kitchen table math, the sequel

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

My son needs to take Calc Honors next year (sr year of high school).  He took Trig Log Calc Honors as a sophomore, but it was a waste of a year.  Catherine's son had same teacher -- so it was same problems, but then add to that that he was out sick with mono for 6 weeks.

He did not take Calc this year (i.e. Junior Year) because it would have been the same teacher, and we could not live through that again.  Plus, he didn't want to take that risk for  his junior year transcript.

Instead, he took AP Stat - which is fun for him, but not really a math class.

So, I need to get him ready this summer to take Calc in the Fall.  First, I'm not sure if he should take a pre-calc class -- or take a calc class and just audit it, my thought being that he'd  pre-learn the material before school starts.

So that's my first question: Calc class vs Pre-Calc. class

Here are the 3 options I was thinking about:

1) MIT Online math class.  I haven't looked into, but I understand they have High School courses.  He loves this idea.  I wonder whether an online course could work for him.

2) Local community college (Mercy).

3) Local Private school summer course (Horace Man was recommended).  IT needs to be some place he can get to via public transportation, as I can't drive to from every day.


The other thing to add is that he has a summer job lined up (90%) -- so the most time-efficient way for him to do that is a variable too.  i.e MIT sounds appealing because it's on his own schedule.  Horace Mann sounds good because it's only 6 weeks (I think).  I haven't checked into Mercy College schedule.

Ah, the conic sections. Like many of the topics in the Pre-Calculus curriculum, this topic can take as much or as little time as you like. The basic ideas are a focus on the four conic sections Parabola, Circle, Ellipse, Hyperbola. There are many different ways to consider the shape and defining components of these mathematical objects.

Conics as a Slice of a Cone

The Greeks originally conceived of these as the shapes generated by slicing through a right circular cone. In A History of Greek Mathematics Vol. II (1921), Sir Thomas Heath says:

The question arises, how did Menaechmus come to think of obtaining curves by cutting a cone? On this we have no information whatever. (pg. 110)

Lost in the mists of time!



More after the jump...

I don't know if any of you are around during the day, but if you are I would be grateful for a quick read on this issue.

C. is set to move to a different math class.

He was planning to move to "Stat Honors," which unfortunately is the only class that fits his schedule. (Otherwise, we would have had him move to AP Stat.)

The calculus teacher, whose class he is leaving, thinks he should move to precalculus instead.

That makes sense except for the fact that he took precalculus last year, ending the year with a B average. The only precalculus course available is a lower level course than the one he's already taken.

That raises the college transcript issue: how do college admissions officers interpret a transcript showing two consecutive years of precalculus, with the second year being a lower level course than the first?

I have no idea.

There's also the question of the teacher. We don't know the teacher of the precalculus class; we do know the teacher of the stats class. C. took geometry with her, and my friend D's son, who is a math kid, is taking his 2nd course with her now. We are confident that C. will learn the material she's teaching.

We don't know the precalculus teacher.

One last thing: Ed is strongly opposed to hiring any more tutors. Precalculus tutors around these parts charge $150/hour. If you need a tutor to get your kid through 24 weeks of precalculus, you're talking about another $3600 on top of tuition.

I'm flummoxed, and I guess we need to make this decision today --- any thoughts?

Thank you!

I recently added some links for problems to my previous two posts on algebra review and rational roots. Today I'd like to post about functions.

The appropriateness of when to introduce the function concept varies depending on the views of the individual instructor and the student population one is working with. I value the importance of the function concept and notation, but feel that teaching the function notation and the “vertical line test” without follow-up material relating to curve analysis (max, min etc.), composition and inverse, and transformation of functions is of questionable value particularly in the case of a gen. ed. math course that is to be taken by a broad range of students.

I've found that it is possible to discuss many topics related to the behavior of polynomial, rational, exponential and logarithmic functions without the additional two or three days of material required to introduce the function notation to a gen. ed. student population.

Given time, I like to cover functions by introducing the basic notation, followed by curve analysis in which the concepts of increasing and decreasing behavior, as well as maximum and minimum values are discussed. I then like to address function composition and inverse, the transformation topics (which can be somewhat tricky for students) and finish with a unit on the application of functions to a range of optimization problems common in many Calculus courses. I typically have the students use the TI 83/84 calculators to find the maximum and minimum values that will be found algebraically in a Calculus course.

So, I typically spend four to five weeks covering

1) notation

2) increasing, decreasing, max, min

3) composition, inverse

4) transformations

5) applications/modeling

More after the jump...

Continuing in the project to blog the major topics in the Pre-Calculus curriculum (as I see them) brings us to finding the rational roots of polynomial functions.

The topics involved in studying the Rational Roots Theorem seem to me to fall into two major categories – polynomial factorization and the analysis of polynomial functions and their roots.

More after the break...

I recently wrote about Algebra Review for Pre-Calculus and discussed some problems that I typically assign in the College Algebra courses I teach. Here is a link to some examples of those problems including algebraic simplification, simplifying rational expressions and solving equations involving rational expressions.

