Introduction: Math Course Pathways The 2022 draft California math framework for public schools Second Field Review (SFR) guides for the removal of the math course pathway that enables a student to reach a college level Advanced Placement (AP) Calculus course in high school. Taking AP Calculus in high school creates career enabling opportunities for a Science, Technology, Engineering, and Math (STEM) student. Students in districts that adopt this SFR draft framework may therefore not have these opportunities available to them, and will be disadvantaged in pursuing a STEM major or career relative to students in districts that do not adopt this SFR draft framework --- and disadvantaged versus students in other states and countries that maintain pathways, encourage students to accelerate, and complete Calculus in high school. The SFR draft framework fails STEM students. Currently, there are several pathways available to a student as shown below in figure 1 (a) and if a student wishes to take an AP Calculus course in high school 12^th grade, they typically follow the topmost course pathway. The pathway to AP Calculus is typically a five-year course sequence that begins with the student taking Algebra 1/Integrated Math 1 in 8^th grade.

Caption Fig. 1. a) Current Typical Math Classes and Pathways Available. b) SFR draft framework’s proposed math pathway. (Note: The course pathway of Integrated Math 1-3 ”is an existing reorganization of this pathway“ (Algebra/Geometry/Algebra 2), covering the same material, just in different order, so districts offer only one of these pathways).

Figure 1 (b) shows the math course pathway the SFR draft framework guides for. The SFR draft framework proposes one pathway and Algebra 1 is delayed until 9th grade, leaving only 4 years to cover 5 year's worth of material, making it very difficult for a student to reach AP Calculus in 12th grade without going outside the public school system. Students in districts that adopt the SFR draft framework will therefore have obstacles created inhibiting them from reaching AP Calculus in high school.

The Benefits of Taking Calculus in High School

Taking Calculus in high school is very important, to the point of being imperative, for a student whose goal is a STEM major. For college admissions, a student who has not completed Calculus will be disadvantaged because they will be competing against students who have. More to the point, the math learned in Calculus is foundational for upper division courses that follow in all STEM majors. It is for this reason that, essentially all STEM college majors start with a required Calculus course sequence. For example, at University of California at San Diego (UCSD) as of 2022, 78 different majors covering 16 different fields start with a required calculus course sequence. Taking an AP Calculus class in high school enables the student to pass out of part of or all of the first year of the college Calculus sequence. A student taking AP Calculus AB can pass out of part of the sequence, while a student taking an AP Calculus BC class can pass out of all the sequence. This enables a student to move earlier onto the technical electives offered by the major, and therefore take more of them, which are the advanced classes more directly related to the major. Taking more of these, or completing them earlier, means that a student will more likely have time available to them in their senior year for research opportunities such as internships, collaborations with a professor, independent study projects, study abroad programs, an honors thesis, or double major. All of these help build connections which can be vital for a STEM career. Additionally, they help prepare for graduate school, bolster resumes, be better prepared for interviews, and have better letters of recommendation. On top of this is the cost savings that occurs with being more likely to graduate in a timely fashion. These benefits are well known to STEM professionals and are eluded to in an Open Letter (Open Letter on K-12 Mathematics) signed by 1739 and counting STEM professionals:

“College students who need to spend their early years taking introductory math courses may require more time to graduate. They may need to give up other opportunities and are more likely to struggle academically.”

This is elaborated upon in the analysis that accompanies this Open Letter:

“…students with AP Calculus credit can place out of the Calculus I and II requirements. Students who place out of requirements have an advantage of lighter workload, and the opportunity to take advanced courses early, opening up internships and research options not otherwise accessible until later.”

The benefits of taking Calculus in high school and using it to pass out of the introductory college sequence are tangible and significant. The SFR draft framework refers to “the traditional pathway for taking Calculus in high school or in their first semester of college (as is sometimes expected for many STEM majors)” (Ch. 8 line 919), but closer to accurate would be “as is always expected for all STEM majors”. As a math curriculum framework, it should be giving accurate guidance on this point.

