kitchen table math, the sequel: Designing a High School

Monday, November 10, 2008

Designing a High School

A parent group at the charter school my children attend in Colorado is investigating adding 10-12th grades. I volunteered to work on the curriculum committee even though I doubt my middle school sons will attend. In order to be approved by the district to extend the charter, the school must provide an education not available at other local schools so they are looking at a math & science focus.

State and district standards in math and science are pretty basic, trig concepts, bio, chem and physics. The high school would be added to a k-8 school with a core knowledge curriculum that also teaches Singapore Math and Latin. An overwhelming majority of the students are in Algebra by 8th grade. My 7th grader is in Algebra and there is at least one 8th grader taking differential equations at the community college. (FYI- This school uses New Elementary Math as a pre-algebra course)

The committee is set on requiring math through Calculus and the basics in science. We're wondering about what to add as required: engineering courses, advanced biological science courses or physical science courses?

Suppose you were given the opportunity to devise a math & science high school. What type of coursework would you require?

49 comments:

RazzyHENZ said...

I wouldn't have the least idea of how to design a math and science high school.

I just wanted to mention an interesting charter school that we have locally. The students take their core classes at the local community college, but they take them together as a group, not with the college students. Most of the classes are taught by the regular college instructors. When they graduate from high school they also graduate with an associates degree from the community college. They still attend their regular high school for electives and sports.

CassyT said...

razzy-
We were thinking of hooking up with CSU. Can you tell me the name of the high school? I searched the top 100 high schools on Newsweek and US News & World Report and they have similar partnerships.

The way charters are set up in Colorado, students can participate in sports at any high school in the district. My boys played soccer on campus, however there was a group of kids who played tennis for a nearby middle school.

Jean said...

My husband will write up something coherent later--he's working too much today--but here are his basic thoughts. (He co-owns a small software company that designs scientific software, such as radiation therapy dosage calculation.)

Yes, then I'd require calculus, chemistry, physics, biology (with emphasis on genetics), and a FREAKING CLASS ON CRITICAL THINKING

Logic, basic computers--everyone in the sciences needs basic programming

RazzyHENZ said...

The charter school is called Success Academy. Their website is at http://successacademyonline.com. They are partnered with two different colleges and two different school districts.

Anonymous said...

First, I would hope the committee has a curriculum mission statement that says "we build toward mastery."

Mastery of the basics is FAR more important than the number of advanced courses you offer in teaching your students. It may not be in terms of getting your charter, but it will be the reality.

So, a sequence of the basic courses with 10th grade chemistry, 11th grade bio (that can build on the chemistry), and 12th grade physics. The chem course has algebra 1 as a pre req and alg 2 as a co req means you have to have had alg1 and geometry 1 already by then, and be in alg2. The bio course has enough detail that you could really understand the krebs cycle because you understood how to write teh chem equations.

The physics course does NOT need to have calculus under the belt of the students. Just studying pure mechanics without calc to mastery is good enough. one year of mechanics.

Each course should have a real lab component, not some silly one hour a week lab. Try to have 3-5 hours of lab per course.

Then, for teh additional stuff, i'd suggest:

1. a science fair course, or other science project course. Each year. A course where students design an experiment from scratch, test the hypothesis, report the results. most local science fairs are in March, with those winners feeding into the state and big international competitions, like the westinghouse or intel ones. it's unreasonable to support such a course inside the basic science course at the high school level, so an additional course would be fantastic. this course would help students find topics, refine topics, build their experimental apparatus or whatever, learn about data reduction, etc.

I suggest teaching a real computer science course. Not a programming course, but a course in how to think about computer science. The gold standard is a course in Scheme, based on Simply Scheme, which is a simplified version of the first few weeks of UC Berkeley's version of MIT's SICP: structure and interpretation of computer programs.

then, for additional courses: a 2nd physics course that taught mechanics or E&M with support for calculus, if the studetns have had it.

(more thoughts later).

CassyT said...

Allison said:
So, a sequence of the basic courses with 10th grade chemistry, 11th grade bio (that can build on the chemistry), and 12th grade physics.

I'm hearing more about sequencing in science, however typically the *new* sequence is physics, chem, bio. Thoughts?

I joined the committee to learn more about high school non-math curriculum. The k-9 school that is already in place was one of the 53 National Charter Schools of the year in 2007 and is fairly rigorous.

Lsquared said...

Physics first seems plausible, but make sure you have algebra 2 as a co-requisite. Physics without algebra isn't worth doing.

Doug Sundseth said...

I'd think about the following:

* Deductive Logic
* Rhetoric and Logical Fallacies (to possibly include debate)
* Research (to include writing a real research paper)

BTW, if you don't mind answering, whereabouts in CO are you? (For reference, my family lives in the northern Denver suburbs.)

Anonymous said...

lsquared is right. unless they are at least at the algebra 2 as co req and geometry as pre req level, physics is worthless.

also, there's a reason to teach chem to help bio: knowing enough chemistry to understand redox in biology is really helpful; knowing how carbon and oxygen and hydrogens affect respiration is vital. knowing how proteins are formed, maintain shape, denature, etc. is critical to understand genes.

but how classical mechanics works matters not one whit to how stoichiometry works. and the physics you need to understand chemistry at the periodic table level is some elementary quantum physics--you can teach all the quantum you need inside that chem class. teaching real mechanics or E&M wouldn't teach that level of quantum anyway either. and teaching real quantum at h.s. level is impossible.

SteveH said...

I agree with Allison. Define the basics well and then worry about the needs (glitz?) to get your charter.

Also, students should not be rushed to get to advanced courses like differential equations. If a student is advanced enough to get to this course before college, then a local community college course is not going to be proper. Alignment with a major university is a better approach, if it can be done. There is plenty of time in college to get all of the advanced courses you need unless the goal is to minimize college costs or if the student is truly advanced.

There is a big push around here for "Physics First". I think the idea is that physics leads to a better understanding of chemistry, which leads to a better understanding of biology. This might be great for chemistry and biology, but not for physics. The problem is that you really want math first, and that can't happen. The more math you have, however, the better physics will be. Physics First seems to place physics lowest on the totem pole and turns it into a conceptual, hands-on, low-math course.

