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Posted at 5:00 AM ET, 01/14/2011

Math class: What’s the right order?

By Valerie Strauss

In what order should students take math? Should Geometry come before or after Algebra 2? Here are some of the ways schools are working through the issue.

Below are responses school counselors around the country to a request for information about the order that high school students take math classes. The query was issued by Jean Crowder, director of the Mathematics, Engineering, Science Achievement (MESA) Program at Sacramento State University and University of California at Davis. The responses were sent to her via the e-list of the National Association for College Admission Counseling. She gave me permission to publish the responses (keeping them anonymous), which I found revealing about how schools experiment with math.

The original query to counselors from Crowder: "I am working with a local high school site that is considering altering the enrollment/completion order for mathematics for students. This possible new completion order would be as follows: Algebra I, Algebra II, Geometry, and higher/other courses that follow geometry at the 11th grade year. I am interested in comments on the proposal from both the secondary and post-secondary levels."

The responses from counselors (I've left in the abbreviations as written by the authors):

1. We went in that direction and then changed it back to Alg 1, Geom, Alg 2 (and then either College Alg or PreCalc, Calc, APs etc.) because we found that students were not getting the Geometry they needed for the SATs in time.

2. My own children attended a charter school that used your proposed sequence, while my school of employment uses the sequence of Alg. I, Geometry, Alg. II, PreCalc/Trig, Calculus. From a personal and professional viewpoint, most students just aren’t developmentally ready for a rigorous Algebra II as sophomores. The retention is not there, which makes the PreCalc / Trig course senior year even tougher.

3. I can see a good argument being made for taking the alternate approach that puts Alg 1 and Alg 2 in a direct sequence, followed by Geometry. The Pre-calculus/Trigonometry course that follows all of these will use a lot of Algebra and also will mix in some elements of Geometry. But the Calculus will initially be much more dependent on a good background in Algebra than in Geometry. (second year Calc and MV calc will not be so forgiving.) So if the argument is to get students into Calculus earlier, I could see an advantage in making the sequence Algebra 1, then Algebra 2, then Geometry at a later point. The question begs whether it is wise to do so - if the student is ready developmentally for the meaty Algebra 2 class so early in the high school career. I could anticipate some students not having the study habits needed to fully master that topic (Algebra 2) which is an important stepping stone on the way to the Calculus.

4. We used to use this format, however, the math teachers found that by the time the kids were in Pre Cal, Alg. II was too far away. There was a lot of re-teaching Alg. II concepts. We are not, Alg. I, Geometry, Alg. II.

5. We’ve been teaching Alg 1, Alg 2, Geom, Precal, Calc as our math sequence for 40+ years. Never had a problem. Math teachers like not having a year break between Alg1 & 2.

6. My daughter goes to one of the Quaker prep schools in the area and that’s their sequence. Only slightly different than the Alg geom. Alg I did so many years ago and she seems to be doing fine. Having worked with public schools they do the more traditional Alg geom. Alg but again ... no difference.

7. I teach math as well as being college counselor. I taught Algebra 2 for 18 years. I now teach a transition course we call Algebra 3 for students not ready for precalc. There is a lot of merit in the traditional order of Alg 1, Geom, Alg 2.

a. Just to be pragmatic, there is a lot of geometry on the PSAT and juniors who have not finished geometry are at a disadvantage.
b. There are certain topics in algebra 2 that assume students have some background in geometry. Algebra 2 textbooks often are written with the assumption that student have taken geometry.
c. Some younger students are not developmentally ready for the abstract concepts in Algebra 2. The second part of the course really demands that students integrate previously learned concepts and skills and apply them. It is not memorizing how to solve a particular type of problem (at least it should not be that way).

8. As a math instructor. I think that would be great, but where does Trig fall in? Usually, the higher level students take alg2/Trig following Geometry.

9. I work at a small private school... We ordered our math classes like this for YEARS and then with the California standard change, we just recently changed it to Algebra 1 (8th), Geometry (9th), Algebra 2 (10th), Pre-calculus, Statistics (11th, 12th). I personally think that the way we ordered classes before was more developmentally appropriate, but that’s my personal opinion.

