1. Creating Active Thinkers
Taking the
Ultimate Journey
Kitty Rutherford
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2. Who’s in the Room”

3. Norms Listen as an Ally Value Differences Maintain Professionalism
Participate Actively
Thumbs up if you agree with these norms. Are there other norms we need to add so that we have the best possible learning experience for all?
7/4/2018 • page 3

4. Let’s Define the Problem.
We in the mathematics world are all about problem solving. If we want to follow best practices, develop the mathematical Practices, help students develop 21st century skills, we have to move beyond the traditional teaching model.

5. High School
Rows of 5, all eyes on the chalk board, blue overhead marker smeared from palm to elbow…. Students asleep or praying for a fire drill.

6. First Grade The Leader The Ethics Police
The “I’m Finished First” Winners
The Do-Overs
read

7. Instruction Must Change
TIMSS and other international measures
Common Core State Standards
N.C. Teacher Evaluation Process
Add TIMSS slide, equality chart, read from NCTEP

8. Types of Math Problems Presented
How Teachers Implemented
Making Connections Math Problems
Types of Math Problems Presented
To highlight how the math task framework plays out in the United States – this TIMSS research shows what types of task we use in the United States as well as how they are implemented. Although we are in line with other high achieving countries in terms of the number of high level tasks we use, we do not implement any of them at a high level. We tend to take the struggle out of the mathematics in our country.
The results of the recent TIMSS video study provide additional evidence of the relationship between the cognitive demands of mathematical tasks and student achievement. In this study, a random sample of 100 8th grade mathematics classes from each of six countries (Australia, the Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland) and the United States, were videotaped during the 1999 school year. The six countries were selected because each performed significantly higher than the U.S. on the TIMSS 1995 mathematics achievement test for eighth grade (Stigler & Hiebert, 2004). The study revealed that the higher-achieving countries implemented a greater percentage of making connections tasks in ways that maintained the demands of the task. With the exception of Japan, higher-achieving countries did not use a greater percentage of high-level tasks than in the U.S. All other countries were, however, more successful in not reducing these tasks into procedural exercises. Hence, the key distinguishing feature between instruction in the U.S. and instruction in high achieving countries is that students in U.S. classrooms “rarely spend time engaged in the serious study of mathematical concepts” (Stigler & Hiebert, 2004, p. 16).
Approximately 17% of the problem statements in the U.S. suggested a focus on mathematical connections or relationships. This percentage is within the range of many higher-achieving countries (i.e., Hong Kong, Czech Republic, Australia).
Virtually none of the making-connections problems in the U.S. were discussed in a way that made the mathematical connections or relationships visible for students. Mostly, they turned into opportunities to apply procedures. Or, they became problems in which even less mathematical content was visible (i.e., only the answer was given).
Other findings from the TIMSS research are addressed on the next slides.
SAS Secondary Mathematics Teacher Leadership Academy, Year 1

9. Lesson Comparison United States and Japan
The emphasis on skill acquisition is evident in the steps most common in U.S. classrooms
The emphasis on understanding is evident in the steps of a typical Japanese lesson
Teacher instructs students in concept or skill
Teacher solves example problems with class
Students practice on their own while teacher assists individual students
Teacher poses a thought provoking problem
Students and teachers explore the problem
Various students present ideas or solutions to the class
Teacher summarizes the class solutions
Students solve similar problems
Think about the practices!

10. US Data / Hong Kong US students ranked near the bottom.
US students ‘covered’ 80% of TIMSS content.
US students were outperformed by students not taught the same objectives.
Hong Kong had the highest scores in the most recent TIMSS.
Hong Kong students were taught 45% of objectives tested.
Hong Kong students outperformed US students on US content that they were not taught.

11. Why is change necessary?

12. 8 + 4 = [ ] + 5
Think for a minute about your answer to this problem, and what students in 1-6 grade might think the answer is.
What goes in the box? What might students say?
Children’s Mathematics: Cognitively Guided Instruction (CGI), by Carpenter, Fennema, Franke, Levi & Empson, 1999

13. Percent Responding with Answers
8 + 4 = [ ] + 5
Percent Responding with Answers
Grade 7 12 17
12 & 17
1st - 2nd
3rd - 4th
5th - 6th
Across the top you see the various answers students offered: 7, 12, 17, 12 & 17.
How did they get each of these responses?
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003

14. Percent Responding with Answers
8 + 4 = [ ] + 5
Percent Responding with Answers
Grade 7 12 17
12 & 17
1st - 2nd 5 58 13 8
3rd - 4th
5th - 6th
We can see that 5 percent of 1-2 graders produced the correct answer.
However, 58 percent thought the answer was 12.
How did they get that?
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003

15. Percent Responding with Answers
8 + 4 = [ ] + 5
Percent Responding with Answers
Grade 7 12 17
12 & 17
1st - 2nd 5 58 13 8
3rd - 4th 9 49 25 10
5th - 6th
Now we look at % more were right.
Why do 12 % more students think 17 is correct?
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003

