The Benefits of Taking Calculus in High School

To prepare students to take Advanced Placement (AP) Calculus in high school, public school districts typically offer accelerated pathways to advanced mathematics beginning in 7th grade. With the 2022 draft California math framework for public schools Second Field Review (SFR) guiding to remove accelerated pathways, and therefore remove these pathways to AP Calculus or at least place significant obstacles in the way of a student to reach AP Calculus, it becomes important to remind ourselves both why taking AP Calculus in high school is beneficial, as well as analyze the data the framework is using to justify removing it.

Taking Calculus in high school is beneficial for many reasons. Some are:

1. Calculus is foundational for all Science, Technology, Engineering, and Math (STEM) majors and is therefore a required introductory course sequence in all STEM majors. Even majors such as Economics, Psychology, and Data Science require a Calculus course sequence. Taking Calculus in high school benefits students in these majors for the reasons below.

2. Performing well on the AP Calculus test enables a student to pass out of the initial required Calculus college sequence and move onto the upper division technical electives of the major, which in parallel enables many research opportunities.

3. College level Calculus can be a significant step up in rigor in comparison to a high school course. Students who “re-take” Calculus in college are therefore not re-taking the same course, but taking a far more challenging course. It is found that students who took Calculus in high school measurably benefit from the exposure to Calculus they received by taking it in high school.

4. The majority of students who took Calculus in high school benefit from the experience.
   

1) Calculus is Foundational for All STEM Majors and is a Required Introductory Course Sequence

There are many reasons why taking Calculus in high school is very helpful and beneficial for the career of a student. The first reason for this is that the mathematical principles learned in Calculus are foundational for the more advanced courses encountered in Science, Technology, Engineering, and Math (STEM) college majors. It is important for parents and students to understand at the outset that the breadth of majors classified as STEM fields is very large (as shown below). In fact, many colleges have formally reclassified Economics majors as STEM majors. Since the mathematics learned in Calculus are foundational for STEM majors, one of the first set of course sequence requirements given a student in a STEM major is a Calculus sequence. To demonstrate this point with an example, Figure 1 shows screen shots describing course requirements for various majors at University of California at San Diego (UCSD), with some superimposed notation. In particular, what is seen here is that the course sequence Math 20A/ Math 20B/ Math 20C is required for all these STEM majors. UCSD defines the classes Math 20A and Math 20B as “Calculus for Science and Engineering”, while Math 20C is “Calculus and Analytic Geometry for Science and Engineering”.

Caption for Figure 1 (a-e). Lower division or Freshman year course requirements for various majors at UCSD. These are screenshots from the UCSD site describing their required courses (with some superimposed explanatory notation). Note that UCSD uses the quarter system. UCSD defines the classes Math 20A and Math 20B as “Calculus for Science and Engineering”, while Math 20C is “Calculus and Analytic Geometry for Science and Engineering”. (a) Bioengineering: Biotechnology. (b) Computer Science. (c) Chemistry. (d) Oceanic and Atmospheric Sciences. (e) Aerospace Engineering. (f) Data Science (Note” Math 31BH is defined as “Honors Multivariable Calculus” which is a course that “covers the topics in Math 20C, but also many more topics”).

These examples show that a widely diverse set of STEM majors commence with a Calculus sequence. To continue with this point, Table 1 shows all the UCSD majors that require starting with a Calculus course sequence. This table was generated by inspecting all the UCSD majors.

Caption Table 1. Listed above are all the UCSD majors which require starting with a Calculus course sequence. These 78 majors are broken down by the 16 fields in which they reside. These majors use the course sequence UCSD labels as Math 20A/ Math 20B/ Math 20C. UCSD defines Math 20A and Math 20B as “Calculus for Science and Engineering”, while Math 20C is “Calculus and Analytic Geometry for Science and Engineering”. The majors denoted with an (*) require Math 10A/ Math 10B/ Math 10C, which UCSD defines as “Calculus”, and the majors denoted with a (**) require Math 10A and Math 10B.

