• Acomparison of Mathematics Teachers' and Professors' Views on Secondary Preparation for Tertiary Calculus

      Wade, Carol H.; Sonnert, Gerhard; Sadler, Philip M.; Hazari, Zahra; Watson, Cherrie; Florida International University; Harvard University; The College at Brockport (2016-01-01)
      This article compares the views of teachers and professors about the transition from secondary mathematics to tertiary calculus. Quantitative analysis revealed five categories where teachers and professors differed significantly in the relative frequency of addressing them. Using the rite of passage theory, the separation and incorporation phases were investigated by carrying out thematic analyses on these five categories. For the professors, the analysis revealed specific content within algebra and precalculus that they viewed as vital preparation for students’ tertiary calculus success. For the teachers, the analysis highlighted the classroom environment realities of teaching in the separation phase. The rite of passage and professional turf theories are used to discuss and interpret the findings.
    • Are Homeschoolers Prepared for College Calculus?

      Wade, Carol H.; Wilkens, Christian P.; Sonnert, Gerhard; Sadler, Philip M.; Harvard University; The College at Brockport (2015-01-01)
      Homeschooling in the United States has grown considerably over the past several decades. This article presents ?ndings from the Factors In?uencing College Success in Mathematics (FICSMath) survey, a national study of 10,492 students enrolled in tertiary calculus, including 190 students who reported homeschooling for a majority of their high school years. The authors found that, compared with students who received other types of secondary schooling, students who homeschooled: (a) were demographically similar to their peers, (b) earned similar SAT Math scores, and (c) earned higher tertiary calculus grades.
    • FOUR COMPONENT INSTRUCTIONAL DESIGN (4C/ID) MODEL CONFIRMED FOR SECONDARY TERTIARY MATHEMATICS

      Wade, Carol Henderson; Wilkens, Christian P.; Sonnert, Gerhard; Sadler, Philip M.; Harvard-Smithsonian Center for Astrophysics; The College at Brockport (2020-12-01)
      Cognitive load theory (CLT) was introduced in the 1980s as an instructional theory based on well accepted aspects of human cognitive architecture (Sweller, van Merriënboer, & Paas, 2019). A major premise of the theory is that working memory load from cognitive processes is decreased when domain specific schemas are activated from long term memory. Comprehension, schema construction, schema automation, and problem solving in working memory often create high cognitive load. Hence, schemas transported from long term memory into working memory support learning and transfer of learning (Ginns & Leppin, 2019). One of the key developments from CLT has been the Four-Component Instructional Design (4C/ID) Model generated from evolutionary theorizing (Geary, 2008; Ginns & Leppink, 2019). Since its creation, the 4C/ID Model has been successfully applied to instruction that requires the learning of complex tasks. Van Merriënboer, Kester, and Paas (2006) defined a complex task as having many different solutions, real world connections, requiring time to learn, and as creating a high cognitive load. Based on this definition, the instruction and learning of mathematics is a complex task. For example, different solutions are algebraic, analytic, numeric, and graphic. Relative to real world connections, mathematics is one of the domains in the broader science, technology, engineering, mathematics (STEM) field and is regarded as the language of the sciences. Regarding taking time to learn and creating a high load on learner’s cognitive systems, mathematics teachers deal with the tension between covering all the required standards and taking the time to teach for understanding. Teachers face challenging decisions about instructional approaches, materials, productive struggle, and the amount of classroom time spent on various standards. Better models for instruction that support transfer of learning could help teachers improve instructional decision making. Although the 4C/ID Model has been used in secondary mathematics education (Sarfo, & Elen, 2007; Wade, 2011), it has never been confirmed as a mathematical instructional theory. The purpose of this research report is to present an empirical confirmation of the 4C/ID Model, using data from the Factors Influencing College Success in Mathematics (FICSMath) project from Harvard University.
    • High School Prpearation for College Calculus: Is the Story the Same for Males and Females?

      Wade, Carol H.; Sonnert, Gerhard; Wilkens, Christian P.; Sadler, Philip M.; Harvard University; The College at Brockport (2017-01-01)
      Usingdatafromthefirst national study on high schoolpreparationfor college calculus, the Factors Including College Success in Mathematics (FICSMath) project, this paper connects males’ (n53,648) and females’ (n52,033) instructional experiences from their high school precalculus or calculus course to their college calculus performance. A hierarchical linear model identifies several significant instructional experiences that predict college calculus performance. Our findings show that high school instructional practices affect college calculus performance similarly for males and females.
    • Instructional Experiences that Align with Conceptual Understanding in the Transition from High School Mathematics to College Calculus

      Wade, Carol H.; Sonnert, Gerhard; Sadler, Philip M.; Hazari, Zahra; Florida International University; Harvard University; The College at Brockport (2017-04-01)
      Using data from the first National study on high school preparation for college calculus success, the Factors Influencing College Success in Mathematics (FICSMath) project, this article connects student high school instructional experiences to college calculus performance. The findings reported here reveal that students were better prepared for college calculus success by high school instructional experiences that emphasized mathematical definitions, vocabulary, reasoning, functions, and hands-on activities. These findings serve to inform high school mathematics teachers about promising instructional practices. They can also inform teacher education programs about how to better prepare secondary mathematics educators to discuss conceptual understanding on the widely used Educative Teacher Performance Assessment (edTPA).
    • The Secondary-Tertiary Transition in Mathematics: What High School Teachers Do to PrepareStudents for Future Success in College-Level Calculus

      Wade, Carol H.; Simbricz, Sandra K.; Sonnert, Gerhard; Gruver, Meaghan; Sadler, Philip M.; Brockport High School; Erie 2-Chautauqua-Ca!araugus Board of Cooperative Educational Services, NY; Harvard University; The College at Brockport (2018-01-01)
      Quantitative analysis of the Factors Influencing College Success in Mathematics (FICSMath) Survey data indicates that high school mathematics teachers’ abilities to teach for conceptual understanding is a significant and positive predictor of student performance in singlevariable college calculus. To explore these findings further, we gathered and analyzed interview data gained from a representative sample of high school precalculus teachers from across the U.S., identified by their students as requiring high levels of conceptual understanding (n=13). Seventeen themes were identified and then combined into five overarching phenomenological themes. These overarching themes suggest that teachers who teach for high conceptual understanding (a) support relational understanding during problem solving, (b) require students to learn how to study to build on prior knowledge and learn from mistakes, (c) use mathematical language and ask critical questions to support learning, (d) focus on content knowledge necessary to make connections, and (e) use technology to support learning concepts but limit calculator use. Comparison of these results to quantitative findings further illuminate that intentional development of disciplinary knowledge, cognition, and language are noteworthy points of intersection for teachers and researchers alike.