July 1, 2024

Cathy Kessel

Back in the 1990s, there was a lot of enthusiasm about discrete mathematics.

Its applications were growing.

During the past 30 years, discrete mathematics has grown rapidly and has become a significant area of mathematics. Increasingly, discrete mathematics is the mathematics that is used by decision-makers in business and government; by workers in such fields as telecommunications and computing that depend upon information transmission; and by those in many rapidly changing professions involving health care, biology, chemistry, automated manufacturing, transportation, etc.[1]

It offered students a fresh start.

Discrete mathematics offers a new start for students. For the student who has been unsuccessful with mathematics, it offers the possibility for success. For the talented student who has lost interest in mathematics, it offers the possibility of challenge.[1]

It offered teachers a fresh start.

Discrete mathematics provides an opportunity to focus on how mathematics is taught, on giving teachers new ways of looking at mathematics and new ways of making it accessible to their students. From this perspective, teaching discrete mathematics in the schools is not an end in itself, but a tool for reforming mathematics education.[2]

Decades later, I see these themes in articles on data science.

Its applications are growing.

. . . jobs are blossoming in areas such as artificial intelligence, machine learning, information security, data analysis and software engineering.[3]

It offers students a fresh start.

[Data science] courses have been gaining in popularity, particularly with high school math teachers. They say the more relevant content offers a highly engaging entry point to STEM, especially for students who have been turned off by traditional math courses.[4]

It offers teachers a fresh start.

“This course transformed my teaching practices and transformed the lives of many students. Special education, English learners and calculus students worked side by side,” Joy Straub, who taught a data science course in Oceanside for six years, told the board. “Students who had a dislike for math suddenly were transformed into math lovers . . . skilled in statistical analysis, computer programming and critical thinking. I saw many students who never would have taken an AP math course take AP Statistics.”^{5}

**Why the enthusiasm?:**

**A conjecture**

Why are teachers, students, and others enthusiastic? Is this due to an intrinsic property common to novel subjects such as discrete mathematics and data science, but lacked by traditional subjects?

My conjecture is that the answer is no—but under current circumstances, it is much easier for teachers to learn to teach novel subjects in ways that engage students.

**Novel subjects in novel courses**

Because novel subjects are novel, that is, new to K–12, teachers often are not expected to know them. This rather obvious observation has the important consequence that professional development may include learning the subject as well as learning to teach the subject. For example, teachers in the Discrete Mathematics Leadership Program at Rutgers attended fifteen weeks of workshops over a period of four years. Half of each workshop day was devoted to learning mathematics by solving problems in sessions that used a variety of instructional formats, and half to using the problems in the classroom. Teachers were expected to attend follow-up sessions and to be part of an active email network.[5]

Similarly, Introduction to Data Science (IDS) teachers are required to engage in 60 hours of professional development. As Robert Gould (principal investigator for the project) describes it, this involves learning the subject: aspects of statistical thinking that are different from mathematical thinking, e.g., “learning to pose statistical questions and understanding, noting, and commenting on the presence of chance variation in data” and what is considered data. Learning to teach the subject involves “understanding students’ prior conceptions, and knowing such fundamentals as which student solutions are strong and which are not.” Learning to teach the subject occurs in a specific instructional context: teachers learn to see “the hidden structure of the curriculum (how small concepts are built so as to lead to bigger ones)” in the IDS materials.[6]

**Novel topics in traditional courses**

I have been puzzled by statements such as “most students’ math experiences remain focused on old-fashioned courses designed 50 years ago”[7] and “the math taught in U.S. schools hasn’t materially changed since Sputnik was sent into orbit in the late 1950s.”[8] This implies:

**Statement S:** The content of Algebra 1, Geometry, and Algebra 2 has not changed for 50 years.

To interpret Statement S, it’s helpful to know that the U.S. does not have a national curriculum. State standards for mathematics specify what students are expected to learn in each K–8 grade and in the course sequence Algebra 1-Geometry-Algebra 2. (Traditionally, accelerated students begin this course sequence in eighth grade rather than in the first year of high school.) State frameworks give instructional guidance. Many states adopted the Common Core State Standards (CCSS) after they were released in 2010, thus recent state standards and courses are quite similar to each other—at least as described in state documents.

According to many of those documents, Statement S is false. One reason is that the organization of traditional content is different. For example, in the old California standards (adopted in 2005), the concept of function is in Algebra 1 and solving systems of linear equations is in Algebra 2. In the new standards (adopted in 2013), both are in eighth grade.

