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PrecalculusOhio
Rethinking how we prepare students to learn Calculus.
Rethinking how we prepare students to learn Calculus.
Ohio has 36 state institutions of higher education. These include universities, community colleges, and their regional campuses and learning centers.
In 1990, Ohio developed the Ohio Articulation and Transfer Policy to facilitate transfer of credits between the state institutions. The Ohio Transfer Module (OTM) detailed the policy. In 2021, the OTM was renamed the Ohio Transfer 36 (OT36).
In 2014, Ohio faculty convened around the need to revise the mathematics pathway and formed the Ohio Mathematics Initiative (OMI).
Together, the OT36 and OMI have created several new collegiate mathematics pathways. In addition to the traditional algebra-to-calculus pathway, college students can now experience mathematical reasoning through Quantiative Reasoning, Data Science, Technical Mathematics, Discrete Mathematics, Mathematical Education, Logic, and Statistics. All of these are available to first-year students and satisfy general education requirements in mathematics.
With the creation of these new mathematics pathways and useful mathematics for non-STEM students, attention has turned back to the traditional Algebra-to-Calculus pathway. Preparation for Calculus is the starting point.
The first step was a change in Ohio's philosophical viewpoint. Previous course descriptions were called guidelines. The previous TMM002 course guidelines listed topics that course materials should include. Guidelines described what a course should cover and the procedures students should be able to perform. Guidelines have been replaced with student learning outcomes.
The TMM002 student learning outcomes describe the experience of a successful student. Learning Outcomes describe how a student uses mathematical tools to investigate and communicate.
The change from guidelines to learning outcomes also provide an opportunity to focus on student needs. By focusing on Calculus preparation, several attention areas for Calculus could be addressed by on-ramps. A Calculus course simply moves to fast for most students. Precalculus as preparation for Calculus could begin a slower introduction to several important ideas.
There is quite a bit of research and evidence that this transition is important for student learning.
Ohio Mathematics Initiative: Ohio faculty convened a summit on the state of mathematical education in their public colleges and universities. They published several documents descibing their conclusions and recommendations.
"Recommendation 1.1: Improve student success in entry-level courses by aligning mathematics to academic programs of study and by improving instructional delivery mechanisms"
"In particular, departments should remove college algebra as the default mathematics course for non-STEM majors"
A Common Vision: This project was a joint effort, focused on modernizing undergraduate programs in the mathematical sciences.
What is Rigor in Mathematics Really?: The Dana Center at the University of Texas at Austin has been a leader in the national conversation about mathematical pathways and rethinking mathematical education. They published a paper on rigor: What is Rigor in Mathematics Really?
Strengthening Ohio High School Mathematics Pathways Initiative: - The Ohio Department of Education and Workforce sponsored a self-examination of their mathematical education practices, which led to the creation of several mathematics pathways all based on rigor.
Publications There are many papers, reports, and publications supporting a change in the preparation for Calculus that we provide our students.
"I have been amazed to discover that across the country it is typical that 25 or 30% of students who take their first calculus course in college fail. It seems to be a national expectation that a significant percentage of students will be lost—indeed, should be lost—from a STEM pathway after taking college calculus. Given those high course failure rates, it is not hard to see that the 25% of college calculus students who never took calculus in high school are at a tremendous disadvantage."
"As a result of the low percentage of exam items at the 'understand' or 'apply understanding' levels of the item orientation taxonomy, we conclude that a large percentage of exam items failed to provide insight into how students understand the concepts on which their computational or procedural work is based. Hence, these results suggest that a large majority of Calculus I final exams being administered in colleges and universities in the United States encourage memorization of procedures for answering specific problem types and do not encourage students to understand or apply concepts of beginning calculus."
"[...]examined 150 randomly chosen Calculus 1 final examinations selected from 246 Calculus 1 final examinations administered at institutions of higher learning across the United States in the fall of 2010. They found that the exams were highly procedural in their focus—87 % of the items were coded as recall a fact or carry out a procedure."
"They also point to the need for further investigation of the content focus and student learning in both precalculus and beginning calculus in the United States."
AP Precalculus - The College Board as created a description for a new AP Precalculus course/exam. Their new direction is remarkably similar to the ideas of TMM 002. Both are focused on function analysis with algebra playing a supporting role. Both include the idea of the derivative as appropriate to their audience and their time frame.
Calculus is an evolution in thinking for students. Calculus introduces a transition in our thinking about arithmetic from the finite to the infinite. The underlying idea of Calculus is that our well-known rules and procedures for arithmetic don't work with an infinite number of numbers. Infinity is not a real number. You cannot calculate with ∞ as if it was just a really big real number. Extending our ideas of arithmetic to the infinite require a new approach, a new language. This new perspective on calculation involves the idea of limits.
Calculus is the wrong place to begin this thinking. There is just too much.
It takes students a long time to comprehend this new paradigm. It takes students a long time to use the language and read the notation. It takes students a long time independently make decisions, carry out plans, and report their conclusions coherently.
If we want students to actually understand the ideas of Calculus and master its tools, then students need a much longer on-ramp to Calculus.
Calculus presents a lot to learn, which is why studnets need a longer on-ramp. The TMM 002 Precalculus learning outcomes acknowledge this upfront:
To achieve this, we need our Precalculus courses to adopt some of the aspects and features of Quantative Reasoning courses.
QR courses are designed and presented and experienced differently than traditional mathematics courses. Students learn by investigating, discussing, and explaining. There is no lecturing. Students familiarize themselves with a situation, including probing inquiries that expand upon the given information.
QR students figure it out.
QR instructors teach students how to figure things out.
We have been presenting mathematics in traditional format for decades. Changes are only coming in steps. Change one thing this semester and another next semester.
For one thing, there are no Precalculus textbooks to support the idea of a Calculus on-ramp. We need to develop our own textbook.
For another, we need to develop our own in-class materials for collaborative learning.
For another, we need to develop our own on-line homework courses.
PrecalculusOhio intends on producing these materials.