I’ve been thinking about the way we structure the flow of high school courses. A typical sequence for the “average” student is Algebra I, Geometry, Algebra II, Pre-Calculus or some type of Trig/Stats class.

As far as I can tell the only difference between Alg II and Pre-Calc is that trig is taught during Pre-Calc and Pre-Calc introduces the concept of the limit. Functions are developed a bit more rigorously too.

The first semester of Algebra II is mostly a repeat of Algebra I as they’ve forgotten it with the year “off” during Geometry.

Why not then teach Geometry first? I’m talking about plane and solid geometry with an emphasis on reasoning, and right angle trig. Obviously there would need to be some supplementing needed (work with radicals, solving equations). Most students have “seen” the solving of equations in 8th grade (Have they mastered it? No, of course not).

I’m thinking that a high school sequence could be structured as: Geometry, Algebra I, Algebra II with Trig, Calculus or Stats.

Analytic geometry could be moved into Algebra II – and there would be time as the “review” of solving systems wouldn’t be needed as there wouldn’t be the year off.

So, I’m looking for some valid reasons why this wouldn’t work. What am I missing? Help me think through this.

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Well, besides the argument that they would best be integrated like in IMP or CMP, I would think state standards tests dictate a lot of what happens here.

In Colorado, for instance, geometry units are added by teachers into the Algebra curriculum before the 9th grade CSAP test–while there is also some Algebra emphasis in the 10th grade geometry classes.

I see no apriori reason why your proposal would not be just as legitimate.

BTW: you’re right about Algebra 2 and PreCalc being very similar. Ever notice how some publishers will have almost identical textbooks in content with different titles. The Larsen College Algebra and Trigonometry book matches their PreCalculus book almost word for word (and problem for problem).

Serving on a textbook selection committee really changed my view of how education works–and not in a good way….

I like the Larson books as well as any textbook, but yes, there for a while (haven’t looked lately) there was a College Algebra, Algebra & Trigonometry, and Pre-calculus book that were only different by a chapter or two. The rigor/complexity of the books goes in the order I listed them in, but I always assumed the major reason was so the title of the book could closely match the title of the class for people who care about such things – thus creating a larger market than simply naming the book Pre-Calculus and letting instructors personally modify the table of contents. Digital textbooks before the digital had completely taken over …

Richard, this is for a school that isn’t ready/willing to consider an integrated curriculum.

Here in Illinois, students are tested at the end of 11th grade for NCLB. We use a combination of scores on the ACT and WorkKeys.

Depending on what kind of students you have, the “minimal” Algebra needed to do Geometry (radicals, equations) might take that whole year. Also, many students are not mature enough to deal with formal proof even directly after Algebra I but can do it later. Success rate at Geometry might go down if students were to do it at en even younger age. On the other hand, success at Algebra might go up if that came later… who knows.

As for the difference between Algebra II and PreCalculus, I’ve just today been looking hard at the CA Algebra II standards and realizing that graphing is really not emphasized nearly as much as the textbook (or I) do. Graphs of polynomials and rational functions are not even mentioned, and I’m not sure graphs of exponential functions are included either (does “understand exponential functions” involve being very familiar with their graphs?). Graphs of conics are explicitly included, though (now, what’s with including graphs of conics but not of polynomials? I don’t get it.) Presumably, graphical analysis of these functions is deferred to PreCalculus – however, oddly enough, this is not really mentioned in the Math Analysis standards either. I’m confused.

At Dan Greene‘s school they do Geometry after Algebra I and II in order to get the Algebra courses in sequence. Dan, why not Geometry first?

In my experience, even the Honors students aren’t mentally ready for the spacial relationships in Geometry when taken after Alg 1. I finally convinced our guidance counselor (when I worked in a private school) to make it Alg 1, Alg 2, then Geom. Now that we are implementing the FCAT 2.0 (end of course exams in public schools in Florida), I may be able to talk them into switching there too.

Algebra before Geometry is most often the wrong approach. Teaching math as any other discipline; it is necessary to take a different approach for levels of reasoning and human retention. Learning Geometry before Algebra can be compared to learning the piano before learning the guitar. Ultimately by learning the piano first you will solidify a much clearer understanding of the guitar and learn at a quicker rate. Now back to Geometry before Algebra… this approach would work because it slowly integrates Algebraic formulas as you progress. Learning Geometry first gives the student a more visual reasoning tool of why Algebra is needed keeping the student interested in learning the math and not becoming frustrated with not being able to visualize the intention. Math and music are accumulative learned disciplines. I feel that this approach would be more confident building for the student as it develops a more visual understanding of math!!

H – I’m glad to hear you’re including the graphs while teaching functions. The graph and the function are related, why in the world do we (sometimes) treat them as separate topics? And yes (to me) “understand exponential functions” means understanding the graph too.

Yes, for some of the students I think the “minimal” algebra might be a struggle. I’m thinking a summer course prior to freshman year (highly recommended) might help with that. I’m still trying to flesh this all out.