However, in practice, differences between old and new courses may be small or nonexistent. This could be due to the belief expressed by some teachers that algebra is unchanging—“algebra is algebra.” Or, it could be due to traditions of “local control” that allow districts, schools, or teachers to choose what and how to teach their courses, ignoring or overlooking any differences between old and new standards. Does this sound implausible? Between 2015 and 2017, 63 randomly chosen eighth-grade mathematics teachers from 34 randomly chosen California districts were interviewed. Most were unable to give a specific and correct difference between old (2005) and new (2013) content standards.[9]

Another reason that Statement S is false is the inclusion of topics in data science and discrete mathematics. Unlike their counterparts in the 2005 standards, Algebra 1 and 2 contain substantial amounts of statistics, and Geometry has some discrete mathematics in the form of combinatorics used for probability. These might be considered instances of novel topics in traditional courses—but they have not been accompanied by the enthusiasm shown recently for data science and thirty years ago for discrete mathematics.

Why the lack of enthusiasm? It may be that these topics do not appear novel to teachers and engaging for students. There is evidence that statistics has appeared in some classrooms and on state assessments before 2012, that is, before many states expected the Common Core Standards to go into effect.[10] Reports from high school teachers in 61 districts in Michigan, Ohio, and Seattle, Washington indicated that almost all Algebra 1, Geometry, and Algebra 2 courses included some statistics and data although, on average, this topic received little instructional time.[11]

Or it may be that those teachers who consider the topics novel do not have the background or support to teach them. In 2013, the National Council of Teachers of Mathematics and the American Statistical Association released a joint statement warning that PK–12 teachers needed substantial professional development in statistics. This statement (reiterated in 2022) seems to have been often ignored. A decade later, teachers are unlikely to have much experience with statistics and data.[12] Thus, it may well be the case that the statistics *taught *in the traditional sequence hasn’t changed much in quantity or quality since the 1950s.

Change in official expectations with little change on instructional style or content is far from being a new phenomenon in U.S. mathematics education. Periodically, observers note that attempts at reform have had little effect on what happens in U.S. classrooms.[13]

In the remainder of this article, I provide evidence for my conjecture: Under current circumstances, it is much easier for teachers to learn to teach novel subjects in ways that engage students. As described above, those circumstances include extensive support for teaching novel subjects but not for reconceptualized traditional subjects or novel topics in reconceptualized traditional courses. In my opinion, although teachers may not need to learn traditional subjects like algebra, they are likely to require support in the form of instructional materials and professional development in order to teach reconceptualized algebra courses.

**Examples of differences in perception**

Part of my explanation draws on notions from cognitive psychology. For the purposes of this article, I describe them in terms of experience, perception, and knowledge, where knowledge takes a variety of forms. To illustrate their meanings and relationships, I give examples from research in cognitive psychology and mathematics education.

A well-known example of how differences in knowledge are associated with differences in perception comes from studies of chess players. After a 5-second viewing of a chess board with a configuration produced by legal moves, for example, a “fianchettoed castled Black King’s position,” skilled chess players were far better able to reproduce the configuration than less skilled players. The difference in performance was considerably diminished however, when the chess pieces were placed randomly, although the more skilled players still performed better than the less skilled players.[14] This study and studies of practitioners in other domains (e.g., history, physics, mathematics, teaching) suggest that experience in a domain affects perception in the domain. For example, when asked to categorize problems from their undergraduate physics text, undergraduates used descriptions such as “blocks on an inclined plane,” but graduate student instructors used principles such as “conservation of energy.”[15]

Ann Dowker studied mathematicians’ ways of estimating the values of products and quotients such as 76 × 89 and 9208 ÷ 32, and how they differed from those used by U.S. undergraduates who were not mathematics majors.[16] Unlike the undergraduates, the mathematicians used a wide variety of strategies and “tended to use strategies involving the understanding of arithmetical properties and relationships rather than strategies involving the use of school-taught techniques,” sometimes making comments such as “I’m sure that’s not how we were taught to do things at school!”[17] For example, one mathematician perceived 12.6 × 11.4 as (12 + 0.6)(12 – 0.6) and used the identity (*a *+ *b*)(*a *– *b*) = *a*^{2}* *– *b*^{2}, calculating 144 – 0.36. Several months later, when some of the mathematicians again estimated the values of the same expressions, their strategies often differed from those in their previous responses.

Dowker refers to 76 × 89 as a “problem,” that is, it means “compute 76 × 89,” “find the value of 76 × 89,” or “76 × 89 = ?” To me, 76 × 89 is an expression and doesn’t demand computation.

