#SlowMath: looking for structure
and noticing regularity in repeated reasoning
How do we provide opportunities for students to learn to use structure and repeated reasoning? What expressions, equations and diagrams require making what isn’t pictured visible? Let’s engage in tasks where making use of structure and repeated reasoning can provide an advantage and think about how to provide that same opportunity for students.
Here’s Jill’s sketch note of our plan:
Dave Johnston (@Johnston_MSMath) recorded his thinking and learning and shared it with us via Twitter.
Crossposted on Experiments in Learning by Doing
]]>Using technology alongside #SlowMath to promote productive struggle
How might we shift classroom culture so that productive struggle is part of the norm? What if this same culture defines and embraces mistakes as opportunities to learn? One of the Mathematics Teaching Practices from the National Council of Teachers of Mathematics’ (NCTM) “Principles to Actions” is to support productive struggle in learning mathematics. We want all learners to make sense of tasks and persevere in solving them. The tasks we select and facilitate must offer opportunities for each learner to develop connections and deepen their conceptual understanding.
Join us to learn more about #SlowMath opportunities that encourage students to persevere through challenging tasks instead of allowing their struggle become destructive. This session will address:
Here’s the agenda:
8:30  Introductions 
8:40  Intent and Purpose

8:45  321 Bridge Visible Thinking Routine 
8:50  Using Structure to Solve a Task – CircleSquare Task 
9:55  321 Bridge Visible Thinking Routine

10:00  Construct a Viable Argument to make your thinking visible: Does (x+1)²=x²+1?