How are you defining “formal” proof? Two column? Actually I’m not a huge fan of those types of proofs. I’d much rather see a paragraph style proof.

Thanks for helping me think this through.

In the CA standards, graphs are explicitly mentioned for linear and quadratic functions, but not for polynomial and rational functions. When I showed one of my concept tests – deduce graphical info such as number of turning points, x-intercepts and end behavior of a polynomial given its equation in factored form – to a colleague, she pointed out that that was essentially a PreCalc skill. I disagreed, but looking at the standards afterward I had to concede that she really was right! I’m not going to stop teaching the graphical representation of these functions – I’ve emphasized it pretty heavily up to now – but it does bother me somewhat that the Standards don’t seem to call for this.

Since I don’t teach Geometry I don’t really have much to add on that discussion… I used to think the entire course was a bit of a waste of time until I taught a very watered down version a year ago and found that the many, many kids really had huge trouble with even distinguishing the concepts of area, length and angle, notions I had considered pre-academic up to that point. After that I’ve had to concede that high school kids really do need a year on surface, volume, and elementary logic.

I’m always ambivalent about high school geometry. It’s great in theory, but I don’t think I’ve seen a single good implementation. Students come away universally detesting it as unduly harsh and rigid.

Even in my equivalent course, the teacher marked me wrong for giving the “wrong” proof for a proposition on a quiz. I couldn’t see why my reasoning didn’t work, but she insisted that it was wrong. It wasn’t until I actually got my hands on the

Elementsthat I realized the proposition was thepons asinorum, that I had rederived Euclid’s proof, and that the book didn’t use that proof. So it was wrong because I didn’t regurgitate exactly what the book said. This was in one of the wealthiest counties in the country with one of the best public school systems anywhere.So what do students take away from high school geometry? Pretty much nothing. I can’t reliably refer to anything more complicated than “a circle is the set of points a fixed distance from a given center” in a calculus class.

And easily the most common worry from students who haven’t even registered for their courses freshman year? “Will we have to do proofs?” Even the top students in the class blanche at the thought of “proofs”, because the only ones they’ve ever seen are the dried-out husks they were assaulted with. Where? In high school geometry class.

/sarcasm

Jackie, how could you even consider teaching geometry at all? Haven’t you heard, these kids are all digital natives and have no use for some silly 4000-year old subject? We should be teaching them how to program video games and look up Wikipedia articles on their cellphones instead.

/sarcasm

My “dream curriculum for high school math would be a year of Euclidean geometry — that’s right, a YEAR — in 10th grade, a year of algebra in 11th grade, and then a year of math in the 12th grade that would be the student’s choice depending on what they wanted to do with themselves after high school. That “elective year” could be a year of trigonometry and precalculus, or a semester of stats plus a semester of discrete math, or something. Geometry should come first because this is where students learn how to reason mathematically, and because geometry is really nice and visual which appeals to younger kids, and because the availability of good, cheap dynamic geometry software makes the hands-on approach really easy to implement.

To reply to comment #3 above, there is not a single mention of algebra in Euclid’s Elements, so I don’t think there is any law that requires there to be algebra in geometry at all if we didn’t want there to be.

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Hee. I registered for three different high schools at one point or another, all of them with different math sequences. (When do you first offer algebra? Geometry between or after algebra? Statistics required? No wonder that last year I couldn’t tell what a course was given only the name “Advanced Math.”

Last year my Geometry students were concurrently taking Prealgebra, Algebra I, and Algebra II. No matter their level, they were more comfortable with basic algebra skills than most of my other students. There may be something to the idea of using skills in a different context to really be confident in them. I’m pushing my department to actually create a sequence, and based on my experience put Geometry in the middle.

JohnI find that very sad for the young John that was. I’m glad that experience didn’t change your perception of mathematics. I detest the way we teach proof: especially the fill-in-the-blank two-column proof. How does that help develop reasoning skills? Give me a logical well-thought out argument. Heck, from a freshman in high school, I’ll take a reasonably thought out argument that we can refine together.RobertYou didn’t specify what you’d do in 9th grade. What are your thoughts there?And I agree that Geometry is a great starting point for mathematical reasoning. I also think it is a natural point to (re)teach working with proportions and solving simple equations. There is a context and reason for doing so! I think it might make more sense for the students.

Sarah, you say you want to put geometry in the middle. In the middle of what? Pre-Algebra and Algebra I or Algebra I and Algebra II? And I’d love to hear your thoughts on why you are advocating for said placement.In 9th grade I would run a yearlong capstone course in all the pre-algebra arithmetic that students would possibly need to know for geometry and above, and make sure that they are rock-solid on all of it. In my experience the weaknesses that I contend with in teaching college calculus all stem from a poor foundation in basic arithmetic — fractions, proportions, absolute values, and so on. Spend a whole year nailing this stuff.

Robert, would you allow me to give specific details in seeking your advise in regard to my oldest son who will be starting 9th grade in the fall? I will proceed with the hopes that you will.