I'll write next about the Rational Roots Theorem and polynomial factorization. Here's a problem I found from a Japanese University Entrance Exam from 1990 that makes interesting use of these concepts. The document I took this from is here - the problem is from the first sample test on page 4.

Suppose the polynomial P(x) with integer coefficients satisfies the following conditions:

(A) If P(x) is divided by x^2-4x+3, the remainder is 65x-68

(B) If P(x) is divided by x^2+6x-7, the remainder is -5x+a

Then we know that a={?}.

Let us find the remainder bx+c when P(x) is divided by x^2+4x-21.
Condition (A) implies that {?}b+c={?}.
Condition (B) implies that {?}b+c={?}.
It follows that b={?} and c={?}

I've changed the notation of the answers a little in the hope of making it less confusing - you can see the original by clicking through the link.

I got pretty tangled up in solving this problem because I had never seen the Remainder Theorem used in quite this way before. The answers for this are on page 5 of the original.

I've been using Michael Sullivan's Algebra and Trigonometry textbook for the last few years to teach College Algebra/Pre-Calculus/Trigonometry on the quarter system, but we're switching to John Coburn's Algebra and Trigonometry next year. They're both pretty good textbooks.

In building Pre-Calculus curriculum I've drawn mainly from four textbooks:

Mary Dolciani - Modern Introductory Analysis (original copyright 1964, my edition is from 1986)

Richard Brown/David Robbins - Advanced Mathematics (this copyright is from 1984, a newer edition of this book is here)

Paul Foerster - Pre-Calculus with Trigonometry (copyright 1987)

Max Sobel/Norbert Lerner - Pre-Calculus Mathematics (copyright 1995)

I had thought about using the Sobel/Lerner book for the Pre-Calculus course I developed for Clatsop Community College, and even e-mailed Max Sobel asking him about a new edition of that text, but he very kindly replied that he had retired (I also have the Harper & Row Algebra I and II textbooks from his series with Evan Maletsky and like these as well).

One of the things I liked about the Sobel/Lerner textbook was the coverage of rational expressions that several of the other books I had considered didn't have. I like rational expressions because this topic requires a firm grasp of many of the most important concepts from algebra – factoring and multiplying of bi- and trinomials, simplifying complex expressions, and combining like terms in the context of manipulating algebraic fractions. If students understand numerical fractions it helps a lot.

When I teach College Algebra courses at Clatsop CC, I often begin with exercises like (x+7)(2x-3) – (x+1)(x+5) to address these topics and prepare the students for when they see this again in working with rational expressions. Another good example is (x+6)(3x+1) – (x+2)^2, to get them used to seeing the squared binomial. ( I make the squared binomial a regular visitor in most of my algebra classes). These expressions often appear as the numerator of a combined fraction in a problem like (x+7)/(x+1) – (x+5)/(2x-3).

Here is a link to a collection of problems I often assign for this topic.

I recently began to reacquaint myself with the College Board Math II Subject Test which I had taken after taking Pre-Calculus in the spring of 1982. The first question on the sample test I looked at was intriguing, and an understanding of rational expressions is really useful in finding a quick solution:

If 3x+6=(k/4)(x+2), then k=

a) ¼

b) 3

c) 4

d) 12

e) 24

In this problem, dividing through by (x+2) so that 3=k/4 (and 12=k) gives the almost instantaneous answer we need for a timed test. This is an interesting problem because it really gets at the concepts involved in working with factors in an equation.

In the Sobel/Lerner textbook, Sections 1.7, 1.8 and 1.9 cover the algebra review (multiplying polynomials, combining like terms, factoring polynomials and rational expressions) necessary to move on. This is where it is important to illustrate these topics with problems, because when I say “combining like terms,” I don't mean 2x+5x. While this type of problem could be appropriate when first teaching the concept, in the context of review for Pre-Calculus, a problem from the Sobel/Lerner text like (x^3-2x+1)(2x)+(x^2-2)(3x^2-2) is better practice for using these skills together.

This is something that I consider extremely important and that textbooks and assessments often don't include enough of – using the skills together. Learning skills in isolation is useful to grasp each skill individually, but to really DO MATH, a student must be able to make decisions about what to do and when. Something that I like about the Pre-Calculus curriculum is that it lends itself well to the type of problem in which the tools of algebra must be applied in a variety of situations.

The Brown/Robbins text covers complex numbers and the quadratic formula in Chapter 1 (1-4, 1-5) and then the solution of equations involving rational expressions in Section 2-2. Chapter 2 goes on to examine the graphing of quadratic and polynomial curves and finishes with material on finding rational roots.

The Sullivan book covers polynomials and algebra in sections R.4, R.5 and R.7, Coburn covers this in sections R.3, R.4 and R.5.

The Foerster and Dolciani texts don't really cover much algebra review at all, but, as a result, they explore a number of topics the other books don't. I'll probably follow a path similar to the Brown/Robbins book and talk about rational roots next.

