1. GROWING ALL STUDENTS THROUGH HIGH-QUALITY ASSIGNMENTS
A Balancing Act: Rigor in Mathematics
July 2017

2. EQUAL ICE BREAKER Balanced Equivalent Even Fair Same
Find your sole mate. One partner stands facing the projector screen, the other away from it. Set the timer for 60 seconds, hit Start, and then the partner can start giving clues. The clue-giver may not say a part of a "taboo" word; for example, using "base" in "baseball" is taboo. The giver may only use speech to prompt his or her teammates; gestures, sounds (e.g. barking), or drawings are not allowed. Singing is permitted, provided the singer is singing words rather than humming or whistling a tune. The giver's hints may not rhyme with a taboo word, or be an abbreviation of a taboo word.

3. The $100,000 Pyramid The Winner’s Circle
Partner switch places. One partner stands facing the projector screen, the other away from it. Set the timer for 60 seconds, hit Start, and then the team can start giving clues. If the category is “Shakespeare Plays” then the student facing the screen would start mentioning things in that category, but in this round they are only allowed to give one clue, then their partner has to guess. Then they can give another clue, and their partner can guess again. So the partner who can see the screen would start saying things like “Romeo and Juliet”, “Hamlet”, etc. until their partner correctly guessed “Shakespeare Plays”. Then move to next category.
The Winner’s Circle

4. $50 Fraction $300 THINGS YOU ANALYZE $100 Mathematical Practices $150
$200
Famous Mathematicians
$250
“What an equation might say”
$50
Fraction
$100
Mathematical Practices
$150
TYPES OF TRIANGLES
60 seconds.

5. Our Norms
P.E.M.D.A.S P – Participate fully, (silence phones please). E – Exchange ideas. M – Make an effort to listen. D – Dialogue equally. A – Ask Questions. S – Share insight, support each other, and self-reflect.
Talk through Session Objectives (1 min). Have norms posted in the room.

6. Session Goals
To clearly define rigor and its role in designing and implementing mathematics assignments.
To analyze examples of mathematics assignments that reflect aspects of rigor.
To refine assignments so that they pursue the balance of rigor.

7. Session Agenda Ice Breaker Norms Rigor in Mathematics
Assignment Analysis
Assignment Refinement
Wrap Up
Talk through agenda. (30 sec)

8. Defining Rigor in Mathematics

9. THINK – PAIR – SHARE
What are the characteristics of a rigorous mathematics assignment?
Share your responses with your partner. Decide on the three top characteristics.
Activity – Participants answer individual, compare and compile characteristics into a list, and post there results. Share out is whole group.
Poll it!

11. What does the research say about rigor in mathematics?
Rigor in Research
What does the research say about rigor in mathematics?
Activity – Rigor in Research
Participants will analysis research statements on rigor to form their own definition.
Tasks must provide entry points for all students, offer them well-defined opportunities to make connections to other mathematics, and include both opportunities and expectations for them to develop deeper understanding. (NCTM. Summing Up, 2013)
Rigor refers to academic rigor—learning in which scholars demonstrate a thorough in-depth master of challenging tasks to depth mastery of challenging tasks to develop cognitive skills through reflective thought analysis problem thought, analysis, problem-solving solving, evaluation, or creativity.
Purposefully extending knowledge on a continuum articulated by both teacher and student which will lead to self-directed
inquiry.
…the depth of interconnecting concepts and the breadth of supporting skills students are expected to know and understand.
Effective, ongoing interaction between instruction and student reasoning and thinking about concepts, skills, and challenging tasks that result in a conscious, connected, and transferable body of valuable knowledge for every student.
Requiring students to demonstrate, justify, and apply mathematical ideas to various situations. Instruction should involve productive struggle and focus on mathematical ideas (Hiebert & Grouws, 2007)
Rigor is a process-not a problem.

12. Rigor in Common Core Math
The pursuit of conceptual understanding,
procedural skills and fluency,
and application
with equal intensity.
~CCSSI
As defined by Common Core…Rigor is one of the components of the “Shifts in Mathematics”

13. Procedural Skill & Fluency Conceptual Understandings
The Pursuit…
Procedural Skill & Fluency
Conceptual Understandings
Rigor
Can you describe an assignment that would fall in the overlap areas of this graphic?
Represents full understanding of a mathematic topic….
Does full understanding happen with just one assignment?...Keep that in mind as we move to the next activity.
Application

14. Rigor Sort – Content Standards
The Standards set high expectations for all three components of rigor in the major work of each grade.
Pay attention to the verbs. Maintain the rigor of the standards.
“The word ‘understand’ is used in the Standards to set explicit expectations for conceptual understanding…” (K-8 Publisher’s Criteria, page 5)

15. Rigor promotes Mathematical Practices
Procedural Fluency
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
Conceptual Understanding
Rigorous assignments provide ample opportunities for students to engage in the Standards for Mathematical Practice.
Application

16. Findings about Rigor (n=1853)
Add numbers of task in the figure. Get the exact numbers of task.

17. Why is theory so different from practice?
Turn and Talk. Then share out.

18. Analyzing Math Assignments

19. Rigor and Representations
Questions for Analysis
Which aspect(s) of rigor does the assignment address?
Does this assignment provide multiple representations of mathematical concepts and/or equations?
Lesh’s Mathematical Representations

20. Discussion (Before Analyzing)
What skills will the student need to know in order to address this question? What should the teacher expect to see in the student’s answer?