My son was home-schooled from the middle of 6th grade to the middle of 8th. He took pre-algebra in 7th grade through a co-op and earned a solid B, somewhere in the 80%. In 8th grade at the same co-op, he began in Algebra (Saxon 3rd edition), and maintained the same percentile. After the winter break, he decided he wanted to go back into a school setting, and began at a small private school where they also used the Saxon math curriculum, however, they were using the newest edition (4th, I believe). He really struggled, I think in part because of readjusting to a class setting and the fact that he is a very social kid. However, I believe it was mainly due to the fact that the book introduced things in a completely different sequence (confirmed by his teacher), and the fact that the class had been using one of those fancy calculators since the beginning of the year and he had never even seen one. Much to my chagrin, we decided to put him back in pre-alg. This really was upsetting to me because he had already done this exact curriculum in 7th grade, as I mentioned before. He still only earned a B the 2nd time around (On a side note, one of his biggest weakness when it comes to math, is his unwillingness to follow the formulas and show all his work).

What would you do if this was your son? I am sure he will go into business (sales) after attending a 4 year university. Should I have him tutored in the concepts that he will need to begin and succeed in Geometry his 9th grade year (he has completed half a year of Algebra)? Or, as you alluded to in your previous post, would it be better to just let him take Algebra 1 his freshman year? We live in an excellent school district and most kids take Geometry in 9th grade (even honors geometry). I hate to have him start off slower than other kids, but that may be my issue and my pride talking. i really want to do what is best for him?

I really appreciate any help you can give. I have not even signed him up for school yet. We have decided to do public, and I missed the boat on meeting with guidance counselors before they went on summer break.

thanks!

Janis

Re Jackie’s QuestionsI was advocating for geometry in the middle of Algebra I and Algebra II.I think what is needed most is the accountability for students knowing material from previous classes. When that’s lacking–which was the case in my situation last year and sounds like what Robert deals with as well–Geometry offers a chance for students to review algebra concepts in situations where they’re not necessarily the main focus.

I wonder how much of what I’m thinking goes back to the argument that you don’t learn arithmetic well until you use it in Algebra. You don’t learn Algebra well until you use it in Calculus.

Re Robert’s proposalIn a situation where we’re not taking an integrated approach, I do like the Euclidean Geometry and the elective year. I like the 9th grade basics, but am tempted to turn it more along the lines of the Numeracy class at Dan Greene’s school. My understanding is that it runs alongside Algebra there. Not necessarily an integrated approach, but working along the lines of making sure you have time to practice basics as you use them. (Though it does reshuffle your classes, double up math in one year, and leave another year for electives.)Re ProofsI actuallyliketwo-column proofs. But then, I was taught very little of them with the fill-in-the-blank method. We had to develop the argument, it was just a way of organizing it. For myself, it’s easier to see how I got from my hypothesis to the conclusion than with paragraph proofs. Whereas with the paragraphs, I’m more likely to skip a step inadvertently.But then, my high school geometry teacher was better than John’s sounds. She would challenge us to find different ways to prove something and meticulously check tests so she could give partial credit.

I like your idea of geometry first. It makes sense historically and can be done without algebra. (As long as we are marching the chillens through 3000 years of mathematical development, doesn’t it make sense to do it in the same order?) It would eliminate the “break” which necessitates reteaching half of algebra 1 to 11th graders.

I think the idea of proof is important and should be introduced far earlier. And no I don’t mean rigid 2-column imitations of proofs, I mean much more informal “What patterns/consistencies/invariants do you notice? Do you think that is always true? Can we hammer out an explanation for why it always has to be true? What ways do we have of convincing ourselves that it’s always true?” Math without proof is, I don’t know what, but not math. Computation and vocabulary, I guess.

Some people advocate Algebra 1, Algebra 2, then Geometry, which gets you more intellectually mature kids for geometry. And if you want to introduce Euclidean axiomatic proof, that’s very helpful.

The main obstacle to rearranging courses (at least in NY, but I imagine in many other states as well), is that the state has handed down a “curriculum” (which is not really a curriculum, it’s just standards) with an order in mind. And they subject the chillens to a culminating examination, which they are expected to take so they can receive a state-sanctioned diploma. Under this restriction, doing the 2 algebras first and then geometry is far more feasible than geometry before algebra.

I always cringe when people suggest adding something to Algebra 2 w/trig. At least in my experience, that course goes at a breathless pace already. If anything, I’d want to subtract.

I don’t get the point of precalc, either.

Well, it didn’t change my opinion, but I was always sort of a special case (and not just as far as math went, though not necessarily so positively). I don’t remember what I was working on myself while walking through the equivalent of that course — I can’t say “high school geometry” since I wasn’t in high school at the time and it wasn’t just a case of going to the high school for that course. Anyhow, the public school math curricula were always the formal hoops someone wanted me to jump through while I did the interesting stuff myself.

The upshot is that I’ve seen the content of this sort of course, both as a student and looking back as a teacher further along the path, but my own personal experience of the course shouldn’t be taken as at all representative.