I prefer putting biology and chemistry first because they require less math than the sort of physics class I would like to see. Algebra II is a minimum, but trig and calculus first would be even better before physics. I don't have a strong opinion about this, but I would never recommend Physics First.

CassyT said...

Doug said:

BTW, if you don't mind answering, whereabouts in CO are you?

Fort Collins - Liberty Common.

Steve H said:
Define the basics well and then worry about the needs (glitz?) to get your charter.

The Poudre district has some pretty good schools, (compared to the rest of the state) and Ridgeview Classical Charter School. (Great school - humanities focus -although they do have a geneticist on staff.) Liberty needs to differentiate to get approval of the charter.

It's a challenge as we started talking STEM, then everyone put in their own ideas. (two languages, band, choir, health, calligraphy, graphic arts & PE for electives) It would be difficult enough to be all things to all people. Creating a science & math program serving 45 tenth graders the first year, then growing each year thereafter seems daunting.

I guess their first try at a curriculum 5 or 6 years ago had almost exclusively AP courses. The district told them it was impossible. All the high schools in Poudre district offer either AP or IB, some offer both.

FYI- I was thinking economics and some sort of science ethics course. (maybe just a semester of ethics) Is that possible at a high school level?

SteveH said...

Is STEM a specific curriculum or is it just a focus? One school I found seems to use an inter-disciplinary, hands-on approach, where the school leader (?) says that STEM is blending practice with theory.

Science, math, and technology in college are not all applied. Even much of engineering is not applied by their standards. Having an integrated, applied approach to these subjects is not a proper preparation for college.

[My middle school son now thinks that science is experimentation. He had to do an test to show that the period of a pendulum is only a function of the length of the pendulum (and gravity). I told him that this wasn't true. Period varies based on the angle. Even though the math is over their heads, the teacher could at least have forced them to do a test accurate enough to discover this truth.]


It's always nice to see applications built on top of the basics (when there is time), but when you design a curriculum around a top-down, or inter-disciplinary approach to the material, mastery of the basics loses out. You don't need an integrated learning environment to get the benefits of application. Students can have separate courses or after-school projects for that.

I suppose this is one of the first decisions you have to make even before putting together an exact curriculum. I would be very skeptical about anything not built around the traditional AP courses.

This brings up another issue; AP, IB, or both. I don't care for IB because it's usually all or nothing, with fewer choices for students. However, My niece's school offers both with three levels of IB. They do this by cross-listing many AP and IB courses. In the end, it seems that the big difference is whether you take Theory of Knowledge or not.


"I guess their first try at a curriculum 5 or 6 years ago had almost exclusively AP courses. The district told them it was impossible."

Impossible schedule-wise, or impossible because they won't allow it?

I think that if you focus on quality AP-track courses, then you can fit other options around that. Since there are so many things to choose from, you should set up a process for adding and deleting electives (over time) based on interest and available expertise.

In the public high school I went to, the music department was huge. This wasn't a top-down decision. They hired a music director who built it from scratch. (I think the school was just lucky.) He got the school system to hire good middle school music teachers and defined a proper, continuous curriculum. He primed the pump, so to speak, and the results were amazing.

A high school could get involved with a robotics competition, the Science Olympiad, or many other projects without having an integrated curriculum. National competitions can be nice because they are formal and force kids to see how they fare regionally and nationally. Competitions can also get you a lot of recognition throughout the state, or even the nation. They inspire kids because they really, really, like road trips. They also allow parents to get involved and provide their own expertise. I think the key to these extras is to hire the right people and provide the proper support. Priming the pump in the lower grades is also good.

Another parent and I were the coaches for my son's FIRST Lego League robotics competition a couple of years ago. The school's science teacher helped out, but didn't put in much time or effort. When we didn't get involved the next year, it all fell apart. Schools have to be willing to provide the framework and continuity. Parents can often provide a huge(!) amount of free support, and many will even spend a lot of their own money. Schools need to tap into that.

Anonymous said...

"I would be very skeptical about anything not built around the traditional AP courses."

I would be skeptical too, if they were replacing general AP courses with flim flam constructivism not recognized as math by a mathematician.

But something to consider (for the discerning skeptic) is that if you want the same old outcome then do the same old AP courses.

Roy Smith, graduate of math from Harvard, now teaching at U of Georgia publically criticizes AP Calculus for ruining honors calculus at the university level. Honors calc is theory based and requires the student to be familiar with proofs while the typical engineering calc is based on teaching the computation necessary for engineering.

A link to his argument is here:

http://myrtlehocklemeier.blogspot.com/2007/04/has-ap-revolution-changed-high-school.html

David Klein last year addressed the lack of theory (mathematicians call this rigor) as well as too calculator use in the AP exam in this report:

http://www.csun.edu/~vcmth00m/AP.pdf

All of the high schools in my suburbian utopia offer AP Calculus. You'd have to do something really special to attract me to a math and science school other than this tired ole same ole. An option for courses with focus on theory rather than application so that my kid might actually have a chance at honors calc might work. Topology, Abstract Algebra, Combinatorics taught by someone who's taken graduate level courses in those topics would be out of the ordinary and would be attractive also.

Anonymous said...

--[My middle school son now thinks that science is experimentation. He had to do an test to show that the period of a pendulum is only a function of the length of the pendulum (and gravity). I told him that this wasn't true. Period varies based on the angle. Even though the math is over their heads, the teacher could at least have forced them to do a test accurate enough to discover this truth.]

Steve,

Do you mean the teacher didn't say "for small angle" ? Or didn't do the experiment except for small angle, and never showed what happened for BIG angle?

I mean, the derivation of the pendulum's independence of theta is beautiful, but it requires sin theta approximates as theta, and that's true for small theta only...I don't know if your son's class is possibly prepared for that--for the approximation of theta, for sin(theta) in the first place, let alone for the double dot of theta that you get when you approx sin theta as theta, but I don't see how a teacher could present it without that. Do you know what she did experimentally?

do you think your son understands that what you meant wasn't what the teacher meant? That omega^2 is g/l is correct--there is no theta in that equation? It'd be awful if your attempt to fix the bad instruction led him to be even more confused....

