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.

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.

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.

Knotty Rope 3-Act: Introducing Division in 3rd Grade

This lesson write-up is for teachers who want to engage their students in exploring division reasoning and problem solving strategies (3.OA.2, 3.OA.3 and 3.OA.7). It’s appropriate to use before and/or after students have explored division and allows for many different conceptual approaches to a solution including using repeated subtraction or repeated addition, equal groups with or without manipulatives, number lines, arrays, bar models, and multiplication or division equations to model a real world problem.

This write-up contains a lesson pathway with specific questions/moves to consider, analysis of the opportunities for student learning, and other wisdoms and insights we learned from teaching this lesson as a part of a lesson inquiry.

Give it a try with your own students. And then tell me how it went. Let’s make it better together.

Array-Bow of Skittles and Multiplication Strategies

This lesson write-up is for teaching a two-digit by two-digit multiplication 3-Act Math lesson where students estimate the number of Skittles in a jar before using information and math to find a more accurate estimate. It uses Graham Fletcher’s Array-Bow lesson and while it addresses standard 4.NBT.5, it’s appropriate for 4th and 5th grade students of all levels. The write-up contains a lesson pathway with specific questions/moves to consider, analysis of the opportunities for student learning, and other wisdoms and insights we learned from teaching this lesson as a part of a lesson inquiry.

Give it a try with your own students. And then tell me how it went. Let’s make it better together.

Proportional Reasoning by Jumping Rope

This lesson write-up is for teaching 6th (and 7th) grade proportional reasoning skills (6.RP.1, 6.RP.2, 6.RP.3) using Graham Fletcher’s Rope Jumper lesson. The write-up contains a lesson pathway with specific questions/moves to consider, analysis of the opportunities for student learning, and other wisdoms and insights we learned from teaching this lesson as a part of a lesson inquiry.

Give it a try with your own students. And then tell me how it went. Let’s get better together.

Decimal Division Strategies and Sense Making for 5th Grade

This lesson write-up is for any 5th or 6th grade teacher who wants to have students explore decimal concepts and refine decimal skills while solving an interesting, low-floor investigation. The lesson covers mostly the decimal division standards (5.NBT.6, 5.NBT.7, 6.NS.3) using Graham Fletcher’s Tomato-Tomato lesson. The activity is accessible to all learners and offers multiple approaches to a solution. The write-up contains a lesson pathway with specific questions and instructional moves to consider, analysis of the opportunities for student learning, and other wisdoms and insights we learned from teaching this lesson as a part of a lesson inquiry.

Give it a try with your own students. And then tell me how it went. Let’s get better together.

How Much Money in the Bowl?

This lesson write-up is for any 5th or 6th (or 7th) grade teacher who wants students to explore decimal concepts and refine decimal operations while exploring a compelling task that is engaging and accessible to all learners. Using an image from Andrew Stadel’s Estimation 180 page, students estimate, investigate, and then calculate the value of a bowl full of coins and demonstrate learning for 5.NBT.2, 5.NBT.7, and 6.NS.3. The write-up contains a lesson pathway with specific questions and instructional moves to consider, analysis of the opportunities for student learning, and other wisdoms and insights we learned from teaching this lesson as a part of a lesson inquiry.

Give it a try with your own students. And then tell me how it went. Let’s get better together.

The Powerful Phrase: “…in a way that makes sense to you.” (Part 1)

I had the opportunity to work with several K-5 instructional coaches last week during a one-day workshop on increasing student engagement and discourse in the math classroom.  (My favorite workshop to lead!)  Leadership in the district is making a focused effort to support teachers in their practice of creating classrooms where Math Practice 3 thrives and students are […]