My own sense is also that the order is flexible material-wise, but less so state test-wise. How students transfer from one place to another is also something to think about, though I don’t think switching the order would make much difference because kids transferring in/out could still take the appropriate class a year earlier/later than what they would have expected.

I spent a semester studying math in Budapest in college, and one of the profound moments there was when the Number Theory teacher said, “You Americans think that Calculus is the be all and end all of everything. Everyone knows that’s really Number Theory.” It was the first time that I realized that, mathematically, there is a LOT of flexibility about what could be taught in high school, and the reason that the curriculum is set up how it is has a political/cultural component as much as a mathematical one.

I’m going to disagree about the two-column proofs being particularly hard, though. I think they ARE hard, but I think they’re hard because proofs are hard. Perhaps some students would gain more from a paragraph proof, but I know some like the structure that two-column proofs provide once they get it. At our college, the math majors take an Intro to Proof course during their sophomore year and even though they’ve been exposed to proofs in high school and in calculus, writing an elementary proof is amazingly difficult for them. For example, we define even numbers (2*integer) and odd numbers (2*integer+1) and ask them to prove that 2n-1 must be an odd number, or that odd+1=even. Many of them — math majors, remember — really struggle with that until a lightbulb goes off, hopefully before the semester has ended and then wonder what was so hard. But with geometry perhaps the talking it through (Jackie’s “a reasonably thought out argument that we can refine together.” with the emphasis of refining it together) is really the key step.

I also brought this up at my school. Other than the proof idea and some of the concepts, most of the geometry material (area, surface area, volume, angle relationships, etc) have already been introduced in middle school. I tried to convince the “powers” that if they would remove the fluff in pre-algebra and 7th grade math and stopped trying to introduce them to everything, we would have a stronger curriculum overall. With pre-algebra if they spent the time really working with positive and negative rational numbers, order of operations, and 1-variable equations, I think we could easily move geometry first. Even NCTM has de-emphasized formal proofs. In my own classroom I don’t do a lot of formal proofs for a variety of reasons, but we do a lot of informal proving and discovering/testing patterns. I agree with you 🙂

Sarah– I think that could work IF there is enough Algebra I in the Geometry course so they don’t forget it. Overall, I think we need to do a better job of integrating the curriculum so students don’t see mathematical topics as isolated skills that have nothing to do with each other.Robert, that’s an interesting idea. I’m not sure how well it would fly with the push to get more students to take Calculus in high school. I’d rather have them doing less topics that they can actually understand rather than rushing everyone through the courses.Kate, I agree that we should not add anything more to AlgII/trig. It isn’t that I don’t like two column proofs at all, I just don’t think we should teach kids that those are the only type of proofs. Paragraph, flow chart, two column,…, whatever makes sense to them to explain their thinking.Ξ, I’m just realizing how deeply what we teach is determined by factors other than what is conceptually/mathematically appropriate for students. I think proofs are difficult for so many because they’ve never been asked to really logically think through a problem. I had a very similar experience in my mathematics courses – the attitude toward proof (from future teachers) was overwhelmingly negative. How do you think their students are going to feel about proofs?DruinHow did that proposal go over? What happened?Jackie, trust me, as a college calculus professor, I have no idea why all these schools want more students to take calculus in high school. All it amounts to are a whole lot of students *re*-taking calculus in college because they had a superficial treatment of it in high school which was preceded by a rushed and superficial treatment of arithmetic and algebra (to say nothing of most students’ non-existent trig knowledge). So I end up with a lot of students who THINK they have had calculus before, but really haven’t. I’d much rather leave calculus to the gifted programs in high school and the colleges, and let high schools really get these kids rock solid on arithmetic and algebra so that we college people have a better shot at actually teaching something. But then again, the school districts don’t often think to talk to us college folks before they decide what’s going to prepare students best for college…

Make that calculus in HIGH SCHOOL there in the first sentence.

I did. (Jackie)Jackie,

I am going to disagree strongly.

There is a question of level of abstract thinking involved in a proof-based geometry. Moving it younger (younger, right, not just “earlier”) pushes the envelope on an already difficult course.

Algebra already gets short-changed. Having a year of algebra, a year in between, and a second bite gives many students a chance to strengthen and improve their understanding. Many students don’t get a firm grasp the first time through. A good precalc, imo, fills in the theorems and adds some non-alg topics to Alg II. Again, the algebraic topics do better with deepening repetition over time. (note, there is no universal definition of precalc, and the course is different in different schools)

Part of this is that we are not in a rush. I agree with Robert’s last comments. We’d rather send a student to college with 3 years of algebra, including some theory, maybe up through limits, and let the kids do well at the next level. But Robert (pretending you represent all post-secondary institutions in this country), colleges encourage kids to take calculus. Math Departments should talk to the admissions people. Seriously.

Also, geometry in the US has become far less proof-based and far more computational. I don’t think the “geometry first” could possibly fit with the typical geometry course as it is taught in the US today.