Anonymous said...

re: topology, abstract algebra, etc. or advanced science:

it all depends on what their foundation is, but you need to define things very carefully. you can't do real science without alg 2 as a co req, and you can't do interesting math without solid proofs from geometry.

fundamentally, can you assume that the 9th graders have had a solid algebra 1 course already or not?

If so, then the best sequence is something like:
geometry for 9th, alg 2 for 10th, precalc for 11th, calc for 12th.

If you want to advance the students more, you could do geo and alg 2 is 9th grade--doing a lot of plane geo in 1 term, and most of alg 2 in another, with trig and precalc in 10th then.

then, you could do some interesting things as electives, perhaps.

number theory could be taught to advanced students--a term of it could get you to chinese remainder theorem, maybe. you could model it on PROMYS or Ross @ ohio state summer programs.

a discrete math course would be really neat--teach some combinatorics, some bayes' rule, some graph theory, recurrence relations, etc. again, they'd need a solid grounding in proofs. perhaps you do this as a 1 term in proofs course, qith geo as a pre req, and then a term of discrete math if they already learned induction.

topology is way too sophisticated for most, because though it sounds like it's about interesting surfaces, it isn't really. it's way too abstract.

but all of this is dependent on a real algebra 1 mastery before they get there. otherwise, you'd need to remediate that, doubling that up with geo 1 on the schedule in 9th.
(cont)

Anonymous said...

now the next issue is textbooks. they are abyssmal, so how do you get these courses designed from textbooks that are garbage?

the Dolciani stuff is the least bad for algebra 1 and alg 2, but by no means is it good. for geometry? it's worse. there's a tiny little precalc book out there that you could design a course around, but only the best math teachers could do that, in that it's too slim to do that as it is. the Kodaira books from Japan are worth investigating, but again, you'd need to have a competent math person design courses from them.

there are no courses for discrete math for high schoolers, I've looked. you'd have to do that from scratch with the help of some folks somewhere. even adapating the Ross or PROMYS materials would take a lot of time, since those courses are designed for several hours a day of study, and working in small groups to do problem sets, not a large classroom.

Anonymous said...

High schoolers should not be doing honors calc in a high school, no way. they aren't prepared, and there are no TEACHERS that are prepared for that either. it's not even a good course in college without an excellent teacher, but it's too sophisticated for any but the srs in high school anyway. the issue is what comes first.

lgm said...

>>The committee is set on requiring math through Calculus and the basics in science. We're wondering about what to add as required: engineering courses, advanced biological science courses or physical science courses?

None of the above as req'd, but all that the faculty and facilities would permit for electives. Communication courses would be next for the req'd component -

Communicating in Engineering & the Sciences
Advanced Composition
Debate
The Art of Selling
I'd also ad:
Design & Drawing (CADKEY and DataCAD are used here)
Intro to Engineering, Intro to the Sciences (even interested students aren't aware of some of the possibilities)

Suppose you were given the opportunity to devise a math & science high school. What type of coursework would you require?

Math, science, psychology, personal finance, fine arts, an overview of how businesses work & are financed, communications, problem solving, and finally some basic electronics, vacuum, and the mechanical technology behind the lab equipment they are using. The geo course would be proof based as it was in my time, which eliminates the need for a separate logic course. Phys Ed would include Orienteering.

-- lgm, an engineer

Barry Garelick said...
This comment has been removed by the author.
SteveH said...

I should have said that this is seventh grade science. I know. What do I expect?

As far as I know, the teacher never said anything about small angles or big angles, and never talked about any equation. This is seventh grade, so the teacher can only go into so much depth. They just make a hypothesis, do the experiment, collect data, and then check their hypothesis. I think this experiment is popular because the results may not be what the kids were expecting, and it's easy to do the experiment.

I couldn't explain to my son that the equation approximates sin theta with theta, but he didn't think there was any approximation or error in his results. That's what first got me started on this. He didn't appreciate the importance of accuracy.

I guess what I don't like is that the teacher leaves it at that. There seems to be no discussion of experimental accuracy or what you might do with the data. Even a discussion of modeling results with equations is not part of the lesson. The kids see science as data collection and graphing.

CassyT said...

Allison said:
If so, then the best sequence is something like:
geometry for 9th, alg 2 for 10th, precalc for 11th, calc for 12th.


No trig? Currently 95%or so of all students take Algebra 1 by 8th grade. There are less than 6 students taking it in 9th grade. The texts for the classes already provided at the school through 9th grade are: 2006 3rd edition Foerster Algebra 1 textbook and the Weeks-Adkins, Geometry, Algebra 2.

The ideas here are informative and helpful. With AP & IB being the expectation in Fort Collins, I think it would be great to do something completely different. If you teach effectively, can't students take AP tests without labeling the classes? Can students still CLEP out of classes at many colleges? (I'm thinking 5 years of Latin ought to count for something!)

My biggest concern is how the committee will be able to put anything together for 10th grade in two years with the addition of only 2 faculty and 2 admin. That means using the current faculty in place for 7th-9th grades for most classes (humanities). I'd cut an admin and add a faculty, but there you go. I'm not a huge fan of bloated admin.

BTW- Block scheduling thoughts?

Barry Garelick said...

the Dolciani stuff is the least bad for algebra 1 and alg 2, but by no means is it good.

Please explain why you believe Dolciani's algebra texts are not good.

Tom Hudson said...

Several of the states have elite residential high schools for science & math - you might look at them. A cousin went to the Illinois school (https://www3.imsa.edu/), and my wife is a North Carolina alum (http://www.ncssm.edu/). Here in NC, the school has a mandate to experiment, so their curriculum changes every few years. I don't see a document on their web site with "lessons learned", but it'd be in line with their mission.

To respond to the one comment above that falls in my sphere of experience: I'd avoid the Scheme-based thinking-about-CS-approach; my understanding was that even MIT has moved away from that particular book. There are a lot of fluffy thinking-about-CS books out there, but I think there are a few good ones available that aren't as hardcore as "the purple book".

Anonymous said...

Yes, MIT has moved away, because some new profs wanted to design a new course. That's what academics do: make new stuff to justify their existence regardless of the old stuff. Why else are there a zillion textbooks in every subject? But Cal hasn't moved away from it, and neither have dozens of other universities.

