by Sugi Sorensen, Jan. 30, 2025
At the regularly scheduled LCUSD Governing Board meeting on Tuesday, January 21, 2025 (agenda here), LCUSD Assistant Superintendent of Curriculum & Instruction and former La Cañada High School (LCHS) Principal Jim Cartnal gave a presentation to the Board on the status of the LCHS math pathways transition.
The agenda item was described as follows:
Background:
The LCHS Math Department, working in close collaboration with site administration, has proposed revisions to the LCHS 7-12 math pathways. Earlier presentations to the LCUSD Governing Board have occurred in the 2022-23 and 2023-24 school years. The essential work to date has been to propose changes to the names of courses, returning them to plain language terms, and to propose to undo the compaction in grades 9-12. Three new courses have been presented and approved to the LCHS Governing Board, two new 7/8 math classes and a high school class called Advanced Placement PreCalculus.Current Considerations:
At the start of the 2025-26 school year, LCUSD proposes to continue with the changes to the LCHS 7-12 math pathway and this proposed presentation will explain the changes that are proposed for next school year and beyond. At the heart of the changes to the LCHS math course offerings are renaming courses in plain language terms and retiring the use of the LC Math terminology and to offer new courses in the high school advanced math pathway.
La Cañada Math Parents (LCMP) has written extensively in the past about the changes afoot in the LCHS 7-12 mathematics pathways:
- The Very Strange, Peculiar Case of the LCHS Math Pathways (Mar. 28, 2018)
- Changes Coming to LCHS Math Pathways (Jan. 13, 2023)
- Recent LCHS Math Pathway Changes Raise Concern (Jun. 06, 2024)
- The Great Reinterpretation (Oct. 16, 2024)
LCMP has supported the goals and broad aspects of these 7-12 math pathway changes consistently since they were first proposed in 2022. The wrinkles encountered along the way are considered minor, and when the transition is completed we hope the end state is an improvement for all LCUSD students. However, the wrinkles must be ironed out first.
Cartnal’s presentation to the Board at the Jan. 21, 2025 meeting, as well as the Q&A that followed, brought a mixture of clarity about the continued pathway transition, ambiguity due to contradictions in Cartnal’s presentation and the 2025-26 LCHS Course Catalog, confusion caused both by misstatements made by Cartnal as well as Board misunderstandings about existing courses and pathway changes, and finally a note of hope and optimism contained in one of Cartnal’s answers to a Board member’s question towards the end of the item discussion.
Reminder About Why the Pathways are Changing
As a reminder, the district is changing the LCHS 7-12 mathematics pathways to achieve five major goals:
- Change the LCHS core mathematics course names to their descriptive names (i.e.. Algebra I versus LC Math 1, Geometry versus LC Math 2, Algebra 2/Trigonometry versus LC Math 3, and Pre-Calculus versus LC Math 4.)
- Undo the 4-to-3 course compaction in the Advanced College Prep pathway.
- Restore the teaching of Algebra 1 to 8th grade to LCUSD students who are ready by compacting the old Math 7 Advanced and Math 8 Advanced courses into a single accelerated Math 7 Advanced course that covers both the 7th and 8th grade math standards.
- Incorporate the College Board’s new (as of the 2023-24 school year) Advanced Placement Pre-Calculus course into both HS math pathways, thus allowing all LCHS students the opportunity to earn an AP extra grade point in their math studies in high school.
- Standardizing LCHS’s mathematics courses making them more consistent with traditional pathway math courses taught in most other public and private schools, thus making it easier to place students transferring into LCUSD, and at the same time allowing students in both math pathways more opportunity to advance their math education and/or level up from the College Prep pathway to the Advanced College Prep pathway by taking standard outside HS math courses during the summer or school year from accredited outside institutions.