Even students who do not use their high school Calculus to pass out of the initial college Calculus class are still found to benefit from having taken Calculus in high school. This point has been studied in detail by Sadler and Sonnert in “The Path to College Calculus: The Impact of High School Mathematics Coursework”. These results among others were also discussed in “The Role of Calculus in the Transition from High School to College Mathematics”. First, it is important to understand that a college Calculus class can be a significant increase in rigor in comparison to a high school class:

“while the AP Calculus curriculum closely follows that of college Calculus I, expectations, especially at the level of practice standards, can be very different.”

This means that taking Calculus again when in college, does not mean the student is re-taking the same class, but instead is taking a class with a significantly higher level of rigor. As shown by Sadler and Sonnert, exposing students to Calculus in high school is very helpful for them to be prepared for the rigor of a college Calculus course:

“Taking high school calculus is equivalent to a boost of students’ college calculus grades of half a letter grade, on average... Students with a relatively weak background in the mathematics considered preparatory for high school calculus appear to benefit even more from participating in high school calculus than do their more well-prepared classmates.”

This led to the authors to conclude that

“being exposed to calculus in high school clearly pays off when students enroll in college calculus.”

What is also important to understand about this study, as the authors point out, is that these benefits are observed over and above the benefits the group students who pass out of the initial Calculus sequence received. Since the study focused only on the group of students who are taking Calculus in college, these results are obtained even though they omit the group of students who passed out of college Calculus through the AP exam, that is the highest achieving students. The bottom line perhaps is that being exposed to Calculus in high school could be the difference between passing the college class and not. Additionally, in a second study of students who took Calculus in high school, it can be argued that the majority of the students who take Calculus in high school benefit from it. In short, STEM major students who have not taken AP Calculus in high school will be at a disadvantage as they will be surrounded by students who have, and who will be taking advantage of the benefits of having taken AP Calculus in high school.

Current Acceleration Pathways Leading to Taking AP Calculus in High School in Various School Districts

Public school districts that are aware of the benefits of taking AP Calculus in high school, design their math courses to have pathways available to AP Calculus for students who are prepared and motivated to do so. For example, Fig. 2 shows the math course pathways currently offered in Irvine Unified School District (IUSD). (Note: the presence of multiple pathways is also called “tracking”).

Fig. 2 Irvine Unified School District (IUSD) mathematics pathway chart. (click image to enlarge)

There are several noteworthy features shown in this schematic.

1) Multiple pathways exist for students to choose from. There is a pathway for students who need more support (top) and pathways for students who are interested in advancement (bottom two).

2) The pathway for arriving at an AP Calculus course in 12^th grade is depicted in the bottom two pathways with acceleration starting in 7^th grade and the taking of Math 1 (Algebra 1) in 8^th grade thus beginning the 5-year long sequence to AP Calculus, with both AB and BC Calculus available to the students.

3) There are options for the students to change pathways. If a student is behind and wishes to switch to an accelerated pathway, they will have to take a bridge course to catch up.

4) There is a pathway for a student who delayed taking Math 1 (Algebra 1) until 9^th grade to arrive at AP Calculus in 12^th grade. This is depicted in the middle pathway. This does require the student to switch to a separate accelerated pathway to catch up on the material they have not learned. This is a demanding pathway as there is a lot of material the student needs to catch up upon, especially if they desire to reach the AP Calculus BC class.

The purpose of multiple pathways is to allow the schools to meet the students where they are in math. Students can follow pathways according to their goals and motivations. Pathways are provided for students who are motivated for acceleration, students who need more support, as well as students who started late and wish to catch up. Students are permitted to change pathways.

Figure 3 shows the math pathways offered in Dublin Unified School District (DUSD).

Fig. 3. Dublin Unified School District math courses and sample pathways. (click image to enlarge)

Similar to IUSD, multiple math pathways are available to students. Options for acceleration begin in 6 ^th grade in this district, which enable the student to reach the Algebra 1 course in 8 ^th grade (bottom two pathways). These pathways enable the student to reach AP Calculus in high school with a pathway of additional acceleration being used to enable the student to reach the AP Calculus BC course in 12 ^th grade (bottom pathway). It is noteworthy that DUSD lists three versions of Calculus (Calculus, AP Calculus AB, AP Calculus BC). Some districts, for example Cupertino Union School District, have middle school pathways with Algebra 1 available as early as 7 ^th grade.