Logo or Scheme are simply brilliant languages for teaching thinking about algorithms. They aren't meant to teach programming per se, and they dont' have to--pick a different language for those. But the coursework exists, free, already, and has been used in high schools before.

re: dolciani: I'd stick with Prof. Wu's comments on Dolciani, but I have to find them. his grade on them is about a C or C-, I think. My short answer: I've used those books personally as a student, and they aren't exactly inspiring, and they have some confusions.

Anonymous said...

re: trig:

I didn't mean to leave trig out. I assumed trig is in the alg 2 and pre calc. I don't know WHAT counts as a pre calc syllabus, but it's got trig in it, de moivre's theorem, analytic geometry I think, etc.

Anonymous said...

Barry, I'll give you a real answer re: dolciani in a day or so when I have had more time to find my examples.

CassyT said...

Allison-
Re: trig

When I went to school it was Alg 1, geo, alg 2, trig, calc 1, calc 2. There was no pre-calc, however I see it listed in place of trig in the district course pathway materials online.

Do schools no longer teach trig? Is it more of a college course or just smushed into pre-calc?

Independent George said...

I'm not sure Calculus should necessarily be a goal in High School; I think linear algebra or discrete math are viable alternatives. They're not any less rigorous, but it's a much smoother transition. The work just "looks" more like the work they'd been doing.

I don't think Calc itself is especially difficult, but it seems does to require an intuitive leap which I'm not sure how to describe. So much of it is about piecing together everything you ever learned in math since the first grade; it's a big step from previous years, and for many students, it's just too much to do in a single year.

Anonymous said...

Cassy, pre calc is trig, plus some proof by induction, or algebraic geometry in most of the syllabi I've seen. it's not a college course at all, it's just mushed into a catchall now called precalc. maybe "trig" scared people, or maybe "precalc" sounds more college prep!

If calc requires an intuitive leap, I can't possibly see how discrete math or linear algebra doesn't. But that's why you've got to have a good solid grounding. It's not too much to do in a single year if you already have solid grounding in limits, rates of change in algebra, plane geometry, algebraic geometry. But most haven't.

Anonymous said...

Hi Barry,

Here are some of the things I dislike about Dolciani et. Al.:

1. their seemingly random usage of axioms, properties, and rules
2. their discussion of the reals without mentioning the rationals when all they need are rationals for the first 200+ pages, and they almost never use anything but an integer anyway.
3. their confusing introduction to inequalities, opposites, and absolute values, and the bizarre way they break up the material on those things and separate it from solving equations.
4. a tendency to explain things using inductive reasoning—as if exploration of an idea doesn’t just build intuition, but actually leads you to know (or have shown?) a truth.

Below are some examples from the Algebra 1 text. They are meant to be illustrative, not exhaustive, but point to my general dislikes that lead me to say that this book is passable, but not good, and I’d be thrilled to find a better text book in wide use today. I haven’t, of course, but that doesn’t mean I have to think well of this one.

Okay, to point 1: In chapter 3, they introduce the notion of an axiom, calling it an “assumption”. Fine. Then they supply the axioms of closure under addition and multiplication. On the same page, they list “the axioms of equality” and then they name 3 properties as the “properties of equality”. Well, which is it? Why suddenly are these axioms properties, but closure is not a property? Further, they don’t define a property. Later, they list rules for addition. They don’t make clear the difference there, either. It’s not clear to a new entrant into algebra what are assumptions and what are derived. The rules, it should be clear, can be derived from the above properties, which were assumed. That’s not stated. Later, they name something the addition property of equality. Huh? Is this an axiom? No, it follows from the closure axiom of addition. These properties are getting hard to keep track of, and I know the field axioms.

(sidenote that annoyed me: Later, they include the axiom of additive inverse, naming it the axiom of opposites. They tell you that you can say opposite of a, the additive inverse of a, or the negative of a, and go to great lengths to say “Caution!” –never call it “negative a”. This is absurd, and much more likely to cause people to get confused about what –a means. )

I could talk more about the reals and rationals, but the point is: they go to great lengths to describe these axioms as being true for the reals, but they almost never use the reals. They do for the inequalities, in that they use the number line occasionally (for equalities, they are always on the integers.) But for the problems, they use integers almost always. Why not introduce rationals? Why not show some non integer problems?

Absolute value: they make this difficult, when this is just such an obvious place to drill home clarity. If you can’t make this clear, you know you’re losing kids immediately. They describe it by saying the positive number of any pair of opposite real value numbers is the absolute value. Fine. They write an numeric example: |5| = 5, |-5| = 5. They leave out writing the variable example, but that’s where it matters most. They never show how to turn an expression containing an abs value into a set of true expressions without the abs value, conj or disj:. i.e. |X| + 3 --> x + 3 and –x +3.This matters little until you hit inequalities, but that’s where the rubber hits the road. So how well do they show this? They have one absolute value inequality example whose answer is a contiguous line segment; this they graph, and at the end, they bother to turn into an equation only barely. They state “another inequality with this graph is …”: as in , they don’t say “this is the same inequality” or “this is an equivalent expression”. They just mention that |x| < 2 happens to have the same picture as another inequality -2 < x < 2.
They show just one example where the inequality is an infinite set of two disconnected pieces; never do they show how this is written without the abs value. They don’t show in the explanation how you’d rewrite it at all, even though they’ve given you all the closure properties you’d need. They never explicitly show you how to see that x < 5 is the same as 5 >x, either, even though turning around the inequality is key to reading the inequalities as presented. Now, this isn’t the chapter where they teach you to solve inequalities (they do that in chapt 5, and there they do explicitly explain the conjunctions and disjunctions), but they present the complicated inequalities in the written exercises here—asking you specifically to SOLVE the equations, something they don’t present until chapter 4.