Rich Beveridge writes:

I suppose that my experience with Pre-Calculus curriculum began in Steve Patterson’s Pre-Calculus class at Briarcliff High School in the 1981-82 school year. Pre-Calculus always stuck out in my mind because it was the only math course that was completely locally developed. Algebra I, II, and Geometry all had Regents exams and the Calculus course was AP Calculus.

I remember studying Conic Sections, Polynomial Long Division and Synthetic Division, the Rational Roots Theorem (and its proof), elementary Discrete Math (permutations, combinations and binomial probability), Polar Coordinate graphing and hand calculating Riemann Sums at the end of the year. I took the College Board Math Achievement Test II (now the SAT Subject Test Math II) after completing the Pre-Calculus course so I recently looked at some current sample questions and saw these same topics – Analytic Geometry, Permutations & Combinations, Synthetic Division, Functions, Sequences & Series.

During the 1999-2000 school year, I taught Pre-Calculus at Maine Central Institute in Pittsfield, Maine. The school was using the Chicago Series text Functions, Statistics and Trigonometry for their Pre-Calculus course. I know that some teachers like the Chicago Series and FST in particular, but I didn’t really get much use out of the textbook, and began to supplement. Standard textbooks can be supplemented quite easily because the order and difficulty level of the topics is often similar. I found that the Chicago Series was very difficult to supplement and just began to create separate materials for the students. I collected these assignments in a binder and showed this to the University of Maine math department when I was interviewing for an adjunct position the following year (yeah - I didn’t stay at MCI very long – they were sticklers for using the approved textbook). I taught as an adjunct at UMaine for two years before beginning their MA program in Math.

I'm hoping Rich will write more posts for us.

Diane Ravitch writing in today's Times:

If every child arrived in school well-nourished, healthy and ready to learn, from a family with a stable home and a steady income, many of our educational problems would be solved.

Waiting for a School Miracle
By DIANE RAVITCH
Published: May 31, 2011

I can tell you definitively that a child arriving at school well-nourished, healthy and ready to learn from a family with a stable home and a steady income does not solve the educational problem of a high school junior needing to learn precalculus inside his actual school.

Approximately $500 for private tutoring at $100/hour.

Now we'll see how much it costs to pay said tutor to re-teach precalculus over the summer.

help desk - precalculus
lsquared on what you must know to take calculus
Crimson Wife's online tutoring recommendation

(lsquared couldn't get this comment to post on Blogger):

Yes, you need to be very good at algebra. Think factoring, equation solving and complex fractions.

You also need to be quite good at trigonometry: knowing right triangle trigonometry is sufficient for almost everything except calculus--for calculus you need to know how to solve equations that have trig functions in them, and you need to know your trig formulas (the double angle formulas are particularly useful). You don't actually do any of the "verify this trig formula" stuff in calculus, but those "verify the trig formula" problems are very useful for teaching the algebraic trigonometry you use in calculus. Oh--and radians. I'm morally convinced that radians were invented to make trig functions work for calculus. Degrees are better for everything else, but in calculus, you have to have radians.

You also need to know logs: solving exponential equations with logs, solving log equations with exponents, and manipulating logarithms.

Back to rational functions. Be able to solve rational equations, simplify rational expressions that are not equations, and graph (by hand, not by calculator) rational functions (using properties of the rational function to graph, not by plotting points) (I may be unusually picky about this).

Sequences and series, including finding sums using the formulas for arithmetic and geometric sequences. Also the binomial theorem.

You don't need matrices, though you do need to be able to solve simultaneous linear equations. If there's a section in the pre-calc book on find partial fraction decompositions, that's directly a calc algorithm.

I think that might be everything you need for pre-calc.

This was an excellent exercise for me. I’m going to have to keep a copy of this to use when counseling students about whether they are ready for calculus or not. Indeed, the homeschooled kid next door wants to get into my calc class in the fall, and I shall be handing this to him I expect. Good luck finding a good class--I think the community college might be a good choice (assuming that they have a pre-calc course).

This year's math course has been a disaster. Beyond disaster, actually. Where math is concerned, this year has been epically^* bad.

Some kind of emergency repair has to happen this summer - but what?

Any suggestions?

I could conceivably sign C. up for an algebra 2/precalculus course at the local community college - but are the teachers there going to be any better? At this point, we desperately need an actual math teacher: a person who can teach math to a student who doesn't teach math to himself.

We could do ALEKS, but ALEKS is super-slow and overwhelmingly procedural; the 1 1/2 courses I took on ALEKS taught me one disembodied procedure after another. At this point disembodied procedures might be better than nothing -- but then again to the extent C. learned anything this year he learned disembodied procedures.

I could insist that the two of us work through Foerster (I own the teacher's edition).

We could work our way through Saxon.

I could advertise for a math teacher or check out the various tutoring companies....

We also have to do serious SAT prep (though that's not going to be onerous & time-consuming).

I'm thinking this wasn't the summer to sign up for the precision teaching institute at Morningside.

^* My neighbor's son, who is a terrific writer, is constantly inventing words he says ought to exist, and having watched him do this a few times, I think he's right. Epic is an excellent word, and its non-existent cousin epically is the word I need today.

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.