2024-25 School Year Updates
When the district embarked upon the grade 7-12 math pathway transition in the 2022-23 school year, they wisely decided to do a phased transition, changing one year’s math per year — retiring one compacted course and introducing its uncompacted replacement per year. The transition is about one-third of the way into its planned deployment, and will be completed in the 2027-28 school year if all goes according to plan.
Changes taking place in the current 2024-25 school year include:
- The first year of Algebra I (8) for 8th graders (and accelerated 7th graders) in the Advanced College Prep pathway.
- This is the last year the LC Math 1 Advanced course in the Advanced College Prep pathway is being offered.
- The LC Math 1 course name was changed to just Algebra I.
And though Cartnal did not mention it in his presentation to the Board, 2024-25 was the first year of the Great Reinterpretation, whereby LC Math 1 Advanced was reimagined as an Algebra II course (it’s not, LCHS staff’s insistence to the contrary), and LC Math 3 Advanced was reinterpreted as a Pre-Calculus course (it also isn’t.)
Cartnal also failed to inform the Board that sometime in the past year someone in the district petitioned the University of California Office of the President (UCOP) to change the course content designations of the LC Math 1 Advanced and LC Math 3 Advanced courses. Fortunately this miscategorization of LCHS math courses in the Advanced College Prep pathway problem will become moot as the LC Math 1 Advanced and LC Math 3 Advanced courses are retired in the next few years.
Announced Changes for 2025-26 School Year
Cartnal also explained to the Board during his presentation the planned changes in the LCHS 7-12 math pathways in the 2025-26 school year, though ambiguities remain because what he presented contradicts what is published in the 2025-26 LCHS Course Catalog. Cartnal reported:
“In the 2022-23 and 2023-24 school years we had begun the process of recalibrating our (math) classes, abandoning the ‘LC’ terminology in the College Preparatory pathway, and adopting plain language names. Courses named LC Math 1 are now called Algebra I. Next year we’re gonna hopefully see Geometry take the place of LC Math 2, and so on.”
This implies that the course names in the lower College Preparatory pathway are being changed to descriptive names one year at a time during the transition, but that is not what was indicated in the slides Cartnal showed:
Page 5 of Cartnal’s slides showed ALL courses in the lower College Prep pathway changing names to their descriptive names next year in the 2025-26 school year.
The 2025-26 LCHS Course Catalog shows conflicting information as well. The Math Courses section of the catalog shows the College Preparatory pathway course names being changed one year at a time on page 17, but not in the “LCHS CSU/UC A-G Course List” on page 6. The Math Course List on page 17 shows LC Math 3, but the actual course description on page 19 refers to the same course as “Alg 2/Trig P – MA 5038.” Even more confusing, the detailed description of the course reverts back to the LC Math 3 name and references to the course in other course descriptions universally call it LC Math 3.
Similarly, page 5 of Cartnal’s slides from the Board presentation shows LC Math 4’s name changed to PreCalculus in the 2025-26 school year, yet the “LCHS CSU/UC Course List” on page 6 of the 2025-26 LCHS Course Catalog refers to the course as LC Math 4 P/ Pre-Calculus P, while the Math Course List overview on page 17 shows the course title as LC Math 4, while the actual course description on page 20 calls it “Pre-Calculus P – MA5036.”
In short, confusion continues to prevail and both LCHS and the district seem incapable of keeping track of their own high school math course names, just as it has for the past nine years. Even after Cartnal’s presentation, it remains unclear whether all courses in the lower College Prep pathway will have their names restored to their ‘plain language’ course content names next school year, or whether they will continue to be changed one year at a time as has occurred so far.
Pre-Calculus in 8th Grade?
Cartnal continued in his presentation to the Board:
“In the Advanced Pathway, we had to develop some new courses. Dr. Gold brought Algebra 8 (sic) last year.1 That course is running right now this year. We’ve made AP Pre-Calc come to the life two years ago. And that class is in its second year of students actually being… And there’s some more things along the way. That’s what this talk is about.
