There are demands to change the precalculus and calculus curricula. Do students still need to learn topics like synthetic division or Descartes's Rule of Signs? Might it be valuable to increase the use of real data analysis in those courses? As we see our colleagues in the social sciences and humanities struggle with the realities of artificial intelligence, how should mathematics instructors change how they manage homework, quizzes, and exams? Of course the pandemic changed everything in education, and the demand for distance learning is clearly not going away, so how does that affect students learning advanced mathematics courses?
If you could change one thing about the precalculus algebra/trigonometry curriculum what would it be?------------------------------Robert CappettaFlorida SouthWestern State CollegeFL------------------------------
I stopped teaching Descartes's Rule of signs years ago. I still teach synthetic division and finding the zeros of polynomials using synthetic division. However, with technology available is this really necessary to do by hand? Especially testing the zeros one by one with trial and error seems very outdated and less efficient. On tests I will give them a starter zero and they need to find the rest by synthetic division. Along those lines, how important do we feel complex zeros of polynomials are?
Do you feel that Newton's Law of Cooling is needed in a precalculus course?
For trigonometry, one thing I think we can change would be for sum-to-product and product-to-sum formulas. I have not seen must need for these.
Newton's Law of Cooling is a nice example of exponential functions. If we could create the model using something like a baked potato while measuring its temperature, it could be motivating for students to use regression analysis. However, the precalculus curricula are overloaded with topics, so removing any given topic must be considered. My greatest challenge currently is how to design a departmental final exam that satisfies all instructors.
The very first thing we need to do is to separate these two topics. Precalculus needs to focus on preparation for Calculus and (College)Algebra/Trigonometry were not designed to do that. College Algebra was initially created as the pathway for students not needing Calculus. And none of it is addressing the quantitative reasoning and communication that all students need. Precalculus needs to begin introduce students to the rigorous reasoning and communication about functions that Calculus wants to develop. On the other side, we need to address the type of mathematical/quantitative reasoning and communication needed in the pursuit of other college degrees.
By insisting that our mathematics courses serve all students, they serve nobody.
Columbus State Community Collge
A question I would have related to assessment, is what are we actually testing for/what is necessary to understand topics in courses to follow. I would argue that we should be testing for "content fluency". I'm not sure if there is a term in the literature for what I'm thinking of, but this is the most accurate way to describe it in a term. Regardless of what we teach, students leave my course I want them to be a reflective level of fluent in the material. In saying that they should have enough knowledge and ability to work through problems orally with scratch paper.
So my question would be how is this assessed. I'm not sold on forcing students to be on camera, unless your institution or colleges that you usually feed into require it.
For precalculus, I do not see the need for students to learn Decartes' Rule of Signs, in this technological era. However, there is a need to instruct students on finding the zeros of Polynomials (still they can do this with technology). There is a need for instructors to teach students the way of using technology to do their Homework, Quizzes, and Exams to advance their learning and boost performance. Distance learning is the future of education and technology plays an important role in distance education with the use of connectivity principles. Moreover, I change the sequence of the topics, not following the sequence of the authors.
I think precalculus should be completely redesigned. We should be looking at what concepts students need to succeed in calculus and those should be what is focused on in precalculus. The main ideas that students need in calculus is what is a function, what is the domain, how to find the range, the long term behavior of many functions, how to use technology to find zeros of functions, how to solve equations with and without technology, and other concepts. We should talk to physics and construction faculty to see what topics they need students to know for their classes, but precalculus should mostly be to help students succeed in calculus. I would also love to see modeling with data in a precalculus class. Functions should have context and data helps provide that context.
I agree with Kate's assessment that everything needs to be changed, as well as with Lee who observed that college algebra currently serves no one. Research in mathematics education offers insight into the ways of understanding function that are important for calculus. I add to Kate's list covariational reasoning and developing an intuition about rate of change for the functions studied in precalculus. Below is a list of several papers and book chapters that have helped me develop learning activities for my precalculus students.
Also, the answer to the question, 'should we allow students to use technology to...?' should be a resounding, 'Yes.' I can't believe this issue has persisted for 30 years.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for research in mathematics education, 33(5), 352-378.
Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students' function understanding. Making the connection: Research and teaching in undergraduate mathematics education, 27, 42.
Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. Compendium for research in mathematics education, 421-456.
With all the math love I can muster in an online post...I say that we need to stop teaching precalculus in a way the ONLY focuses on obsolete computations as if we don't have ubiquitous access to technology. I still observe classrooms where the message is something like this:
Actually, the reality is...
Suppose we are teaching from a historical perspective. We might engage students in mathematical thinking to make sense of obsolete procedures from a purely historical perspective. My dad (long time AMATYC member and former treasurer) and I had a series of presentations at AMATYC Conferences with the theme we called the "Mathematics Attic." We put things like synthetic division in the mathematics attic...but we might sometimes take them out and examine them out of historical interest.
However, in terms of course competencies, learning objectives, assessment, etc., the precalculus course should be focused on supporting student development of ways of thinking and problem solving.
For example, covariational reasoning is a big idea the permeates much of mathematics (read the articles in Ann Sitomer's post to learn more!) and is necessary for a solid understanding of calculus ideas. We might focus on mathematical modeling and developing problem solving skills. This might include work with data analysis, regression, etc. to model real-world phenomena. As we do this, we might consider active learning pedagogies that also help with communication, collaboration, and professional debating skills.
In short...in 2023...we might consider how we can modernize our curriculum and pedagogy to meet the needs of our current world.
Lean and Lively Curricula <o:p></o:p>
In January 1986, Ron Douglas of Stony Brook brought together 25 mathematicians and educators for a four-day conference/workshop at Tulane University in New Orleans. This Sloan Foundation–supported event led to the publication of Toward a Lean and Lively Calculus. This conference and the resulting book written by its participants launched the Calculus Reform Movement in the United States.<o:p></o:p>
The conference called for "a leaner, livelier, more contemporary course, more sharply focused on [its] central ideas." With the advent of the new Advanced Placement® Precalculus course being launched during the 2023–2024 school year, the College Board is initiating a Precalculus Reform Movement. This new AP® course is lean, lively, modern, and squarely focused on the central concept of function.<o:p></o:p>
This new Precalculus course offers other innovations that echo the Calculus Reform Movement. Much can be learned from what took place in the 1980s and 1990s. Conceptual understanding must support procedural fluency. Students need to engage with technology to see the links across graphical, numerical, algebraic, and verbal representations of functions. Instructors must advance student communication and reasoning. And we must address fewer topics in greater depth.<o:p></o:p>
Hurray for lean and lively Precalculus and Calculus!<o:p></o:p>
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