual understanding into students' long-term memory

b) how to keep it there


Here's Stanley Ocken addressing the National Mathematics Advisory Panel:

My second suggestion is that you investigate and make recommendations regarding common sense issues of pedagogy. It's important to think about the sequence of tasks and knowledge that lead to success in algebra, but it is critical and possibly easier to find out why so many entering college students seem to have forgotten the algebra they learned in school. You could begin by stripping away the obfuscating rhetoric of blind rote and drill and kill. Then you might examine the proposition that repetition and practice, properly implemented, are essential to success in mathematics, just as repetition and practice, properly implemented, are
essential to success in music, sports and the study of foreign languages. You could conclude by identifying prior indicators of successful college math students.

Before they got to college, did they experience rigorous and frequent in-class assessments? Were they required, for example, to master the multiplication facts by the end of third or fourth grade, or were their programs grounded in the principle that it doesn't matter if children master the material this year, since they are going to relearn and re-relearn the same elementary material in later grades? In other words, please investigate the role of basic interventions that clarify the scheduling and rigor of learning goals, these may be more effective and easier to implement than complex manipulations of curriculum and pedagogy.

Here’s my third and final suggestion. Enunciate the importance of a coherent K to 16 mathematics curriculum, one grounded in the principle that K to 12 math instruction must permit and encourage students to prepare for the rigors of calculus. To bring that principle to life, we'll need to see fundamental changes in the dynamics of K to 12 curriculum design.
source:
National Mathematics Advisory Panel (pdf file)
Thursday September 14, 2006

I'm fairly certain the only group of people who measure the success of their instruction by student achievement, which means, among other things, student remembering, are the behaviorists. I may be wrong about that, but I suspect not.

calculus textbook recommendation

from anonymous:

You might want to try the Larson 8th ed or the Finney books. Much more "user friendly" [than Foerster] so to speak. I use Larson in my AP Calc (both AB and BC) but supplement from several others. Foerster has a supplement that goes with it called calculus explorations (and one for precal also) that are excellent.

Pre-calculus and Calculus at Key Curriculum Press

Thanks!

Information about Foerster's Calculus book

I'd appreciate any information about Paul A. Foerster's Calculus Concepts and Applications book.

Thanks.


Catherine here - dropping into ec's post

Here's a link to calculus text recommendations at ktm-1. (This may be all of them, but if not I'll try to post the other links later.)

bonus: calculus advice from Rudbeckia Hirta.

Barry's calculus text recommendation.

mother lode

worksheets: arithmetic through calculus

yowza!

Now I can spend the rest of the night downloading stuff onto my desktop instead of revising my chapter.

two birds, one stone

lgm on preparing for calculus

lgm left a terrifically helpful resource on preparing for calculus:

To prepare mathematically, he needs to be ready for calculus.

This canadian site has a booklet "preparing for university calculus" that gives an idea of the math he should be comfortable with before trying calculus as well as good tips on handling a college math course: Preparing for University Calculus

This site has test prep questions: Online Math Tests Homepage

Thanks!

Steve on making it to high school calculus

Here's Steve, responding to Myrtle's comment about set theory:

I remember this when I was growing up. Every year we got a bit of set theory. Now, everyone thinks of "sets" when the old New Math is brought up. However, it did cover a lot of math and required a lot of mastery. With a rigorous course in algebra in 8th grade, I made it to the calculus (pre AP days) track in high school without ANY outside help. You can't do this nowadays. This doesn't mean that there were no problems back then, but at least there was a proper path.

I've been trying to find out whether any kids in Irvington make it to calculus without outside help.

So far I haven't found anyone. The 3 recent graduates with whom I've been in contact all had parents who knew math extremely well. In two cases the parents were employed in math fields. Those parents were actively engaged in their kids' math education. "Actively involved" as in "helping with homework" and reteaching concepts where necessary.

A parent told me the other day that almost no one takes the BC course - perhaps only 10 to 12 kids per class.

This isn't fact-checked, and it may be wrong. Awhile back a student told us that there were 40 students in AP calculus, split half and half between AB and BC. Total class size was 120.

I don't have the patience to try to figure out whether 30% of an Irvington high school class is comparable to figures for same-SES peers across the country.

I should check private schools, actually. That would give me a rough comparison.

I do know that five percent of all high school students take AP calculus (I'll find the reference); don't know off-hand how that 5% is divided between AB & BC. [correction: make that 8% and rising, I presume]

I'll see if I can track that down.


stats on AP calculus enrollment

I've found a couple of statistics:

Once upon a time, calculus was the first college-level mathematics course taken by mathematically talented students. The students in first-semester calculus were mathematically motivated, generally well prepared, and they were seeing these ideas for the very first time. This is no longer true. Most of our best-prepared mathematics students arrive in college with credit for at least the first semester of calculus, many of them with credit for both semesters. Despite steady growth in majors in science and engineering, enrollment in first-semester calculus has been flat or slightly declining at both two- and four-year undergraduate programs. It is the College Board’s Advanced Placement Calculus Program that has been growing steadily at 7–8% per year (see figure 1).

In 2004 over 225,000 high school students took the AP calculus exam. This number is far larger than the number of students who took mainstream first-semester calculus in all four-year undergraduate programs in the Fall of 2000. By the time of the next CBMS survey in 2005–06, we can expect that more students will take an AP Calculus exam than will take mainstream Calculus I in the Fall of 2005 in all 2-year and 4-year institutions combined.

[snip]

The same pressures that are pushing Calculus I into the high school curriculum are doing the same for Calculus II. Traditionally, it was a very elite group of students who took BC Calculus, covering the entire two-semester college syllabus. That group of students also grew by 6–8% per year until the mid-1990s. Over the period 1995–98, the rate of growth of BC calculus accelerated to 10–11% per year, a rate that has held up since then. In 2004, the number of students taking the BC Calculus exam exceeded 50,000. It will likely exceed 60,000 by 2005–06, the year of the next CBMS survey.

source:
The Changing Face of High School Calculus
by David M. Bressoud, Macalester College

Here's more from the National Science Foundation:

Very small proportions of students complete advanced mathematics or science courses that provide college credit (such as AP/IB courses). The most popular category among these is AP/IB calculus; even there, only 8% of 2000 graduates completed such a course.

_______________

Harvard placement in math courses...

BC Calculus course topics