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Comparing an old french mathematics book for grade 9 and a new one, one obvious difference is the ordering of the chapters. In the old book, all the algebra chapters are grouped together then all the geometry chapters so the teacher either should finish all the arithmetics and algebra before moving to geometry then to calculus or the school should have separate weekly periods for algebra and geometry. The new book mixes algebra and geometry chapters.

Is it better to finish algebra chapters before geometry or to mix them? I prefer the first one, I don't like to have geometry chapters between the polynomial chapter and the rational fractions for example or to have algebra chapters between similar triangles and trigonometry.

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    $\begingroup$ I don't understand why you feel the ordering of material in the textbook dictates so strongly the ordering of teaching. If the algebra and geometry are independent, a teacher could switch between the two when they chose. $\endgroup$ – Jessica B Oct 6 '17 at 5:33
  • $\begingroup$ @JessicaB Of course, it's not always easy to change the order of chapters in a book but I'm wondering if there's any benefit from interchanging them. Why do most modern textbooks mix algebra and geometry chapters? $\endgroup$ – Paracosmiste Oct 6 '17 at 10:27
  • $\begingroup$ @whatever It's because most modern algebra and geometry study are interlinked. For a classical example, see straightedge and compass constructions. $\endgroup$ – Chris C Oct 11 '17 at 4:19
  • $\begingroup$ @ChrisC Nothing changed in elementary algebra and geometry in the last 40 years. $\endgroup$ – Paracosmiste Oct 11 '17 at 12:59
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    $\begingroup$ In order to justify the equation of a line is a line, you need similar triangles. To justify that the equation of a circle is a circle, you need the pythagorean theorem. $\endgroup$ – Steven Gubkin Dec 4 '17 at 19:56
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To many students, geometry is very different than algebra. They see geometry as a bunch of facts (we call them theorems) to be memorized (we prefer that they understand). I feel the same things happens with trigonometry. Many students who are successful in algebra struggle with trig and geometry because, I think, it feels so different.

So, I think, that blending the two could help. Blurring the artificial distinction between the two could help students prepare for advanced courses (e.g., in calculus related rates problems and integrals-as-sums-of-tiny-slices problems require geometry) by showing students that they are all related.

This is based on my experience teaching high school geometry for a couple of years as well as high school calculus. I feel that this blending helped students at the school. For examples of the kind of blending I mean, see the first few pages of this.

(As an aside, I also did this blending with algebra. As kids were learning how to solve equations, I asked them to do it in a two column format, where they had to justify the steps they were carrying out)

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The question may well be reversed in places not so far away. Should we finish Geometry before Algebra?

The real news is that once begun in either there is never any finishing. Both topics are vast. You will not need even simple algebra in early geometry but you need to master the theorems. Mastery is why they resort to memorization, the only way I could have managed. Eventually you will have geometry problems that require algebra to solve. These are what got me addicted. That there could be any link between two such odd disciplines, - any, try dozens and dozens - is an awakening you've yet to experience.

I learned very late that math subjects do not truly belong in the layers we find them in school. It's just handy to know algebra before you need it in geometry or you will have to go back and learn it, or go back and learn trig to move ahead in calculus. So long as you can get through all the classes whatever order the teachers are used to using will serve you as well as it has those that have gone before you.

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  • $\begingroup$ This answer is sort of odd. I don’t dispute the points made — that there aren’t really “ends” to, a linear order on, or distinct boundaries between mathematical disciplines — but I’m not sure how much it addresses the actual problem OP is struggling with: that a linear order must be imposed on a curriculum, due to the nature of time. But I guess the advice is to just pick one and go with it :) $\endgroup$ – pjs36 Jan 12 '18 at 7:24
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I don;t really know definitely but maybe the following is helpful.

  1. Discussion of same question:

http://forums.welltrainedmind.com/topic/245197-algebra-ii-or-geometry-after-algebra-idoes-it-really-matter/

  1. My personal experience was that I probably had the brains to go on to Alg2 right after Alg1, I lacked the maturity (too be specific, I was too lazy). Geometry ended up being a good break before slamming into logarithms and the like.

2.5. Also, geometry is a weird beast anyway you cut it (versus the algebra to calculus track). But there is maybe a tiny value in getting exposure to proofs before doing induction in algebra 2. [I think this is a miniscule positive, but including because it occurred to me.]

  1. In the USA, there used to be some benefit in getting Alg1 & Geometry before hitting the SATs. I think there may be some Alg 2 content in there now, but still not much. So you are maybe a little marginally better prepared (earlier) to take the SAT and practice it, if you take geometry earlier. That test loves the darned vertical angle theorem. Um...and adding stuff to 180.

  2. I think in the US and Anglo countries, the more typical fashion is algebra 1, geometry, algebra 2. So there are more books, school systems, etc. geared to it. VHS versus beta. Or maybe there was a reason why they converged to that, not sure.

4.5. I guess traditionally schools (even quite good ones with smart students) didn't cover more than algebra 1 and geometry at all. Algebra 2 was "college algebra". So maybe there is some argument that if you are only going to get two things done, get algebra 1 and geometry.

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  • $\begingroup$ I'm not against doing 6 chapters of algebra then 5 chapters of geometry then... but I don't like doing 1 chapter algebra then 1 chapter geometry then 1 chapter.... $\endgroup$ – Paracosmiste Oct 6 '17 at 10:25
  • $\begingroup$ There are 3ish main choices: (1) most common in US: algebra 1, geometry, algebra 2 [year of each], (2) much less common: algebra 1, algebra 2, geometry; (3) also much less common: mixing geometry in(lecture or chapter at a time with algebra 1, then a year of algebra (this is the Saxon method). My preference is 1 or 2 versus 3 because geometry iis a different sort of thinking (proofs) than solvinv more and more complicated equations (algebra). I would give a mild preference to 1 as algebra 2 can be harder intellectually. $\endgroup$ – guest Jan 11 '18 at 19:50
  • $\begingroup$ Of course arguments can be made for 2 (don't forget algebra 1) or for 3 (make geometry less an oulier to the algebra through calc track). I really do feel it is an oulier and better off just taking the year "pill" and being done with it. [As with many things, it is a tradeoff--which frustrates me on this site when I see answers expecting all good or citing a single downside. Shows the super math types not internalizing the idea of optimization or of multivariable functions into life situations.] $\endgroup$ – guest Jan 11 '18 at 19:54
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I would say yes. I also worry about my (college) students who take time off from math classes, e.g. not taking one their senior year of high school. You can forget a lot in a single year, especially with technical information that you aren't using some other way. The same applies to a two year algebra sequence. It makes sense to move directly from the first year to the second rather than taking a year off to do something else.

In the US, however, this isn't the usual approach. It's usually Algebra 1 -> Geometry -> Algebra 2. The reason for this has more to do with teaching to the test, in this case the SAT, rather than any pedagogical reason. The SAT has geometry so students need to complete geometry before their junior year. Since many students don't take Algebra 1 until their freshman year, this means sliding geometry in between the two algebra courses.

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  • $\begingroup$ The order was common before the SAT existed. $\endgroup$ – guest Jan 11 '18 at 19:44

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