Punch-A-Bunch: A 3-Act Math Lesson on Probability

Welcome back math geeks!  I need your help making a lesson better.
I love Price is Right because many of the games require contestants to make predictions.  This often involves estimating prices of products.  But sometimes contestants have to make choices of a different nature, and these choices are ripe opportunities to think about probability and expected value.  And I love when a fruitful 3-Act Math opportunity presents itself.  (I’ve written about one before here.)

The example I want to share now doesn’t seem to fit a 3-Act format.  Maybe that’s because it’s not truly a 3-Act Math lesson.  But I don’t know what else to call it.  I’m curious about your thoughts on how to make it better. 

Some questions I’m asking: 

Is it too clunky? 
What grade levels will find this lesson useful? 
What concepts/standards does it best target? 
What opportunities did I miss? 
What extensions can be made?

I’m inviting your feedback in the comment section.  Thanks for helping me get better!

Making Data and Stats Matter More in K-6, Part 1

Welcome back math geeks!

I love teaching young students about data and statistics.  And I enjoy finding ways to make data and statistics matter more to young students.  There are two curriculum practices that trouble me about how we teach students to think about data and statistics, especially at the K-6 level.  In this post, I’ll outline one of these troubling practices and my attempt to help to teachers work around this obstacle.

Beyond the Blame Game

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…

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.

What is Argument? And the Rabbit Hole I’ve Found…

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

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.

Seesaw 2: A 3-Act Lesson for 6th Grade Expressions and Equations

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.