But my biggest question is Why? What advantage is gained? Three years of algebra reduced to two, and starting a year later. I see that as an argument against.

Jonathan

Jonathan,

What is a “typical geometry course as it is taught in the US today”? Is there a typical course?

Do you mean an axiomatic proof based course? I’m not sure how typical that is. Nor am I sure that’s the goal for the “average” student. I agree that thinking in terms of why things are true is necessary. I’m just not sure that forcing proofs on students who aren’t conceptually ready for it is the best thing.

As for your why question: I think there are algebraic concepts that can be embedded in the geometry course that allow for algebraic skills to be taught in context. I’m thinking of work with proportions, solving one variable equations, and working with formulas.

Some of the more conceptually difficult geometry items could be worked back in to the 1st and 2nd years of Algebra when students are more ready for the abstract concepts (I’m thinking of analytical geometry and more traditional proof).

I think there is a typical course. Most of the texts I see cover about the same material for the first half: parallel lines, triangle congruence, special quadrilaterals, similarity, with a meager amount of proof. The second term they diverge.

I don’t think I understand what you are proposing. What would be in your geometry course? and, as important, what would not be in? And what advantage would be gained?

Jonathan

Why not teach algebra and geometry simultaneously, as two interconnected areas of mathematics, proceeding from simple to more sophisticated notions and techniques? Separating algebra from geometry, conceptually, and temporally is a self-defeating approach that makes learning much more difficult than it has to be. Than again, in college, the same pattern persists. Linear algebra after calculus and differential equations, etc. It’s like bureaucracy always has an upper hand over common sense. Oh, well, what else is new? Are schools and colleges in business of educating people or selling them diplomas and certificates?

My Christian, private school upbrining rejected the split between the Algebra classes as well. I took Alg I in 8th grade, Alg II in 9th, and then Geometry in 10 back to back with Pre-Cal my Junior year. If I remember correctly, they were pushing up against the whole Geometry is too abstract for certain age groups blah blah, but they say the same thing about Algebra, so who knows. I turned out alright, and I thought the progression made sense.

Jonathan, I think that geometry is more accessible to more students early on. The topics you mentioned, “… parallel lines, triangle congruence, special quadrilaterals, similarity” along with area, volume and surface area would be included. An intro to right angle trig would also be included. What would not be included would be coordinate geometry.

Is there some basic algebra required? Yes, mostly proportions and solving single variable equations. I think teaching these skills in the context of geometry would make more sense. Of course aspects of geometry would be brought back into the Algebra I & II courses where appropriate.

Misha, I agree. I think there are aspects of geometry that are conceptually easier than others. Just as there are parts of algebra that are too. Why can’t the curriculum be designed to move from “easier” topics to “harder” ones regardless of the geometry/algebra distinction?

Irrational – that’s what I’m trying to get at. There are aspects of each that are more abstract. Why not teach the more accessible topics earlier and more toward the abstract?

Why not teach the more accessible topics earlier and more toward the abstract?

Because we lose coherent courses that way. Because we lose the extended example of an axiomatic system that is accessible to many (not all) kids.

Jackie, we just ended a 20+ year experiment with something similar to what you are proposing in New York State. And most of us are delighted, but wish that the break were cleaner.

Jonathan

Jackie,

Some of the coordinate geometry could be done. In most pre-algebra classes, students learn how to plot points, so the midpoint, distance, slope concepts could all be explored in the geometry class. Also, the pre-algebra class covers solving equations in one-variable, so there could be some basic algebra concepts in the geometry course – just not ones that require systems of equations, etc.

When I suggested this at my school, the comment was that students weren’t mature enough for geometry at a younger level and due to the ACT/PSAT, they wanted to cover geometry at the sophomore level rather than do Alg1, Alg2, then Geo. I agree with your reasoning that the kids lose a lot during that Geo year, meaning that Alg2 ends up being more like Alg1.5 because they are having to do so much reteaching becuase of the 15 month hiatus.

Jonathan,

Is there information on what was done in NY for those 20 years? What worked? What didn’t? Why it didn’t?

Heck, is there information anywhere aggregating what has been done (in terms of math ed) in different states/districts and the results?

Druin – Thanks for sharing your thinking. Amazing how the testing is driving instructional decisions isn’t it? Even if it isn’t what may be best for student learning?

New York State used “sequential mathematics” (We called the years Course I, Course II, and Course III) experimentally in the (late 70s or early 80s??) and globally from (the mid 80s???) until they were replaced with Math A and B around 2000 or so. A and B were exams without courses, but most schools and districts continued using (modified) versions of sequential.

I’ve written a bit: here. But maybe the easiest thing to do is to flip through a few of the exams to get some idea about what the content really was: here. Or, the publisher of the most common sequential texts, by far was Amsco, maybe there are summaries or outlines or Tables of Contents available for our old Red, Blue and Green books.

Jonathan

Reading this article over at Washington Post Magazine, I was reminded of the debate going on over here.