10. I have worked in 14 high schools in four states...In my last job before I retired, they taught math in the order that you are considering. I do not recommend it. It you wait to teach Geom in the 11th grade, your students will not have completed Geom when they take their junior PSAT which is the test that determines National Merit finalists (as you know). Since there are quite a few Geometry questions on the PSAT, I think students are at a disadvantage. In the school system that I was in, each school was able to select the order of math they taught, and after some research, I found that the better schools (with higher PSAT scores) chose to keep the Alg 1, Geom, Alg 2 order. Just some thoughts you might consider.

11. Our system, Winston-Salem/Forsyth County schools (WSFCS), probably does it the way you have currently: Alg I, Geo, Alg II, then higher math. I like the geometry between the algebra for several reasons.

a. The jump from Alg I to Geo is “smaller” than the jump from Geo to Alg II from a conceptual perspective. Many students need the “growth” in their brains to handle the higher-order thinking of Alg II.
b. Our geometry curriculum is nowhere near as demanding as the Alg II, so to have Alg I (easy to moderate) to Alg II (very demanding) to Geo (moderate) is not good for students wanting/needing to take an Alg III or precal course their 10th, 11th, or 12th grade years.
c. I find it almost imperative to have the Alg II before taking precal or others, having geometry after Alg II and before precal would/could be disastrous to a student, as the precal is 98% advanced algebra, a year of theorems, proofs, and area formulas could hurt the factoring and function-solving skills.
d. Finally, math is one of those “build upon” courses that should not only increase difficulty and higher-order thinking, but also rigor. If our system changed the order, I think our drop-out rate would go north of 30% (currently at 28%), but that is another matter entirely.

12. Those students who are strong come in with Algebra I in 8th grade. As a result, they take Algebra II in 9th grade without any problems. The kids that are not strong have some trouble but seem to do better with Geometry later. Those who do well in Algebra II in 10th after completing Algebra I in the 9th have an option at our school to take Geometry in summer school between 10th and 11th grade. This route allows them to take AP Calculus in their senior year. I came a large public school so this seemed awkward to me but it seems to work. Those struggling in math struggle no matter what order. Algebra II, at the beginning of the year, is a review so I think no having a year in between seems to help.

13. The University does NOT have a preference regarding the order of mathematics instruction. The pattern you suggest below will be fine with us. – University of California

14. I am now a retired AP Calculus teacher and it has been my experience that the maturity level of sophomores does not support their enrollment into and successful completion of Algebra II prior to the completion of the geometry course.

15. The standard order in the United States is Algebra I, Geometry, and then Algebra II. Outside of the US it is standard to integrate the courses, so that geometry and algebra are included in all three years. There are some good integrated curricula that are used in the US, but they tend to create controversy, because they aren’t what parents expect (because they aren’t what parents had when they were in school).

Placing geometry after both semesters of algebra is sort of the opposite of integration, by totally segregating the geometry from the algebra. I don’t know any schools that follow this practice. When I have heard this discussed, it us usually in response to the fact that when students reach Algebra II, they have forgotten their Algebra I. I would always suggest that if this is the bothersome symptom, then the most likely problem is that Algebra I is being taught as a collection of procedures that are stored in short-term memory, instead of as a connected body of knowledge that is remembered because it makes sense. This syndrome, the teaching of math as a collection of facts and procedures, bereft of understanding, is nearly universal in California these days, because that is the most expedient way to maximize CST scores. If the reason for wanting Algebra II to come immediately after Algebra I is that students forget it, then this is a response to a symptom rather than to the real problem.

