by Scott Adamson, Chandler-Gilbert CC
Background, Meanings, and Conversation Recommendations
Introduction
Try Googling “modernize curriculum in mathematics.” Many of the first page results are in the realm of modernizing mathematics to include topics that students will find relevant in our current culture such as modeling, data science, financial math, and statistics. The traditional geometry and algebra topics that have historically been the focus for high school and two-year college students are recommended to be integrated with the new topics to bring relevance and connections to the real-world. It seems that conversations around modernizing the mathematics curriculum can be categorized in two ways: what topics should be taught (or not taught) and what we want students to learn about these topics.
As this blog is written primarily to an audience of two-year college faculty, we recognize the challenges that exist with any modernization, as curricular decisions are tied to institutionally defined student learning outcomes, departmental final exams, as well as articulation agreements with other institutions of higher learning. Therefore, the focus of this current discussion is on ideas of what should (and maybe shouldn’t) be taught and on how even common topics in the traditional algebra and calculus courses can be taught to promote ways of thinking rather than a primary focus on ways of doing. Part 2 of this series on modernizing the curriculum will provide an example of promoting ways of thinking as a priority over promoting ways of doing using the context of implicit differentiation in a standard Calculus 1 course.
Modernizing Mathematics Curriculum
For the purpose of this blog, modernizing the curriculum is to be thought of as follows:
- Think about what topics we teach. For example, consider why the following procedures are still taught as necessary in the modern mathematics classroom:
- rationalizing denominators
- polynomial long division and synthetic division
- Cramer’s Rule for solving systems
This is not to say that we should avoid certain topics as harmful. Rather, we should consider how they come up and for what purpose.
- Think about what we want students to learn about these topics. Is the primary purpose of the mathematics classroom for students to learn to execute procedures that can be more effectively handled by computers (ways of doing)? Or is the primary purpose for students to develop ways of thinking about mathematical ideas and to develop problem solving skills?
This is not to say that we should never teach students to compute or to manipulate symbolically.
Consider the Active Learning Guiding Principles from the Student Engagement in Mathematics through an Institutional Network for Active Learning (SEMINAL). Specifically, consider Guiding Principle #2: Students’ deep engagement with mathematical reasoning. Modernizing the curriculum may serve to support this guiding principle; considering the following questions may be helpful in starting discussions about modernization:
- What mathematics do students need for success in their careers, considering the ubiquitous access to technology?
- What mathematical approaches can be retired in favor of more modern ideas that are important for an educated citizenry?
- How can we transform course learning outcomes to allow time for students to engage rigorously in developing conceptual understanding, procedural fluency, and problem-solving skills?
For example, the square root algorithm was a course competency at a former point in history when approximating square roots was important for students who had no access to technology. However, it is now extremely rare for any mathematical experience, in K-12 or college, to include computing square roots (not perfect squares) by hand. Within the mathematics education community, this skill has been relegated to technology. Perhaps other computational skills could be similarly retired as well. It’s prime time for the mathematics education community to begin engaging in the challenging discussions surrounding this idea. We see these conversations as occurring in two stages.
Engage in Conversations – Stage 1
In Stage 1, faculty may begin discussing the current course offerings and make decisions about what students need to learn to be mathematically literate in today’s world.
For example, look at your institutions course competencies, learning goals, or student outcomes for a standard college algebra course. In many cases, college algebra course competencies focus on the skill of graphing functions and it is assumed that this is to be done by hand without the use of technology. Often, these graphing strategies involve tedious calculations such as factoring, plotting many points, computing asymptotes, using the Factor and Remainder theorems, and/or requiring long (or synthetic) division. How might the competencies involving graphing be modernized in a way that prepares students for today’s world? How might technology be leveraged so that the tedious process of graphing rational functions by hand, for example, can instead lean more towards important reasoning of the behavior of the function? For instance, the idea of covariational reasoning to make sense of a graph produced by technology (e.g. Desmos) may instead become a focus.
An example is the common practice of communicating to students that in order to graph a polynomial function, we must find its roots – possibly by writing the function in factored form, which may in turn require long or synthetic division. It is just not accurate to communicate that such procedures are absolutely necessary in order to generate the graph of a function. We recommend students learn that graphs can be generated using technology (Desmos), but it is critical for them to develop reasoning abilities around the behavior of the graph. From the graph, we can locate the zeros of the polynomial and thus write the polynomial function in factored form. In written communication, it may be helpful to express a polynomial function in factored or standard form, depending upon context. Within this unit of study, faculty may also choose to facilitate a learning experience where students make sense of historical techniques for finding zeros. How were these zeroes located before access to technology, like graphing calculators and Desmos, was so prevalent? Could the curriculum bring in technology to advance the mathematics not for the purpose of easing the thinking process, but to free up class time to explore more critical facets of the mathematics?