“You would have a hard time finding one math teacher in this county who supports the scope and sequence of the way math is taught.”The evidence here seems to support that claim. We’re wanting to change it. Though again the question seems to be, “But how?”

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Geometry first seems to be working here. I’d like to know more about how their geometry course is structured.

Jackie,

that article seems to say that all the kids take algebra in 8th grade, but they don’t call it Algebra I. In 9th grade they all take geometry. In 10th, those who did well in 8th grade take Algebra II, and those who did poorly take Algebra I, followed junior year by algebra II.

So by giving algebra globally in 8th grade, they’ve sort of disguised what they are doing:

AGA or AGAA

Jonathan

Hmm. Good point

Jonathan. Do you find that most students are taking something that’s Algebra Iish in 8th grade? I’m finding that quite a few of my freshman haveseensome of the Alg I material before (despite what it’s called on their transcripts).However when I ask them what they did in 8th grade, they tell me “ISAT prep”.

It’s the nature of where I teach that all of my students have seen something related to algebra – most of them a full course in NY State “Integrated Algebra” (of which about 50% of the content is algebra) – but few of them have had what we would consider a full course in Algebra.

So we hit them with one or two terms of fairly rigorous algebra, then on to a year of geometry, then algebra II.

In NYC, this is not typical. But among schools with stronger populations, it may be.

Jonathan

Jackie, have you seen a pre-algebra book? It’s pretty much algebra-lite. It is actually boggling to see how much of the material in algebra simply repeats pre-algebra.

I unfortunately have no idea what the ISAT is.

Sorry

Jason, the ISAT is the test students take in Illinois (through grade 8 inclusive).I really wish we didn’t have to “cover” so much material every year. That whole wide vs. deep argument again.

I grew and lived in Chicago and taught 1 year in suburban Chicago before moving to Georgia.

Georgia has been rolling out its new Math Curriculum for the last 3 years. It hit H.S. this year.

I am teaching the first H.S. course, Math 1 it is a combo of Geometry, Algebra I and II. Aside from the fact that there is too much content, “it feels” Ok to have the topics intermingled.

Mathman6293I’m assuming that you mean there are too many topics to cover in one year?Are the topics that you

arecovering developmentally appropriate for your students? Which topics from Alg II are being taught freshman year? I’d love to hear more about how the new curriculum was designed and how the implementation is working.One more thought…I believe Algebra I is a pre-requisite for Chemistry…so if you wait too long for it, the other subjects may be thrown off course, so to speak.

I need some advice from the math pros here. My son is in 3rd grade and taking courses in school with Stanford and John Hopkin’s gifted programs. His math tutor will have him take the PreAlgebra and Algebra I Honors courses this term. If he has the choice with this type of arrangement at school for math, should he take the Algebra II course or the Geometry course next? As for any college professors with experience with students taking these accelerated programs, what happens to them after they have exhausted all the math courses before college (he definitely is doing this for his LOVE of math and has specific colleges in mind already). This all happened so suddenly and I am getting worried about how to foster his love for math and handle his educational needs too when they are so outside the norm already.

We are going to pilot a new remedial mathematics sequence at our school next year. We’ve found that the retention between Algebra 1 and Algebra 2 is pretty poor with Geometry in the middle. Instead, we’re going to go Algebra 1-Algebra 2-Geometry. We’ll then try to integrate as much algebra (linear, quadratic, systems, etc.) into the Geometry as possible and treat it as an applications class. Because we’re waiting until Junior Year for Geometry, we’re actually going to use the non-remedial curriculum beefed up with extra Algebra and less “proof”. Hopefully this will translate into stronger mathematicians!

You’re right

Beverly, any changes need to be discussed with the other departments to see how things affect their curricular work. I know we’ve moved some topics around in pre-calc so that students had seen vectors in math prior to using them in physics.AdamI like the idea of integrating the algebra topics into geometry (especially with the ACT/PSAE junior year). Do you have a blog? I’m interested in hearing how this works for you.I am the mom of a 14 year old 8th grader currently doing b work in algebra 1. We are faced with a math dept at the high school who is experiencing in-fighting about whether geometry should come next as a freshman (the guidance dept holds this stance), vs algebra 2 coming next (the math dept stance). She is not the strongest math student…any advice?

The Algebra 1 – Geometry – Algebra 2 sequence is successful for the majority of kids because it “works”. What I mean by this is that the regular students who have no problem with mathematics courses do fine with this sequence and score high on the ACT.

I believe that the reason why Geometry is second, however, is because until the past few years, Illinois and most states only required 2 years of mathematics and exposing students to Algebra and Geometry makes sense. This is the traditional sequence of events.

Would Geometry be better before Algebra or after Algeba 2? I can see the value of both stances since Geometry is a lot of problem solving and applications. The logic building would be helpful before Algebra but the application would tie all that Algebra together. If your 8th grader successfully completes Algebra 1 during 8th grade, he/she’ll do fine regardless of what your high school decides.