So why are we changing this? A couple things. From the students’ perspective, in that Advanced Pathway the LC Math 1 Advanced and (LC Math) 3 Advanced courses were dense. And they were really, really difficult — difficult to remember a lot of the things… we were compacting some Pre-Calc concepts that would have normally been taught in a senior (or) junior class, down into the 9th grade, and in some cases in the 8th grade.”
This last statement is patently untrue. No math topics from Pre-Calculus were ever brought down into the nominal Math 8 or Math 8 Advanced courses that LCMP is aware of.
The misstatements continued:
“…And we also had this situation where because each of the Advanced pathways had a course year’s worth of stuff…curriculum…plus a third of the standards from another class compacted into it, it made it really difficult for students who wanted to maybe say ‘I’m really enjoying math now — I wish I could level up to an honors or an advanced class!’ That certainly was a challenge with the older pathway.”
Cartnal’s hypothetical makes no sense. The only students who could ‘level up’ were students in the lower College Prep pathway. But those classes were uncompacted — they did not contain another third of standards in them. Students in compacted courses in the old Advanced Pathway were already in honors and advanced courses, and there were no other honors or advanced classes they could ‘level up’ to, except the next in-sequence course in the Advanced pathway. If Cartnal meant that students in the lower College Prep pathway could not ‘level up’ to the Advanced College Prep pathway in the following school year, that also isn’t true because it was possible if the student took a bridge course during the LCFEF Summer School.
He was not even correct that ‘leveling up’ in the old Advanced College Prep pathway was a challenge. It was impossible because students did not take summer math courses for credit because of the punitive LCHS math department policy that states that, “Students taking math during the summer may not enroll in the advanced level of the next in sequence class.”2
Board Misconceptions in Abundance
During the Q&A after Cartnal’s presentation on agenda item 11.g at the LCUSD Governing Board meeting on Jan.21, 2025, a flurry of misstatements and misconceptions were tossed about, adding to the confusion.
Board member Josh Epstein asked Cartnal:
“I have a few questions and maybe one of those should be saved for later. And the question is what is going to be the differentiation between, for instance, Geometry and Honors Geometry, and the same thing with our Honors Algebra?”
Epstein, and many others, wrongly believe honors versions of all LCHS math courses will be available for students to take. This is incorrect. In the revised math pathways, there is no Honors Algebra I course offered or planned. This seems an oversight since there are honors versions of Geometry, Algebra II/Trig, and Pre-Calculus. Only Honors Algebra I is missing. If the absence of Honors Algebra I is intentional, then the justification is baffling.
LCMP brought this seeming oversight to the attention of LCHS administration and certain Board members in 2023, but no explanation or rebuttal to the concern has ever been offered. It is possible that Board member Epstein and district staff and administration believe the Algebra I (8) course is an honors Algebra I course, but the course outline approved by the Board in April 2023 indicates it is not — it contains the same topics and material taught in the regular College Prep Algebra I course, as well as the old LC Math 1 course.
The reason this matters, as LCMP has stated in previous writings on the math pathway changes, is that the current Algebra I (8) course is considered too easy for an honors track math course. About half the material in Algebra I (8) is a repeat of topics covered in earlier Advanced College Prep math courses.
Cartnal’s answer to Epstein’s question about what differentiates a regular math course in the lower College Prep pathway versus an honors level course in the Advanced College Prep course elicited more confusion. Cartnal said:
“…It’s important to note that the abandonment of math compaction at the high school means in essence that it’s a year’s worth of course curriculum taught in a year, as opposed to what we had been doing, which was 1.3 standards taught in one year. And so as it regards the basic standards, Geometry and the Geometry Honors class will cover the same standards.
The primary differences between the Honors Geometry and Geometry class will be that the way that the California Common Core math standards are written is that there are star standards that are part of each of those curricula or subject domains. And the honors class will engage with the star standards whereas the College Prep class may or may not, certainly not at the level or depth that they are going to do (in the Honors class.) And I think it’s also the fact that the application and the kind of orientation of the classes are similar to be sure, but are different, and I mean concretely, that it would be the output or the degree of engagement that the students engage with.”