Finally, figure 4 shows the math pathways at Huntington Beach Union High School District (HBUHSD).

Fig. Huntington Beach Union High School District Math Course Pathways.

The pathway for reaching AP Calculus in high school in HBUHSD is a 5-year pathway, shown as the two rightmost pathways. There is an additional pathway with further acceleration used to enable a student to reach the AP Calculus BC course (far right). Also noteworthy, is the district offers a Data Science course. As can be seen, this course does not enable a student to cover the material required to reach Precalculus in high school, yet alone Calculus, and is therefore not a pathway a student would take if they wish for a STEM college major, even a major in Data Science. This is a concern because the SFR draft framework guides that some students “may be better served by a data science course as a culminating high school mathematical science experience” (Ch. 5 Line 1595). This very concern is very well articulated in an Open Letter entitled Data Science and the High School Math Curriculum.

Figure 5 shows the recommended course offerings in the 2013 California Common Core State Standards for Mathematics (CCSSM). As can be seen, even there, guidance is given to offer Algebra 1 or Mathematics 1 to students as early as 7^th grade.

Fig. 5. 2013 California Common Core State Standards for Mathematics math courses by grade level. P 58.

This guidance is consistent with the California Mathematics Placement Act of 2015 (EDC § 51224.7 , EDC § 51228.2) which states:

“Placement in appropriate mathematics courses is critically important for a pupil during his or her middle and high school years. A pupil’s 9th grade math course placement is a crucial crossroads for his or her future educational success. Misplacement in the sequence of mathematics courses creates a number of barriers and results in pupils being less competitive for college admissions, including admissions at the California State University and University of California. The most egregious examples of mathematics misplacement occur with successful pupils…”

“a school district maintaining any of grades 9 to 12, inclusive, shall not assign a pupil enrolled in any of grades 9 to 12, inclusive, in a school in the school district to a course that the pupil has previously completed and received a grade…”

It is clear that this Act envisions that one student’s proper math placement in 9th grade may differ from another student’s, depending upon the individual student’s level of math mastery and preparation, and that a district may not force students to repeat a properly accredited course that they have already completed to the satisfaction of a post-secondary institution. Students who have satisfactorily completed properly accredited math courses cannot be forced to retake such a course, meaning these students may move ahead of their grade level peers into a separate math class, rather than remain in the same common math classes with their peers.

The SFR Draft Framework’s Guidance for Math Courses

In contrast to the math pathways offered in these districts, recommended in the CCSSM, and dictated by the California Mathematics Placement Act, figure 6 shows the “possible pathways for high-school coursework” (Ch. 8 Line 854) guided for in the SFR draft framework.

Fig. 6. 2022 SFR draft framework “possible pathways for high-school coursework”. (Ch. 8 Line 870)

First, it is important to understand the pathways Integrated Math 1-3 and Algebra/Geometry/Algebra 2 cover the same material, just in different order, so districts offer only one of these pathways. This leaves only two pathways. With the MIC 3 course, there is little detail given about what content is actually covered in the course. Some descriptions given about the course are:

“MIC 3 has many student investigations driven by data. Students generate questions, design data collection, search for available existing data, analyze data, and represent data and results of analysis. They use powerful technological tools to help with all of these tasks. Much of the content in all Content Connections is situated in stories told through data.” (Appx. A, Line 760)

“To respond to varying student interests, distinct MIC 3 courses can be designed by emphasizing data-driven investigations…” (Appx. A, Line 770)

“In MIC 3’s data-driven investigations, students begin by searching for, gathering, or examining data about their authentic questions…” (Appx. A, Line 901)

These are consistent with the SFR draft framework’s view of reducing math to being about “telling stories with data” (Appx. A, Line 783):

“Content Connection 1: Telling Stories with Data” (Appx. A, Line 783)