JoAnnDid either department provide a rationale for their argument as to what should come next? Do the two different options lead to two different course sequences (or do they end with the same course)? Also, you say she’s doing “b” work, to me that means she has a better than average grasp of the concepts? Is she working really hard to get this “b”? How long is she spending on homework? Is she getting outside tutoring?AdamGreat point about the historical aspect of the course sequencing. I hadn’t thought of that.We are behind. My state has a new growth model, which makes it more difficult for students in the 9th grade to “grow” than it does for that same child when they are in the 10th grade. My proposal is to allow those students who would take algebra 1 in the 9th grade to take geometry instead and then take algebra 1 in the 10th grade. I am being shot down with, that is the way it has always been done. I was just wondering how it was working in those places where geometry was taught prior to algebra 1. I really need some feedback for my district.

What is used for state testing? If it is a state generated test, they may lock you into a course sequence simply so your school can maximize it’s score potential. I know it is a growth model but if freshmen are all tested on Algebra 1 and you don’t teach that until Sophomore year, you may be in some trouble.

I think one way you could sell the Geometry first is to insure that you’re incorporating Geometry throughout the three year sequence. Teach all the Geometry you can first year but then continue to give students Geo-friendly word problems in your Algebra courses. Many of the ACT math questions are Geometry based Algebra problems with 1-2 steps involved. These could be your “A” student problems on exams, for example.

Lastly, if you teach Geometry first and you have to filter out some of the heavy Algebra involved, you may consider a semester Geometry and a semester problem solving instead of the traditional Geometry year course. The problem solving could then also be carried through the 3 year sequence. I’ve seen presentations by speakers on problem solving courses and they seem to be successful.

Jackie, I agree 100%.

I recently saw a question in the Mathematics category of Yahoo!Answers: a student could not draw a line of slope 1 through a given point. How come, I thought? What stands between a teenager and an ability to envision several lines passing through the same point, and to choose the line with the right tilt? The only explanation that comes to mind is the practice, common in U.S. high school education, of teaching algebra 1 before geometry. This order is beyond me. Consider the facts. Geometry is our connection between observing nature (experiment) and understanding it (model). This connection is, on the one hand, natural (a 4-year old can tell a circle from an oval from a square) and, on the other hand, deep (geometry is the indispensible apparatus of classical mechanics and other physics). Algebra has developed to serve the computational needs of geometry and physics. Why is algebra being torn out of the context that gives it meaning?

Please, when you think HS math, think of functions. Functions is why you NEED to learn in algebra, 1 or 2, it doesn’t matter. In alg. 1 they are linear, in alg. 2 non-linear. Everything else you learned in 1 and 2 is related to functions, period. All this is guiding you to calculus. Now, why then geometry? Properly taught, geometry is a very simple, enjoyable, graphical course where all the shapes analytically taught in alg. 1 and 2 are covered. Because the jump from alg. 1 to alg. 2 is actually a leap, in the middle geometry should allow the student to transition. Therefore, if you have the right teachers, the alg. 1-geo-alg. 2 sequence makes sense. You may forget geometry, but please, don’t forget functions!

Sarah says that you don’t learn algebra until you need it for calculus, and you don’t learn arithmetic until you need it for algebra.

So doesn’t it follow, if we want students to learn arithmetic in elementary school, that we introduce them to algebra in elementary school? And if we expect students to master algebra in high school, then we should introduce them to calculus throughout their high school mathematics studies?

How well I remember hearing in elementary school that you can’t subtract 5 from 3. And who didn’t encounter open sentences like “6 + __ = 8” en early grades? But I never say anything like “6 + x = 8” in school until I took algebra. O,r in high school, who didn’t learn that velocity is the rate of change of position with respect to time, and that acceleration is the rate of change of velocity with respect to time, and instantaneous velocity is velocity “at a point”, and similarly for instantaneous acceleration. But mention that those fuzzy concepts would not only be clarified, but would be, in face, a main part of calculus which we would study in college??? Not on your life.

If we believe that an important reason for studying algebra is as preparation for calculus, or that study of arithmetic is important (in part) because it is necessary for the study of algebra, then we should say so to students. Tell them frankly why they must study what they must study, and give them a hint of the exciting things to come!

I know this is an old post, but for other people who stumble upon it like I did … I’m greedy/selfish … I always advocate for low to high-average students to be encouraged/recommended/required to take both Geometry and Algebra 2 during junior year. The first two years are a firm foundation in algebra that includes a mix of pre-algebra, algebra, geometry, probability, and statistics – these courses can be called Pre-Algebra and Algebra 1 or Algebra 1A and Algebra 1B depending on local preference (we’re even calling a course Algebra 1.5 right now with the move to Common Core Standards). So basically, two years worth of integrated math then two classes during the third year, one focusing on geometry (including right triangle trigonometry) and one on algebra (including an introduction to the unit circle and trigonometric functions) – this gets them as ready for the ACT/SAT as they’re going to be. Seniors then take either Pre-Calculus, dual credit College Algebra (“Algebra 3”), or a transition/remedial course if the exam scores from junior year were low. Depending on clientele, a senior year offering on practical math (prob, stats, financial, etc.) is probably also worth considering. Theoretically, above average students work through this sequence a year ahead of schedule to open up AP Calculus and/or AP Statistics. But no, that’s not exactly what we’ve ever gotten working – but I’ve tried and will keep trying – it just seems like it should work ‘on paper.’

