### How Can (an Why Should) PD Drive Mathematical Justification and

```Julie Fredericks
Teachers Development Group

Definition and Purpose


Challenges & Strategies



What is a mathematical justification and what
purposes does mathematical justification serve in the
K-12 classroom?
What are the challenges that the teachers we work
with face regarding mathematical justification?
Specific examples of tools and strategies we use in
our work
Our Wonderings and Future Plans
An evidence-based premise of the studio work is the
notion that orchestrating productive mathematical
discourse increases students’ opportunities to learn
and, in turn, raises achievement and participation
levels in mathematics.
Embracing this premise requires developing
teachers’ knowledge, skills, tools, and disposition for
building classroom communities of mathematical
discourse.
Leahy, Lyon, Thompson, and Wiliam, 2005; Yackel & Cobb, 1996; HufferdAckles & Sherin, 2004; Stein, Engle, Hughes & Smith, 2008; Weaver & Dick,
2006






Student mathematics achievement will improve if teachers
consistently use research-based instructional practices to
develop both computational fluency and a deep understanding
of mathematics concepts by engaging all students consistently
and effectively in the following mathematical practices:
Providing Explanations
Making Justifications
Formulating Conjectures & Generalizations.
Using Multiple Representations
Engaging in Metacognition
Making Connections
Bransford et al, 1999; Cohen, 1994; Donovan & Bransford, 2005; Franke et
al, 2007; Kilpatrick, 2001; Lotan, 2003, 2006; Stein et al, 2000; Common
Core State Standards Initiative, 2010
Structuring the discussion.
 Find a partner at your table and switch papers.
Review your partners work WITHOUT talking.
 First partner shares their interpretation of other
partners work. Other partner clarifies.
 Second partner shares their interpretation of
first partners work. First partner clarifies.
 Both discuss compare and contrast their
solutions.
Prove that the sum of two odd numbers is even.
After you have solved the problem, think about
what is mathematical justification? At which
points in your work have you provided
mathematical justification.
Structuring the discussion.
 Find a partner at your table and switch papers.
Review your partners work WITHOUT talking.
 First partner shares their interpretation of other
partners work. Other partner clarifies.
 Second partner shares their interpretation of
first partners work. First partner clarifies.
 Both discuss compare and contrast their
solutions.

What are some of the possible purposes that
justification could serve in the K-12
mathematics classroom?
“One hallmark of mathematical understanding
is the ability to justify, in a way appropriate to
the student’s mathematical maturity, why a
particular mathematical statement is true or where
a mathematical rule comes from.”
-Common Core State Standards for Mathematics
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving
them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the
reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated
reasoning.



Teachers recognizing mathematical
justification.
Students recognizing mathematical
justification.
Choosing when to press for mathematical
justification.
Teachers recognizing mathematical justification
 Teachers often struggle with the differences
between mathematical justification,
explanation, and verification.
Providing opportunities for teachers to discuss
mathematical justification.
 “[I am] still trying to understand what justify means in
the context of 8th grade algebra vs. a general problem
solving task vs. high school geometry. Not always sure
when students are verifying vs. justifying.”
 “First grade has been working on word problems. We
want our students to be able to solve a word problem
Student Mathematical Discourse Observation
Tool
 Watch a video clip of a 3rd grader.
 As we watch, we are going to code the student
discourse. Particularly with an eye for
justification, conjectures and generalizations.


5 minutes PTT – Use the provided transcript of
the video to code the student mathematical
discourse from this episode.
At your table, share one place that you
identified a justification, conjecture or
generalization.

Students recognizing mathematical
justification.

When first pressing for justification in the classroom,
students have trouble understanding what you
want.
A Looks Like/
Sounds Like
Chart for
Mathematical
Justification from
a classroom
Structured Student Math Talk


Using the structure to press students to share and
compare their thinking .
Today we used the Interpret and Compare routine
which is outlined on the second page of the tool.

Knowing when to press for justification

Deciding which questions are appropriate for grade
level
 “I am finding that my students can explain how they
solved a math problem, but I am stumbling when it
comes to explaining why (justification). Right now we
have a standing question in our class: Can you
generalize a rule about how diameter and
circumference are related and justify why your rule
would work with any circle.”

Pressing for justification on every question
regardless of mathematical goal of lesson.

Spend explicit time during lesson planning for
mathematical justification.
For which questions would pressing for justification
be most important? Why?
 What does are the acceptable justifications for this


“It seems as though this is starting to become one of the
norms in our math conversations. I can think of at least 4-5
different occasions in collaborative meetings, with different
teachers, that someone said "what would you want kids to
do/show for their justification?" This seems to be becoming
more of a norm.
It is a simple, yet complex, question and really forces the
to know and what will be "acceptable justification." Now, if
we can all calibrate our own definitions of "acceptable
justification" we will really move forward as a school. This is
where I think it is necessary for us to continue to work
together, refine our ideas in order to come to these shared
understandings. This cannot be done in isolation. “

Continuum of justification
 How can we evaluate teachers growth?
 How can teachers evaluate their students growth?
Thank You!!
```