This is in stark contrast to the math covered in Algebra 2, which is as described in the SFR framework:

“Building on their work with linear, quadratic, and exponential functions, students in Algebra II extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions 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…” (Appx. A, Line 273)

Given that there is little information about what content is in the MIC 3 course, let alone what content is removed from the Integrated Math 3 course it is based on to make room for these data analysis projects mentioned above, there is certainly not enough information given in the SFR draft framework to determine with confidence that the MIC 3 course suitably covers the material from Integrated Math 3 or Algebra 2 for a student to continue onto and be prepared for Precalculus, the next class in a STEM student’s pathway. The significant concern with this very point is the opening line in the Open Letter entitled Data Science and the High School Math Curriculumsigned by 372 and counting “academic staff at 4-year colleges and universities in California”:

“We write to emphasize that for students to be prepared for STEM and other quantitative majors in 4-year colleges, including data science, learning the Algebra II curriculum… in high school is essential”

Therefore, taking the MIC 3 pathway could be catastrophic for a student who wishes to pursue a STEM college major and is definitely not a path a STEM student would take. This leaves only one pathway for a STEM student, whichever remains of the Integrated Math 1-3 or Algebra/Geometry/Algebra 2 pathways. In fact, perhaps more disconcerting, the draft framework itself guides “high schools will typically offer one of the first-two-years pathways (Integrated, MIC, or Traditional)” (CH. 8 Line 856), implying districts will choose between providing a STEM math pathway and the MIC math pathway for their students. The draft framework states that its math pathway schematic figure 6 is “reflecting a common ninth- and tenth-grade experience” (ch. 8 line 854), and later states “this framework recommends that all students take the same, rich mathematics courses in kindergarten through grade eight.” (Ch. 10 line 1147). This means that the draft framework is guiding for a one-size-fits-all approach to math classes until 11^th grade. Until 11^th grade, students who need more support are grouped with students who are prepared and motivated for more advancement, along with all students in between. Another term for this is “heterogeneous” classrooms. Since the draft framework describes the figure as “reflecting a common ninth- and tenth-grade experience”, it is guiding for all students to take Algebra 1/Integrated Math 1 in 9^th grade. It is noteworthy that this guidance is contrary to what is directed in Appendix A to the California Common Core State Standards for Mathematics (CCSSM-A)^3:

“…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.”³

The MIC 3, Integrated Math 3, and Algebra 2 courses are defined as third-year courses in the SFR draft framework from statements: “If students take another third-year course (besides MIC 3, Integrated Math III, or Algebra 2)” (Ch. 8 Line 917), and “In addition to offering Integrated III, Algebra II or MIC 3, districts have the flexibility to offer other third-year courses” (Ch. 8 Line 877-879). Therefore, in order to reach Calculus in 12^th grade, students will have to combine the content of Algebra 2 and Precalculus into one year. Doing this is also contrary to the direction of the CCSSM-A:

“…mathematical concepts are likely to be omitted when trying to squeeze two years of material into one. This is to be avoided, as the standards have been carefully developed to define clear learning progressions through the major mathematical domains.”

It is especially noteworthy that there is no mention of AP Calculus in the math course offerings in the figure shown in the SFR draft framework. In fact, the Calculus course guided for by the framework shown in the figure is “Calculus plus Trigonometry”. Trigonometry is a topic that comes before Calculus and is typically associated with Algebra 2, as shown for example in the DUSD math course offerings chart in figure 4. This indicates the Calculus course the SFR draft framework envisions for students is a less rigorous version of Calculus than AP Calculus AB or BC. In summary, the draft framework is guiding for all students to be taking Algebra 1/Integrated Math 1 in 9^th grade and thus not arriving at AP Calculus in 12^th grade. It is guiding for universal deceleration in math. The guidance of the SFR draft framework is contrary to what districts currently provide for their high achieving students, contrary to what is guided for in the CCSSM, and in opposition to the directives of the California Mathematics Placement Act of 2015. Since the SFR draft framework’s guidance does not enable a student to reach AP Calculus in high school, the SFR draft framework fails each and every aspiring CA STEM student in the state.

