My New Favorite Term: Abductive Reasoning

What is Abductive Reasoning?

I’m going to share my new favorite term:  abductive reasoning.  Maybe you’ve known about it for years and never told me about it.  (If that’s the case, you might be a jerk.)  Or maybe it’s new to you too.  (If that’s the case, let me know because I’m a little embarrassed I haven’t learned about abductive reasoning until recently.)

To recap, deductive reasoning is about making specific conclusions from general statements (like a math proof).  Inductive reasoning is about making generalizations about specific observations (like a science experiment).

By comparison, abductive reasoning is about making your best prediction based on incomplete information.

Abductive reasoning?!?!?!  Where have you been all my life?  Welcome to my lexicon.  Have a seat front and center and let’s talk.

What is Argument? And Emerging from the Rabbit Hole

Welcome back math geeks!  Last week, I was preparing for a workshop facilitating the learning of a dozen elementary teachers as they prepared for a 6-week interdisciplinary math/science summer academy.  I chose to focus their thinking on “argument.”  More specifically, I wanted teachers to internalize how making arguments based on reason and evidence is a […]

Access, Equity, and the Bias of Brilliance

A Confession

As a student, math class was mostly about taking notes, practicing algorithms and getting right answers. And I loved it. I really did. I could hide. It was safe and without risk. I was comfortable. “I do; we do; you do” was my jam as a student. More importantly, in this teaching style, I identified as a really strong and talented math student because I learned algorithms and got right answers. That’s what good math students do in class, right? I never had to share my thinking, never had to learn from the thinking of others, never had to challenge the dimensions of my adolescent bubble of insecurity. I never had to share or listen to a classmate share an interesting math question or an elegant solution. I never had to be vulnerable.

When I started teaching, I created the same classroom culture. I was safer as a teacher as well. I could hide. Never take risks. Be comfortable. I was the author of culture in my classroom. They were the factory workforce with one job to do: consume the algorithm and produce right answers. And my teaching aligned to this goal. Employing this style of pedagogy created a silent pact in my classroom: If you sit there and do your job and let me do mine, I will never ask you to take a risk, to challenge the dimensions of your adolescent bubble of insecurity, to share anything unique or interesting about you or your thinking. Time will pass and so will you. We might be bored, but we’ll get through this together without ever knowing each other or our vulnerabilities. Now let’s open up our textbooks and get to work.

Why Lesson Study?

Imagine a football team operating like a faculty at a school site.

Players (teachers) gather at the beginning of the season (school year) for some team-building and some pep talk (fall PD) about goals and visions for improvement from their coach (principal).

The players then study a playbook (curriculum) and some plays (instructional strategies) and maybe they practice them. Maybe they don’t. Once the season starts, they hardly ever observe each other run drills (routines). They definitely don’t scrimmage together. The coach may walk around once or twice, check some boxes on a list, and give that feedback to a player, but rarely does the coach model techniques or facilitate collaboration and discussion between players.

The players practice all year for one single game (student testing) that they don’t even believe is worth playing but everyone makes them prepare for it anyway because how else could we measure our effectiveness except through standardized test data. The players won’t find out until 4 months later how they did and how they compared to other teams (schools) in the league (district). Except by then, the offseason has happened, players have shifted teams, new playbooks have been adopted, perhaps new coaching has been hired, and it’s time to start the whole process again.

The season concludes without any player ever watching another player play.

How stupid is that?

Numberless Word Problem 3: Data Exploration in 2nd Grade

I worked with a team of amazing 2nd grade teachers this week as a part of an ongoing lesson study. They were in the latter chapters of their curriculum where the Measurement and Data content is often stuffed away as an afterthought because they aren’t “Focus Standards.”

And it’s a drag too because there’s so many rich opportunities for meaningful student discourse about data. That is, if it’s done right. Most textbooks suck all the life out of the content. Students need to understand that data tells a story; it has contextual meaning that is both cohesive and incomplete. Students need to learn how to ask questions about data and to learn to identify information gaps. In other words, students need to learn to be active mathematical agents rather than passive mathematical consumers.

We’d like to share with you what we learned about using Numberless Data Problems and crafting an open investigation into bar graphs that is engaging for all students. As always, feedback welcome. Let’s get better together.

Numberless Word Problem 2: Data Exploration in 1st Grade

Have you seen the amazing visuals over at Number Talk Images? These pictures are ideal for any teacher looking to get all students talking about numbers and mathematical reasoning, regardless of ability levels. We used this image as a number talk to launch a lesson that focused on first grade students making statements about a data display. Inspired by the work by Brian Bushart and Regina Payne, we used a numberless word problem approach to build and structure discourse about a data display.

I hope that there are other 1st (and 2nd) grade teachers out there that might find this analysis useful if they are looking for strategies to get students talking about their mathematical thinking. We wanted students to produce mathematical thinking, not just consume it. Here’s what we created.

The Objectives of Objectives, Part Three: Joy

One of the reasons why we teach is because we want our students to experience the JOY of mathematics. Mathematics should be about questioning, wondering, and the joy of discovery…and math classes should leave students wanting to know more math and do more math thinking. We cannot build an appreciation of math through content standards alone. Math classes should be filled with opportunities for students to have voice and a choice. At the very least, they need a voice in making meaning of problems and a choice in how they go about seeking a pathway to a solution.

But sometimes we (or our textbooks) squash all the joy out of a math lesson. We rob them of their right to notice math things, wonder about math ideas, or do messy math stuff. And lessons that focus on “measurable outcomes” with “explicitly defined objectives” often euthanize mathematical curiosity.

The Objective of Objectives, Part Two: Headache First

Welcome back Math Geeks! I’ve been thinking a lot about the objectives of lesson objectives, and I’m committing to writing a series posts to spark a conversation. I’m curious about how you frame lesson objectives to maximize student thinking, and I invite you to tell me. If you missed my first post about ways to make the learning objective an invitation and not a mandate, I encourage you to check it out here.

Dan Meyer has written extensively about the importance of creating intellectual need in the mathematics classroom. If we are going to ask students to use mathematics to solve problems, we need to let students internalize problems through inquiry and exploration BEFORE we teach them the mathematics. As Dan suggests, if math is the aspirin, then how do we create the headache?

The Objective of Objectives, Part One: Invite

Why we teach must align with how we teach. In other words, our professional identity (the reasons why we became teachers in the first place) must be congruent with our choices and our practices in the classroom. When purpose and practice are not aligned, both teachers and students waste energy fighting needless friction in the classroom and learning suffers.

One habit where I continue to see a misalignment between purpose and practice centers on how we post, frame, communicate lesson objectives to our students. All teachers want students to be inspired, motivated, engaged, and curious, yet I’ve witnessed a lot of teachers euthanize student intellect by spending the first 5 minutes of a lesson reading aloud and unpacking a lesson objective that is written on the board.