I think geometry is easier to understand simply because it is easier for visual learners. It makes more sense because it has to do with shapes that people use in everyday building costruction, packaging design, and engineering problems. For me, the concepts are easier and more straightforward. In advanced algebra it is difficult to connect problems with real life situations.

Hello everyone. I am just a theologian, not a school teacher. I barely passed school algebra, but got a B in geometry, basically by teaching it to myself, thinking it out visually in terms of ratios. I was therefore annoyed to find out, years later, that that is how geometry developed in the first place. Why can’t maths be taught in the order of its historical development? It would certainly make it more comprehensible for arts students. (I might even have managed to understand algebra if I had been taught geometry first!) Anyway, having missed that opportunity at school, I would be grateful now if anyone could tell me WHETHER THERE IS A BOOK THAT TEACHES GEOMETRY AS GEOMETRY, without algebra. Given that I write about Christian Platonism and the geometric principles behind Byzantine iconography, this would be of real help.

The best bet for you is probably Elements by Euclid, translated by Sir Thomas Heath. There’s no algebra in Euclid’s books because algebra itself wasn’t developed for another 1000 years or so. That’s why I’ve never understood people who say you need algebra to understand geometry. You might need algebra to understand a particular geometry textbook, but our ancestors understood geometry just fine having never known algebra.

Thanks for the auspicious writeup. It actually was once a leisure account it.

Look complex to far brought agreeable from you! However, how could we keep in touch?

I home school and have a student whose very visual and artistic. She just finished a curriculum that alternated units of geometry concepts with algebra concepts. So we’d be better doing geometry first? Then algebra? And which curriculum to use?

Any advice would be great. I learned a lot from the cents posted. I’m not a at whiz but we had a great time using Connected Mathematics that’s written by the University of Michigan.

Do we need to do a prealgebra book? With one student is that necessary? Is there a text that teaches all of algebra so we don’t get it in pieces from pre,1 and 2?

I am a high school math teacher of 23 years, having taught everything from 6th grade math through the traditional math course sequence up to AP Statistics (and some day, AP Calculus). I am also a homeschool dad, teaching my kids at home along with my wife. The other day, while talking with one of my kids, the idea struck me about reaching students and my own kids who are more creative and artistic, that starting with formal geometry before “Algebra I” makes a lot of sense.

I’m intending to research this more to see if this has been implemented, but just based on my experience, not only is this viable, but might help to reach students who struggle with mathematics by giving them a class with logical thinking, tangible mathematics and application, before a more formal, abstract approach to problem solving through algebraic thinking. Plus, with so much overlap between Algebra I, Algebra II, and Pre Calculus, the three courses taught sequentially, with geometric ideas supplemented to make the connections between algebra and geometry could be condensed into two courses, though keeping them separated into three would allow for a more natural development through abstract algebraic thought.

I have always intended to write my own curriculum someday, and this idea of geometry before algebra is inspiring. The historical development of mathematics just helps to reinforce this potential method… just look at historical writings of Euclid, Newton, Copernicus, Kepler, etc to see just how heavily they relied on geometry and blended it with algebra to develop mathematics in the directions we see today. Most of the algebra we see in geometry is equation solving (covered in pre-algebra), and is mostly contrived to help reinforce algebra concepts because its a good and natural fit. But, design a curriculum with the idea in mind of reaching visual, creative learners and those wondering where do we use this by introducing formal geometry (plane and solid, with a touch of analytic) with a well grounded introduction to logical thinking, and yes, proofs… which we math teachers should really emphasize all throughout the math course sequence and not just in geometry, then calculus.

It was good to come across this blog post written several years ago and hopefully others too will come across it who have similar thoughts.

I would like to find an English translation of The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Musa al-Khwarizmi. I am having a hard time finding a copy. From the examples I have seen it’s interesting how practical it was. He intended his book to be for practical use. If I had learned algebra this way maybe it would have made sense.

Have you had any luck with research on a Geometry, Algebra I, Algebra II…sequence? It certainly seems to have merit. With the fundamentals of algebra (proportions, simple equations, graphing) largely introduced in the middle school, it would seem that the topics in current Geometry curricula could be explored with minimal supplementation. The benefits of running back to back years of algebraic sequence without interruption would alleviate the huge amount of review required to bring students’ skills back to functioning level for use in Algebra II. With the cram packed Algebra II curriculum, anything that could be done to allow more time for the ‘new’ material presented would be of definite benefit. Higher thinking skills and problem solving, now often cast aside because of the brutal pace, could come to the forefront to allow better conceptual understanding and meaningful links throughout.