Book Study Post 2 as it relates to “linking expressions and functions,” MORE THAN MEETS THE EYE (Example 9, pages 38-40) – a resource for promoting mathematical practices:
#1 – Attend to precision;
#2 – Construct viable arguments and critique the reasoning of others.
Generally, in viewing the 22 examples posted by Focus in High School Mathematics Reasoning and Sense Making I must say that I was drawn to the ones that had scaffolded teacher/student questions/responses. This scaffolding provides a prototype of the kinds of questions to ask, as well as, possible student comments that promote mathematical discourse. Given that the examples are from Number and Measurement, Algebraic Symbols, Functions, Geometry, and Statistics and Probability the mathematical content supports the Core Content Standards while the questioning prototypes promote the Mathematical Practice Standards. All of the examples make connections across the content; some promote inquiry, risk-taking, and metacognition where teachers and students alike make their thinking public; and other examples promote quantitative literacy (see NY Times Opinion ) with real-world applications of the mathematics.
The structure of these scaffolded examples reminded me of one of the most rewarding professional times I have had (both for my own professional growth and student achievement) back in the late ’80s at Fairdale High School using the four part series, Middle Grades Mathematics Project published by Addison Wesley and written by the Glenda Lappan team from the University of Michigan. This National Science Foundation-funded teacher resource helped to develop problem-solving and critical-thinking skills for grades 5 – 8. A team of three colleagues used the series with low performing 9th and 10th graders and experienced significant gains in student achievement. The materials are still timely and certainly promote the content and thinking required of the Core Content Standards.
The reason I recall this experience is that this teacher resource also scaffolds the questions to pose to students and was strategically sequenced through: Teacher Action, Teacher Talk, and Expected Response. The content and reasoning skills were so profound in these resources and for first time instruction, it was very helpful to have this scaffolding provided and to use these actions/talk for promoting conversations about the key mathematical concepts and for making connections. As the FHS team contextually used the resources we began to alter and add to the discourse possibilities.
I see a comparable use of these examples from Focus in High School Mathematics Reasoning and Sense Making where teachers use the teachers’ questions to guide the facilitation and then use their context with teachable moments to alter and add to the questions provided.
For Example 9, pages 38 – 40) MORE THAN MEETS THE EYE:
Within the linking expressions to Polynomial Functions example, the key instructional aspects illustrated that provide opportunities for attending to precision and constructing viable arguments and critiquing the reasoning of others are as follows.
The use of Multiple Representations (multiple representatives – see other CTL Blogs) is critical as students make sense of/visualize the connections amongst the representations – numbers/tables, algebra, graphs, sentences/words. From the teacher/student discourse I noted that students: visualized the intersection points of the two functions; solved the algebraic equations that equates = ; graphed the functions; and verbalized what those points of intersection mean, as well as, other values within and beyond what is visualized.
Attending to Precision is evident throughout by evaluating each function at particular points, graphing accuracy, simplifying the expressions, solving the equations.
The opportunity for constructing viable arguments and critiquing the reasoning of others through mathematical discourse (mathematical discourse – see other CTL blogs) is throughout with the use of the scaffolded questions and responses to questions. Given individual classroom makeups and as teachers use the questions and experience the lesson, teachers will add to, and expand upon the questions. A connecting quote states that, “Building fluency in working with algebraic notation that is grounded in reasoning and sense making will ensure that students can flexibly apply the powerful tools of algebra in a variety of contexts both within and outside mathematics.”
Further extensions to this example would: area under the curve; investigating other functions; contextual questions added to the mathematical discourse.