A primary reason for this recommendation is that when students are so focused on the procedures required to meet the student learning outcomes, they often neglect to understand the mathematical concepts being studied. They often mimic the procedures in hopes that they are able to earn a passing grade, as the assessments (exams, quizzes, and/or homework) often just test student proficiency in these same procedural methods.
For example, when students study rational functions and are asked to find horizontal asymptotes, they can work to memorize correct steps and may even write the answer in proper form (e.g. y = k). However, when probed to explain what a horizontal asymptote is, they have great difficulty articulating a meaningful response. Ideally, students should think about a function as a covariational relationship between two quantities. To understand the horizontal asymptotic behavior of the function, students might describe the following: as the input quantity gets larger and larger (in magnitude), the output quantity gets closer and closer to a constant value. Again, students might start with a graph generated by technology, recognize asymptotic behavior, and then use the graph to explain procedural methods that confirm their observations and graphical analyses.
Or, consider a rational function where slant (or oblique) asymptotes are present. Without pre-teaching the idea, we imagine students encountering a rational function situation that is different than previous situations involving only horizontal and vertical asymptotes. Rather than focus on the details of using polynomial long division to identify slant asymptotes, students might investigate the graph of such a rational function and then describe, covariationally, what they are observing. In the case of
students would be confronted with the graph shown.
Now, as the input quantity gets larger and larger (in magnitude), the output quantity gets closer and closer to a linear function rather than to a constant value! One might wonder, “what linear function is this?” Polynomial division, using technology, can help students to identify this linear function serving a slant asymptote. Note that the use of technology shifts the thinking from remembering, in this case, obsolete procedural techniques to interpreting the output from the chosen technological tool.
We will show the output from wolframalpha.com starting with how the polynomial division was entered into the tool:
WolframAlpha outputs many things. The student must now work to choose what output is most useful and relevant to the question at hand: what linear function is serving as the slant asymptote in this case? We find this among the other outputs from WolframAlpha:
With discussion and collaboration, students can learn that
where the linear function y = 3x + 3 serves as the slant asymptote. The intent of this blog is not to unpack the details of this truth. Rather, the intent is to highlight how the curriculum can be adjusted to focus on a different aspect of learning where interpretation of the computer output might take priority over the “old school” skill of procedurally dividing polynomials.
We like to emphasize here that it may very well be appropriate to allow students the opportunity to work through the polynomial long division by hand as well given that some students might wonder, “how did WolframAlpha produce this result?” It is our opinion that mastering this skill is not important in the current day and age but interpreting and understanding the result of a computer algebra system is important.
In general, does this mean that students should never have some learning experiences with historical methods such as synthetic division or even a square root algorithm? Not necessarily. Faculty might consider creating learning experiences where students make sense of methods used in the past before technology was available. Students need not master these skills, but learning historical mathematical methods may serve to enhance the mathematical maturity of students. Students can develop stronger algebraic skills, mental math strategies, and deeper understanding of mathematical ideas when these historical methods are experienced (not necessarily to a mastery level) in conjunction with conceptual understanding and the ability to make sense of important mathematical ideas.
Engage in Conversations – Stage 2
In Stage 2, faculty may advance discussions involving alternate mathematics pathways. For decades, students have been asked to navigate the “STEM pathway” consisting of algebra (introductory, intermediate, college, and precalculus), trigonometry, and calculus (for some). Might students be better served, considering our current societal needs, to engage in a rigorous pathway focused on quantitative reasoning, statistics, and data analysis? Many colleges do have alternative pathways for students to consider, but we encourage math faculty and other stakeholders to consider ways in which students are able make a well-informed choice of the best pathway for them. It is likely that many more students could benefit from a quantitative reasoning-focused pathway. In May 2022, David Kung of the Dana Center (University of Texas at Austin) wrote the following:
Our curriculum is woefully in need of updating. The college math curriculum in the U.S. was largely designed to produce a small number of STEM professionals who could beat the Soviet Union to the moon, armed with pencils, paper, and room-sized computers. The challenges we now face require a much broader range of mathematics knowledge. Statistics is more important than ever—in nearly every field of study. We are awash in data, which requires different mathematical (and programming) tools to wrangle. And two years of a global pandemic has exposed the importance of quantitative literacy in society at large.
Conclusion
To summarize, we recommend a two-fold approach to modernizing the curriculum: 1) focus on what topics should be taught and 2) focus on what we want students to learn even with traditional topics. We might begin with what will likely be the long process of discussing the questions posed in this blog and begin making moves toward creating course learning outcomes that prepare students for the current, technologically advanced world. These discussions involve both updating current course learning outcomes and increasing student understanding and informed selection of alternate pathways including courses focused on quantitative literacy and statistics. In the meantime, we might consider how teaching the current course outcomes can be accomplished in a way that recognizes the historical context from which they were created and teach big mathematical ideas with the appropriate use of technology.