This list shows that Calculus is required for a widely diverse set of majors, not just mathematics, physics, and engineering as is sometimes thought. A total of 78 majors in 16 different fields all begin with a Calculus course requirement. It is seen to be required in fields ranging from Bioengineering to Chemistry, from Economics to Psychology, and even Data Science. Again, the reason for this course requirement is that an understanding of the mathematical concepts learned in Calculus is imperative for studying the concepts taught in the Upper Division courses that follow in the major. Although it is beyond the scope of this discussion to perform a similar analysis for all universities and colleges across the country, the Calculus course requirements at UCSD are representative of similar majors in all institutions. If this is in doubt, choose an institution of interest, and scan through the majors. As a final example, here is a statement related to Calculus in a discussion of admissions to Massachusetts Institute of Technology (MIT) showing that all students must take Calculus at MIT:

“All MIT students, regardless of intended major, must pass two semesters of calculus, plus two semesters of calculus-based physics. The substance and pace of these courses are both very demanding, and they culminate in long, challenging final exams that students must pass⁠ to proceed with their education.⁠ In other words, there is no path through MIT that does not rest on a rigorous foundation in mathematics, and we need to be sure our students are ready for that as soon as they arrive.”

2) Taking AP Calculus in High School Enables a Student to Pass Out of the College Calculus Sequence

Students who completed calculus in high school could possibly pass out of the initial college Calculus course requirement. This is accomplished by either by scoring high on the Advanced Placement (AP) Test or by taking a test at the Institution they will be attending demonstrating proficiency. Figure 2 shows screens shots with explanatory highlighting from a UC university, a private university, and a service academy describing how a student can pass out (or validate) some or all of this initial Calculus sequence from the high school AP Calculus class.

Caption Figure 2. Screenshots with highlighting from university or service academy websites showing how an AP Calculus high school courses can be used to pass out or validate the initial Calculus course sequence at the university or service academy. a) UCSD. b) Stanford University, Stanford, Cryptography. c) United States Naval Academy (USNA). The links where this information was obtained are specified.

The first example is for a UC university, where in Fig. 2a the prerequisites for the Calculus 20 series at UCSD are shown. The figure describes that if a student earned a 4 or better on the Calculus AB exam, or a 3 or better on the Calculus BC exam, the student can pass out of Math 20A and start with the Math 20B class. Also, if a student earned a 4 or better on the AP Calculus BC exam, they can pass out of Math 20A and Math 20B and start with Math 20C. The second example shows a private university, Stanford University, where similarly AP Calculus credit can be applied toward the Mathematics major as shown in Figure 2b, and the AP Calculus class can be used to pass out of the Stanford Calculus course sequence as shown for the Cryptography major. At Stanford, as well as other universities, a student will have to demonstrate competency of the AP Calculus material they learned in high school through a university administered test as well. Finally, the third example is for a service academy, where Figure 2c describes how a student can pass out or validate the initial Calculus sequence at the United States Naval Academy (USNA). For all these examples, note that the AP Calculus BC is a more advanced version of Calculus which is why performing well in the BC test enables passing out of more classes than the AB test. Three examples of using the AP Calculus class to pass out of a university’s or academy’s initial Calculus course sequence requirement are shown here, shown for a UC university, a private university, and a service academy. This policy is common in education as colleges have no interest in forcing a student to repeat a class they are already fluent in.

Being able to pass out of part or all of the initial Calculus sequence holds great benefits for the students. These cannot be underestimated. First, it enables the student to move onto and take more of the technical electives that are offered in the major. Taking more of these technical electives means that a student will be better prepared for a STEM career, more experienced for interviews, and have a better resume. Passing out of the initial Calculus sequence means that there will more likely be time available in the students’ senior year for research opportunities such as internships, collaborations with a professor, independent study projects, or an honors thesis or double major. All these opportunities build connections which can be vital for a STEM career. They also enable the student to have more impactful letters of recommendation. A student who does these things will be better prepared for a STEM career or graduate school. Highly competitive study abroad programs become more likely for a student as it will be easier to match courses at another school if most of the major’s requirements have been completed. It is also important to understand that the technical electives are why the student has chosen the major in the first place. These are the exciting and interesting part of the major. The more of these classes that the student is able to take, the more likely they will be engaged with the major. A final very important reason is cost. Passing out of this initial sequence means that it is more likely a student will graduate in a timely fashion. The high cost of college education is well known and having to take an extra year to graduate can be an added financial burden, a burden especially difficult for a student with less resources. In short, take Calculus in high school while it is free.

From above it can be seen how passing out of this initial Calculus sequence opens many doors for a student. An Open Letter on K-12 Mathematics opposing the draft framework signed by 1739 and counting STEM professionals shows awareness of these points in the following statement:

“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.

3) Taking Calculus in high school benefits Students even if they do not use it to pass out of Calculus

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 other 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 implies that even when a student decides to “retake” the Calculus class, they are in fact taking a class of significantly more rigor and therefore not “retaking” the same class. The experience they had with Calculus in high school is therefore helpful, and in fact measurable, as stated in the study by Sadler and Sonnert “The Path to College Calculus: The Impact of High School Mathematics Coursework”:

“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.”

Thus, taking Calculus in high school potentially could be the difference between passing and not passing the class, or the difference between receiving a C or a B grade, or a B or an A grade leading the authors to conclude:

“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 since it is studying the only 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.

In “The Role of Calculus...“ the study continues with statements:

“Even students with relatively weak preparation in mathematics appear to benefit from taking a calculus course in high school. While they may not learn all that much calculus (or earn a high grade), the course can bolster their understanding of concepts and build skills that will be used later in college calculus.”

“Those who do well on an AP Calculus exam are significantly more likely to return for a second year, take more mathematics courses, and pursue a mathematically intensive program.”

“This means that access to calculus is important.”

Statements in the California Common Core State Standards for Mathematics Appendix A (CCSSM-A) directing math education for students K-12 shows that they are clearly aware of these benefits:

“...students taking high school mathematics in the eighth grade are expected to take Precalculus in their junior years and then Calculus in their senior years. This is a good and worthy goal...”.

“there is also powerful research showing that among academic factors, the strongest predictor of whether a student will earn a bachelor’s degree is the highest level of mathematics taken in high school.”

Knowing how beneficial taking Calculus in high school is for a student, and knowing what is stated in CCSSM-A, it is therefore baffling that the SFR draft Framework would guide toward removing the pathway to AP Calculus. This philosophy of the draft framework is made abundantly clear by a statement by one of the draft framework’s authors 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)

Also, another member of the CFCC, perhaps unaware of how ubiquitous Calculus is today in the pursuit of a STEM major, as shown for example in Table 1 and discussed above, stated of this traditional pathway to Calculus:

“I just keep wondering about this traditional pathway, I know its something we have to consider per the guidelines, but I keep asking myself why are we looking at an antiquated pathway when that pathway was developed for the needs in the 1800’s maybe early 20th century.” (at 1:53:44)

It is for all these reasons that STEM professionals, people who are especially aware of the role Calculus currently plays in a STEM career, point out the significant problematic issues of the SFR draft framework in two clearly articulated Open Letters entitled Open Letter on K-12 Mathematics and Data Science and the High School Math Curriculum.

4) Taking Calculus in High School Benefits the Majority of Students Who Take it

A heavily referenced data set, both in the study discussed above (The Role of Calculus in the Transition from High School to College Mathematics) as well as in the SFR draft framework (Ch. 8 Line 1039), analyzes “the breakdown of the first college mathematics class for the 800,000 students who took calculus in high school”. These students are broken down into five groups which are discussed below.