Unsubstantiated Guidances in SRF Draft Framework

The SFR draft framework presents guidance in numerous instances which are self-contradictory, misrepresented, confused, or not substantiated. Problematic guidance related to math pathways are:

1) SFR Draft Framework Guidance on Acceleration

The SFR draft framework presents self-contradictory guidance on acceleration. Sentences claiming acceleration pathways “may” exist are sprinkled in amidst a sea of discussion guiding against them. An example of one of these sentences comes in the paragraph discussing high school math course pathways (Ch. 8 line 852). This paragraph begins with the phrase describing high school pathways “reflecting a common ninth- and tenth-grade experience”, clearly guiding for placing all kids in the same class, but later states “advanced students may complete that pathway in an accelerated fashion to access additional advanced mathematics courses” appearing to guide for the presence of multiple pathways. However, it then quickly qualifies this with “or, as described in chapter 9, they may be offered additional or supplemental challenges within or beyond the courses they take in their pathway” which is in line with what the framework guides for, that is that advanced coursework should occur within the confines of the one-size-fits-all classroom model. Then, to make sure it is clear the SFR draft framework is not guiding for accelerated pathways, it shows a math pathway schematic with no accelerated pathways present (Fig. 6) eliminating the possibility of acceleration. First of all, it must be made perfectly clear, that these statements are self-contradictory. They are incompatible. Guidance placing all students in the same class, that is a “common… experience” with just one pathway available to students, contradicts guidance for accelerated pathways, which involves multiple pathways available to students. To understand the origin of this conflict, it is necessary to be aware that the previous draft of this framework made public in April 2021 guided for denying access to advanced classes. This was met with widespread condemnation in both news coverage, and most notably two Open Letters (Open Letter on K-12 Mathematics, Replace the Proposed New California Math Curriculum Framework) signed by thousands of STEM professionals explaining to the writers of the framework in no uncertain terms that doing this leaves students unprepared and unqualified for STEM college majors and careers. To assuage these criticisms, apparently the authors believe that inserting sentences stating multiple pathways are possible, even though they are obviously contrary to what the SFR draft framework is guiding, will distract STEM professionals from condemning the SFR draft framework. It appears as if the authors are trying to stay true to the original intent of denying access to advanced classes, while at the same time hoping to circumvent criticism. It must be understood, however, that guiding for both a common pathway AND accelerated pathways cannot exist simultaneously. They are mutually exclusive.

The draft framework guides that the presence of multiple pathways for students is exclusionary because they “exclude most students from studying the higher-level courses” (Ch. 9 Line 90), and it “filters most students out of high-level mathematics” (Ch. 9 Line 94) and it “reflects a long history of inequality” (Ch. 9 Line 155). The SFR draft framework accuses districts that offer multiple pathways as drawing “upon the incorrect idea that some students cannot learn higher level mathematics”. However, as a solution to these issues, instead of guiding for ways of getting more students into these pathways, it guides to remove these pathways for all students, such that no students receive them. This disadvantages students in districts that adopt the framework in comparison to students in districts that do not and will still have these pathways. To be sincere about enabling students to reach STEM careers through enabling accelerated pathways, the framework would need to remove guidance for one pathway with a “common… experience”, and remove guidance for arranging math course structures “without blunt methods of tracking”( Ch. 9 Line 130) which is guiding for removing accelerated pathways. Then, written into the framework would need to be clear guidance describing the benefits of accelerated pathways. This is easy to do since there are so many studies on the topic of acceleration that there is a meta-analysis, or a study of studies, on the topic, with the results summarized here:

“A review of 380 studies revealed that almost all forms of acceleration result in growth in achievement.”

Finally, the framework would need to include a math course pathway chart showing multiple pathways available to students, as all the other districts shown above clearly do, the CCSSM guides for, and the Math Placement Act directs. As the SFR draft framework stands, simply claiming they are allowing for accelerated pathways, while guiding to the contrary, is insufficient. The SFR draft framework guides against the presence of accelerated pathways.

