Change is happening in Pre-College Mathematics! Pressure is mounting to get students into certification and degree bearing tracks. The GED now demands more conceptual math understanding as well as more algebraic content. How Can Faculty Address These Shifts? After a brief overview of institutional responses, Carren Walker of Collaborative for Ambitious Mathematics presents online resources to support teachers who seek to change both content and pedagogy in their courses, with a focus on active learning and formative assessment and specific examples of tasks and approaches. Watch the Blackboard Collaborate Recording of "Transforming the Classroom through the Standards for Mathematical Practice."

1. SBCTC Webinar: Transforming the Classroom through the Standards for
Mathematical Practice
Carren Walker
May 19, 2014

2. Brief Orientation
• This is on-going dialogue
• This session will be recorded

3. Overview
Review of Changes in GED
• Curricular
• Assessment
Review of Institutional Response
• Structural Change and Curricular Change
What about Pedagogical Change?
• Practices
• Norms
• Important Ideas
• Getting Started
• Content Resources for Assessment and Learning

4. GED shifts in Curriculum
• Algebra – More of it!
• A lot more– graphs, lines, polynomials,
inequalities, radicals…
• Requires foundational conceptual understanding
• Big Idea Understanding Necessary
• Demands Symbolic Knowledge as well as Vocabulary

5. GED Shifts in Assessment
• Conceptually demanding Problems
• Foundational Skills assessed only in the
accomplishing of Problem Solving Tasks
• Calculator Use – YES! (five no calculator problems)
• Calculator on-computer with user guide
• Formula sheet available
• “tips and tricks” will not be enough
• This might feel overwhelming!
LOTS of people and institutions are Working on
Responses!

6. Structural Responses
• Delivery of learning
– Multiple options for the diversity of lives
– Accelerated Programs (Many examples)
– Learning labs with trained tutors
• Re-Define of Basic Skills Math
– High School 21 and I-Best
– Modular Approach (ie. Virginia)
– Side by Side support and Just in Time learning (Valencia)

7. Curricular Responses
Maricopa – Scottsdale College
• Faculty created-Creative Commons
• Attention to vocabulary and constructs
– Multiplication is “copies”
– 5/7 is 5 copies of 1/7
Pathways
Dana Center
Non Stem Quantitative Reasoning/Literacy

8. Mathematical Practices
Standard 1: Make sense of problems and persevere in solving them
Standard 2: Reason abstractly and quantitatively
Standard 3: Constructviableargumentsandcritiquethereasoningofothers
Standard 4: Model with mathematics
Standard 5: Use appropriate tools strategically
Standard 6: Attend to precision
Standard 7: Look for and make use of structure
Standard 8: Look for and express regularity in repeated reasoning
http://www.insidemathematics.org/index.php/mathematical-
practice-standards

9. First Steps..
What could a Practice Standard Look Like?
Demonstrate that:
• Student Thinking Matters
• All student thinking is valued
• Student voices and ideas will be heard
• Ideas will be valued
• NUMBER TALKS
– Public thinking about a do-able problem
– Example of a Number Talk

10. Formative Assessment
• Questions that inform your teaching
• Questions that reveal student understanding and emergent
learning need savvy design.
Example from Dylan William:
Which fraction is smallest?
a) 1/6, b) 2/3, c) 1/3 , d)1/2
Success Rate of 88%
Which fraction is largest?
a) 4/5, b) 3/4, c) 5/8, d) 7/10
Success Rate of 46%, 39% chose b.

11. Shifting Norms: Questioning Patterns
When students share their thinking, the classroom
script changes.
IRE/IRF
Not Working for Students
Probing Into Student Thinking and
Pressing students for ideas.
Feedback instead of Evaluation:
Inquiry: What is half of a fourth?
Response: 1/8
Evaluation/Follow-Up: Right.
Research says when the follow up is
evaluative, student engagement and
class participation lessens.
T: What is half of a fourth?
S: 1/8th
T: How do you know?
S: I just do
T: I want to understand about your
thinking.. Why 8ths?
S: I cut the fourth piece in half
T: How is that an 8th?
S: Now there are eight pieces
T: So why is eight mean eighths?
S: The whole has eight pieces and I
picked one of them. So an eighth.

12. Shifting Norms: Learning Focus
Observed “REMEDIAL PEDAGOGY”
Grubb, et al. (2011) Working Paper 2
Pedagogy Re-imagined
•Parts to Whole
•Procedures
•Rules
•Isolated Ideas
•Repetitive Practice of Skills
•Whole to Part
•Concepts
•Themes
•Context
•Problem Solving using
foundational skills

13. First Steps: Practices in the Classroom
Consider choosing one or two of these ideas to
try out in your classroom:
• Number Talks at the beginning of Class.
• Which of these are Equivalent?
• Card Sorts
• Pattern Recognition
• Meta-Cognitive Questions
• Limiting Scaffolding

14. First Steps..
What could a Practice Standard Look Like?
Equivalent to .3? Yes? No? Explanation
30/100
30%
0.300
.3%

15. Card Sorts
• Students sort cards in to sets of Equivalent
groups (Start with a total of 4 groups of 4
cards.
• Great work for groups of two.
• Ask groups of two-four to verify solutions and
be ready to explain sorting rule.
• Ask: What was easy about the sorting? What
was hard?

16. Scaling Up Card Sorts
• Introduce a card with an intentional error.
• Introduce blank cards. Students fill in.
• Card sorts that have 2 reasonable sorts
possible.
• Have students make card sorts as a study tool.