21. Analyzing Assignments
This is the anchor task for this section. Participants will refer to this task as each factor of analysis is discuss. Participants will use the task to cite evidence of the aspect of rigor.

22. Multiple Representations (MP1)
The Lesh Translation Model
The Rule of Four
Lesh’s Mathematical Representations connected to the language of MP1. Participants will look for evidence of representation in the anchor assignment.
(Lesh,1979)
(The Harvard Calculus Consortium, 1994)
“Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem” (CCSSI).

23. Procedural Skills & Fluency
Skills in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Computational fluency with or without thin context (MP6)
Use of tools like calculators, computers, and manipulative materials (MP7)
Procedural fluency using informal reasoning strategies and the properties of the four arithmetic operations (MP8)
Memorization, mnemonics, or tricks

24. Application
Independent use of appropriate concepts and procedures in “real world” situations.
Single-step and multi-step contextual problems (MP4)
Students make an assumption in order to model a situation (MP4)
Solid conceptual understanding and procedural fluency (MP4)
The use of tools, models, calculator, or technology (MP5)
Interpretation of mathematical results in the context of the situation and reflection on whether the results make sense (MP1)
Phony context (context that is both unrealistic and also plays no role in helping students understand or make sense of the mathematics)

25. Conceptual Understanding
An ability to demonstrate understanding of concepts from
one or more perspectives
Concrete and Pictorial Materials (MP2)
Conceptual Questioning (MP3)
Writing and Speaking about Understanding (MP1, MP6)
“Show your work” for procedural problems
-Conceptual understanding can be promoted in a variety of ways including use of concrete and pictorial models, conceptual questioning, as well as writing and speaking about understanding.
-Concrete materials give students an experiential understanding of concepts. Pictorial representations offer greater flexibility than concrete models, challenging student understanding at a deeper level while maintaining their connection to the contextual situation. Without the concrete or pictorial models, operations become disconnected from meaning, rendering students unable to judge when and where they apply.
-Reflections:
Identify conceptual questioning in the example.
Compare and contrast the impacts and advantages of conceptual questioning with the example of concrete and pictorial models.
-Speaking about understanding can occur among students debating a problem or with a teacher questioning students individually or as a group.
Writing about understanding can occur at the board, on worksheets, on homework, or in student journals.
Both speaking and writing are valuable ways to consolidate learning and reveal students’ current level of understanding.

26. Procedural Skill & Fluency Conceptual Understandings
Putting It Together
Procedural Skill & Fluency
Conceptual Understandings
Rigor
Traveling to School
Application

27. Analyzing Assignments

28. Analyzing Assignments

29. Analyzing Assignments

30. Analyzing Assignments

31. Refining Assignments

32. Rigor Ramp-up Work with a partner to select a assignment to modify.
Discuss the expectations of the assignment.
Refine the assignment so that it exemplifies at least two aspects of rigor.

33. Share one strategy you used to ramp-up the rigor in your assignment.
Rigor Ramp-up
Share one strategy you used to ramp-up the rigor in your assignment.

34. Rigor Ramp-up Strategies
Two sure ways to ramp up the rigor.
Effective assignments builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems (Diane J. Briars, NCTM President, 2016).
Effective assignments build fluency with procedures on a foundation of conceptual
Understanding.
Effective assignments build conceptual understanding by using multiple representations.

35. Rigor in mathematics assignments is instructional balance over time.
Remember...
Rigor in mathematics assignments is instructional balance over time.
All three aspects of rigor do not always have to be presented together, just as they do not always have to be presented separately.

36. Rigor is a process, not a problem.
Remember…
Rigor is a process, not a problem.
Rigor Promotes Mathematical Practices!

37. Success Criteria
1 - Not yet With help Got it! I can lead this work
I can define rigor and its role in designing and implementing mathematics assignments.
I can identify aspects of rigor in a mathematics assignment.
I can modify or create assignments so that they pursue the balance of rigor.

39. Thank You!
References
Lesh, Richard, Tom Post, and Merlyn Behr. (1987). “Representations and Translations among Representations in Mathematics Learning and Problem Solving.” In Problems of Representation in the Teaching and Learning of Mathematics, edited by Claude Janvier, pp. 33– 40. Hillsdale, N.J.: Erlbaum
The Harvard Calculus Group (1995). Rule of Four Principles to Action. Retrieved from
Common Core State Standard Initiative. (2013). Mathematics Standards. Retrieved from
Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. London, England: Routledge.
The National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM