This document discusses strategies for helping students construct mathematical arguments and critique the reasoning of others. It introduces the idea of having students determine whether mathematical statements are always, sometimes, or never true and provide arguments to support their categorization. Video examples show students attempting to prove statements about addition and the equal sign. The document suggests moves like examples, counterexamples, and describing concepts without numbers that help build convincing arguments. It encourages teachers to have students practice these argument skills and reflects on how the strategies support mathematical standards and practices.

1. Constructing Mathematical
Arguments
Jana Banks
Karen Brown
Lucy Marincel

2. Purpose
- Students were explaining their thinking but it felt like something
was missing.
- How can we support students to push past the word “efficient?”

3. Purpose
• What could help students construct viable
arguments and critique the reasoning of
others (SMP.3)?
• How can we make the “prove it” more
natural?

4. Always, Sometimes, Never

5. Mathematical Statements
• Statements we tried on:
– When you add, you can take an amount from one
addend and give it to the other without changing
the sum.
– The equal sign means the answer comes next.
• As you watch these videos, think:
– How are students trying to prove their thinking?
– What student moves support SMP.3?

6. When you add, you can take an amount
from one addend and give it to the other
without changing the sum.
*Audio clips of students have been removed.

7. When you add, you can take an amount
from one addend and give it to the other
without changing the sum.
*Audio clips of students have been removed.

8. Modeling of an Argument
The equal sign means the answer comes next.
What do you notice? Wonder?

9. Supporting SMP.3
• How are students trying to prove their thinking
• What student moves support SMP.3?
• A few moves our students have already combed out
that help convince others:
– Examples
– Counterexamples
– Describe what is happening without using numbers
– Background information
• Other thoughts?

10. Try it out!
• Please choose geometry and/or fractions. The point
isn’t to rush but to develop rich arguments and try to
convince your partner if and when you disagree.
• Work with a partner and decide where to tape your
statement.
• Feel free to write all over your paper!
• Note: This work is messy! We may not know if we
are right or wrong.

11. Think back to standards
• How does this work support SMP.3?
– Create viable arguments and critique the
reasoning of others.
• How does this work support SMP.4?
– Model with mathematics.
• How does this work support SMP.7?
– Look for and make use of structure.

12. Teachers as Learners
• How did this feel?
• What struggles did you have?
• What statements were tricky? Easy?
• What helped you to support your opinion?

13. Reflections
• How could this be used in your classroom?
• What are you excited about?
• What did you notice about your own thinking?
• What did you notice about your partner’s
argument?
• What are you still wondering about?
• What are you curious about?
• Can this concept be used in other content areas?

14. Resources
• There is a crowd-sourced list on Google Drive of Elementary Always,
Sometimes, Never statements. I can share with you!
• Includes resources of how other teachers are implementing this
work in their classrooms.
• Jana Banks
• Karen Brown
– karen.brown.dcps.4@gmail.com (send emails here for
resources)
– Twitter: @kiby33
• Lucy Marincel