My preference for distinguishing between an expression and its value in school arithmetic reflects, I think, my repeated engagement with instances of this idea in Chinese and Japanese school arithmetic[18] and my concern about the transition from arithmetic to algebra. I’ve come to think that if 5*x *and *ab* are called expressions, then 76 × 89 should also be called an expression. This terminology is part of an approach that connects arithmetic problems and algebraic expressions—which are often perceived as disparate by U.S. students—and puts more emphasis on work with arithmetic expressions.

**Perception and fluent retrieval**

Learning to drive a car provides a good example of fluency and automaticity. When first learning, novices cannot drive and simultaneously carry on a conversation. With experience, it becomes easy to do so. Similarly, novice readers whose ability to decode words is not yet fluent are unable to devote attention to the task of understanding what they are reading. . . . An important aspect of learning is to become fluent at recognizing problem types in particular domains—such as problems involving Newton’s second law or concepts of rate and functions—so that appropriate solutions can be easily retrieved from memory.[19] — *How People Learn*

Teachers’ experiences include teaching particular topics, using specific instructional materials, to specific students. These experiences shape not only their perceptions of the topics (their subject matter knowledge, SMK) but also their knowledge of how to teach those topics (pedagogical content knowledge, PCK). The latter includes knowledge of typical student responses in a given situation, knowledge of student understandings and misunderstandings of topics, ways of presenting concepts, and addressing student misunderstandings. Teaching experiences as well as learning experiences shape a teacher’s perception of classroom topics and situations. With experience, a teacher acquires classroom routines that are fluent and almost automatic.[20]

Miriam Sherin studied high school teachers’ actions in their classrooms and their descriptions of those actions. She found that, like the experienced chess players who deploy a particular strategy in response to a familiar configuration of chess pieces, experienced teachers deploy a particular routine in response to what they perceive as a familiar situation involving familiar content.

Sherin writes:

When teachers consider a particular topic, they do not think just in terms of their subject matter knowledge or their pedagogical content knowledge; instead, they tend to call on both types of knowledge. As a result, teachers almost automatically apply the pedagogical routines associated with a particular piece of content. For example, if a teacher has taught linear functions many times before, when it is once again time for the lesson on slope, the teacher will draw on his or her familiar pedagogical strategies to teach the lesson.[21]

Sherin’s studies of high school teachers document how such ties between their perception of a topic and their knowledge of how to teach it can affect their interpretations of novel curricula for that topic. “Precisely because subject matter knowledge and pedagogical content knowledge are connected, once a content knowledge complex is accessed, teachers’ actions are constrained, and they tend to implement established pedagogical routines.”[22]

## Perceptions of traditional topics

I was reminded of Sherin’s finding when I came across a statement implying that:

In California, synthetic division is part of Algebra 2.[23]

In California, my home state, synthetic division is not in the current standards, the current framework explicitly discourages it, and it was not in the previous standards.[24]

So why would someone believe synthetic division is part of Algebra 2? A speculation based on Sherin’s finding: Algebra 2 does include standards for arithmetic with polynomials and rational expressions, so for some teachers the topic “polynomial long division” is so closely tied to a teaching routine that includes “synthetic division” that they interpret the former as the latter.

**Concluding remarks**

Based on the ideas and evidence discussed above, my current collection of conjectures about past enthusiasm for discrete mathematics and present-day enthusiasm for data science is as follows.

For an experienced teacher to teach a familiar subject that has been reconceptualized, it is necessary for the teacher to perceive how the reconceptualization is different and its implications for instruction, for instance, novel student responses.[25] This may require abandoning or revising well-established instructional routines associated with familiar topics, e.g., topics in arithmetic or algebra. As Sherin’s work shows, changing one’s perception of familiar topics and associated instructional routines is not always straightforward, even with the support of instructional materials and professional development focused on those materials. Without such support, teachers must not only understand reconceptualization of topics as described in standards, but express that understanding in their selection and sequencing of instructional materials.[26] Small wonder then that teaching appears unchanged.

It may be easier for a teacher to adopt different ways of teaching for subjects not previously taught or learned, e.g., discrete mathematics in the 1990s or data science in the present—particularly, when given substantial and continued support for learning and teaching. In that sense, data science is the new discrete mathematics.

**Endnotes**

[1] Rosenstein, 1997 as quoted by DeBellis & Rosenstein, p. 48, 2004, “Discrete Mathematics in Primary and Secondary Schools in the United States,” *ZDM.*

[2] Rosenstein as quoted by DeBellis & Rosenstein, pp. 48–49.

[3] Boaler, 2022, https://www.latimes.com/opinion/story/2022-03-14/math-framework-california-low-achieving.

[4] Burdman, 2023, https://hechingerreport.org/opinion-our-students-need-up-to-date-approaches-to-math-education-for-a-quickly-changing-world/.