They make more of a hash of the inequality stuff later in chapter 5. Here, while trying to teach you that the numbers are well ordered, and that comparison works, they introduce “addition and multiplication axioms of order” for inequalities. This is bizarre, and unnecessary. These aren’t axioms, in that they don’t need to be assumed, they follow from the properties of closure addition. Why here name them axioms? Worse, they seem to be an attempt to relate to inequalities to closure equalities for addition. In fact, they come out of nowhere trying to explain how to think about the statement
-5 < 2 if you happened to add the same number to both sides. Here’s the thing: the expression isn’t an equality. It’s true, and it’s true even if you add a number to both sides, but WHY they are doing this is frankly untethered since they are making as if they are solving an expression. They try to help with a picture—see, -5 was on the left of 2, and now when we shift both by c, -5 +c is still left of 2 + c. Then they generalize to “what if you did multiplication by a negative?” Now, they are trying to show that the signs change, or equivalently, you flip the inequality. But they aren’t showing what the heck it MEANS to multiply, let alone by a negative. The comparison to addition on the number line is limited, and their picture doesn’t help. The addition showed that you didn’t change the order—great, okay. But multiplication doesn’t shift the number line, it stretches and collapses it in some odd ways. Multiplication by a negative rotates it around the 0. none of this is said, so it just comes out of nowhere. There’s a bitty picture, but I doubt anyone would figure it out if they didn’t know already what was going on.

Okay, now to a problem I have that requires a bit of subtlety: they seem to think inductive reasoning is a helpful way to introduce ideas. Their intro to closure and equality says “after checking many examples, you would no doubt assume that the product of two whole numbers is a whole number.” Later they give another example and state “these thoughts suggest the addition property of equality”. Yes, it might suggest that. But I find that this level of subtlety, where we are trying to be illustrative and then jump into properties is troubling. It seems to suggest we can look at examples and leap to properties. It confuses, I think, when examples are general enough to turn into properties. I see what they are trying to do, but I think this is really undermining of a mathematical way of thinking. YMMV.

Anonymous said...

Do any of you have experience with the Art of Problem Solving books? How does their algebra book compare to other textbooks?

Anonymous said...

"High schoolers should not be doing honors calc in a high school, no way. they aren't prepared, and there are no TEACHERS that are prepared for that either...?

From Stuyvesant High School:

"Multivariable Calculus
MC5
Multivariable Calculus (MC5) is a second or third year college level course in advanced mathematics. The course extends the limit, differentiation, and integration concepts of first year calculus to functions of more than one independent variable, which occurs in economics, statistics, physics, chemistry, and biology. There is a more theoretical treatment of the limit, derivative and integral. These concepts are applied to work, projectile motion, cycloids, optimization, related rate, volume, normal curve, mass, density, and moment of inertia problems. Vectors are used in three-space for analytic geometry and for rates of change of functions in all directions "

They have both students who complete it successfully and teachers with the math background to teach it properly.

My guess is that the prerequisites to this class weren't taught out of a Saxon math book.

To prepare a student for rigorous math, you condition them through use of rigor in their prior classes, not by teaching "intuition".

bky said...

You cannot use the best high school in the world (Stuyvesant) as an example of how to construct a curriculum. They have skimmed the cream of the cream to populate their one- (okay, one or two) of-a-kind school.

Also, as someone with an advanced degree in math, I dispute the claim that to prepare for rigor in future work you need to emphasize rigor in high school. Intuition is more important than rigor in the early stages, and is indispensable at later stages. Rigor can be introduced gradually over high school and college and comes easier if the student already has some facility.

Notice that in the course description from Stuyvesant, they are doing a more theoretical treatment of limits, etc. in what is in effect a 2nd or 3rd year college course, which means that, regardless of whether we are actually dealing with high school or college kids, this sort or rigor is being introduced to kids who already know a lot of calculus. They have the intuition and now they are ready for the rigor.

You don't have to prepare students for rigor in mathematics by starting them with rigor in mathematics.

SteveH said...

A lot depends on your raw material coming into 9th grade. After all I've seen and heard about education and standards, I guess I automatically focus on a good coverage of the basics. I suppose I could come up with something that's better than standard AP classes, but I don't know how that would fly in reality. I've been trained to be happy if the curriculum is not complete crap.

I would like to see rigor, but that might mean something completely different for a school that teaches the top of the top. On one end, we have schools that think all learning can be done in a hands-on, thematic fashion, and on the other, we might have a school that preps the top one percent. When I hear words like intuition, it makes me first think of hands-on, not theory. Sorry. This comes from reading too much education talk on critical thinking, like trying to solve a simple two equation and two unknown problem with guess and check.

I do think there are different levels of understanding (rigor?) and that it's OK to introduce topics without achieving a full understanding every step of the way. I found it perfectly fine to have a course in linear algebra without having a formal introduction to linear spaces. When I got that instruction much later, I never thought that I should have had it before. But, then again, the devil is in the details and where you're going.

I read Cassy's search for a good high school and I realize that finding, let alone designing a top-end high school is quite unlikely. It just tends to set my sights lower. But it's still miles higher than what's going on. Talk of parking, rather than academics, would cause my head to explode.

If I were part of a curriculum committee for a new school, I would focus on getting the basics right and providing a structure and process that allows the curriculum to evolve. It doesn't have to be a one shot deal.

Anonymous said...

--David Klein last year addressed the lack of theory (mathematicians call this rigor)

"theory" is not "rigor". Mathematical rigor means sophistication in mathematical topics. A rigorous math problem requires the synthesis of many ideas from many different points of view, the ability to adapt argument to problems you've not seen before, and the ability to turn the crank--a lot of turning the crank. it's computationally intensive, for whatever kinds of computations you're doing.

Many mathematicians would think of rigor as synoymous with complexity, and elegance as a kind of foil. While not opposites, elegant proofs are usually not viewed as rigorous, but seem to fall out of the sky with simplicity. Rigorous proofs involve a lot of grinding away at a problem, working on several fronts, et.

Both are necessary. But the more you are comfortable with the concepts, the easier it is to be rigorous. the more you've handled the simple stuff to mastery, the less the rigor will scare you.

Building intuition to mastery in high school is more important that slogging through rigorous courses. Calculators don't teach mastery, but well designed problem sets do teach to mastery and improve intuition at the same time.

Adrian said...

Also, as someone with an advanced degree in math, I dispute the claim that to prepare for rigor in future work you need to emphasize rigor in high school. Intuition is more important than rigor in the early stages, and is indispensable at later stages. Rigor can be introduced gradually over high school and college and comes easier if the student already has some facility.

Notice that in the course description from Stuyvesant, they are doing a more theoretical treatment of limits, etc. in what is in effect a 2nd or 3rd year college course, which means that, regardless of whether we are actually dealing with high school or college kids, this sort or rigor is being introduced to kids who already know a lot of calculus. They have the intuition and now they are ready for the rigor.

You don't have to prepare students for rigor in mathematics by starting them with rigor in mathematics.


I'm afraid I'm going to have to call bullshit on that.

It is true that there is a certain school of thought that is responsible for the general curriculum nearly universally employed today and for the last close to a hundred years now. That school of thought is a radically empiricist view of the world including math that seems to think that the "real reason" for the truth of the theorems in mathematics is based on their application to the physical world or some sort of other purely *heuristic* rationale. Nothing could be more absurd or further from the truth. The real reason for anything in math is the best completely rigorous proof of it. If it were any other way, then math would be done completely differently at the professional level, and graduate students and math majors (or mathematicians for that matter) would just simply never have to deal with rigor at all. Rigor really would be pointless -- a kind of a game like the people (such as Morris Kline or W W Sawyer or a good many rank and file applied "mathematicians") that denigrate it so much would like to make it out to be.

But, rigor is not a game, and intuition and heuristics are nothing more than one man's fallacious way of thinking about things that, at best, serves as some sort of mnemonic device for remembering the actual reason -- the real reason -- for why the theorems are true, namely their rigorous, a priori, mathematical proof. I, too, have my own heuristics -- my own way of thinking about things. But, no one's not even a Fields Medalist's nor a Millenium Puzzle Solver's nor Euler's nor Gauss' nor Archimedes' heuristics can ever be a substitute for the actual rigorous mathematics. And none of that, and quite specifically none of one's calculus (and, it is a shame that I must use that name to refer to the abomination of freshman college math) will ever get you past that qualifier in real analysis. All calculus does is teach you to think fallaciously.

It is widely known that what prepares students for the abstract algebra and real analysis that they take as senior math majors is that combinatorics class specifically set up as the bridge-to-higher-math-course. It might be some other course -- usually it is combinatorics -- but almost every department in the country has that bridge course. And, they specifically use it to get you something like (again quite specifically) exposure to Mathematical Induction and a little naive set theory (that is why discrete math/combinatorics is usually used) before you hit Abstract Algebra. And, then, it is typically advised to do analysis after algebra.

Your calculus isn't going to help you in real analysis. Instead you are going to have to forget everything you only think you know about calculus and actually do it the right way when you get to real analysis. And, everyone that has been through the program knows this, so let's just cut the crap. What calculus is for is so that the engineering and business majors can really superficially cover the material in a very short amount of time and short cut their way through it so that they can cover a crap load of physics based on that and then do their engineering or business statistics or something. Calculus is quite literally the Royal Road to Geometry. What actually ever prepared anyone for rigorous math was rigorous math -- the baby proofs they did in their combinatorics class and then the slightly harder and faster material they covered in abstract algebra. And, then the hope was that they were ready for real analysis in a few short classes. And, the hope was that that, in turn, would prepare them for graduate school which they probably arrived at and found out that they weren't really prepared for. There is just massive attrition starting at or before that senior real analysis class. And, it is really all because you should have spent the last eight years slowly building up to harder and harder problems with complete rigor rather than just jumping into rigor all of a sudden in the last two or so.

So, calculus might be just grrrreeeeeaaaat! -- a f-ing fabulous course! Fine. But, don't act like we do it for the math majors. We are specifically screwing them over to accommodate the physicists, engineers and business folks. (And, frankly, it's probably a false economy, at best, even for them.)

********

For anyone that wants to know what to do for a supped-up Science and Math curriculum, my advice is just forget the math and just do science. Science is what everyone wants anyway. So, just do it -- more and more of it. Extra biology and extra chemistry and computer science. Do some sort of high school version of snap circuits. Teach it all in a practical, hands on, heuristic and empirical fashion. And, just forget about the notion that throwing someone in the kind of differential equations classes that undergraduates take is actually doing more math. It is just methods of the physical sciences. That's all it is. If you want to really do the math for something like that, it starts with rigorous calculus which you are never going to be able to pull off in that context.

But, I have taught an average twelve year old axiomatics. He has definitely made some remarkable progress on a path specifically aimed at Spivak's calculus which is the only truly rigorous modern development of that material for freshman college students that I know of. So, it's not like it's impossible. You just have to make some choices that most (even technically literate) parents aren't willing to make. (And, of course, you have to search really, really hard for materials to do something like this with.)

Anonymous said...

---Roy Smith, graduate of math from Harvard, now teaching at U of Georgia publically criticizes AP Calculus for ruining honors calculus at the university level.

funny to criticize a COURSE. I mean, seriously, this is putting the cart before the horse, and then being mad that the cart isn't doing the pulling.

the criticism should be levelled at universities. it was their CHOICE to accept AP credit. if Stanford is idiotic enough not to offer Apostol vol 1 to its students, BLAME STANFORD. Likewise, blame U Georgia. they know that a 5 on the BC test doesn't teach proofs, but they DO NOT CARE. Universities could push back on the College Boards folks any day and STOP GIVING OUT CREDIT for these courses. "parents expect it"--so FIX their expectations! (MIT never before let anyone out of calc for taking the AP test. do they now? Certainly not for Apostol!)

--Honors calc is theory based and requires the student to be familiar with proofs while the typical engineering calc is based on teaching the computation necessary for engineering.

and honors calc is beautiful and meaningful for math majors and physics majors and a handful of folks, and that beauty is only really meaningful if they have enough intuition to make headway in it. this applies to a tiny tiny fraction of kids. Are we losing them? Yes, we are losing the best and the brightest from math and physics, it's true. The problem isn't that AP calc doesn't teach that--it's that nothing ELSE in high school does either. so abolishing the AP course might work at TJ or Stuyvesant, where the other courses are good enough to teach proofs to mastery, but you aren't learning that EITHER in standard urban or suburban high school.