There are several potential problems I see with having geometry follow Algebra II.

a. Geometry feels like a very different course to students, because of its emphasis on reasoning, and postponing that might make that feel even starker to students.
b. There might be some aspects of Algebra II that make more sense to students if they’ve already had geometry. The texts for Algebra II would be written with the assumption that students know some geometry.
c. When students go into precalculus, if it has been a year since they’ve had Algebra II, perhaps the "forgotten algebra" problem will be worse. This has implications for college-readiness.
I would also point out that California is going in the opposite direction with its new math standards. California’s old standards taught Algebra I to students without a background in similar triangles, which has always been a problem for students in understanding the meaning of the slope of a line. Under our new standards, congruent and similar triangles precede Algebra I, so here is a specific case in which geometry topics are being moved earlier in the curriculum, because of its importance as background to facilitate the understanding of the algebra.
California hasn’t yet decided what its high school math courses will look like, using the new standards. When decisions are reached about the assessments for those courses, the decisions about the courses will be implicit. But we are using the Common Core standards as the basis for our new standards, and work is underway at describing pathways through the high school material that will try to provide the focus and coherence that the Common Core standards create in grades K-8. That work is called the Pathways Project, which is being done by Achieve. I would suggest that people who might consider a new and non-standard arrangement of high school material should first look at the Pathways proposals, as it is likely to represent the directions that will be taken nationally.

18. We tried it and it did not work. We returned to the standard order of Algebra I, Geometry and Algebra II.

Jean Crowder concluded:
The consensus appears to weigh on the side of the traditional order of sequence for completion.


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By Valerie Strauss  | January 14, 2011; 5:00 AM ET
Categories:  Math  | Tags:  STEM, algebra 1, algebra 2, algebra II, calculus, geometry, high school, high school math, math classes, order of math class, pre-caculus  
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Perhaps this question would be better answered with a review of commonality and differences. If there is loss between subjects then maybe a streamlined course would better serve students. Simply because "we've (always) done it this way" does not (always) mean it was successful.

Posted by: jbeeler | January 14, 2011 8:12 AM | Report abuse

I am a math teacher in public high school who has taught A1, Geo, and A2, a private tutor in all those subjects and a college admissions test prep instructor (SAT, ACT, and Math 2c are the relevant ones here). I've tutored kids through the alternate sequence, as well as taught them SAT prep.

The experts have hit the two huge reasons why the traditional sequence is best. The kids who take the alternate sequence are paying the piper in SAT scores, and developmentally, A2 is brutal for all but the most academically advanced.

Posted by: Cal_Lanier | January 14, 2011 9:16 AM | Report abuse

I am not a math teacher (I teach history and government) but I cannot help but be struck by how instructive it is to have actual math professionals -- veterans and AP teachers and other assorted experts -- weigh in on this issue. Even in disagreement there is cogent thought on the why, whether it be developmental considerations, the curricular relationship between classes, the assumptions of textbooks, or preparation for the PSAT. I learned a lot.

If only such wisdom was soliticed from such bona fide experts (i.e. experienced practitioners) on other matters of education policy...

Posted by: joshofstl1 | January 14, 2011 12:00 PM | Report abuse

Thanks for this article on order of Math courses. I am not sure what order they should be in -- it may depend on how the topics are divided up. A2 is usually more rigorous than the others but it can depend on the school/class/state.

It would benefit students to learn and know the Math on the ACT/SAT/GED/ACCUPLACER (placement tests used by colleges). Many students have not seen the topics enough times, or have had the topics slivered (and are unused to multiple topics on the same exam), or have not developed the speed that will help them problem solve 20 questions in 25 minutes.

While the Common Core are under development, we already have these standards at the high school and college level.

Studying multiple choice items can improve metacognition due to compare/contrast and by studying "good wrong answers" (for example, exponent rules questions always have "good wrong answers"!!).

Students, teachers and parents can use the free SAT Question of the Day (and other free or reasonably priced resources) to better scores and knowledge and skills!

Perhaps, we can bring academic and cognitive abilities up to the level of respect that athletics commands.

Robin Schwartz
Author, Build Math Confidence e-newsletter

Posted by: mathconfidence | January 14, 2011 12:37 PM | Report abuse

Apparently mathematics is not being taught in the public schools.

What is being taught is memorization. Memorize these facts and pass the test. A year after the test the students on a new test are found to need remedial work.