2) SFR Draft Framework’s Justification for One-Size-Fits-All Approach to Math Classes

To justify its guidance for a one-size-fits-all classroom, also called a heterogeneous classroom, the draft framework first cites what occurs in high achieving foreign countries. The framework rightly points out that “the US now ranks about 32nd in the world in mathematics on the Program for International Student Assessment (PISA)” (Ch. 1 Line 177). Then it points out that “mathematics in the highest-achieving countries is typically taught in heterogenous classrooms prior to tenth grade” (Ch. 8 Line 346). However, the draft framework makes no mention of the specific nature of the curriculum in these countries, i.e., which specific content standards or material are taught in which specific grades. Studies analyzing the nature of education in Japan, the country which scored highest on the PISA test (Ch. 1 Line 190), such as “The Comparison of Japanese Mathematics Education and United States Mathematics Education” and “Mathematics and Academic Diversity in Japan” conclude:

“ Rather than slowing down the curriculum or directing struggling students to alternative programs of instruction (e.g., Title I services, special education, low track classrooms), students are encouraged to put in extra effort to keep up.”

“Japanese high school students who take all of the courses offered will know more mathematics than many U.S. students do when they graduate from college .”

“ U.S. eighth graders are being taught at a seventh grade level compared to many of their international counterparts.”

From this, it is observed that the realities of what is happening in Japan are opposite to the guidance of the draft framework. One deduces that a Japanese student is working with accelerated curriculum relative to a US student for a given grade. Thus, if the SFR draft framework is proposing to implement what successful countries, such as Japan, are doing, they should be recommending acceleration for all.

A second study the draft framework quotes to guide for heterogeneous classes is from Burris et al where the SFR draft framework states:

“The researchers found that the students who learned in the heterogeneous classes took more advanced math, enjoyed math more, and passed the state Regents test in New York a year earlier than students in traditional tracks.” (Ch. 9 Line 295)

However, when analyzing what occurred in this study it is found that:

“Rockville Centre NY implemented a universal (detracked) > math acceleration intervention in grades 6-8, with all students completing Algebra 1 in grade 8> , with the express goal and metric that students would complete certain advanced math courses (Algebra 2, Pre Calculus, and Calculus) in high school.”

Again, this is opposite to the guidance of the draft framework. Placing all kids in the same class and giving them accelerated curriculum is completely different than placing all kids in the same class and giving them decelerated curriculum as the draft framework guides for. These are completely different classroom environments with completely different dynamics. The success of universal acceleration does not at all imply the success of the universal deceleration.

3) SFR Draft Framework’s Justification for Delaying the Taking of Algebra 1 Until 9^th Grade

The argument the draft framework uses to justify its guidance of delaying the taking of Algebra 1 until 9^th grade is that it claims taking Algebra 1 in 8^th grade “has led to many students missing the structured content of middle school mathematics” (Ch. 8 ), that students who take Algebra 1 in 8^th grade “may miss foundational learning” (Ch. 8), and implies that there is a “rushing through courses with a superficial understanding” (Ch. 8 Line 806) occurring as well as “rushing to Algebra in middle school” (Ch. 8 Line 818). Although it is perhaps a statement of the obvious that students who are not prepared to take Algebra 1 in 8^th grade should not be forced to, the draft framework presents no data showing that the above comments are the case for students who are prepared and motivated to take Algebra 1 in 8^th grade and earlier. It is important to realize that the corollary to this is that forcing kids to spend more time to studying the mathematics before Algebra 1 means that there is less time to go into depth in the advanced mathematics courses that occur after Algebra 1. This implies less depth of coverage in those courses, courses that a student is taking en route to and including AP Calculus. This is perhaps a more salient point. Roughly 1 out of 4 students in the U.S. take Algebra in 8^th grade so the SFR draft framework’s guidance to delay the taking of Algebra I until 9^th grade potentially affects a tremendous number of students. The reality is that for students who are prepared and motivated to take an Algebra 1 course in 8^th grade and earlier, the phrase best suited to describe their mathematics education covering material before Algebra 1 is that they have