Cartnal misspoke about the California Common Core State Standards in Mathematics (CCSS-M) use of so-called ‘starred’ standards. While starred standards do exist in the CCSS-M, they “indicate a modeling standards linking mathematics to everyday life, work, and decision making.”3 Standards marked with a (+) symbol indicate additional standards “that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics…”:
Starred modeling standards, contrary to Cartnal’s assertion, are not extra standards for honors students heading to take more advanced courses. They are expected to be taught to all students, even those not aiming for advanced courses. Cartnal probably meant (+) standards. Even then, (+) standards are typically not the distinguishing difference between honors level and regular high school math courses. For example, in the CCSS-M’s Geometry standards in the Higher Mathematics Courses: Traditional Pathway section of the standards document, there are 48 regular standards listed, of which 17 are starred, and 8 (+) standards. In the Modeling with Geometry (G-MG) domain of the CCSS-M Geometry standards, all three listed standards are starred:
Clearly all high school Geometry students, whether in a regular pathway or an honors level course, are expected to know and be able to apply these standards.
In reality, the differences between a typical high school Geometry and Honors Geometry course are more substantial.4
The 60/40 Claim
Later in the Q&A after Cartnal’s presentation, Board member Epstein asked Cartnal what percentage of LCUSD 8th graders are in Algebra I. Cartnal answered succinctly and assuredly, “60/40”, presumably meaning 60% of current 8th graders are currently taking Algebra I (8). It is unconfirmed but assumed that Cartnal meant 60% of LCHS 7/8 8th graders are taking Algebra I (8) or higher, meaning there are 8th graders who are accelerated in math and known to be taking LC Math 1 Advanced or LC Math 2 Advanced currently.
Thereupon ensued a significant number of follow-up comments and questions from both other Board members as well as cabinet members commending the district for such a remarkable achievement. For example, Board member Octavia Thuss paid the following compliment:
“I was going say, historically, prior to Common Core, we had Algebra I in the 8th grade and kids in California were doing quite well. And there was a lot of frustration when Common Core came into play that there was a loss among students who were having a great progress and success with having Algebra I (in 8th grade.) It’s really great to see it back in the 8th grade. It’s quite remarkable to hear that 60% are taking it. I myself, not so much. And it’s wonderful for those kids and I’m glad to see it come back.”
The problem is the 60/40 claim is overstated. For at least the last nine years, LCMP annually reports to its members and the wider community on the number and percentage of LCUSD students in 7th grade taking regular Math 7, Math 7 Advanced, and beyond. LCMP obtains the data annually from LCHS 7/8 Principal Jarrett Gold. Gold reported to LCMP August 20, 2024 the following breakdown of 8th graders by enrolled math class:
Note that the number of 8th graders in the four classes as reported by Gold does not sum to 344 — it totals only 339. It is not known what accounts for the missing five students. They could be in an RSP math course or on an IEP, or Gold could have misreported one of the other totals. Regardless of the reason, as can be seen only 42.44% of LCUSD 8th graders are in Algebra I (8), or 53.77% if you include 8th graders accelerated in classes above Algebra I (8.) LCMP reached out to LCHS 7/8 Principal Gold on 01/28/25 for updated numbers, but as of the publication date of this article he has not responded. It is possible, though highly improbable, that 21 students in Math 8 regular were moved up to Algebra I (8) or above since Gold reported the data to LCMP in August.
The Danger of Taking Algebra I by 8th Grade?
Later in the Q&A, Governing Board member Dan Jeffries expressed concern about students “rushing to Algebra I”:
“Can I ask a follow up kind of to Josh’s question — is something if you are new to our district — you may have not heard of us talking about it for the last few years…this whole idea that Algebra in 8th grade — you know there are some districts around the state who are saying you should not take Algebra until 9th grade or higher. In fact, you know, the state guidelines on it…and your response to Josh was 60% of our 8th graders are taking Algebra makes it clear that that’s not what’s happening here. Do you…can you just make it clear to people why are we not doing that?”
