ANALYSIS AND CRITIQUE OF CALIFORNIA MATH FRAMEWORKS REVISIONS (CMF)

See also blog post and open letter.

The California Department of Education recently published proposed revisions to its framework for mathematics K-12 education. These revisions can be found here (see also copies here).  

While the proposals are broad and encompassing, and have some positive aspects as well, below we provide a critical analysis of the proposals for grades 8-12, specifically focusing on the following aspects of these revisions:

We believe these recommendations are well-intentioned but misguided. While below we focus on the direct negative consequences of these revisions, we remark that they will likely have indirect negative consequences. Specifically, as has already happened in San Francisco, more affluent students will be able to still take Algebra I in middle school, by either attending a private middle school or taking private afterschool or summer courses. Also, because the framework is not legally binding, more affluent school districts will continue to offer Algebra I in middle school, exacerbating the gaps between different districts.

In short:

  1. Despite state guidelines to the contrary, the new revisions are clearly biased toward pushing students to the data science path. This is evidenced both by the text as well as by recorded assertions of authors and committee members during public meetings.
  2. The proposed data science path is woefully inadequate in terms of mathematical preparation for college STEM degrees, even for careers in data science itself!
  3. The revisions do not provide students who still want solid mathematical foundations with a viable path towards taking AP calculus. The proposed path requires either compressing two years into one, using an out-of-school course such as a private summer or after-school program, or skipping a prerequisite class.
  4. The evidence on which these drastic changes are based upon is spotty at best, and often literature is invoked selectively or in ways that misrepresent the content of the citation.

BACKGROUND

The traditional pathway in US mathematics education in grades 8-12 is the following:

  1. Algebra I (taken mostly either in 8th grade or 9th grade, with 24% of students nationally taking it before 9th grade). It has been a longstanding goal of the US Department of Education and many organizations to increase the number of students taking Algebra I in 8th grade.
  2. Geometry.
  3. Algebra II.
  4. Precalculus.
  5. AP Calculus.

Most students don’t take all of these courses, but most schools in the U.S. offer them (though not all schools, and those that don’t tend to disproportionately serve lower income or minority students). Research has shown that taking Algebra I before high school (8th grade or earlier) can have lasting positive effects and that students who take trigonometry, precalculus, or calculus in high school are more likely to enter STEM fields in college. Indeed, extending the reach of Algebra I in 8th grade is precisely the focus of Bob Moses’s Algebra Project.

Integrated Math (IM) is an existing reorganization of this pathway. It consists of three courses Math I, Math II, Math III which together cover Algebra I, Geometry, and Algebra II (with trigonometry) in an integrated way. Math III is fairly similar to Algebra II.

The  revisions encourage postponing Algebra I to 9th grade for everyone. They also propose an alternative “data science” pathway for grades 9-12, which is called  “Mathematics: Investigating and Connecting (MIC)”. It has the following form (see Chapter 8):

  1. MIC 1
  2. MIC 2
  3. Either MIC 3: Modelling with functions or MIC 4: Data Science or both (one after the other)

The MIC1 and 2 courses are claimed to cover the content of Math I and Math II, but they also contain significant additional topics on “data science” (indeed this is the point). A lack of sufficient details makes it unclear what topics are removed from Math I and Math II to make room for those.  MIC 3 and MIC 4 are optional courses and either one can be taken in 11th grade.  The CMF  (Chapter 8, page 32) states that with either of these 11th grade options, students in the MIC pathway have “the full-range of 12th grade options” open, including calculus. But it is wishful thinking: these do not cover even all of the Algebra II material (e.g., logarithms), let alone precalculus. This puzzling statement is claimed to be backed up by research, but it is not (see Section 4).

1) BIAS TOWARDS DATA SCIENCE

Although the CMF is supposed to be agnostic about the different pathways, the text clearly pushes the “data science” pathway. It specifically associates “data science” with equity, stating:

“The data science field provides opportunities for equitable practice, with multiple opportunities for students to pursue answers to wonderings and to accept the reality that all students can excel in data science fields”  and that the non-traditional pathways are “focused on the use of inclusive teaching practices … allow more equitable access to authentic mathematics for all students, and necessitate a view that mathematics is a beautiful and connected subject, both internally and to the greater world around it.”  (Chapter 8, page 19)

To put it mildly, there is no basis for the assertion that “data science” is inherently more equitable than algebra or calculus, and many documented uses of  “data science” amplify inequity. Data science is also not the only field of math that is connected to people’s lives - good math teaching has always included real-world motivation, examples and applications. A proper preparation for data science is not easier or more accessible than calculus and algebra (see also Section 2 below).

The bias towards the MIC pathway was even clearer in public discussions of the revision. In a meeting of the Curriculum Framework and Evaluation Criteria Committee (CFCC), a co-author of the framework said  that the effort to introduce vibrancy and excitement into the curriculum was only for the data science pathway “which I’m sure we all appreciate is the mathematics students should be working on. I worry about making the traditional pathway just as vibrant, and interesting, and illustrated because it would give the message that both of these are equally important for kids.”

Another CFCC member called the traditional pathway and calculus “an antiquated pathway”, and remarked that “we may not be able to change it this round [but when using] 21st-century teaching and learning, that traditional pathway does not lend itself to it whatsoever.” These comments display ignorance of 21st century science. The recent breathtaking developments in artificial intelligence are built on the foundations of linear algebra and multivariate calculus. To take part in these, students need these so-called “antiquated” foundations taught in Algebra II and AP calculus.  