[5] Rosenstein & DeBellis, pp. 420–422, 417–418, “The Leadership Program in Discrete Mathematics.” In Rosenstein, Franzblau, & Roberts (Eds.), *Discrete Mathematics in the Schools*, 1997, American Mathematical Society and National Council of Teachers of Mathematics.

[6] Gould, section 3.1, “Toward Data-scientific Thinking,” 2021, *Teaching Statistics,* https://onlinelibrary.wiley.com/doi/10.1111/test.12267

[7] De Loera & Su, 2022, https://www.sacbee.com/opinion/op-ed/article260529232.html.

[8] Boaler & Levitt, 2019, https://www.latimes.com/opinion/story/2019-10-23/math-high-school-algebra-data-statistics.

[9] Polikoff, 2021, pp. 66, 77–78, *Beyond Standards: The Fragmentation of Education Governance and the Promise of Curriculum Reform, *2021, Harvard Education Press.

[10] Schmidt & McKnight, 2012, Figures 4.5, 5.6, 8.2, Table 8.2, *Inequality for All: The Challenge of Unequal Opportunity in American Schools, *Teachers College Press.

[11] Schmidt & McKnight, 2012, p. 94, Figure 4.5.

[12] Gould, 2021, Section 3.1; McCrone et al., 2020, p. 61, *Mathematics Education in the United States 2020: A Capsule Summary Fact Book, *https://www.nctm.org/uploadedFiles/Grants_and_Awards/Other_Grants/ICME-14%20Factbook%2016018.pdf; Sparks, 2023, “Are Students Getting All the Math They Need to Succeed?,” *Education Week*, https://www.edweek.org/teaching-learning/are-students-getting-all-the-math-they-need-to-succeed/2023/07.

[13] Cohen, 1990, pp. 311–312, “A Revolution in One Classroom: The Case of Mrs. Oublier,” *Educational Evaluation and Policy Analysis*; Polikoff, 2021, p. 109; Stanic & Kilpatrick, 1992, pp. 415–416, “Mathematics Curriculum Reform in the United States: A Historical Perspective,” *International Journal of Educational Research*; Stigler & Hiebert, 1999, pp. 12–13, *The Teaching Gap*.

[14] Bilalić, McLeod & Gobet, 2008, “Expert and ‘Novice’ Problem Solving Strategies in Chess: Sixty Years of Citing de Groot (1946),” *Thinking & Reasoning*, 10.1080/13546780802265547.

[15] Bransford, Brown, & Cocking, 2000, Chapter 2, *How People Learn: **Brain, Mind, Experience, and School*, National Academy Press, https://nap.nationalacademies.org/read/9853/chapter/1.

[16] Dowker, “Computational Estimation Strategies of Professional Mathematicians,” 1992, *Journal for Research in Mathematics Education*; Levine, “Strategy Use and Estimation of College Students,” 1982, *Journal for Research in Mathematics Education.*

[17] Dowker, pp. 45, 52, 53.

[18] See discussions of *shi *and *shiki *in, respectively, Ma, 2001, “Arithmetic in American Mathematics Education: An Abandoned Arena?” National Summit on the Mathematical Education of Teachers: Meeting the demand for high quality mathematics education in America, https://www.cbmsweb.org/archive//NationalSummit/PlenarySpeakers/ma.htm; and Watanabe, “Translating Elementary School Mathematics Curriculum: Isn’t School Mathematics Universal?,” in Usiskin & Willmore (Eds.), *Mathematics Curriculum in Pacific Rim Countries—China, Japan, Korea, and Singapore: Proceedings of a conference,* 2008, Information Age Publishing.

[19] *How People Learn*, Chapter 2.

[20] Sherin, p. 125, “When Teaching Becomes Learning,” 2002, *Cognition and **Instruction*.

[21] Sherin, p. 124.

[22] Sherin, pp. 129–130.

[23] Burdman, 2023, https://hechingerreport.org/opinion-our-students-need-up-to-date-approaches-to-math-education-for-a-quickly-changing-world/.

[24] In discussing the Remainder Theorem (A.APR.2), the 2013 framework says, “This topic should not be simply reduced to ‘synthetic division,’ which reduces the theorem to a method of carrying numbers between registers, something easily done by a computer, while obscuring the reasoning that makes the result evident” (p. 489).

[25] For example, see the section on reconceptualized topics in the introduction of *Progressions for the Common Core State Standards*, https://commoncoretools.me/.

[26] See the section on “Encouraging Teachers to Make Better Textbook Use Decisions,” Polikoff, 2018, “The Challenges of Curriculum Materials as a Reform Lever,” https://www.brookings.edu/articles/the-challenges-of-curriculum-materials-as-a-reform-lever/.