by Sugi Sorensen
October 16, 2024
Perhaps most notable for its high turnover rate — thirteen teachers retired or resigned in the last seven years1 — the La Canada High School (LCHS) math department deserves notoriety for its recent creative reinterpretation of the content of its math courses.
Without any announcement to the community, LCHS directed the University of California’s Office of the President (UCOP) to re-certify two of its high school math courses as different than they’ve been certified since they were first introduced when the California Common Core State Standards in Mathematics (CCSS-M) were adopted in LCUSD beginning in 2014.
LC Math 1 Advanced, formerly (wrongly) classified as a Mathematics I course since the 2015-16 school year, is now classified as an Algebra II course. It should have been classified as an advanced Algebra I course given the district has always taught traditional pathway mathematics classes when deciding how to meet the CCSS-M standards. For whatever reason LCUSD staff or the UCOP committee that certifies courses at the time classified it as an integrated Mathematics I course and LCUSD didn’t notice the error for a full seven years. The debacle that ensued when LCUSD chose to name its traditional pathway math courses with integrated course names has been noted by LCMP as far back as 2018.
Similarly, LC Math 3 Advanced, originally judged by the UCOP course review committee to be an Other Advanced Mathematics course when LCHS first submitted it in 2017, has miraculously been re-imagined to be a Pre-Calculus course for the 2024-25 school year. Again, no content was changed from the 2023-24 school year to the present 2024-25 school year.
Were these courses re-classified because their content or rigor were increased? Hardly. No substantive course content was changed since the courses were originally classified by the UC. In fact, the LC Math 1 Adv course has had material removed from it since it was first designed by then-LCHS math teacher Brian McDermott in 2015. Additionally, the 2024-25 LCHS Course Catalog still describes LC Math 1 Advanced as an Algebra I Advanced course and LC Math 3 Advanced as an honors Algebra II/Trigonometry course:
These unexpected and creative re-interpretations of foundational high school mathematics courses are perplexing given the lack of change in course content. Though the impact of the change is diminished by the fact that the present 2024-25 school year is the last year both LC Math 1 Adv and LC Math 3 Adv are being taught, the fact they were reinterpreted is cause for serious concern about LCHS’s math department leadership, its understanding of the math it is charged with teaching our students, the district’s administrative leadership to approve such a dubious act, and the Governing Board since they have been warned about these changes since LCMP first learned about them in 2022 and have done nothing in response.
As La Canada Math Parents (LCMP) observed in our January 2023 article — “Changes Coming to LCHS Math Pathways”, certain LCHS math teachers first presented the novel theory that Math 8 Adv was really an Algebra I course, and LC Math 1 Adv was really an Algebra II course, in December of 2022. LCMP responded strongly to the novel assertion2, informing LCHS and LCUSD staff that they believed their reinterpretations were mistaken. However, allowing for the possibility that LCMP might be wrong in its initial response, LCMP informed LCUSD staff that we wanted to conduct a more in-depth analysis of the course contents of both LC Math 1 Adv and LC Math 3 Adv, and compare them to the Common Core Math Standards before providing a definitive response. That analysis was conducted over the 2023 winter break and following its completion was forwarded to then LCHS Principal Jim Cartnal. LCMP concluded that while the LC Math 1 Adv course indeed included some Algebra II and Pre-Calculus topics pulled forward into its course outline as would be expected for a compressed pathway math course, it was still mostly an Algebra I course. Here is a summary of the analysis of LC Math 1 Adv showing which standards map to which subject topics in the CCSS-M standards:
As can clearly be seen, three-fourths of the topics in LC Math 1 Adv map to Algebra I CCSS-M standards, with some advanced topics in Algebra II and Pre-Calculus compacted into it. To see the full analysis of all topics in the LC Math 1 Adv course outline and how they map to CCSS-M standards, see LCMP’s detailed analysis here.
What’s in a Name?
So what makes an Algebra I an Algebra I course and not a Pre-Algebra or Algebra II course? The CCSS-M straightforwardly addresses this question on page 60:
The fundamental purpose of the Algebra I course is to formalize and extend the mathematics that students learned in the middle grades. This course includes standards from the conceptual categories of Number and Quantity, Algebra, Functions, and Statistics and Probability. Some standards are repeated in multiple higher mathematics courses; therefore instructional notes, which appear in brackets, indicate what is appropriate for study in this particular course. For example, the scope of Algebra I is limited to linear, quadratic, and exponential expressions and functions as well as some work with absolute value, step, and functions that are piecewise-defined. Therefore, although a standard may include references to logarithms or trigonometry, those functions are not to be included in course work for Algebra I; they will be addressed later in Algebra II.
Note that the Math 8 Advanced course, which is no longer taught in the LCHS 7/8 middle school as of the 2024-25 school year, and the revised Math 7 Advanced compressed pre-Algebra course, do not teach quadratic and exponential expressions or equations, nor do they delve into absolute value, step and piece-wise functions. All of these extra topics are covered in LC Math 1 Adv.
