1. Building Conceptual Understanding through Progressions
“Differences among students; not enough time”

2. Who am I? 8th Grade math – Fairmont
Prior 6 years – Orosi: 8th Algebra/7 CCSS
Low performing on CSTs – moved to CCSS early
Department chair during transition to Common Core
ALWAYS wanted to see Phil Daro speak (in person)

3. Who area you? Name Grade/Subject and Site
What do you hope to get out of this session?

4. CMC Central Symposium
I will be using slides/information from Phil Daro’s presentation both during the time that I saw him, as well as others accessed on the internet
Focus of talk; Many students aren’t “ready” for the math we are supposed to be teaching
Main country of comparison mentioned was Japan

5. Answer Getting vs. Learning Mathematics
USA:
How can I teach my kids to get the answer to this problem?
Use mathematics they already know. Easy, reliable, works with bottom half, good for classroom management.
Japan:
How can I use this problem to teach the mathematics of this unit?
(5:25)
(3:55)

6. Consider the Videos… Pick one to answer: What struck you?
What did you find most interesting?
What are the implications for us as math educators?
What questions arose from these ideas?

7. Comparing Mindsets ALL students start a lesson “all over the place”
America: Mathematical illness called gaps
Japan: Focus on future mathematical health of student – connect lower grade level methods of approach to grade level methods (Progression) – depth of understanding

8. “Four Levels of Understanding”
(stop at 6:33)

9. You - we – I designs better for content that depends on prior knowledge
Student A Student B Student C Student D Student E
Lesson START Level
Day 1 Attainment
Day 2
Target

10. Start with a “Transparent” Problem
A bullet train goes 100 miles in 50 minutes.
Add a question to turn this situation into a math problem
*Redirect “answer-getting” attitude: Start without a question!

11. Start with a “Transparent” Problem
What options did we come up with?
Pick one (for now) to work on as a class
*Be strategic – have an idea of what you are looking for before you start. You want a problem that can be solved in as many ways as possible and is understandable to everyone (how we get the “wavy” start line).

12. Illustration:
A bullet train goes 100 miles in 50 minutes. How long will it take the train to go 700 miles?
Student prompt: You may solve using ANY method you choose, but you must have a clear explanation and justification of your answer that everyone in class can understand. All students must be prepared to present your solution strategy and justification of answer.
For you: What are the possible ways to solve this problem? Think of what you would expect to see from students

13. As students are working…
It is important that students have at least a couple minutes to get their own thinking on paper, but pairs or groups can offer support
Circulate, observe strategies, probe thinking with questions
Assess approaches. Decide who will present and in what order. You want the presentation to go in order of complexity (easiest to understand  most complex).

14. What strategies would you expect to see students use?
Picture w/ skip counting (2nd grade)
Table of Values/Scaling up (4-6th grade)
Scale Factor (5th grade)
Unit Rate (6th grade)
Misconception: Additive thinking (“I have to add 600 to get from 100 to 700, so if I add 600 to 50, I get 650 minutes”)
*DON’T LEAVE OUT MISTAKES!

15. Video: Mistakes in Japan vs US
Start at 3:50, end at 5:45

16. Reflection…
How would this shift in how we view mistakes impact the experience in our classrooms?

17. So what do we do with all these strategies? -
Picture w/ skip counting (2nd grade)
Table of Values/Scaling up (4-6th grade)
Scale Factor (5th grade)
Unit Rate (6-7th grade)
Misconception: Additive thinking (“I have to add 600 to get from 100 to 700, so if I add 600 to 50, I get 650 minutes”)
Have students present in order of complexity – question and push them to make connections between each strategy

18. Teacher’s Focus for Presentation:
Does each explanation make sense to the rest of the students
Can the presenter and other students find/make connections between strategies and representations?

19. Summarize
Have a summary prepared (basic sketch from lesson goals) to tie the strategies together – quote student work, show progression, then push them to see why they NEED the grade level approach: “What does ‘2’ represent in the context of the problem? How could we use this strategy to find any time given distance? Distance given time?” Etc…)
Students add key points of the summary to their explanations

20. At the END of this Process…
You have a better idea (diagnostic) of where students “are” on the progression in terms of their thinking with this type of problem
You can push students towards a more complex example (What would happen if the distance travelled was 725 miles?) or representation (Could you create a graph from your table of values? Could you write an equation to represent the situation?)
You can do direct instruction because you have GIVEN all students “prior knowledge”

21. Prior knowledge
There are no empty shelves in the brain waiting for new knowledge.
Learning something new ALWAYS involves changing something old.
You must “upgrade” prior knowledge to learn new knowledge.

22. “Upgrading” Prior Knowledge
“Mathematical concepts do not exist in isolation. Mathematics is a set of systems. Arithmetic of whole numbers is a self-contained, logical system until division is introduced. Division produces rational numbers that are not whole numbers, a new kind of number. The system has to be upgraded to encompass all rational numbers. Learning a concept in mathematics is almost always an upgrade of prior mathematics knowledge. Comprehending explanations in mathematics requires a strategic perspective of repairing and upgrading prior knowledge.” (Daro, Mosher, & Corcoran, 2011)

23. Where could you use this?
Think of a topic/unit of study where this would be valuable
Best if it is a MAJOR area of work (8th grade: Linear Relationships – not simplifying square roots!)
Best BEFORE grade level instruction
Can you write a transparent problem with multiple angles of approach that leads to this grade level topic/unit?

24. Next Step: (after problem formulation):
Think through possible student approaches and misconceptions
What questions could you ask in each situation that make them think and don’t just “give it away”?
Order the possible methods by complexity and outline main ideas of summary
Where will you go from here? – Perhaps have some ideas for questions to push students on

25. Sources and Resources:
Phil Daro’s original PPTs, from which I stole and edited slides:
cmc-math.org/temp/wp-content/uploads/2013/05/bakersfield.mar12.pptx
Quoted document (upgrading prior knoweldge):
HUGE resource for more info:
Principles to Actions put out by NCTM. Also… See their PD website to accompany this book which has tons of classroom vignettes with lesson plans in all grade bands:

26. Summary of Key Points
The following slides are stolen from the original talk I attended and summarize the key points from today…

27. Differences among students
The first response, in the classroom: make different ways of thinking students’ bring to the lesson visible to all
Use 3 or 4 different ways of thinking that students bring as starting points for paths to grade level mathematics target
All students travel all paths: robust, clarifying

28. Classroom culture: ….explain well enough so others can understand
NOT answer so the teacher thinks you know
Listening to other students and explaining to other students

29. Questions that prompt explanations
Most good discussion questions are applications of 3 basic math questions:
How does that make sense to you?
Why do you think that is true
How did you do it?

30. …so others can understand
Prepare an explanation that others will understand
Understand others’ ways of thinking