APs aren't a panacea. But blaming them is like blaming a fire cracker for the fire Mrs. Oleary's cow started.

Adrian said...

Mathematical rigor means sophistication in mathematical topics. A rigorous math problem requires the synthesis of many ideas from many different points of view, the ability to adapt argument to problems you've not seen before, and the ability to turn the crank--a lot of turning the crank. it's computationally intensive, for whatever kinds of computations you're doing.

Many mathematicians would think of rigor as synoymous with complexity, and elegance as a kind of foil. While not opposites, elegant proofs are usually not viewed as rigorous, but seem to fall out of the sky with simplicity. Rigorous proofs involve a lot of grinding away at a problem, working on several fronts, et.


No. Elegant proofs are rigorous. That is what is so nice about them -- that they are completely rigorous and yet that simple. Rigorous proofs are quite frequently not at all computational. Probably your typical point set topology involves NO computations. (It's just set theory -- you don't even have a metric yet to do computations with.) A rigorous proof rarely involves "turning the crank". That is what you do in heuristic classes like calculus where you are given a formula to apply. You go forth and turn the crank on 50 nearly identical problems for homework and then complain to the professor that the nearly identical 51st problem he gave you on the test was "...COMPLETELY DIFFERENT!! I WAS NEVER TAUGHT THAT!!!"

A rigorous proof actually takes skill.

Anonymous said...

Well, it definitely takes skill. but gosh, the rigorous proofs I've done in number theory turned a lot of cranks. now, I wasn't very good, but my diff geo proofs and topology proofs were a lot of crank turning too. my definition of "computation" didn't mean number calculations or heuristics. That's hwy I said "for whatever calculations are", because when I'm proving things about metric spaces it's grungy in different ways than when I'm proving things in combinatorics. and when I had to do miserable point set topology, it was ALL turning the crank on the definitions. you might not think of that as turning the crank, but all i did was crank through the defnitions, over and over again. I found it utterly uninspired.

Adrian said...

For instance, you think the Heine-Borel Theorem is just a matter of cranking through definitions? I don't think so. The Intermediate Value Theorem in calculus is a topological result that is thought to be so hard that it is universally skipped in freshman calculus. If it was just a matter of turning the crank on definitions, then everyone would at least just give the rigorous proof of it in the text. In fact, the chapter on topology is usually the hardest chapter in classical real analysis texts. Baby Rudin's last problem is: if G(n) is a sequence of dense open subsets of k-dimensional Euclidean Space, then the infinite intersection of all G(n) is not empty. Crank through the definitions on that one!

You can't just mindlessly apply definitions -- not even in a field more plausibly like that like Abstract Algebra. But, certainly issues like compactness or whether a space is lindelof or not are not at all like turning a crank. You might find it uninspiring, but that's the true nature of mathematics, and it is infinitely harder than coming up with some heuristic rationale for a bunch of theorems someone else had to prove. (Although, on the other hand, if all you have are heuristics, then you are easily duped by someone who knows the rigor, so maybe it is harder to rely solely on heuristics.)

Rigor is not just some game that pure mathematicians play in order to vex the uninitiated, nor is it just a trivial matter of attending to the details once you've got the "concepts" all figured out. It is the hard part. It is what makes the difference between getting and not getting it.

Anonymous said...

Adrian,

we've clashed before. I've already told you you're a better mathematician, but if you think that you've never done turning the crank work in proofs, then you're the only person I've ever heard of who thought that.

the straw man you're fighting isn't me. you didn't agree with my words. fine. our definitions don't match for what "rigor" "theory" or "turning the crank" mean. i've never talked about heuristics, and never alluded to doing numeric computations at all.

and yet you agree with me: you solve real problems by cranking through all of it.

rigor is the hard part. that's why intuition matters first, because it makes it possible to tackle rigorous work. the kind of crank turning i'm talking about IS hard, that was the point, and that's why rigor is about mathematical sophistication, not just a synonym of "theory".

Barry Garelick said...

I agree with Allison. Rigor is the hard part; I believe intuition comes first and leads to the more thorough examination; i.e., rigor.

An excerpt from a conversation with a PhD mathematician which relates to this subject (original conversation was about how much theory do you teach kids):

"It is been my experience in teaching college level mathematics that it is sometimes good to hold off on explicit definitions and proofs and just allow students to get comfortable with calculations first. (One most often sees this in the way that Calc I is taught: most professors wave their hands at the definition of limit, get students to understand the concept using pictures, and then proceed to calculations of derivatives. We demand that undergraduates
understand careful definitions and proofs in a junior or senior level class after they have had 4 or more semesters getting comfortable with calculating derivatives first.)"

Another mathematician I know, who teaches at a small college, (one of his classes is real analysis) talks about "mathematical maturity" that students must have before they can start tackling rigorous proofs. There is a differentiation (or there used to be anyway) in lower level calculus classes. I went to school at U of Michigan. Those students who had calculus in high school and a qualifying score on an AP exam in calculus were allowed to take honors calc, which was a more theoretical version of the subject than the entry level version that hoi polloi such as I had to take. (Ironically, the textbook I had in freshman calc was by Edwin Moise, which had a lot of point-set topological explanations in it; the book had other bigger problems with it, however, and it was dropped after that one year. I still have the text. It's a good book if you've been through calc.)
Anyway, there were two honors level calc classes. The top honors class used Apostle's Calculus.

Adrian said...

For a hundred years we have taught the freshman calculus. One might think that this is a pretty credible well-established thing to do since it is nearly universally done throughout the world and has been done so for as long as anyone can remember. However, for centuries prior to that, calculus was universally deemed an advanced topic not suitable for freshman. One might imagine that, then, it must have been some sort of evolved outcome -- that we used to do that until we figured out how to do it better with the calculus sequence.