Jeffries assertion that there are state guidelines warning students off of taking Algebra I in 8th grade (or earlier) is only partially correct. The root of the problem here is that the CCSS-M, and the 2013 California Mathematics Framework (CMF) that guided implementation of the standards, gave conflicting advice about Algebra I by 8th grade. As Jeffries and other LCUSD Board members and staff seem to believe, the 2013 CMF recommended delaying Algebra 1 until 9th grade for most students. What they do not appear to know, however, is that the CDE gave contradictory advice and left the door open to Algebra 1 earlier in middle school buried in an appendix.
The language permitting acceleration into Algebra 1 in middle school is contained in Appendix A of the CCSS-M and is not directly accessible from the California Department of Education’s (CDE) standards website. Appendix A is only accessible via a search on Google or the CDE’s search feature if you know the appropriate terms to search. The exact language in Appendix A is:
“…the Achieve Pathways Group endorses the notion that all students who are ready for rigorous high school mathematics in eighth grade should take such courses (Algebra I or Mathematics I), and that all middle schools should offer this opportunity to their students.” (CCSS-M, Appendix A, p.80)
Most California school districts, including LCUSD, missed this very clear and very important language in the Standards, and instead followed the 2013 CMF’s conflicting but stern warning, “decisions to accelerate students into the Common Core State Standards for higher mathematics before ninth grade should not be rushed. Premature placement of students into an accelerated pathway should be avoided at all costs.” (2013 CMF, Appendix D, p.829)
As I have written about extensively in other places, this mistaken notion that Algebra I in 8th grade is somehow dangerous to students is based on a grievous misreading of California CST data in a key research study (i.e. Williams (2011).)5
As if to confirm the state and district’s misbegotten fear of Algebra I in 8th grade, Cartnal echoed Jeffries’ concern in his response to Dan’s question:
“Well when we implemented Common Core State Standards back in 2015 we absolutely took that seriously and we had Algebra classes in 9th grade and found that our students, when we looked at it…faced, like I said, a compaction in the Advanced pathway that was very, very difficult 9th and 11th grade years. 10th grade wasn’t a walk in the park either.”
Cartnal went on to conclude that contrary to the phantom fear raised by Jeffries, restoring Algebra I to 8th grade in LCUSD has been a success, “And so as we’ve kinda unbuilt that compaction and came up with this idea let’s see how they do in 8th grade, and can it work? The early returns are very, very positive. The 8th graders are doing well.”
Increasing Opportunities to Move Between Pathways
A theme repeated by Cartnal during his presentation and subsequent Q&A with the Governing Board was that one of the major reasons for modifying the math pathways is to make it easier for students of varying abilities and math maturity to move between math pathways and to allow students, independent of pathway, to access at least one AP math course by the end of high school. Quoting from Cartnal:
“Achieving our goal that at La Canada every single student, independent of pathway, has the ability to move depending on how hard they want to work and their aptitude and interest, has the ability to take Advanced Placement class by the time they graduate. In the College Preparatory pathway: AP Precal(c) is a senior, though not a lot of seniors this year are doing that, or AP Statistics, and in the Advanced pathway, exceeding what most schools in the state of California offer — AP Calculus AB and BC — through a partnership with Pasadena City Colleges — offering Linear Algebra and Multivariable Calculus. There’s a lot of great stories to tell in the LCHS math department I’m sure.”
Board members Epstein and Joe Radabaugh interrogated this theme with several questions about moving between math pathways. The prevailing attitude from the Board, and Cartnal as well, is that the pathway changes will provide more opportunities for all LCHS students, not just math precocious students, to move up and down pathways, and take advantage of math courses outside of school. Cartnal added one word of caution to this newfound flexibility:
“Please be careful about the prospect of trying to test out of the entirety of the 7th and 8th grade standards. But I do believe that again another innovation that’s present is presumably you don’t do that. You listen to the caring and possibly meddling advice of a well-intended administrator, counselor — you could take a summer school class, call it Geometry, and you’re in the exact same place. And that’s the good news. That’s what we are trying to do.”