What makes an Algebra II course different than an Algebra I course? Again, look to the CCSS-M for guidance:
Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include logarithmic, polynomial, rational, and radical functions in the Algebra II course. This course includes standards from the conceptual categories of Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability. Some standards are repeated in multiple higher mathematics courses; therefore instructional notes, which appear in brackets, indicate what is appropriate for study in this particular course. Standards that were limited in Algebra I no longer have those restrictions in Algebra II. Students work closely with the expressions that define the functions, competently manipulate algebraic expressions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms.3
LCMP’s analysis of LCHS’s honors Algebra II course, deceptively called LC Math 3 Adv, revealed the following mapping of course topics to CCSS-M standards:
As can be seen, LC Math 3 Adv is mostly an Algebra II course with a fair amount of review of Algebra I topics, and a handful of Pre-Calculus and other advanced math topics beyond Algebra II.
What about Pre-Calculus?
Unfortunately, one of the major weaknesses of the CCSS-M standards is that it does not include Pre-Calculus as a model course among its Higher Mathematics Standards. Thus it is difficult to map topics from any extant high school math class outline or syllabus to CCSS-M standards for Pre-Calculus.
Absent guidance from the CCSS-M standards document, a concise definition of what should be covered in a CCSS-M aligned Pre-Calculus class can be gleaned from the California Mathematics Framework (CMF). Recall that a curriculum framework is intended as a guide to schools and district for implementing the state standards in the classroom. Given the California Department of Education’s recently-approved 2023 CMF did not provide any guidance on Pre-Calculus, we have to look to the previous 2013 version of the CMF for help:
Precalculus combines concepts of trigonometry, geometry, and algebra that are needed to prepare students for the study of calculus. The course strengthens students’ conceptual understanding of problems and mathematical reasoning in solving problems. Facility with these topics is especially important for students who intend to study calculus, physics, other sciences, and engineering in college. The main topics in the Precalculus course are complex numbers, rational functions, trigonometric functions and their inverses, inverse functions, vectors and matrices, and parametric and polar curves. Because the standards that comprise this course are mostly (+) standards, students who enroll in Precalculus should have met the college- and career-ready standards of the previous courses in the Integrated Pathway or Traditional Pathway. It is recommended that students complete Precalculus before taking an Advanced Placement calculus course.4
Given there is significant overlap of material between Pre-Calculus and Algebra II, it is informative to see what the 2013 CMF says about what students learn in Pre-Calculus:
Students in Precalculus extend their work with complex numbers, which started in Mathematics III or Algebra II, to see that complex numbers can be represented in the Cartesian plane and that operations with complex numbers have a geometric interpretation. They connect their understanding of trigonometry and the geometry of the plane to express complex numbers in polar form.
Students begin working with vectors, representing them geometrically and performing operations with them. They connect the notion of vectors to complex numbers. Students also work with matrices and their operations, experiencing for the first time an algebraic system in which multiplication is not commutative. Additionally, they see the connection between matrices and transformations of the plane— namely, that a vector in the plane can be multiplied by a matrix to produce another vector—and they work with matrices from the perspective of transformations. They also find inverse matrices and use matrices to represent and solve linear systems.
Students extend their work with trigonometric functions, investigating the reciprocal functions secant, cosecant, and cotangent and the graphs and properties associated with those functions. Students find inverse trigonometric functions by appropriately restricting the domains of the standard trigonometric functions and use them to solve problems that arise in modeling contexts.
Although students in Precalculus have worked previously with parabolas and circles, they now work with ellipses and hyperbolas. They also work with polar coordinates and curves defined parametrically and connect these to their other work with trigonometry and complex numbers.
Finally, students work with rational functions that are more complicated, graphing them and determining zeros, -intercepts, symmetry, asymptotes, intervals for which the function is increasing or decreas- ing, and maximum or minimum points.
In the old LCHS mathematics pathways, Pre-Calculus did not exist as a separate course in its Advanced College Preparatory Pathway. Pre-Calculus was subsumed within the three courses LC Math 1 Adv, LC Math 2 Adv, and LC Math 3 Adv. LCHS administrative staff have insisted since the CCSS-M standards were first adopted by LCUSD in 2014 that the Advanced College Prep Pathway represented a 4-to-3 compaction of the lower College Prep Pathway:
Keep in mind that LC Math 4 has always been described as a Pre-Calculus/Trigonometry course. Here is the course description for LC Math 4 from the 2024-25 LCHS Course Catalog:
LCHS Staff Defend The Great Reinterpretation
LCHS staff and administration defended its claim that LC Math 1 Adv is really an Algebra II course by stating that teachers who teach LC Math 1 Adv teach mainly from the Glencoe Algebra II textbook, and the rest of the time from the Glencoe Algebra I textbook. This is a flimsy basis of defense because there is significant overlap in mathematical topics between the two Glencoe Algebra textbooks. For example, the LC Math 1 Adv pacing guide provided to LCMP by then LCHS Principal Cartnal shows that the first three units are taught from the Glencoe Algebra 2 textbook:
- Unit 1 – Equations and Inequalities
- Unit 2 – Linear Relations and Functions
- Unit 3 – Systems of equations and Inequalities
These three topics are also presented in the Glencoe Algebra 1 textbook, and in fact are taught in the Math In Focus – Course 2 Accelerated textbook used in the Math 7 Advanced course taught to about half of LCUSD students in 7th grade. In summary, calling LC Math 1 Advanced an Algebra II course merely because you teach some of its topics from an Algebra II textbook is not a sound basis for claiming that the entire course is an Algebra II course.
