A friend and I were reflecting over a beer at Twitter Math Camp in July about how to get more elementary teachers to attend this amazing conference. (Click here to know more!)
He’s an inspirational colleague with a background in special education at the elementary and middle school level. We were talking about content knowledge. He said, “My ability to teach math has always been limited by my lack of content knowledge beyond middle school.” After pondering a beat, I replied, “Me too.” Knowing my teaching experience, he leaned back with a skeptical smirk and looked askance at me. I continued…
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.
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 […]
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.
I went searching for more knowledge about argument. I fell into a rabbit hole. Help me make sense of this. Do you agree? Disagree? What are your thoughts?
“…there is a difference between mathematical arguments and scientific arguments. The difference is that scientific arguments are always based on evidence, whereas mathematical arguments never are. It is this difference that renders the findings of science provisional and the findings of mathematics eternal…Blurring the distinction between mathematical and scientific arguments leads to a misunderstanding of what science is about.” -NGSS Appendix L
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?
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.
How do we invite 6th (and 7th) grade students to authentically engage with an equation in a way that invites students (1) to appreciate how the structure of an equation models a context and (2) to dive deeper in to the meaning of the relationships between variables?
Instead of teaching students how to use the properties of equality to solve “one-step” equations first (which is like using a bazooka to kill a cockroach by the way), I’m wondering if there’s a way to start the exploration of equation solving by inviting students to experience the dynamic relationship between variables first.
Here’s my thinking on one way to do that. I’d love to hear your thoughts so we can get better together.
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.
I’m on a mission. And I invite you to join me. I’m on a mission to tear calculus down from its ivory tower on the math landscape. Even if you don’t know calculus, you can still join me because this mission is also for you. Here are a few myths that I would like to dispel on […]