Note: This post is a third in a series about my learning and thinking about argument and how it relates to our work in the math classroom. Click here to read the first post. Click here to read the second.
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.
Why Abductive Reasoning Matters
Just to clarify my own understanding, reasoning abductively is what we do when we don’t have enough evidence/information/data to reasoning inductively.
In my last post about argument, I argued that inductive reasoning belongs in the math classroom even if pure math arguments are deductive in nature. Lessons should be designed to invite students to inquire and investigate math concepts first. From this investigation, student intuition and observations can be used to generalize their thinking into more formal procedures.
I’d like to add abductive reasoning into the mix because I think it encapsulates a lot of what we in the #MTBoS community are trying to create: learning opportunities that compel students into making predictions or decisions based on incomplete information.
The human brain is a predictive machine. It’s drawn to missing information and wants to fill it in. It wants to make meaning of the world it lives in. It is engaged in learning when it perceives something it does not know, recognize, understand and realizes a genuine need to know it.
But most print textbooks don’t leverage abductive reasoning well. Like, not at all. Dan Meyer and others in the #MTBoS community have been talking about this for years. Textbooks often offer problems where all the needed information is given, no unnecessary information is added, and a pathway to solution has been scaffolded for them. Teachers teach the textbook. Students complete the workbook. And we try to do math without ever doing much thinking and wondering.
We need to reclaim the vitality of math learning from the lifeless grip of math textbooks. And incorporating opportunities for students to reason abductively should be a pedagogic routine and a lesson design principle as teachers work to liberate their teaching from the textbook page.
And this is what #MTBoS members (and others) have been doing for years and why our work engages students in mathematical wonderings, discourse, and explorations before formalizing understanding of structures and algorithms.
Leveraging Abductive Reasoning
So, how do we remove information from math problems so students are encouraged to make predictions, offer estimations, and reason intuitively? As Dan Meyer suggests, how do we become less helpful? Here are three examples of work from the #MTBoS world that showcase the power of using abductive reasoning to engage students in mathematical reasoning.
3-Act Math lessons
Act 1 in a 3-Act math lesson is designed to compel students to use abductive reasoning to wonder about a visual, ask questions, identify missing information that might be useful, make predictions, and develop their understanding of a problem through inquiry and structured, but also informal, discourse.
Think about Andrew Stadel’s iconic File Cabinet task. Here’s the Act 1 video.
A textbook might offer the same surface area problem that looks like:
A sticky note is 3 inches square. A file cabinet measures 18 inches by 36 inches by 72 inches. How many sticky notes would it take to cover the lateral sides and the top of the file cabinet? (Note: The file cabinet is a rectangular prism.)
Andrew’s approach invites abductive reasoning. Brains are excited and are compelled to ask: How many sticky notes will it take? And from there a need for clarity about the problem and an avenue to inquire about missing information.
The textbook approach just sucks.
(Want to know more about 3-Acts? There are a lot of folks that have amazing 3-Act Math lessons. You can find some of my favorite folks on my home page. You can find my small bank of 3-Acts and useful background information here. Check out Dan Meyer’s 3-Acts for middle and high school students here and Graham Fletchers 3-Acts for elementary students here. )
Numberless Word Problems
Here’s a way to start thinking about a word problem: Suzy needs some apples to make a few pies. What do you wonder?
Here’s another way to approach that problem:
Here are two pictures to get students talking about data and asking/answering questions.
Here’s another way to get students to try and talk about data:
The textbook problems above miss the opportunity to let students understand the problem and reason about a solution using abductive reasoning. The numberless presentation of problems invite students to fill in the missing information and leverages abductive reasoning to promote student thinking and engagement.
(Brian Bushart and Regina Payne have done a lot of work on Numberless Word Problems. Check out Brian’s amazing stuff using images too. If you’re interested, you can find out more about my work with numberless word problems here, here, and here.)
If you haven’t see Andrew’s Estimation180 activities, you should check them out. They’re amazing for leveraging abductive reasoning as students use context clues to improve their estimating skills and number sense.
Here’s a simple example that asks students to put these cups in order from least to greatest in terms of their capacity.
You can’t help but wonder about which of these glasses might hold the most (or least) water. And indeed, there’s no firm evidence to use. We must fill in the information by making thoughtful guesses and estimations about the dimensions of each glass. Students can start talking about volume without having to formally talk about volume. It makes math thinking more accessible.
An Invitation to You
Help us get better together.
- How do you leverage abductive reasoning in your math classes to engage students?
- What questions do you have about abductive reasoning?
- What opportunities am I missing in the conversation about abductive reasoning?