16. Percent Responding with Answers
8 + 4 = [ ] + 5
Percent Responding with Answers
Grade 7 12 17
12 & 17
1st - 2nd 5 58 13 8
3rd - 4th 9 49 25 10
5th - 6th 2 76 21
Now 5th -6th grades. The good news is that very few still think there are 2 answers. The bad news is that we are down to 2 % getting the right answer. Procedures memorized but not understood are getting in the way.
Barbara Bissell used this in Charlotte as a benchmark assessment and only 25% of 3rd and 4th grade students got it right.
Dr. Drew Polly at UNCC replicated this study for 6th grade, and none of the students got the correct the answer.
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003

17. Estimate the answer to (12/13) + (7/8)19 21
Only 24% of 13 year olds answered correctly. Equal numbers of students chose the other answers.
NAEP

18. Students were given this problem:
4th grade students in reform math classes solved it with no problem. Sixth graders in traditional classes responded that they hadn’t been taught that yet.
Dr. Ben Klein, Mathematics Professor
Davidson College

19. Research Students are shown this number
Research Students are shown this number. Teacher points to the 6 and says, “Can you show me this many?” 16
Constance Kamii has done extensive research on how young children learn mathematics. Here is a task frequently used to assess understanding of tens and ones. Students usually are successful, and count our 6 blocks.

20. Research When the teacher points to the 1 in the tens place and asks, “Can you show me this many?”16
Kamii found that essentially no first graders could correctly complete this task.

21. Research By third grade nearly half the students still do not ‘get’ this concept.16
Many 3-5 graders still do not give the correct answer.

22. More research - It gets worse!
A number contains 18 tens, 2 hundreds, and 4 ones.
What is that number?
2824
1824
Grayson Wheatly’s research with 5,000 middle-school students were given the following task:
Some students gave this answer.
Others knew the tens had to be in the middle, so…..
Many gave this answer, knowing about decimals, and that you could only have 3 digits if a number was in the hundreds.
Around 50% of the middle school students gave the correct answer.
384
218.4

23. We know “What” Students Need…
21st Century Skills, critical thinking and problem solving, collaboration and leadership, agility and adaptability, oral and written communication, accessing and analyzing information.
Tony Wagner, Rigor Redefined
Tons or research has told us what students need for success in the future.

24. But Not “How” to Meet Their Needs
Common Core Standards for Mathematical Practice
Read several smp, several nctep, character traits.

25. Standards for Mathematical Practices
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.
Survey group for familiarity.
Refer task completed in algebra
Find this in your standards
Carry across all grade levels
Describe habits of a mathematically expert student
The Common Core proposes a set of Mathematical Practices that all teachers should develop in their students. These practices are similar to the mathematical processes that NCTM addresses in the Process Standards in Principles and Standards for School Mathematics.
These can be implemented now even before content goes into effect.

26. Creating Active Thinkers
Do You Value Thinking? Turn and Talk with your shoulder partner about your Teacher Test.
This is an amazing book, one I have used personally. Let’s make sure you are ready for this. Turn to the last page of your Case Study handout. Test time.

27. The First Step
“Before all else, a classroom environment that fosters complex thinking must be predictable and safe.”
Creating Active Thinkers, page 34
The Case Studies all refer to ways character education supported their work in creating a safe environment.

28. The Next Step
“Complex thinking is developed in students primarily through the careful planning and teaching of lessons.”
Creating Active Thinkers, page 37
The book includes useful guidance on this topic, but is not the main emphasis for today. Needless to say, there has to be something to think and talk about.

29. The Famous Horse Problem
A farmer buys a horse for $60.
Later he sells it for $70.
He buys it back for $80.
Finally, he sells it for $90.

30. Shoe Store Problem
A man walks into a shoe store and buys a pair of shoes for $5. He pays with a $20. The store owner goes next door to the baker to get change for the $20, returns, and gives the customer his change. That afternoon the baker shows up with a police officer, declaring that the $20. was counterfeit, and he wants his money back. The shoe shop owner returns his money. How much did he make or lose?

31. Jigsaw on Teacher Strategies

32. Find Your Teacher Strategy #

33. Find Your Teacher Strategy Color

34. Student Responsibilities
“The student takes his or her cues from the teacher.” Include your students in the journey. Meet some of your students Creating Active Thinkers, page 97-99
Take a look at some student types. How many do you know?

35. Student Behaviors
Read the student behaviors on page 101. Compare student behaviors to the Standards for Mathematical Practice. Surprise?
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36. Developing These Behaviors
The first step is to let students in on the game. They must be explicitly taught about these nine behaviors.
I believe students meet our expectations, expectations they infer from observing us. Andrew Carnegie said, “ As I grow older, I pay less attention to what men say. I just watch what they do,”

37. Self Assessment
Students are amazingly honest when assessing themselves. Creating Active Thinkers, page 117 – 121;
Kathy Richardson’s self assessment 0,1,2,3

38. Self Assessment Doesn’t Always Work
The last pages contain Observation Forms, to help you see what your students and others see.
Creating Active Thinkers, Appendix C
These are designed for PLC work during which teachers would observe each other. Students from grades 4 and up could also take turns recording. This really builds a community of learners.

39. For all you do for our students.
Special Thanks To:
YOU
For all you do for our students.

40. Contact Information
Kitty Rutherford
Website:

41. QUESTIONS