But, for one thing, if you go back to the very first textbooks on the subject you will find something that I think is rather remarkable. They are just like they are today! Well, not exactly -- at least on the surface they look quite different. The older texts such as this one by F L Griffin called An Introduction to Mathematical Analysis (don't let the name foll you -- this is just a normal calculus text) is a lot smaller and without nearly as many diagrams and so on. No glossy pages, etc. Nevertheless, it covers the standard freshman calculus almost by the numbers. It starts with functions and graphs -- typical fare for a chapter 0 on precalc. Then, it heuristically talks about limits and instantaneous rates of change using graphs of functions and lists of values at various points. Then, it proceeds to discuss differentiation and then integration., all with lots of exercises that ask the student to apply these concepts to problems from science and engineering. Sound familiar? The next two chapters cover log and trig functions in a fashion that appears to be what you would see these days in special precalc courses. (I guess they didn't have those as much back then, so it had to be covered here.) But, then it gets back to the calculus in a chapter that continues talking about the log and exponential functions, introducing e as the limit of (1+1/k)^k as k -> infinity. Then it goes on to discuss rectangular and polar coordinates, newton's method for finding the zeros of a funciton, calculus of trigonometric functions, using integrals on surfaces rotated about an axis, for instance, series,.... There is a chapter there on permutations -- I don't think you would see that in a typical calculus text today. But, this is almost lock step playing by the same script we do today. And, this is like the first American text like this copyrighted in 1921.

Calculus is not some kind evolved outcome. It is a very specific concoction that has barely changed since it was first introduced almost a century ago. In fact, there are good reasons that extend well beyond the field of mathematics as to why it may have seemed appropriate to do something like that back then but I won't digress into that any further.

The point of all that is to help understand the evidence and/or source of this notion that "intuition comes first". Essentially, I am going to say that no it doesn't. There is no such thing as this kind of "intuition". There is no heuristic or intuitive understanding of mathematical concepts or ideas that stands on its own in any epistemologically legitimate way when the true nature of those ideas is formal and a priori like that. In fact, it is the definition of sophistry that it consist of persuasive but not sufficient justification for its assertions. And, that is exactly what these heuristic rationales -- this intuition -- is by its very design. It really doesn't matter that the assertions they purport to defend are true. In fact, all the worse that those assertions are true -- especially after years of this treatment of things in mathematics, even the brightest and most strong minded, independent thinkers will eventually have to succumb to the program and be conditioned into employing this so called "intuitive" approach that will ultimately fail them.

And, I will say another thing: proving theorems and carrying out calculations are really entirely different activities. Doing the latter doesn't really train you to do the former. Decades of "bridge courses" to higher math and virtually every preface to every real analysis text out there explaining just how completely unprepared students are for the course are proof enough of that.

CassyT said...

Well, here's what the committee settles on after an hour or so of discussion:

Science Required Courses –
40 credits required for graduation

ICPE
Biology
Chemistry (Advanced)
Physics (Advanced)


Science Electives

AP Biology
AP Chemistry
AP Physics

Other Potential Science Electives

Computer Programming (C++ or other highly common language)
Anatomy / Physiology
Introduction to Engineering (Semester)
Drafting / Graphics / CAD (Semester)
Energy (Semester)
Climate / Meteorology (Semester)
Internships - based on success in semester class with local business / university

Mathematics Required Courses
40 credits required for graduation

Algebra I
Geometry
Algebra II
Pre- Calculus (Trigonometry)
-----------------------
AP Calculus AB

Math Electives
Statistics
Discrete Math

Finance falls under the Humanities committee, and I believe they are using the course to meet a "life skills" requirement.

Year long courses are 10 credits, semester courses are 5 credits. District requires 20 credits in each math and science, however local universities (CU, CSU) require 3-4 years.

Anonymous said...

i'll try one more time, Adrian, and then you can have the final word.

1. Have you ever successfully proved a theorem true that you believed false? Or did you not have to come to believe it was true first?

2. have you ever been afraid? most people can't do pure research because they are afraid of the risks--the risks of looking dumb, the risks of being wrong, the risks of being in uncharted territory. they are afraid to ask the interesting questions, afraid to work on that blackboard alone. they know how to answer when someone else lays the breadcrumb trail, but they need to get passed that on their own eventually.

the kind of sophistry you are talking about and railling against is the kind that helps them start walking that trail by themselves. Even if it IS a fiction, it is an essential element of the authentic description of proving a mathematical theorem. part of the authentic experience is first a bunch of inauthentic attempts--the part where you write the problem down again, doodle some, go get coffee, go reread the book, get distracted, write down some definitions because you don't know where else to begin, etc. the inauthenticity IS some place that students can get stuck, and if they never get out of it, they'll never be professors in math depts. but even math department profs start proofs this way. it's just a question of how long you stay in that inauthentic ground before you work up the existential courage to do the real work.

so the intuitive approach is a way to get rid of their fear, because fear prevents having the ideas. and of course, refined intuition that is more often CORRECT is even better. but that comes with practice, too.

Linda Seebach said...

Adrian said, "For a hundred years we have taught the freshman calculus ..."
but a hundred years ago, and far more recently than that, that was true only for a small number of freshmen. I entered Gettysburg College, a moderately selective school, in 1957, and I was only the second freshman ever to take calculus (and I married the first, who was two years ahead of me).

Our text was Granville, Smith and Longley, which dated from around 1904 (Granville was the president of Gettysburg from 1910-1923, and it was suspected royalties were involved). Even then it was an antique, full of hand-waving about "infinitesimals." No rigor at all, no matter what definition you prefer.

Even today, what percentage of first-year college students start with calculus? Maybe 20, 25 percent?

Catherine Johnson said...

good grief - 46 comments?

I'm behinder than I thought.

OK, I'm throwing in the towel for the night; must do GrammarTrainer with Andrew!

Catherine Johnson said...

Still haven't read, but I just saw the reference to Stuyvesant.

C's school is filled with kids who didn't get into Stuyvesant!

Of course the major population there is kids who didn't get into Regis.

Being from the boonies, C is in the category of kids who didn't get into Fieldston & Rye Country Day. He wasn't eligible to apply to Stuyvesant or to Regis (only Catholics may apply to Regis).

Of course, no one got into Fieldston & Rye Country Day this year. It's unbelievable out there; people are fleeing the public schools in droves.

Every time a district dumps its AP courses parents scram out if they can.

That's life here in the leafy suburbs!

Catherine Johnson said...

Razzy Henz --- where is the school you're talking about!?

That sounds fantastic.

A brilliant work-around.