I started reading my e-copy of the book last night and was struck its seemingly simple goal, to make reasoning and sense making a cornerstone of high school mathematics instruction. Content isn’t the goal with this book; instructional approach is.
As I quickly looked through the contents of the book, I saw the intentional discussion of reasoning in all the high school content areas; number and measurement, algebra, geometry, and statistics. Yes, we can reason in all areas of mathematics and more importantly we should! The approach the authors took seemed very comforting to me: laying out a theoretical framework in section 1, providing extensive examples in section 2, and detailing issues of implementation in section 3.
The progression is very typical for mathematics teachers, because we want to know the theory, but we crave to see the examples that accompany the framework. It is now up to us to identify the importance of the approach to instruction, understand the goals of the examples, and apply those to all of the activities in our classrooms, not just these examples. I look forward to journey of understanding the framework better, and developing routines that I can work with teachers to implement.
Let’s go! The theoretical framework starts as it should by defining reasoning and sense making:
Reasoning: “…in mathematics is often understood to encompass formal reasoning, or proof, in which conclusions are logically deduced from assumption and definitions. However, mathematical reasoning can take many forms, ranging from informal explanation and justification to formal deduction, as well as inductive observations.”
I get that we are looking at the progression of logical arguments with reasoning but it’s the last piece of this quote that I think has HUGE implications for high school mathematics instruction. We don’t start with formal reasoning, but we start with observations. I don’t think the authors are arguing that reasoning is limited to mathematical induction as much as they are arguing that we have to have students making observations in mathematics to begin with and develop their skills from there.
I have long felt that students have the ability to make observations if we let them. Besides, if we wait until students have a strong enough background in underlying structures then students will never get to make observations. There will always be more and more things they need to “know” before they can make observations about the new content they are studying today. I love that the book espouses making observation a corner stone of instruction in the mathematics classroom from day 1!
It doesn’t say or mean to say (I think) that we stay there, but that we start there and develop student capacity to develop more and more complex observations and arguments along the continuum of their high school career, and this quote from the National Academies book, “How Students Learn: Mathematics in the Classroom” came to mind, “Knowledge-centered and learner-centered environments intersect when educators take the idea that students must be supported to develop expertise over time; it is not sufficient to simply provide them with expert models and expect them to learn.” If we want them to learn effectively we have to have a dual focus on content and the structures for learning.
Sense Making: “…define sense making as developing understanding of a situation, context, or concept by connecting it with existing knowledge.”
This may be one of my favorite quotes of the book so far (I’m only on page 4). It is simple and plays on everything that I know about how we learn. We have to pay attention to student preconceptions and begin instruction with what students think and know.
Connections to Problem Solving and Proof
The authors quickly connect reasoning and sense making to problem solving and proof. When a student makes mathematical decisions while solving problems or making proofs they must make connections between concepts, represent their understandings, and communicate their thoughts.
What strikes me about this discussion is that it is predicated on students communicating their ideas effectively, which is a critical component to reasoning and sense making. We have to create structures in our classes were students are communicating with each other, and with us, but first and foremost they are communicating in a variety of situations and in a variety of methods.
If we are going to develop students who can reason mathematically, we are going to have to develop their abilities to communicate mathematically verbally and in writing.
I absolutely loved Chapter 2 Reasoning Habits.
The list of reasoning habits that get the chapter started was impressive and a model for what I would like to see in any classroom. It isn’t about just having conceptual knowledge or procedural knowledge, but the overlap of the two and being able to apply appropriately to any given situation, and it is important that “…students are expected to move naturally among various reasoning habits as they are needed.” I read that as we need students with flexible knowledge.
I don’t think students will be able to accomplish that on the first day of class, but we have to make that the goal for the day they graduate from our high schools. With that in mind, I need to think about what habits of mind I want to develop in my students from day 1 of class their freshman year, and map out those instructional goals.
Additionally, I need to think about how I enable students to work through the progression of reasoning from empirical to preformal to formal, and I don’t believe that it is a linear progression. There will be considerable backtracking as concepts get more complicated students will need to regress back to empirical observations but they can and need to be expected to provide formal reasoning throughout their high school career. That is not a trivial expectation. It is a daunting task to consider as a classroom teacher but a worthy one.
The authors outline several tasks that can enable students to develop reasoning habits in the classroom. Here are a few that resonated loudly with me:
- Provide complex tasks that require students to figure things out for themselves (Give students tasks worthy of their time and efforts)
- Ask students questions that will prompt their thinking and expect them to make that thinking visible as a constant expectation (when students struggle don’t give them the answers, but give them questions that help them communicate their thinking, make it about their thinking and not the thinking of the well intentioned teacher)
- Provide adequate wait time after a question for students to formulate their own reasoning (if we ask simple questions then we don’t need as much wait time, BUT if we ask complex questions and want more complex answers, then we have to provide students with more time! Not anything new, but just a good solid idea)
The chapter finished with descriptions of how important it is that we provide opportunities through statistical analysis, problem solving and modeling, and the use of technology that develops our students’ abilities to reason for themselves, and to make sense of situations and communicate their understanding effectively to us and to others.
I’m looking forward to reading the rest of this book and hope you’ll push my thinking here. Jo Ann and I are going to be flexible in how we organize the book study and are not in a hurry to plow through the book, as much as we are hoping to develop conversation about the merits, intricacies, and values of making reasoning and sense making a corner stone of instruction in a high school mathematics program.