Could it be that the exhausting debate about whether reasoning, problem-solving and conceptual understanding need some serious attention in math education is near consensus? Do we have enough believers in our midst that we can move on from selling the product of a student-centered classroom and now move on to the “how”…while maintaining balance?
After spending 2 days with some of the highest quality math education professionals in our nation, I'm posting my take-aways. It’d be an interesting survey to collect your “noticings” of these from this weekend’s CA Math Council – South.
Blessed to get CMC started with my own session, Promoting Depth through Students’ Representations, we honed in on the intricacies of connecting representations. Using the premise proposed by CCSSM author Phil Daro, in Four Levels of Learning, teachers compared these 2 visuals. What is direct instruction good for...and when?
Inspired by Dan’s talk on Deleting the Textbook, and through a killer Numberless Word Problem adapted from van de Walle that exposed the progression of thinking that leads to the “invert-and-multiply” algorithm for dividing fractions, teachers experienced the student end of the 5 Practices for Orchestrating Productive Discussions. However, no teacher presented their way of thinking. Rather teachers were asked to analyze different aspects of different representations, and tried to infer and interpret what the creator of that representation was thinking, and why.
Hence, we took a deeper dive into examining what conditions should exist in the classroom that would help a teacher determine whether to have the student who created a representation “present” his or her way of thinking to the group, versus having the teacher displaying the representation and asking all students to engage in discourse about the features of that representation, BEFORE having the creator of the representation speak about it.
Can you make the connection between the visuals above and this critical distinction to consider when connecting representations? Who should be talking and when? Can you please dig into the intricate decision-making when connecting students representations that move towards learning new and important mathematics?
The current, and likely temporarily and incomplete, conclusion came as follows:
I buzz over to Tim McCaffery’s session, who’s promoting the 5 Practices through engaging in a system of linear equations task, and exposing to a large group there is a difference between having students present, and asking others to interpret what that person was thinking...ideas we've examined together over the past few months.
Steve Leinwand once again lights up the room, but really re-exposes the importance of an Exit Ticket…marking evidence of students' journey on their progress from conceptual understanding to procedural fluency.
Next, John Stevens and Matt Audrey inspire dozens to take a risk to give choice in students as to how they can demonstrate their understanding…and procedural fluency.
Michael Fenton lays out a case to vary the verbs in what we ask kids to do, with a general sequence of asking them to observe, estimate, sketch graphs, explain, interpret and then become increasingly more formal and precise, but only AFTER students' reasoning and ideas are out.
Bill McCallum clearly emphasized that at the lesson and unit levels, there is a great need to move from informal thinking and reasoning, over time, to the more formal, abstract and procedurally fluent ways of thinking.
Andrew Stadel then goes right at it…and exposes the powerful Math Teaching Practice #6 from Principles to Actions (#nctmp2a) stating procedural fluency happens after conceptual understanding, using estimation, number lines, and clotheslinemath to illuminate that this transition happens "over time".
For the grand finale, Dan Meyer takes the stage…repeatedly emphasizes the need to for us to help students develop the intellectual need for the conventions, procedures and skills that we want kids to have…moving them from very informal and conceptual ideas, to reason that these things (variables, arrays, grids, parentheses) actually help us, not give us more work to do. Concluding that we should be letting kids discover why we have these by briefly experiencing what mathematical life is like without them…HARDER, and do so before using them.
Please take a moment to reflect on my take-aways above, and try to connect them to the chart below from Principles to Actions. Then think about where you are, the teachers you work with are, and how we can keep the momentum moving.
CALL TO ACTION
I propose that we now have many believers in the power and importance of conceptual understanding (first) as a foundation for procedural fluency, but now is the time for us all to dig deep into the intricate “how” to make that the reality using Principles to Actions and the Standards for Mathematical Practice as our foundation.
Then, let's talk some more. In Talking Points style, do you agree, disagree, or are you unsure...and why?