Ensuring Rigor in First-Year Mathematics Courses Joan Zoellner, Course Program Specialist March 21, 2019
Outcomes Participants will: • Explore the meaning of rigor in mathematics. • Discuss ways to promote rigor in the first year mathematics/quantitative reasoning courses. • Engage with resources from the field and the professional associations. 4
Why are we exploring rigor? These are some of the things we hear: 1) Concerns about whether it is realistic for students with weak math backgrounds to pass a rigorous college-level math course within their first year. 2) Questions about the curricular choices offered to students under math pathways (e.g. the belief that offering students statistics or quantitative reasoning, rather than a calculus- prep algebra course, is weakening the degree). 3) Speculation that offering stretch courses or support courses will lessen the rigor of the gateway math courses. 5
Concerns about rigor Has the concern of rigor come up in department and/or division meetings? 1) Yes, as it relates to algebraic-intensive courses (Pre- Calc, College Algebra, etc.) 2) Yes, as it relates to non algebraic-intensive courses (Quantitative Reasoning, Statistics, etc.) 3) Yes, as it relates to offering stretch or support courses. 4) Yes, but for another reason. (Please share!) 5) No, concerns about rigor have not been raised. 6
Preparation Reflection At your tables, discuss your responses to the reflection prompts: • Do you have an internal definition of rigor? Can you put it into words, or is it a feeling that you can recognize it when you see it? • We have found that there is “no consensus on a definition of rigor to apply in addressing the effectiveness of mathematics education" - Why do you think is this the case? Why has no formal definition been previously agreed upon? 7
Preparation Reflection • What was your biggest take-away from reading “What is Rigor in Mathematics Really?” 8
Dana Center’s Understanding of Rigor Rigor in mathematics is a set of skills that centers on the communication and use of mathematical language.
Towards a practical view of rigor • We should attend to all of our math courses, whether it be statistics-, modeling- or algebra-based, to ensure that they are all taught with rigor. • To learn mathematics, all students must have the opportunity to tackle rich problems and productively struggle with them. • They must not only solve those problems but also be able to articulate the basis of an argument at a level of precision appropriate to the course. • Math departments should play an essential role in determining the content of their introductory courses in conjunction with the views of the professional associations and the needs of the institution’s various programs of study. 10
Rigor as a Core Course Component 11
Components of Rigor Communication 12
Any Pathway Can Be Rigorous The 17 professional associations of mathematicians which comprise the CBMS have endorsed the idea that there are many areas of mathematics that, when well taught, can serve as appropriate introductions to college mathematics and mathematical thinking and work. http://www.cbmsweb.org/ 14
Is the following a characteristic of a rigorous course? Provide connections among mathematical ideas No Yes
Is the following a characteristic of a rigorous course? Require memorization of rules and procedures and use of a rote procedure to solve problems Yes No
Is the following a characteristic of a rigorous course? Require memorization of rules and procedures and use of a rote procedure to solve problems -What modifications could you make to turn this into a characteristic of a rigorous course?
Is the following a characteristic of a rigorous course? Teachers doing the work while students watch No Yes
Is the following a characteristic of a rigorous course? Teachers doing the work while students watch -What modifications could you make to turn this into a characteristic of a rigorous course?
Is the following a characteristic of a rigorous course? Students know how to perform a list of algebraic tasks such as: multi-step factoring, rationalizing n th roots, completing the square, etc. Yes No
Is the following a characteristic of a rigorous course? Students know how to perform a list of algebraic tasks such as: multi-step factoring, rationalizing n th roots, completing the square, etc. -What modifications could you make to turn this into a characteristic of a rigorous course?
Learning Experiences Learning experiences Experiences that do that involve rigor … not involve rigor … are more “difficult,” with no challenge students purpose (overly-complicated polynomial long division) require effort and tenacity by require minimal effort students focus on quantity (more pages focus on quality (rich tasks) to do) include entry points and are offered only to gifted extensions for all students students https://www.nctm.org/News-and-Calendar/Messages-from-the- President/Archive/Linda-M_-Gojak/What_s-All-This-Talk-about-Rigor_/
Learning Experiences Learning experiences Experiences that do that involve rigor … not involve rigor … provide connections among do not connect to other mathematical ideas mathematical ideas contain rich mathematics that is contain routine procedures with relevant to students little relevance develop strategic and flexible follow a rote procedure thinking require memorization of rules encourage reasoning and sense and procedures without making understanding expect students to be actively often involve teachers doing the involved in their own learning work while students watch https://www.nctm.org/News-and-Calendar/Messages-from-the- President/Archive/Linda-M_-Gojak/What_s-All-This-Talk-about-Rigor_/
Challenges with Ensuring Rigor Thinking about your college’s gateway courses what are the biggest challenges to ensuring that they are rigorous? 1) Making explicit connections between concepts 2) Using relevant mathematical scenarios 3) Helping students develop strategies that make sense to them, rather than relying on memorization of rote procedures 4) Encouraging students to work actively and take control of their learning 5) Other (please share!)
Activities and Assignments That Promote Rigor: § Encouraging alternative approaches. § Asking students about the reasonableness of their answers. § Asking students to make explicit connections between multiple representations. § Including new situations where student need to extend their understanding. § Demonstrating that premises of the course are solidly based. § Expecting students to use precise mathematical language along with understanding. § Giving students feedback about the clarity of their reasoning.
Connected Learning
Using Rich Tasks to Create Rigorous Learning Opportunities Rich mathematical tasks include: • Students as the workers and the decisionmakers • High-level thinking and reasoning by students • Discussion, collaboration, or active inquiry Process • Multiple layers of complexity Standards Academic Content • Multiple entry points Standards Rigor • Multiple solutions and/or strategies Rich Tasks 33
Planning Rigorous Content 34
Planning Rigorous Content The following table summarizes data from the Trust for Public Land on park area and spending for five large cities. Based on this data, which city appears to have the most resources devoted to public parks? State your answer in complete sentences and include quantitative measures to support your conclusion. 35
Planning Rigorous Content 36
Increasing Rigor in Co-requisite Courses 1. [Closely] aligning developmental course content with college-level course expectations 2. Providing consistent opportunities for students to construct knowledge [including problem solving, critical thinking, reasoning, and making predictions], and 3. Making struggle a part of the learning process – Barragan, M., & Cormier, M. S. (2013). Enhancing rigor in developmental education. Inside Out, 1(4)
Planning Rigorous Co-requisite Courses College-Course Preparation Support Course Content College-Course Content Homework Calculate probability of independent Convert probabilities to a “1 in ___ Operations with fractions events involving “and” and “or” chance” statement statements Determine simple and conditional Chance and probability; probability Calculate conditional probabilities probabilities of events; dependent notation for dependent events and independent events Conversion factors Dimensional analysis Using conversions to compare data Reference values; comparing values Make/justify decisions and evaluate with percentages; reading Calculate cost of living averages claims using index numbers spreadsheets Percentages of the whole; Use weighted averages to analyze calculating percentages with Mean and weighted average data and draw conclusions spreadsheets Population data and percentages; Sum and mean of a data set; Expected value; making predictions spreadsheet calculations percentages based on data analysis 38
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