“had an elegant sufficiency; any more would be a superfluity.” ¹

4) SFR Draft Framework’s Nonsolution for Getting Students to Calculus After Being Delayed

To enable students to reach Calculus in 12^th grade who have had Algebra delayed until 9^th grade the framework suggests removing material from classes, that is “eliminating redundancies in the content of current courses” (Ch. 8 line 788). It is noteworthy that this is opposite to what is done in IUSD where a separate accelerated pathway is created for those motivated students (Fig. 2). Furthermore, this is contrary to what is found, as districts that enable their students to reach AP Calculus BC typically use pathways of additional acceleration, that is more material added with more depth of coverage, over and above the 5-year pathway starting with Algebra 1 in 8^th grade to ensure their students reach AP Calculus BC. Examples of these districts with these pathways are shown in figures 2,3, and 4. It is also noteworthy, that this suggestion implies that districts who have been successfully enabling their students to arrive at AP Calculus BC for many decades have not carefully thought through their math pathways, a point that would be rather surprising. More specifically, the SFR draft framework states:

“One possibility could involve reducing the repetition of content in high school, so that students do not need four courses before Calculus. Algebra 2 repeats a significant amount of the content of Algebra 1 and Pre-calculus repeats content from Algebra 2. While recognizing that some repetition of content has value, further analysis should be conducted to evaluate how high school course pathways may be redesigned to create a more streamlined three-year pathway to pre-calculus / calculus or statistics or data science, allowing students to take three years of middle school foundations and still reach advanced mathematics courses.” (Ch. 8 Line 808)

After making this statement, the framework makes no attempt at describing in detail specifically what material might be removed, and what specifically a course structure could be that they are referring to. What more appropriate place is there to describe in detail a math curriculum that achieves this stated goal, than a 1000-page math curriculum framework. Thus, this suggestion remains unsubstantiated.

To demonstrate what the SFR draft framework focuses on instead of going into detail on this topic, it spends an 86 page chapter discussing a data science class, a class which the Open Letter “Data Science and the High School Math Curriculum“ warns: “students who take a data science course as an alternative to Algebra II in high school will be substantially underprepared for any STEM major in colleges”. The SFR draft framework also devotes more space to a financial Algebra course (Ch. 8 Lines 879-896) than to discussing this point of how to get students to Calculus. In the end this lack of focus on Calculus is not a surprise as one of the SFR draft framework authors stated in an August 6, 2020 meeting of the Curriculum Framework and Evaluation Criteria Committee (CFCC):

“If you can point to anything in the document that you read as still a push to calculus it would be great to know because that is absolutely not what we intended in writing it.” (> Link: at 1:54:43> )

However, if the SFR draft framework’s only ambition is for students to arrive at the “Calculus plus Trigonometry” course they envision (Ch. 8 Line 854), and not AP Calculus BC, then perhaps less material would suffice. More realistically, this is what is at play here. Simply put, getting students to AP Calculus in high school is not a priority of the SFR draft framework.

5) SFR Draft Framework’s Guidance on Cloudy Math Pathways

Instead of showing clearly defined pathways, the framework places 12^th grade classes in a cloud (Figure 6). This does not hide the fact that delaying Algebra 1 until 9^th grade means that there is no path to AP Calculus in the SFR draft framework’s course pathways. The danger of leaving math pathways ambiguous as the SFR draft framework does is demonstrated in the draft framework itself. The draft framework describes six sample pathways for specific cases that it envisions for students (Ch. 8 Lines 943 – 987). Two are non-STEM pathways, Political Science and Construction. In a third, the student plans to work in a “fabrication shop” directly after high school, although it is unclear as to what type of position would be available to them without a college degree. A fourth desires an “accounting degree”, but took Financial Algebra after Algebra 2, skipping Precalculus leaving them unprepared for Calculus in college, a significant concern as an Economics degree may require Calculus. A fifth is interested in a STEM career in “robotics engineering” and was fortunate enough to be able to take Integrated Math 1 in 7^th grade and AP Calculus in high school. This student is clearly not in a district where “all students take the same, rich mathematics courses in kindergarten through grade eight” as the SFR draft framework guides for, as the SFR draft framework does not even hint at the idea that this double acceleration, that is of being able to take classes two years ahead of their peers (something required if a student is to take “Integrated Math 1 in 7^th grade”), would be an option let alone desirable. This student is clearly not in a district implementing the vision of the SFR draft framework shown in figure 6 where there is no acceleration or AP Calculus. In other words, this student must be in a district that does not adopt the draft framework, but rather in a district that arranges its math pathways along the lines of Figures 2,3, and 4. The sixth and final pathway is described here:

“Inez likes digital photography, so was planning on majoring in graphic design at a university, a degree not requiring calculus. As Inez is completing her third-year course in Data Science, however, she found herself enjoying using the software and various applications to work with the data sets and create captivating data displays. This, combined with her interest in creating mods for her favorite video game, has her now thinking about pursuing computer science coursework at a university. So, in her fourth year, she enrolls in her school’s precalculus class, along with a half-semester support class her school offers for students whose interest in mathematics grows late in their high school time. She enters her university well-prepared to take freshman calculus and the programming classes she hopes to pursue alongside additional work in data science.” (CH. 8 Line 971).

A Data Science course does not cover the content of an Algebra 2 course. Even assuming that this “half-semester support class her school offers” does exist, even though there is no guidance for or description of this course in this 1000-page SFR draft math curriculum framework, a Precalculus course assumes the student has completed Algebra 2 in its entirety. Therefore, these courses, a “half-semester support class” which would need to cover the portion of Algebra 2 Inez missed and Precalculus, cannot be taken in parallel. Inez would be lost entirely in the Precalculus course and as soon as Inez chose the Data Science course, her future was limited. She would not be prepared for “pursuing computer science coursework at a university” as she wishes. It needs to be made abundantly clear to both students and parents that once they choose the Data Science course, they are essentially giving up a STEM college major. This is another point made in the Open Letter “Data Science and the High School Math Curriculum”. The math pathways in this draft framework are so cloudy (pun intended), that even the makers of the framework are confused by them. This shows the need to make math pathways clear to the student as shown in the math pathway schematics of other districts (Fig. 2,3, and 4).

Conclusion

The SFR draft framework’s guidance removes the pathway to AP Calculus and thus fails STEM students. Since a framework is guidance and not a mandate, there will be many districts which realize the importance of AP Calculus and will maintain the pathway to it, while districts that adopt the framework will remove the pathway per its guidance. This systemically disadvantages students in districts that adopt the framework, meaning there will be a system in place that creates obstacles for students to reach AP Calculus in districts who follow the SFR draft framework guidance. These obstacles are not in place in districts that do not adopt the framework. As has been pointed out, “because the framework is not legally binding, more affluent school districts will continue to offer Algebra I in middle school”. Thus, the SFR draft framework creates inequities and exacerbates disparities. The draft framework states that “equity influences all aspects of this document” (Ch. 1 Line 239) --- However in its pursuit of equity, the SFR draft framework has lost sight of the consequences of taking away math pathways for students, and fails the very group it purports to help. The SFR draft framework fails STEM students. Furthermore, the SFR draft framework presents self-contradictory, misleading, misrepresented, and confused guidance. Therefore, not only does SFR draft framework fail STEM students, but it fails as a framework. This SFR draft framework should be rejected.

Final Thought

Parents can ask themselves which district in figure 7 below would they wish for their child? The one on the left not following the SFR draft framework guidance which has clear pathways and options for all students (repeating Fig. 2), or the one on the right that the SFR draft framework guides for with a “common ninth- and tenth-grade experience” (Ch. 8 line 854) and Math 1/Algebra 1 delayed until 9^th grade (repeating Fig. 6)? If parents could choose, which would they choose? If parents could vote on this, which would they vote for?

Fig. 7 Comparing a district not using the SFR draft framework’s guidance (left) to the district the SFR draft framework guides for (right). (Ch. 8 Line 870). (Click on figure to expand view).

References

1. Fred Chappell. I Am One of You Forever. Louisiana State University Press p. 51