For example ask students what x to the -1 is and x to the zero and they will be lost if they have forgotten the answers they were forced to memorized for an examination.

Ask a student the logic of why x to the zero is 1 and the student will probably be totally lost since they were expected to memorize the answer and not understand the concept.

If mathematics tests in high school where changed to essay questions it would be quickly evident that the large number of students had no concept of mathematics.

As for the question whether geometry should be taught before or after algebra, this is dependent upon whether you are teaching modern algebra and want to teach geometry as an algebraic system.

Posted by: bsallamack | January 14, 2011 2:41 PM | Report abuse

I agree with the order of Alg.I, Geometry and Algebra 2 for on level kids for purposes of the SAT.
I have taught all of the subjects and am presently teaching Honors Geo. and IB Pre-Calculus. My 2 cents would be to spend time practicing the Algebra skills that are embedded in Geo. It might even make the transition from Alg. to Geo. a little less painful because we are starting with what they know.
Our IB program has students double up on math their freshman year, for those who have taken Alg. I in eighth grade. They take Alg. 2 and Geo. at the same time. For many kids this works well and I would like to see more of it done for Honors students.
My IB Pre-Calculus is all 10th graders, they will take AB Calculus their Junior year. However, although most of my students do well they are still lacking the maturity to truly apply themselves, the pace is what they have a hard time with, it's a very big leap.

Posted by: ananna | January 14, 2011 2:48 PM | Report abuse

@bsallamack, yes mathematics as a series of interconnected ideas is being taught right here in DCPS in my classroom. Now if the powers that be would only develop a testing instrument to measure this type of thinking and learning, that would be a step in the right direction. Next, hold students accountable to pass such a test, even if it takes a few tries, before they can move on to the next grade/course.

Posted by: pat1117 | January 15, 2011 9:29 AM | Report abuse

Now if the powers that be would only develop a testing instrument to measure this type of thinking and learning, that would be a step in the right direction.
Posted by: pat1117
The problem is that mathematics is not being taught but rather memorization is being taught.

For example children have to memorize the multiplication table supposedly to learn multiplication. The result is that they do not fully really understand multiplication since they are tied down in their effort to memorize the table.

Why not provide each child with a copy of the multiplication table and then ask them to use this to do multiplication problems? The table shows sequence and relationships. These concepts are not evident when children are simply expected to memorize the multiplication table. By using a multiplication table over time children will start to memorize the table naturally.

At some time there has to be recognition that ideas regarding numbers took thousands of year to develop and that these ideas can not be simply thrown out to children with the expectation that they will understand them by memorization.

I am sure that currently children are being taught that -1 * -1 is 1 with no mention that this is simply a convention and not some inherently logical outcome that can be explained.

I wish that mathematics teachers in public schools would be honest and review the material that they teach with a determination of whether the material is teaching concepts or simply memorization.

By the way geometry at one point was taught on the basis of proving statements based upon the axioms. Classes in geometry were an introduction to logic and reasoning. Everyone talks about critical thinking skills and simply ignores that geometry is one of the few subjects that can be taught to help the development of critical skills. Based upon the old form of teaching geometry it should be taught as early as possible. For some strange reason trigonometry used to not be taught with geometry. This is strange since trigonometry is so dependent upon the relationship of the lengths of the side of triangles and angles. But my memory is fuzzy and perhaps it was taught simply as an afterthought at the end.

My memory of Algebra in school was that it was poorly taught. In working with equations there was the idea of reduction and not the idea of equality and the operations that could be performed to retain equality. I am fairly certain that teachers are still teaching tricks so that students can deal with tests. It is only natural that over time the student forgets the trick since the student has no idea of the underlying concepts.

Teachers really have to be honest with themselves and review whether they are teaching concepts or simply tricks.

Given the current testing environment it is totally understandable that teachers in mathematics are simply teaching tricks.

Certainly glad that I am not a teacher in a public school.

Posted by: bsallamack | January 15, 2011 7:17 PM | Report abuse

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