There are two problems with this word of caution and the baseline sentiment that the new pathways offer new flexibility.
First, the supposed newfound flexibility of moving up and down math pathways in high school is essentially handcuffed by the present LCHS math department policy that states that, “Students taking math during the summer may not enroll in the advanced level of the next in sequence class”:
This punitive policy should be rescinded immediately. LCMP has been making this point for several years, since before the 7-12 math pathways idea was first presented to LCMP representatives in December 2022. LCMP has asked district staff about rescinding the policy several times since then, and each time district staff has responded that the policy will be revoked, yet it has still to be accomplished. This punitive policy remains in place as of the publication date of this article. If the district truly wants to reap the benefits of the pathway changes, they must remove this policy.
Second, Cartnal’s warning to students to be careful about trying “to test out of the entirety of the 7th and 8th grade standards,” through the middle school math placement test offered annually in May/June and again in August is misguided. This statement implies that students can test out of just the 7th grade standards. They currently cannot. The middle school math placement exam is only for students who want to try to skip Math 7 Advanced in the Advanced College Prep pathway. There is no opportunity to test out of just Math 7 regular in the College Prep pathway.
Over 80% of students fail the annual middle school math placement exam offered to 6th graders in the Spring primarily because of a lack of district transparency with parents. Uninformed parents blindly send their students into the middle school math placement exam not knowing that the exam tests students on their mastery of 7th and 8th grade math standards. As Board member Radabaugh related during the Q&A:
“But the reality is there are still people in town that…and it’s a content…I think I’ve told the story. I had my daughter try to take that test and she came back furious at us because she’s like, ‘Dad, I’ve never even seen this content!’ And so that was just…I just thought that magic, through some osmosis, she might somehow have picked it up and spring board! But the reality was she hadn’t seen the content.”
This annual tragedy could easily be avoided if the district was more transparent with families. In fact, this lack of transparency is why LCMP has been offering a presentation annually to district parents about the middle school math placement exam since LCMP was founded in 2016.
A Better Way…
An even better solution that would multiply the benefits of the district’s 7-12 math pathway change is if the middle school math placement exam allowed students to skip just Math 7 in the lower College Prep pathway. That simple change would enable even more students to reach Algebra I by 8th grade. This could be easily achieved by returning to a two-part middle school math placement exam as was used in the district prior to last year.
Currently students are only allowed to accelerate in middle school math in the Advanced College Prep pathway. Why? If the district truly wants a more equitable state of affairs that offers more opportunity for students to move ahead in math at later times in middle or high school, and wants to legitimately achieve 60% or more of district students reaching Algebra I by 8th grade, they should lower the onramp at the beginning of middle school. As Board member Radabaugh commented during the Q&A when talking about moving between math pathways, “There’s a maturity issue. You do it then when you’re in 10th grade (instead of) trying to do it in 6th and 7th grade…” This could be enabled and the transition to the upper pathway made more achievable if students could start middle school with Math 8.
1 – The course is actually called Algebra 1 (8) in all slides used in Cartnal’s presentation and in previous years’ presentations.
2 – Quoted from the LCHS Math Department web page (accessed on 01/28/25 — it’s been this way since the Common Core State Standards in Mathematics were implemented in LCUSD starting in 2015.)
3 – See California Common Core State Standards in Mathematics (adopted by the CDE 2010 and modified Jan. 2013), page 5.
4 – For a more thorough explanation of what differentiates a typical California high school Geometry and Honors Geometry course, see this explanation from ChatGPT 4.0 (generated 01/30/25):
The main differences between a typical high school Geometry course and an Honors Geometry course in California generally lie in the depth, rigor, and pace of instruction, as well as the level of mathematical reasoning expected from students. Below is an outline of the typical differences based on California’s Mathematics Framework and common honors-level practices.
