Research-Based and Research-Proven Results
Pathways courses are designed based on decades of research on the ideas and ways of thinking most critical for students’ success in calculus and other STEM fields and how students learn, both generally and relative to specific mathematical ideas.
We have observed and documented multiple benefits to students taking Pathways courses.
Preparedness for Calculus
The Precalculus Concept Assessment (PCA) is a validated instrument designed to assess students’ readiness for calculus. The PCA includes 25 multiple choice questions related to the ideas most correlated with students’ success in calculus.
During the validation process it was shown that the PCA is a strong predictor of students’ success in calculus with 13 out of 25 as the cutoff score. 77% of students who scored 13 or above at the beginning of calculus earned an A, B, or C in Calculus I. 60% of students who scored 11 or below received a D, F, or withdrew. (In subsequent years data suggests the prediction potential may be even stronger with 83% of students scoring a 13 or above going on to earn an A, B, or C in Calculus I and 85% of students scoring an 11 or less going on to earn a D, F, or withdraw from Calculus I. Nine sample scores from five universities and four high schools are given in the table. These represent average pre- and post-test scores for students before and after taking Pathways precalculus. You can see that Pathways students, on average, complete precalculus well-prepared for calculus.
| PCA Pre-Test Score (out of 25) | PCA Post-Test Score (out of 25) | Pre-Post Gain | |
| University 1 | 6.2 | 14.2 | 8 |
| University 2 | 6.9 | 14.7 | 7.8 |
| University 3 | 7.1 | 14.9 | 7.8 |
| University 4 | 9.1 | 14.3 | 5.2 |
| University 5 | 6.5 | 14.2 | 7.7 |
| High School 1 | 9.4 | 14.8 | 5.4 |
| High School 2 | 11.1 | 16.2 | 5.1 |
| High School 3 | 11.3 | 18.9 | 7.6 |
| High School 4 | 7.8 | 14.1 | 6.3 |
Note that the highest ever recorded post-test average score for non-Pathways users is 10.1, indicating that typical precalculus courses do not adequately prepare students for Calculus I. It is most common that the pre- to post-test average gains are only one to two points for non-Pathways courses.
Success in Precalculus and Calculus
Universities who have tracked Pathways students’ success into calculus courses have found that Pathways precalculus students perform better than students from other calculus preparation programs.
For more information, see our section on research related to the Pathways program.
- Passing rates in precalculus are higher for Pathways students, and they succeed in calculus at a higher rate. (In other words, more students have the opportunity to take calculus, and those who do are more likely to pass calculus.)
- At at least one site, passing rates in precalculus improved by more than 25% (from less than 50% to over 75%) after adopting Pathways precalculus.
- Multiple universities have compared Pathways students’ performance on common exams with non-Pathways students from the same course level and found that (i) Pathways students performed significantly better on conceptual questions and (ii) performed at least as well (and sometimes better) on procedural questions when compared to non-Pathways students.
Performance on Learning Assessments
The Calculus Concept Readiness Instrument (CCR) and the Algebra and Precalculus Concept Readiness Assessment (APCR) are attempts to expand the usefulness of the Precalculus Concept Assessment (PCA) by creating tests for two different levels of students and by including content (such as trigonometry) not included in the PCA. These assessments have been administered at numerous schools and universities across the country to students using a variety of curriculum materials. Time and again, Pathways students overperform relative to their peers on these assessments. Three sample questions are provided along with student performance metrics to illustrate this point.
Impact on Students’ Affective Characteristics
Student surveys indicate significant positive shifts in students’ confidence and beliefs on items similar to the following.
- I am confident in my mathematical abilities.
- I see myself as a strong mathematics student.
- I am good at answering word problems.
- I am interested in taking more math course.
- When confronted with a challenging problem, I stick with it until I get an answer.
- I can become good at mathematics by working hard.
- I am confident in my ability to verify the correctness of my answers.
- I enjoy working on challenging mathematics problems.
Impact on Teachers/Instructors
Pathways teachers exhibit (and report that they experienced) significant transformations in their teaching practices; away from showing students steps for getting answers to engaging students in “meaning-making” activities.
Pathways materials have been used at multiple universities to help train preservice mathematics teachers.
Prior to working with Pathways materials, less than 8% of preservice math teachers showed a disposition toward reasoning about quantities and relationships between quantities and less than 8% could provide a working definition for and examples of quantitative and covariational reasoning.
After working with Pathways materials, more than 84% of these teachers showed a disposition toward reasoning about quantities and relationships between quantities and more than 90% were able to provide a working definition for and examples of quantitative and covariational reasoning. These preservice teachers also increasingly interpreted state mathematics standards using these working definitions and examples.
Beliefs surveys indicate that teachers using Pathways materials make significant shifts:
- from viewing teaching as being about helping students learn to work specific problem types to supporting students in constructing an understanding of key concepts,
- from viewing student success as being predominantly dependent on their ability to master skills and procedures to viewing student success as dependent on the teacher’s ability to engage students in sense-making,
- from viewing a teacher’s role as presenting material and students’ roles as getting right answers to viewing students’ success as dependent on applying mathematical reasoning to solve novel problems, and
- from responding to students’ questions by showing solution methods to responding to student questions with question prompts intended to reveal students’ thinking.
Student and Instructor Testimonials
“Before this class I thought math was just about memorizing rules. This is the first time I’ve been able to understand the word problems. This class is making me a better thinker in all my classes. I now expect ideas to make sense and know that I am smart enough to figure things out instead of waiting for teachers to tell me what to do.” – Pathways Student
“I really liked the investigations we did during class. The questions kept me learning and engaged. When the teacher lectured I was able to understand. I also got more and more comfortable in explaining my solutions and liked seeing mathematics from different viewpoints.” – Pathways Student
“Teaching with Pathways I came to the realization that it is not my job in the classroom to teach students to think the way I’m thinking. Rather, it is my job to help students learn how to do mathematics in their own way—using structured activities that are providing content, but at the same time are providing leeway for them to make their own meaning from the material.” – Pathways Instructor
“The pathways materials provide an instructor an opportunity teach in ways that ask students to think about a mathematical concept, to form their own understandings of it, and express those understandings to themselves and others. And then, the instructor can take those understandings and use the students expressions of their understandings as a way to advance the lesson towards a particular mathematical goal. You have to modify traditional textbook lessons in order to do that. So for me, it’s more work to try to teach with a traditional book because I have to somehow figure out how I’m going to modify it to help get the students ideas on the table.” – Pathways Instructor and Precalculus Course Coordinator
“The TAs teaching the courses would kind of catch that spark and get enthusiastic about this different way of teaching things and some of them relayed to me afterwards that it completely changed the way they thought about teaching. They would never go back to teaching these ideas in like they used to.” – Precalculus course coordinator
“The Pathways materials revolutionized the way I think about teaching. In the past there were so many things that I just had not thought about, you know? When looking at exponential functions or trigonometric functions, I realized when I was teaching out of Pathways that there was just an awful lot that I had never really thought about. I just accepted procedure-based approaches and just used them, and had never really given a lot of thought about, “Okay, I can prove that these things work, but why should they work? What’s the motivation behind the way we define the sine function? Or what’s the motivation behind the differences between exponential growth and linear growth?” And it’s just, it was a real revolution to me, and kind of humbling, too.” – Pathways Instructor