This blog is one of five posts for the CTL book study, FOCUS IN HIGH SCHOOL MATHEMATICS REASONING & SENSE MAKING. To date, We’ve looked at (see other posts): Linking Expressions and Functions, More Than Meets the Eye (Example 9, pages 38-40) – a resource for promoting mathematical practices that relate to attending to precision, constructing viable arguments, and critiquing the reasoning of others; and Examples 5 & 6 from that same Chapter as they relate to Distribute Thoroughly and Horse Shoes in Flight – an application of the quadratic function.
For Chapter 2 of FOCUS IN HIGH SCHOOL MATHEMATICS REASONING & SENSE MAKING – ‘Reasoning Habits’ discusses an instructional structure for supporting a process of thinking that develops over time and assists students in understanding and using the mathematics needed for the 21st century. It is a way of thinking about the mathematical situation/problem. The recommended progression of thinking as students approach a problem is: analyzing the problem, implementing a strategy, seeking & using connections, and reflecting on a solution – this process for me is nonlinear and triangulates a way of thinking about a problem.
Within the Chapter, is a listing of instructional prompts/ examples for curriculum integration of these reasoning habits as students solve problems. What I like the most is the list of ways to provide engaging tasks and questioning that create the opportunity for students to develop/practice mathematical reasoning habits. Many of the suggestions we have written about and are similar to other CTL blogs (See – Implications for Modeling and Reasoning from Common Core State Standards for Mathematics and Response to Intervention Research, Series #4: Recommendation 3 – Model problem solving – Through verbalizing thought processes).
Within this list of tips are suggestions like: require students to figure things out (without a lot of scaffolding); ask students to verify their answers through prompts such as, how do you know, can you show me another way; provide wait time allowing for students to process and formulate a response; create an environment where it is safe for student risk taking.
The Chapter also lists the strands of mathematical proficiency and competence (conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition) from ADDING IT UP, National Research Council. The interdependence of these strands only further undergirds the importance of engaging in mathematical reasoning and sense making for critical content understanding. Again, we have research that supports that students cannot learn content unless they engage in mathematical reasoning and sense making.
In viewing Chapter 2 of FOCUS in HIGH SCHOOL MATHEMATICS REASONING and SENSE MAKING, “Reasoning Habits,” I immediately connected to several useful instructional resources, one of which is FOSTERING ALGEBRAIC THINKING, a GUIDE for TEACHERS GRADES 6 – 10, by Mark Driscoll. Much like FOCUS in HIGH SCHOOL MATHEMATICS REASONING and SENSE MAKING, FOSTERING ALGEBRAIC THINKING also guides us through developing habits of mathematical thinking that support reasoning, discourse, and problem solving – those “habits of mind that develop as the thinker pays attention, over and over again, to what works.” (Note that the publication date for FOSTERING ALGEBRAIC THINKING is 1999 and is as timely a publication as the more recent FOCUS in HIGH SCHOOL MATHEMATICS REASONING and SENSE MAKING – they both lend instructional support for helping students garner those skills necessary for thinking like a mathematician.)
Additionally, NCTM’s PRINCIPLES and STANDARDS for SCHOOL MATHEMATICS, published in 2000, is also as timely with the “reasoning and proof” Standard for Grades 9 -12. That Standard along with the connecting Standards of problem solving, communication, connections, and representations provide those opportunities for students to develop the process of thinking like a mathematician rather than one who memorizes and regurgitates algorithms only to be forgotten, by most students, after the chapter test is over. The quote that resonant with the instructional practices in FOCUS in HIGH SCHOOL MATHEMATICS REASONING and SENSE MAKING is:
As in other grades, teachers of high school mathematics should strive to create a climate of discussing, questioning, and listening in their classes. Teachers should expect students to seek, formulate, and critique explanations so that classes become communities of inquiry. (pp346)
Lastly, on a colleague’s wall was a poster from National Geographic listing the ‘strategies good readers use.’ Are these reasoning/thinking skills familiar? What connections can we make to the ‘reasoning habits’ of thinking like a mathematician?
Maybe we should combine reasoning habits, habits of mind, and strategies of good readers to
Habits That Good Thinkers Use To Make Sense of the World.
What are your reactions/connections to Chapter 2, “Reasoning Habits?”