17. Pattern Recognition
• Number talks with Patterns
• Focus on how students see the pattern
• Values multiple ways of seeing
• Builds to generalization
• What is the 3rd Stage? The 10th, The 100th?
Stage 1 Stage 2 Stage 4

18. Meta-Cognitive Questions
• What would someone have to know to answer this question
efficiently and accurately?
• How confident are you that you can solve this problem? Why?
• What do you understand now that you did not know at the
beginning of the course?
• What advice would you give someone enrolling in this course?
• Which of the questions on this worksheet are similar?
• Write another problem this is like this one. How do you know
they are the same?

19. Limit Scaffolding
• Find out what students know – not what they
don’t know.
• Opportunity for formative assessment. The
instructor observes student knowledge as well
as problem solving dispositions.
• Invite student knowledge and experience into
the classroom.

20. Time in Hours Since Vehicles Started
VolumeofGasinTankinGallons Driver A
Driver B
Driver C
Driver D
What Can you Determine from this Graph?
4 cars started with full gas tanks and drove 60
miles per hour until the car ran out of gas.

21. Questions for Reflection
• What can you Determine?
• What Questions Do You Have?
• What Else Do You Want to Know?
Limit Scaffolding Prompts:
General questions meant to provoke student reflections opposed to calling upon
procedures.

22. Time in Hours Since Vehicles
Started
VolumeofGasinTankinGallons
(0,42)
(4,12)
Driver A
What can you determine given this Data?
Limited Scaffolding Extension 1:
a)Student Private think time. c) Share out in group discussion.
b) Work time with partner. d) Public charting of ideas, knowledge and wonders.

23. Based on what you know about Driver A’s vehicle, add ordered pairs to
the graph that make sense to you.
Time in Hours Since Vehicles
Started
VolumeofGasinTankinGallons
Driver A
Driver B
Driver C
Driver D
Limit Scaffolding Extension 3
This Extension might allow a revisit to the problem or an extension for
student groups who are ready for a new question.

24. Questions for Reflection:
Limiting Scaffolding Wrap Up: A teacher can facilitate the making of explicit
connections. The summary and reflection tasks are the opportunities for students
to reflect and summarize what they know and how they know it.
What math did you use today?
Where did you start in your thinking?
Where did you get stuck?
How did you get unstuck?
What are some general ideas about graphs do you think you
understand now?
How do you know you understand those ideas?
Did you learn any knew math skills while thinking about this
problem?

25. Providing Authentic Relevance
Dan Meyer –A math experience in three acts –
more than watching a video and seeing what
happens. His blog articulates his intentions:
http://blog.mrmeyer.com/2013/teaching-with-three-act-tasks-
act-one/
Examples:
http://www.101qs.com
This website has a selection of searchable “experiences”
Nana’s Milk is an example that requires thinking about
ratios. Search: Nana’s Milk and watch the 48sec clip.
See ACT 2 and ACT 3 on the web site below the video.

26. Pause
• Changing your Pedagogy is Hard, rewarding
work.
• You want a plan: change one thing at a time
• You want a buddy: plan to adopt a classroom
innovation idea with a colleague
• Visit each other’s classroom
• Commit to getting better.
• Remember to consider student buy-in

27. You are establishing
Sociomathematical Norms
• Students ask each other questions that press for
mathematical reasoning, justification, and understanding.
• Students use mathematical arguments when they explain
solutions.
• Mistakes are opportunities to rethink ideas around
mathematical concepts. Does this always work? Why or why
not? Mistakes lead to new learning about mathematics.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms,
argumentation, and autonomy in mathematics. Journal for
Research in Mathematics Education, 27, 458-477.

28. Providing Authentic Relevance
http://www.gapminder.org
These are videos representing socially relevant
data using colorful information rich charts.
• What are the unit rates being used in this video?
• What does it mean if a circle is:
– Larger?
– Moves up or down?
– Moves to the left?
• What else do you want to know? Make a graph of your
own collected data.

29. Example
• Here is a “graph” from Gapminder
One Example
What questions might students ask?
What learning outcomes could you expect?
What skills do you need to use a resource like
this in class?

30. Interactive Resources
• Online manipulatives
– http://nlvm.usu.edu/en/nav/vlibrary.html
• Supporting “hands on learning”with online versions of common manipulatives
– http://www.geogebratube.org/?lang=en
• Geogebra has an abundance of user contributed resources, both worksheets and online tools.
It takes a little sifting to find appropriate tools
• Animated lessons
– http://www.ccsstoolbox.org/standards_content_mathematics.html
• Common Core lessons
• Online graphing Calculator

31. Lesson Plans
Illustrative Math Project
• http://www.illustrativemathematics.org
Growing site with lessons coordinated to Common Core Standards and Practices
Inside Mathematics
• http://www.insidemathematics.org
Tasks coordinated to the Common Core Standards supported by the Dana Center at UT Austin and
the Silicon Valley Mathematics Initiative
Scottsdale Community College
• https://score.scottsdalecc.edu/
Creative Commons Open source books
Math Assessment Project (MARS Tasks)
• http://map.mathshell.org/
Assessments, tasks, professional development support
Southern Regional Education Board
http://www.sreb.org/
Text available for download and use.

32. In Summary
• Responding to change requires dialogue.
• You are going to have questions and wonders.
• Create a professional community of practice.
• Elicit Support
Handouts: http://bit.ly/CarrenW_Practices
Email: carren@collaborativeforambitiousmathematics.org
Carren Walker