10:25  321 Bridge Visible Thinking Routine

10:30  Close 
Here’s Jill’s sketch note of our plan:
Dave Johnston (@Johnston_MSMath) recorded his thinking and learning and shared it with us via Twitter.
And, a little more feedback from Twitter:
Crossposted on Experiments in Learning by Doing.
]]>In my Qualitative Research class, we watched Big Fish to think about the difference between perspective seekers and truth seekers. We have been talking about the difference between truth and reality.
How do we teach our students to construct a viable argument and critique the reasoning of others? It takes time to distinguish truth from fiction. It takes time to figure out from whose perspective something might be real. It is faster to refute claims without asking why, when, how.
In “Trial Before Pilate” from Jesus Christ Superstar, Pontius Pilate ponders “But what is truth? Is truth unchanging law? We both have truths. Are mine the same as yours?”
How do we teach our students to construct a viable argument and critique the reasoning of others? It takes time to determine the conditions for truth. It is faster to assert a statement as always or never true than figuring out whether and when it is sometimes true.
Dan Meyer, Shira Helft, Juana de Anda, and Fawn Nguyen presented the CMC North keynote in December 2016. I would encourage you to watch the whole talk. If you are a beginning teacher and/or you support beginning teachers, you might be particularly interested in Shira’s part. I am particularly interested in Dan’s question: How do we help people believe fewer lies?
How do we teach our students to construct a viable argument and critique the reasoning of others? It takes time to distinguish fact from fiction. It is faster to press share without checking primary sources.
In Visible Learning for Mathematics, Grades K12: What Works Best to Optimize Student Learning, Hattie et al. assert the importance of expecting students to engage in accountable talk in our classrooms and emphasizes the role that the teacher plays in ensuring that happens. Teachers must consistently exemplify accountable talk. The authors share examples of Accountable Talk Moves for teachers to relentlessly use in conversation with students, with the expectation that students, too, will assimilate into accountable talk in their conversations with others (2017, p. 144).
In a Slow Math classroom, we take time for students to learn to construct a viable argument and critique the reasoning of others. Even though it is faster not to.
Possible Resources for Continuing the Conversation:
Hattie, J. (2017). Visible learning for mathematics, grades K12: what works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.
Interpreting Data: Muddying the Waters
Why Facts Don’t Change Our Minds, from the New Yorker, and referenced in Dylan Kane’s blog post On Changing Minds.
Teaching Why Facts Still Matter, included in the January 31 2017 edition of the NBCT Accomplished Teacher by SmartBrief.
For EdTech Company Newsela, ‘Fake News’ a Big Challenge – and Opportunity, included in the February 2 2017 edition of Education Week Digital Directions.
]]>Dylan Wiliam’s assertion from Embedded Formative Assessment resonated with me and the teachers with whom I work: “Sharing highquality questions may be the most significant thing we can do to improve the quality of student learning.”
When your team plans together, plan questions to ask. When you find the question that makes a difference in knowing what students are thinking, don’t keep it to yourself – share it.
As I continue to teach, though, I’ve decided that my most important work happens during the lesson – in the moment – making decisions about what to do and ask next based on how students respond.
I’m reading Hattie, Fisher, and Frey’s Visible Learning for Mathematics. I paused when I read the following paragraph.
Slow Math isn’t just for students. It’s for teachers, too.
“Give yourself permission to stop and think about the ‘right’ question to ask at any given point in the lesson.”
I’ve also heard this advice from Elham Kazemi in the form of teacher time outs. Team teaching is such a good opportunity to practice good questioning. Even if you’re alone, though, give yourself permission to take a teacher time out. Slow Math is taking time to think about the “right” question to ask.
Hattie, J. A., Fisher, D. B., & Frey, N. (2016). Visible learning for mathematics, grades K12: what works best to optimize student learning. (p. 112). SAGE Publications. Kindle Edition.
Wiliam, D. (2011). Embedded formative assessment. (p. 104). Bloomington, IN: Solution Tree Press.
]]>I ran across Coyle’s blog post recently: There are Two Types of Coaches. Which are You?
Coyle offers a few statements for us to consider to figure out which type. I’ve taken the liberty of replacing the people/workplace/other language with teacher/classroom/student language.
So what do you think? Is your focus as a teacher on building skill? Or is your focus as a teacher on building students?
Would your students agree with you?
See Coyle’s blog post to find out your official results on his unofficial quiz and check out The Talent Code to read more about becoming a master coach – a builder of people.
]]>… We might lift up the teacher as an example of patience. A good teacher knows that finally you just can’t impose the answer in a student’s brain as much as you might want to. You have to wait for that student to do that work herself, or not. This is tough, tough work, but finally, there can be no hostile takeover of the mind and will of a student. Learning is voluntary; it’s not mandatory. You have a classroom discussion, and you hear a “wrongheaded answer” (Kenneson). You want to jump in and fix it. But you might kill the thing that is fermenting there if you rush it. You cannot take over that process. You can only make the invitation, and then wait to see if the student will do the work and make her own connections. Teaching takes patience, or it’s not teaching …
]]>Let’s take a look at a round between SO and SA. SO selected a hexagon. SA asked:
SO answered no.
SA eliminated one.
SO answered no.
SA eliminated two.
SO answered no.
SA eliminated three more.
SO answered no, and SA eliminated all but one.
What a great way for students to learn how to practice MP6: attend to precision. In a whole class discussion, we talked about what it meant for a polygon to be regular. We talked about convex and concave. We talked about symmetry. It turns out that the hexagon SO chose actually does have rotational symmetry – it just didn’t have line symmetry like the rest. My students and I have so many opportunities to learn from each other when we take time to slow down, share our thinking, and listen to other’s thinking.
After a round of Polygraph last year, one student reflected that he learned that he could ask questions to find an answer.
Which has me thinking more about Slow Conversations. The Polygraph practice round celebrates the beauty and diversity of all of our students.
How might we teach our students to embrace that diversity by not only asking questions to identify and learn about each other’s uniqueness but also listening to each other’s responses? That’s where Slow Math intersects with Slow Conversations.
]]>Slow Math is about asking questions. #AskDontTell is one hashtag I regularly use that describes my teaching. But how often does my perspective make me think more about the questions I ask than the questions my students ask?
e e cummings wrote,
always the beautiful answer
who asks a more beautiful question
In “A More Beautiful Question”, Warren Berger tries to figure out why children start school asking hundreds of questions a day but then their questioning “falls off a cliff” as they go through school.
In a Slow Math classroom, questions are not only welcomed – they are sought.
]]>Slow Math is not just about how to do math … it’s about valuing why the math works. I am grateful to work with teachers who agree. And I am grateful for students who recognize and appreciate that why is valued.
]]>One question I have is whether student voice = student choice.
I wonder what student voice in a math class has to do with slow math.
I looked back at a lesson from last year when students had the opportunity to select the task they wanted to work.
We were practicing show your work, and we used Jill’s leveled learning progression to monitor student progress.
Students looked at the first task (Circles and Squares) and noted what they wonder about how the figures are related to each other.
We watched Dan’s video for the second task (Some Really Obscure Geometry Problem), and I sent a Quick Poll to collect their best estimates of the area percentage for each region.
The third figure came from the Mathematics Assessment Project, but it is no longer available.
All of the tasks provided students the opportunity to practice MP7 look for and make use of structure and think about area ratios in figures.
Each team selected the task they wanted to spend more time working.
Students had choice in geometry that day. Did students have voice?
Was that a class period well spent, even if we didn’t synthesize ideas as a whole class? Would it have been better (and worth the time) if students have reviewed the work of those who worked on a different task? What difference does providing #slowmath student voice opportunities make for students?
And so the journey continues …
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