LCHS administration also claimed that no topics taught in LC Math 3 Adv would be missed by skipping the class and going straight from LC Math 2 Adv into AP Pre-Calculus. LCMP believes otherwise. Our analysis shows 22 Algebra II topics that would be missed by skipping LC Math 3 Adv.
The last reason used by LCHS administration to defend the great reinterpretation actually contradicts other claims they make in their advice to LCHS students. Former Principal Cartnal told LCMP, and tells its counseling staff, that LCHS students aiming to take AP Calculus BC in a later year at LCHS should take AP Pre-Calculus instead of LC Math 3 Adv, but students aiming to take AP Calculus AB, Financial Literacy or AP Statistics should instead take LC Math 3 Adv after LC Math 2 Adv:
Implicit in this advice is the logical argument that there is material in LC Math 3 Adv that is worthwhile to students completing LC Math 2 Adv who are aiming for higher math courses like Financial Literacy, AP Statistics, and AP Calculus AB. If that is true, then why shouldn’t that also be true for students heading into AP Calculus BC?
Why Should Anyone Care?
So who cares whether the UC system considers two courses in their last year at LCHS as Algebra I, Algebra II, or Pre-Calculus? Several issues come to mind as it potentially affects students in LCUSD schools:
- A cohort of students was advised to skip LC Math 3 Adv and go from LC Math 2 Adv directly into AP Pre-Calculus. In fact, the LCHS counseling department and administrative staff are still advising LCHS students in the Advanced College Prep Pathway to skip LC Math 3 Adv. According to LCMP’s course analyses, those kids will miss important Algebra II topics that were taught in LC Math 3 Adv and are not taught in AP Pre-Calculus. This will make AP Pre-Calculus for students who skipped LC Math 3 Adv unnecessarily difficult. LCMP has already received reports of exactly this phenomenon happening to students currently in AP Pre-Calculus who took LC Math 2 Adv last year.
- The UC and Cal State University (CSU) system’s admission requirements in the area of mathematic (Area C) are 3 years of mathematics in high school between grades 9-12, four years recommended, including Algebra I, Geometry and Algebra II in the traditional pathway, or Mathematics I, Mathematics II, and Mathematics III in the integrated pathway. A cohort of LCHS students has been told to skip Algebra II. That jeopardizes their ability to apply to CSU and UC campuses.
- The SAT and ACT standardized college admissions tests include mathematics up to and including topics in Algebra II. Students who skip LC Math 3 Adv and go from LC Math 2 Adv into AP Pre-Calculus may miss important Algebra II topics unless they supplement or learn skipped topics on their own.
The most concerning issue that comes to mind is how well do LCHS teachers, administrators and LCUSD cabinet understand the mathematics they are entrusted with teaching LCUSD students? Only one of the two following propositions can be correct: either the great reinterpretation is correct, which means LCUSD was mistaken when they originally classified LC Math 1 Adv as an advanced Algebra I class, LC Math 3 Adv as an advanced Algebra II/Trigonometry course, and Math 8 Adv as the second half of Pre-Algebra, and if that’s true then why did they not recognize the problem for 7+ years and teach hundreds of students under a misbegotten understanding? Or else it must be true that the great reinterpretation is wrong, in which case can we trust our leadership and teachers with such important decisions?
- LCHS 7-12 math teachers departed in the past seven years include: Annie DiFiore (2017), Andrew Kang (2017), Jeenie Lee (2017), Charisse Bontemps (2018), India Dastic (2018), Jessica Quinn (2019), Elizabeth Gruss/Akiyama (2020), Megan Dever (2022), Dana Leu (2022), Michael Jardini (2023), Paul Cyhaniuk (2024), Andrew Mendoza (2024), and Rebecca Hassan Elsaadany (2024.) ↩︎
- In our first report to LCMP members about the claim, we called the reinterpretation of Math 8 Adv as an Algebra I course “ludicrous.” Here’s the summary comparison of LCMP’s winter 2022 analysis of LCHS math courses that mapped course content to the CCSS-M standards:
As can be seen, LC Math 1 is an almost “pure” Algebra I class. It follows the Glencoe Algebra I textbook in lockstep from chapter 1 through chapter 11. Math 8 Adv, on the other hand, is 3/4ths an 8th grade Pre-Algebra class and only includes about 1/5th additional Algebra I topics. ↩︎ - California Common Core State Standards in Mathematics, p. 77. California Department of Education (2010, modified 2013.) ↩︎
- p.1, Precalculus Chapter of the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve, California State Board of Education (Nov. 2013.) ↩︎