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Friday, November 5, 2010
Algebra II Before Geometry
This is happening in our area (not at my son's HS), but I'm not sure why. Is it being driven by the CCSSO standards? Do high schools want to get most students through the material (pseudo-Algebra II) before it's too late or while the algebra "iron" is hot? If this is true, then the top math track in high school is under attack. It's one thing to channel students off to integrated math, like Core Plus, but with the current emphasis on the traditional sequence of math classes, there seems to be only one direction for the content to go. Traditional math won, but ends up losing. Is that what will happen in lower schools if they decide to use Singapore Math? Perhaps I'm reading too much into this. I don't have any strong feelings about whether geometry or algebra II should come first, but I do have strong feelings about rigor. What is the real driving force behind the switch? Do they use the same textbooks? One comment I heard once was that this would allow the geometry classes to delve more deeply into proofs because the students would be more mature and would have more math background. I don't buy it. What direction would algebra II go, even for the honors version?
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SAT and other Junior year tests had had more Geometry than Alg2 emphasis. Hence the sequence Alg1 Geom Alg2. Many have not like how this disrupted the "Algebra flow" and made it more vulnerable to textbook changes.
My observation is that it's driven by the science sequence. Algebra II is typically a pre- or co-requisite for chemistry. In my alma mater, they kept the traditional sequence but shifted Algebra I from 8th/9th (honors/regular college prep) down to 7th/8th.
This is often a response to the fact that an increasing number of students come to high school having taken a weak "algebra I" course in junior high. They don’t want to retake algebra 1 in high school (since they and their parents think that they’ve already “had it”), but they don’t know the material well enough to dive into a real Geo or Alg2 course. So, the next course winds up spending a *lot* of time “reviewing” algebra 1. That’s often easier to do in the Alg2 context than in the Geo context, so some high schools flip the order.
Not happening here; the weird thing here is that a high school student can take Chemistry concurrently with Geometry, before taking Alg. II. (Chem and Physics are offered every year, not alternating years here).
In my time Geometry and Alg II could be taken concurrently if one was a math brain. That practice is expressly prohibited here.
Our math curriculum change last year was to add "Bridge to Algebra II" for the politically chosen (my term) that were allowed to take 8th Alg I and 9th Geo and leaned so heavily on a afterschool help & a tutor that they couldn't maintain an 85 in Geo. 8th Alg so far is a lot more rigorous than 9th Alg...could be the individual teachers but more likley the dept chair's decision on what units will be taught to what students.
My older kids attended a suburban high school that has a strong national reputation in math and science, although it's not a magnet. The honors/AP sequence was (and I think still is) algebra I (only offered at honors level) in 8th grade, followed by honors geometry, honors algebra II with trig, honors elementary functions with analytic geometry and AP calculus BC. The equivalent science sequence was freshman honors lab science, sophomore honors bio and often honors chemistry, junior honors chemistry and/or physics and/or AP chem or AP bio, then AP physics (concurrent req AP calc) or AP chem. All AP sciences were double-period, every day, and require successful completion of the corresponding honors course.
I can't help thinking that the push to remove homogeneous grouping in ES-MS and push everyone into 8th-grade algebra has resulted in a watered-down algebra I. Unsurprisingly, there used to be a strong positive correlation between 8th-grade algebra and HS success/good SATs/etc. because 8th-grade algebra was then taken only by the top students (same deal for Latin, debate etc). Putting kids who haven't mastered the basic facts and operations, let alone decimals and fractions, into "algebra" amounts to educational malpractice; for those kids and for the kids who are prepared and are being deprived of the real algebra they deserve.
The high school in question is medium sized (900 kids), and offers TERC and then CMP in the lower grades. If kids do well enough on the state math test in 8th grade, they are allowed into the (now before geometry) algebra II class as a freshman. Imagine going from CMP into algebra II. Most end up in a regular algebra course, and some end up in an algebra course (with lab). Ironically, around these parts, a math "lab" means that the lab focuses on mastery of basic skills rather than tackling traditional lab-type problems. Some of it is just moving chairs around on the Titanic, but I can't imagine that CMP kids can handle a properly-rigorous algebra II class.
