Abstract
In the fall of 2011 five secondary level biology teachers in the northeast United States implemented an experimental instructional module that challenged their students with a design problem. This challenge required students to perform both mathematical analysis and the engineering application of biological concepts in order to reach a resolution. Specifically, given the parental genotypes of two gecko parents, students were tasked to: (a) mathematically represent the relative frequency of all possible offspring genotypes; and (b) design a systematic breeding program for the geckos that would consistently produce a rare and highly desired genotype as a result. Presented here is a study of how the participating teachers made sense of the mathematics and engineering design applied to the biological process of inheritance, and their reflections on their own implementations of the instructional module. Emergent themes dealt with the limitations of mathematics in teachers’ own biology education, their lack of experience with either engineering or design, and their efforts to help students address similar circumstances.
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Appendices
Appendix A
Protocol for first interview regarding experimental biology unit questions: teachers reflecting on mathematics proposed for inheritance instruction. We realize that experimental content might work for some students and not others. Please tell us the weak points as well as the strong ones.
Category 1: Personal Justification for Increasing Mathematical Exposure/Awareness/Mastery in General Studies, and in Biology Specifically
What math are you comfortable using off the cuff? Is the math you’re using for the unit inside or outside your zone of comfort? [prompts: algebra and variables; geometry and progressions]
In your opinion, what place does math have in biology instruction? [prompts: on a continuum from good to neutral to bad, say, or with good being an important tool for understanding biological processes and their range and limitations]
In your opinion, what place should math have in biology instruction?
You can think of these next questions as ones of did you: learn and then retain the math through reuse; learn and then forget from disuse (certainly my case); or were you never exposed to it?
How was math used to define inheritance concepts when you were:
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A student in secondary school and university
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Learning to teach
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Since you’ve been at the present school [prompt: depending on who sets policy, well-defined administrative or departmental item?]
The National Research Council says this as part of its framework: Mathematics serves pragmatic functions as a tool – both a communicative function, as one of the languages of science, and a structural function, which allows for logical deduction. Mathematics enables ideas to be expressed in a precise form and enables the identification of new ideas about the physical world. Does that support how you feel about introducing math into biology? (2012, p. 64) [National Research Council of the National Academies (2012). A framework for K-12 science education: Practices, crosscutting concepts, and core ideas. Washington, DC: National Academies Press.]
Does this support what textbooks show or say about use of math in biology?
How did you use math in inheritance instruction before BLOOM? For example, did you use math to explain, calculate, or verify inheritance concepts for yourself before BLOOM?
Was it necessary for you to relate the math you used then to actual biological concepts and processes, or was it sufficient to find a reliable widget for calculation (e.g., a Punnett square) without investigating its limitations as a representation of a biological processes such as independent segregation, independent assortment, gamete formation?
Did you use math on any assessments when teaching inheritance before BLOOM implementation?
Category 2: Reflection on Interaction with Unit Content
When did you need to rely on math during the implementation: can you remember when math was helpful or any times when it was harmful to students’ progress or understanding? [prompts: defining combinations; making combinations; counting combinations; predicting combinations, comparing combinations expected theoretically versus observed empirically]
Did you recognize any difficulty that the materials introduced or made worse, that might have gotten in the way of student understanding?
Did you include any items related to math on assessments subsequent to the implementation, and why?
Do you anticipate any circumstances that would cause you to include such items or revise the structure of your exam? [prompts: response to standardized testing of science, administrative or departmental directive]
How do you make sense of the concepts and the sequence of presenting rules in the BLOOM materials? [prompt: inheritance, combinations, expression, design as plan with scientific explanation]
Appendix B
Protocol for second interview regarding experimental biology unit questions: teachers reflecting on math proposed for inheritance instruction. We realize that experimental content might work for some students and not others. Please tell us the weak points as well as the strong ones.
Category 1: Triangulation of Data Analysis
Please look over the section for which your pseudonym is indicated. What do you think is inaccurate?
How would you change that to be accurate?
Category 2: Self-assessment Using the Design Challenge
At what stage of the implementation did you understand what the design challenge was asking students to do? [prompts: professional development, review on my own, while helping students, never really sure]
At what stage of the implementation did you feel confident in answering the design challenge yourself?
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Cox, C., Reynolds, B., Schuchardt, A., Schunn, C. (2016). How Do Secondary Level Biology Teachers Make Sense of Using Mathematics in Design-Based Lessons About a Biological Process?. In: Annetta, L., Minogue, J. (eds) Connecting Science and Engineering Education Practices in Meaningful Ways. Contemporary Trends and Issues in Science Education, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-319-16399-4_14
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