Group 1: “About 150,000 will receive credit for calculus when they get to college and use that credit to enroll in Calculus II or higher”. As discussed above, this is potentially a huge benefit for these students. It can be career enabling. Therefore, this is success of public school education and the goal should be to enable more students to be able to take advantage of this. The study goes on to comment that this number might be as high as 200,000. Therefore, at least 1 in 5 students who take calculus in high school take advantage of this, and this number might be as high as 1 in 4 students.

Group 2. “About 150,000 retake Calculus I and earn an A or B.” The important point to understand here is that as discussed earlier, a college Calculus class can be a significant step up in rigor in comparison to a high school Calculus class. Therefore, “retaking” means taking a class with a significantly higher level of rigor and not just repeating the same material studied in high school. Receiving a grade of an A or a B in a college level Calculus course is an accomplishment. It enables the student to continue the path to a STEM career and is therefore a beneficial for the student. As discussed above, a student’s college grade was measurably improved by having taken Calculus in high school such that for these students the difference between receiving an A versus a B, or a B versus a C, could have been the experience gained by taking Calculus in high school. The bottom line is that if there is a pathway towards receiving an A or a B in a college level Calculus course, and this pathway includes being introduced to Calculus in high school, then take advantage of this pathway. For this group of students it can be readily argued that they profited significantly from their high school Calculus experience.

Group 3. “About 150,000 students start with a non-mainstream calculus course such as Business Calculus or Statistics, or take no mathematics when they get to college.” There is nothing wrong with this. There are many non-STEM careers for students that are obviously very exciting pathways. Not every student should be forced into a STEM major. It can be readily argued that that a student’s high school Calculus experience will help them in whatever career they end up pursuing and these students benefited from their high school Calculus classes. Furthermore, later in life their interests may change and they may wish to return to a STEM career. By having taken Calculus in high school, this option will be more readily available to them.

Group 4. “100,000 students retake Calculus I and receive a grade of C or lower.” There is some cause for concern with this group. There is certainly some indication that students in this group need or needed more support. However, it is important to again bear in mind that a C is still passing, which means that the student can continue with the STEM major. Again, since a student’s high school Calculus experience measurably improves the grade of the student, it is possible that for many students in this group, the high school Calculus experience was the difference between them passing and not. Therefore, the high school experience was beneficial for them. The student’s first semester at college is usually the first time a student is attempting to adapt to life away from home. Therefore, there may be quite a few other significant issues the student is grappling with at this time. If a student’s high school Calculus experience enabled them to get through this trying time, then it was beneficial. Therefore, it can readily be argued that for many of the students in this group, being introduced to Calculus in high school was beneficial as well.

Group 5. “Around 250,000 will need or choose to take precalculus, college algebra, or even remedial mathematics as their first college course.” There is certainly cause for alarm with this group. However, the key here is the difference between “need” and “choose”. There are all kinds of reasons a student may choose to take an easier class in their first semester at college if a major does not require otherwise. There could be reasons like lightening the work-load so they can focus on something else, something as simple as sports for example, or perhaps they are trying to boost their GPA. There is certainly some fraction of this group that needed more support that the system did not provide. What fraction of this group fall into this category is impossible to tell.

In conclusion, from the discussion above, it can be readily argued that a large fraction of these students were very well served by their high school pathway to Calculus, and having Calculus available to them in high school was very beneficial. To reiterate, it is concluded in the same study discussing this data that “this means that access to calculus is important”. Yet, from this same data set, the SFR draft framework comes to a puzzling conclusion guiding to remove the path to Calculus for everyone.

Conclusion

In summary, completing Calculus in high school is very beneficial. Students who are prepared and motivated to do so must be afforded this opportunity. The goal should remain to figure out how to get more students, and especially students from Socio-Economically Disadvantaged areas, into the Calculus pathway so they are not limited later. The SFR draft framework’s guidance to remove this pathway for all students, will only severely disadvantage students in districts that adopt the framework.