Does anyone know whether these schools use the same textbook for algebra II as before? I'm still thinking that this change has more to do with the "flow" issue. I don't think it's done to make the material more rigorous.
They also do "physics first", but the course is called "Physics First and Earth Science", which is just a fancy way of trying to meet the state's low requirement for science. It has no meaning for the students who are trying to get to the AP science courses. My impression is that the algebra II & geometry swap is also driven by the need to meet the pseudo-algebra II state standards before the state test in the junior year.
Since this topic has the word geometry in it, I'll ask a question I've been mulling over:
Can anyone recommend a good high school proof oriented geometry textbook that doesn't use Euclid's axioms, but instead uses more modern (and non-broken) axioms?
-Mark Roulo
I took Algebra II right after Algebra I. I took Algebra I as honors in the 8th grade, and the honors 9th grade class was Algebra II. I think this was because 9th grade was at the junior high, so to teach geometry as the honors 9th grade course would required getting a high school teacher geometry teacher to come over for one period a day which wasn't practical. I don't think it matters so much the sequence the courses are taught in as much as the rigor of the courses.
Can anyone recommend a good high school proof oriented geometry textbook that doesn't use Euclid's axioms, but instead uses more modern (and non-broken) axioms?
Adrian mentioned Birkhoff's Basic Geometry on the Well-Trained Mind boards a few times - it's published by AMS and uses Birkhoff's axiom system, which he put together in the 20s/30s. Unlike Euclid's, they use the real numbers and their properties. I'm not sure if they are the most up-to-date axioms, but they are far more so than Euclid ;), and the book is accessible for high schoolers, which seems to be a rare thing with more modern axiom systems. Anyway, the book, teacher's guide, and answer key for Birkhoff are all on Amazon.
One good reason to postpone Algebra II until after Geometry is that there are plenty of kids who do better at Geometry (highly visual, highly spatial) than at either of the Algebras. The good experience of Geometry sort of breaks up the more difficult experience of Algebra, and keeps them in the game.
Thanks, forty-two! I will look into this.
-Mark Roulo
Recent math article in the NYT may be of interest:
http://www.nytimes.com/2010/11/07/education/edlife/07books-t.html?ref=edlife
- Hainish (now blogging)
(Hainish, thanks for the pointer to the article. My complaint is about the article, but I thank you for pointing it out to us.)
From the article:
[A certain math book] is written clearly and covers the expected topics. But it also falls into the common trap of ridiculous word problems:
A tank contains 20 gallons of antifreeze solution...
No, the author falls into the common trap of thinking that a word problem is about its surface features, not it's abstract "underlying concepts". By this logic, Singapore math is more about training grocers than budding professionals because all you learn about is mangoes and papayas.
The point of word problems in math class as opposed to, say, physics, is not to study the real world, but to learn how to see through the real-world to the abstract world beyond.
HS Geometry books: I got some good use out of Moise & Brown in terms of axiom systems. It's old, but worth a look.
For me, SteveH's post prompts some questions:
(1) Is there a generally accepted definition of where Algebra I ends and Algebra II begins? What about the boundary between pre-algebra and Algebra I?
(2) I know what algebra, geometry, and trigonometry are. What is "pre-calculus"?
(3) Does geometry deserve a full year? I have used it much less than algebra, trig, and calculus. Geometry is often recommended based on the training in logic afforded by geometry proofs, but one could do proofs in other areas of math.
The Saxon and Singapore curricula integrate geometry into algebra and have an optional separate course in geometry, according to
http://www.learningthings.com/blog/index.php/saxon-geometry-singapore-math .
"Geometry is often recommended based on the training in logic afforded by geometry proofs, but one could do proofs in other areas of math."
I think the question is whether one can build up a 'proof structure' in other areas of math that is as nice, large, and reasonably well defined as the proof approach using geometry.
Number theory, for example, is full of proofs, but I don't get a sense that there is a 'tree' of them that you build up. Maybe this is incorrect.
The nice thing about doing proofs with planar geometry seem to be:
(a) The students can visualize/understand what is being proved,
(b) The proofs start small and build on each other,
(c) The proofs all 'fit together'
If another area of math provided these, I don't see much reason to favor geometry proofs.