1. Content Differences (Standards & Topics)
Both regular and honors Geometry courses follow the California Common Core State Standards for Mathematics (CA CCSSM), but honors courses extend beyond them in depth and scope.
Concept Regular Geometry Honors Geometry Euclidean Geometry Covers basic postulates, theorems, and proofs related to points, lines, planes, angles, parallel lines, and polygons. Emphasizes formal proof writing, including indirect proofs and more complex multi-step proofs. Transformations & Congruence Focuses on rigid transformations (translations, rotations, reflections) and proving triangle congruence using SSS, SAS, ASA, AAS. Adds more emphasis on the algebraic representation of transformations and extends to complex plane geometry. Similarity & Trigonometry Covers triangle similarity (AA, SSS, SAS), Pythagorean Theorem, and basic right triangle trigonometry. Introduces Law of Sines, Law of Cosines, and applications of trigonometric ratios in non-right triangles. Circles Covers basic properties of circles, central and inscribed angles, tangents, and arc length. Delves into deeper theorems, such as cyclic quadrilaterals, power of a point, and more challenging constructions. Coordinate Geometry Includes finding distances, midpoints, slopes, and equations of lines. Includes proofs in coordinate geometry, conic sections, and equations of circles and parabolas. Three-Dimensional Geometry Covers surface area and volume of 3D figures. Introduces cross-sections, Cavalieri’s Principle, and more challenging 3D visualization. Logic & Proofs Covers direct proof techniques, two-column proofs, and simple deductive reasoning. Includes conditional proofs, proof by contradiction, and deeper logic-based reasoning.
2. Depth of Rigor
- Regular Geometry focuses on basic understanding of concepts, using hands-on activities, visual models, and scaffolded problem-solving.
- Honors Geometry demands abstract thinking and formal proofs, including writing complete deductive arguments.
- Honors students often apply geometry in unfamiliar or non-routine situations, making use of more challenging problem-solving techniques.
3. Pace of Instruction
- Honors courses move much faster, sometimes covering additional topics typically found in Precalculus (such as vectors and more advanced trigonometry).
- The regular course allows for more review and step-by-step guidance, whereas honors students are expected to absorb material more quickly.
4. Emphasis on Proof & Reasoning
- Regular Geometry teaches students how to construct basic two-column and paragraph proofs.
- Honors Geometry incorporates indirect proofs, coordinate proofs, and non-Euclidean geometry concepts in some cases.
5. Additional Topics in Honors Geometry
Some honors programs include:
- Conic Sections: Equations of parabolas, ellipses, and hyperbolas.
- Advanced Transformations: Compositions of transformations, matrices, and dilations.
- Vector Geometry: Introduction to vectors in 2D space.
- Non-Euclidean & Projective Geometry: Concepts that extend beyond Euclidean postulates.
6. Problem-Solving & Application
- Regular Geometry problems tend to be straightforward, focusing on practicing skills.
- Honors Geometry problems often integrate algebra, trigonometry, and deeper problem-solving techniques, including proof-based contest problems (e.g., AMC 10, AIME-style questions).
7. Preparation for Higher-Level Math
- Regular Geometry prepares students for Algebra II and Trigonometry.
- Honors Geometry is designed to prepare students for Honors Algebra II, Precalculus, or even AP Calculus AB/BC more quickly.
Conclusion
Honors Geometry in California builds upon the same standards as the regular course but introduces greater abstraction, faster pacing, deeper proofs, and additional enrichment topics. The course serves students aiming for higher-level mathematics and STEM pathways, often incorporating problem-solving skills needed for math competitions or advanced placement courses.
5 – See my article co-authored with David Margulies, “Algebra I in 8th Grade Makes a Comeback,” Cryptomeric Thoughts, (May 09, 2024.)