Soon thereafter, a staff and member of the State Board of Education reminded the committee members (see this video for around 2 minutes, and this video for around 3.5 minutes) that state guidelines prohibit any display of a preference among the pathways.

We do think data science is an important discipline worth studying, and believe that there are appropriate and beneficial ways to incorporate data fluency in the K-12 curriculum. But these should be in addition to basic mathematical foundations, and not replace them.

2) THE DATA SCIENCE TRACK PROVIDES INADEQUATE MATH PREPARATION

The CMF refers to an 11th grade data science course currently offered in some districts as an example for a potential MIC 3 course. The curriculum can be found here. It is a good course for its goals, but does not offer the material covered in Algebra II (nor is it intended to), and it is not designed to prepare students for progressing to precalculus and calculus. Other “data science” courses and lessons such as this one, which (Chapter 5, page 10)  is also one of the primary resources informing data science as proposed in the CMF,  cover even less mathematical content.  Despite that, the course page states that it  “can lead to a pathway in calculus, statistics, …”.  We find this statement to be misleading, because this is not the case when taken as suggested  “as an alternative to Algebra 2”, which is a prerequisite to AP statistics (and certainly AP calculus).

The framework also discusses data science as follows:

“In a high-school data-science class students can learn to clean data sets, an important part of the work of a data scientist. High school students can also learn to download and upload data, and develop the more sophisticated ‘data moves’ that are important to learn if students are tackling real data sets.”

While data cleaning, downloading, and uploading skills are useful to have (and can be taught even below the high school level), these and similar shallow skills are no replacement for the mathematical foundations required for students to pursue STEM in college. This is even true for students who will want to become data scientists!

There is no universally agreed upon definition of “data science”, but the academic fields most related to it are Statistics and Computer Science.  The American Statistical Association’s curriculum guidelines for undergraduate programs in statistical science states that single-variable calculus and linear algebra should be required for majors, while multivariate calculus is recommended. The ABET accreditation criteria for computer science programs requires that they include “at least 15 semester credit hours (or equivalent) that must include discrete mathematics and must have mathematical rigor at least equivalent to introductory calculus. The additional mathematics might include course work in areas such as calculus, linear algebra, numerical methods, probability, statistics, or number theory.” Indeed, the most data-intensive aspects of computer science such as Machine Learning are also those that use the concepts from (multivariable) calculus and linear algebra the most.

UC Santa Barbara offers a “statistics and data science major”, which lists the following as recommended preparation: two years of algebra, courses in plane geometry and trigonometry. Notice that they do not recommend courses on data science or statistics. Indeed, once students have the foundational mathematical knowledge, they can learn statistics and data science in college.

A high school data science course will not get credit in Berkeley’s data science major, but 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.

Hence, even students that want to pursue a career in data science (let alone those who are interested in other aspects of STEM) are better-served by the traditional or integrated pathways than the proposed “data science” pathway.

3) OBSTACLES TO TAKING AP CALCULUS UNDER THESE REVISIONS

Taking AP calculus is valuable preparation for STEM majors. Regardless of admissions requirements, students who have learned the material of AP calculus are at an advantage in STEM programs, since they can often place out and enjoy a lighter workload and/or increased opportunities. Indeed, this is why the National Society of Black Engineers (NSBE) set as a goal to double the number of African Americans taking AP calculus.

By discouraging the 8th grade Algebra I option, the new revisions make it very difficult for students to be able to reach calculus in 12th grade.The CMF authors say that  “Many students, parents, and teachers encourage acceleration beginning in grade eight (or sooner) because of mistaken beliefs that Calculus is an important high school goal.” and so did not design the revision with a goal of responsibly enabling students to reach calculus (not surprising, since a framework co-author said on video, “we know that the current pathways, particularly the push to calculus, is deeply inequitable”).

Indeed, the way to reach calculus within the CMF design is by either skipping pre-calculus (only feasible for very few advanced students) or taking a “compression course” that squeezes both Algebra II and pre-calculus into a single course. Trying to squeeze two challenging courses into one is doomed to fail for all but the most exceptionally talented students. Indeed, when the compression approach was tried in San Francisco, it led to a 13% decline in the number of students enrolled in AP calculus. It is unknown how many of the students that did enroll took advantage of the various pay-for workarounds.

Chart from mathpathsf.com. The only paths to reach AP calculus available to students that didn’t attend a private middle school or took a private online course are to “double up” (take concurrently two courses intended to take sequentially), or take a compression of two courses into one.

4) RESEARCH EVIDENCE

The CMF repeatedly cites research papers in a way that is inconsistent with the actual findings: (This is not a judgement about the papers themselves, only about the way they are used to claim the recommendations are evidence based.)

In order to support the viability of taking calculus in 12th grade following the MIC pathway or a traditional/integrated pathway with Algebra 1 taken in 9th grade, the CMF cites the following (Chapter 8, page 32):

Research in education is hard, perfect randomized controls are hard to come by. But, as the open letter says, it is irresponsible to make such radical (and detrimental) recommendations for the education of students in our largest state based on inconclusive or cherry-picked evidence.