-Mark Roulo
The pre-calc class I took was trig, analytical geometry, and towards the last part of the year, a gentle intro to differential calculus. I think that's fairly typical.
I was skipped from 7th grade math to Algebra I in the 8th grade so I never took pre-algebra, and did just fine in algebra. In high school, I wasn't able to take the pre-calculus course offered by my school that followed the trig course I took. In college, I initially signed up for pre-calculus thinking I needed it before taking calculus. The first day of class I came in, took one look at the syllabus and realized I had learned everything in my trig class, so I dropped the class and went right to calculus. I think pre-algebra and pre-calculus are pretty much useless terms. Pre-algebra is just a fancy term for 7th or 8th grade math, and pre-calculus is just whatever math didn't fit into the other courses.
When I was in school, it was called trig, not pre-calculus. Pre-calculus seems to be a fairly common name now, but you have to look at the actual textbooks to compare content.
The same is true for pre-algebra. I don't remember what it was called when I was in school, but now it's a common name and that's what the textbooks are called. My son had the Glencoe Pre-Algebra textbook and it was fine.
My rule of thumb is that if the textbook is called just Pre-Algebra or Algebra I, then it's the most rigorous version offered by the publisher. If it adds something like "Tools for a changing world", then you know it's not for the top students.
That's why I would like to see which textbooks schools use for Algebra II when it's scheduled before geometry. My fear is that it will geared to the lower algebra II expectations of the CCSSO standards. In my son's high school, they use the same textbook for both the regular college prep and the honors course. That will work only up to a point.
I haven't seen a pre calc or calc class that has taught analytic geometry in the last 20 years. It isn't taught anymore, as far as I can tell. Go find a Thomas calc book, 2nd edition, and you'll see chapters and chapters of real work in analytic geometry. None of that left now in whatever edition they are at.
Steve, what does this mean? "he lower algebra II expectations of the CCSSO standards. " Can you tell me what's lower?
My AP calc textbook was titled The Calculus with Analytic Geometry. I thought it was awesome.
(And yes, this was slightly under 20 years ago.)
Allison wrote, "I haven't seen a pre calc or calc class that has taught analytic geometry in the last 20 years. It isn't taught anymore, as far as I can tell."
There is a book "Analytic Geometry", 7th edition, by Tarwater (published by Addison Wesley in 1993). On Amazon, there is an interesting review by Craig Frutiger recommending the book and echoing Allison's complaint that analytic geometry is no longer taught.
"Can you tell me what's lower?"
It would help if I was more organized. I can't find my notes. I remember looking at their expectations and comparing them with my son's Algebra II text. It seemed to follow their ideas for algebra I. Things were missing.
I also remember reading how the algebra II requirements were based on their "workplace analysis" of what people could expect on a typical job. I thought it was odd that they didn't work backwards ... e.g. This kind of job requires this sort of degree, and the degree typically requires the student to pass these math classes. They could at least have looked at the requirements for the SAT or the ACT to define their content, but they didn't. The problem with that approach is that you might have good enough scores to get into a college, but not be prepared for the degree you want.
Then, there is my issue about whether a course is treated like a lifetime terminal course in math or whether it will form the base from which to tackle advanced topics. I don't want to see the traditional AP math track dumbed down. To some extent, this can be dealt with using a regular versus honors version of each course, but it might depend on whether the regular course is a simplified version of the rigorous course or whether the rigorous (?) course means they cover a few more chapters in the dumbed down textbook.
I checked the website of the high school my older kids attended and the honors/AP sequence seems to be the same as the one (see previous post) my older kids had, with the exception of the name of the course between honors algebra II with trig and calc BC. When they took the course, it was called Elementary Functions and Analytic Geometry and widely called E-FAG. I know that title was changed, for PC reasons, and the course is now called Precalculus with Analysis, but I think the course is essentially the same.
It would've been 16 years ago that I took a pre-calc class that covered analytic geometry. I don't remember which specific text we used, but the title definitely mentioned both trig and analytic geometry. I had no idea this was no longer the standard for pre-calc courses.
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