1. Service With Honour
MAKING THINKING VISIBLE FOR SECONDARY MATHEMATICS LESSONS USING THINKING ROUTINES

2. FLOW OF PRESENTATION
At the end of the sharing, participants will be able to see how thinking routines are incorporated into lessons.
Introduction of Thinking Routines
See, Think, Wonder
- Written Assignment: Types on Numbers
- E-Learning: Probability
- Hands-on Activity for You!
(3) Claim, Support, Question
- Activity: Sieve of Eratosthenes
- Activity: Types of Numbers
- Activity: Geometric Properties of Circles

3. WHAT ARE THINKING ROUTINES?

4. Learning from the experts..
HARVARD
GRADUATE SCHOOL OF EDUCATION
Examine the development of learning processes in children, adults, and organisations.
Today, Project Zero’s work includes investigations into the nature of intelligence, understanding, thinking, creativity, ethics, and other essential aspects of human learning.
Highlight the part about Project Zero’s multitude of works and show website.

5. TOOLS FOR FOR THE TEACHERS, HABITS FOR THE STUDENTS
Tools for the teachers, habits for the students
Thinking routines are simple structures that students can settle down to during lessons.
Thinking routines can be applied across disciplines and grade levels.
Each routine targets a different type of thinking and by bringing their own content, teachers integrate the routines into the fabric of their classrooms.

6. Learning from the experts..
EDUCATIONAL LEADERSHIP
(1) By making thinking visible, we watch, imitate and ADAPT TO OUR OWN STYLES.
(2) Visible Thinking includes a number of ways of making students' thinking visible to themselves, to their peers, and to the teacher, so they get more engaged by it (THROUGH FORUMS)
When thinking is visible, it becomes clear that school is not about memorizing content but exploring ideas. Teachers benefit when they can see students' thinking because misconceptions, prior knowledge, reasoning ability, and degrees of understanding are more likely to be uncovered. Teachers can then address these challenges and extend students' thinking by starting from where they are.
The emphasis here is on “Thinking”; any responses given after giving thought to it will be considered a good response –we are not looking at the correct answer.
They begin to display the sorts of attitudes toward thinking and learning we would most like to see in young learners -- not closed-minded but open-minded, not bored but curious, neither gullible nor sweepingly negative but appropriately sceptical, not satisfied with "just the facts" but wanting to understand.
Visible Thinking emphasises several ways of making students' thinking visible to themselves and one another, so that they can improve it.

7. Tools for the teachers, habits for the students SEE, THINK, WONDER
* Originally conceived as an exercise to garner interpretations on an image/video on humanities or art. For Math when we use the routines, we are looking at students explaining, comparing and contrasting, drawing connections and of course, when the design is sufficiently open, looking at creativity.

8. SEE, THINK, WONDER SEE, THINK, WONDER Purpose
Encourages students to make careful observations and thoughtful interpretations.
Stimulates curiosity and sets the stage for inquiry.
Application: When and Where Can It Be Used?
Use the routine
at the beginning of a new unit to motivate student interest
with an object that connects to a topic during the unit of study
near the end of a unit to encourage students to further apply their new knowledge and ideas.
* Originally conceived as an exercise to garner interpretations on an image/video on humanities or art. For Math when we use the routines, we are looking at students explaining, comparing and contrasting, drawing connections and of course, when the design is sufficiently open, looking at creativity.

9. “See”: the ability to notice details
SEE, THINK, WONDER
ASSESSMENT
“See”: the ability to notice details
“Think”: how students support their interpretation and assertions
“Wonder”: questions that are broad and adventurous rather than those that require specific factual responses
from Making Thinking Visible

10. WRITTEN ASSIGNMENT: TYPES OF NUMBERS
SEE, THINK, WONDER
WRITTEN ASSIGNMENT: TYPES OF NUMBERS
JH1 MA100 Written Assignment 1: Questions

11. E-LEARNING: PROBABILITY
SEE, THINK, WONDER
E-LEARNING: PROBABILITY
JH2 MA203 E-learning (June Holiday)

12. SEE, THINK, WONDER
Video introduces the founding of Probability by a Gambler-Physician-Mathematician Cardano, the calculation when rolling two dice, and also the idea of randomness.

13. SEE, THINK, WONDER

14. THE DESIGN: FACTORS TO CONSIDER
SEE, THINK, WONDER
THE DESIGN: FACTORS TO CONSIDER
The video or object must offer a rich context or content such that details or interpretations can emerge after examination or thinking.
While addressing a particular topic, there should be a scope for linking up different areas or fields.
A platform must be provided for students to engage in group activity where participants build on the responses offered by others.

15. SAMPLE RESPONSES FROM STUDENTS
SEE, THINK, WONDER
SAMPLE RESPONSES FROM STUDENTS
Students’ thinking is made visible so that teacher can work on misconceptions. One thing to note about STW is the word “THINK”. By asking students what they “Think”, this conditional suggests possibilities and openness rather than absolutes. This encourages the sharing of tentative ideas, or innocuously, sharing of “comments”.

16. SEE: NOTICING DETAILS Student C.C.Y.
SEE, THINK, WONDER
SEE: NOTICING DETAILS
Student C.C.Y.
The dots on a die are created by removing the material used to make the die.
Between the sides with numbers '1' and '2', the side with number '2' will have more material removed, thus the die will be biased to one of the sides.
Therefore, the outcomes will not be equally likely. 
Demonstrating prior knowledge

17. WONDER: DRAWING BROAD IDEAS
SEE, THINK, WONDER
WONDER: DRAWING BROAD IDEAS
Student L.K.Y.
Something else that I like to say about the video is that the terms brought out are very insightful. One thing that I have learnt is that in math or in anything actually, we should always look at the big picture rather than the individual jigsaw puzzle pieces.
Draw broader ideas or lessons.

18. WONDER: DRAWING BROAD IDEAS
SEE, THINK, WONDER
WONDER: DRAWING BROAD IDEAS
Student M.E.Q.
The fact that the human brain favours specific numbers and patterns also fascinates me.
What causes the human brain to think that certain numbers and patterns are more random than others?
Is it an unconscious action ingrained into our brains or is it caused by something else...(Any answers?)

19. IN RESPONSE... Student T.Y.F.
SEE, THINK, WONDER
IN RESPONSE...
Student T.Y.F.  
I'm not exactly sure about a good answer, but humans have a cognitive bias towards certain patterns,
which is "a pattern of deviation in judgment, whereby inferences of other people and situations may be drawn in an illogical fashion." (quote Wikipedia)
If you wish to inquire further, a good place to start would be about pseudo-randomness, though I'm not too sure if it is related.

20. THINK: UNSUPPORTED ASSERTIONS
SEE, THINK, WONDER
THINK: UNSUPPORTED ASSERTIONS
Student H.W.G.
Probability is basically based on a fair situation under fair conditions.

21. IN RESPONSE... Teacher When you read the papers and people talk about
SEE, THINK, WONDER
IN RESPONSE...
Teacher
When you read the papers and people talk about
odds and probability, are the events/outcomes
all fair?

22. SEE, THINK, WONDER: HANDS-ON ACTIVITY
Materials:
Scissors
Paper
Wondering mind

23. CLAIM, SUPPORT, QUESTION

24. CLAIM, SUPPORT, QUESTION
Purpose
Helps students develop thoughtful interpretations by encouraging them to reason with evidence.
Learn to identify truth claims and explore strategies for uncovering truth.
Application: When and Where Can It Be Used?
Use the routine with topics in the curriculum that invite explanation or are open to interpretation.
The questions can challenge the plausibility of the claim, and often lead to a deeper understanding of the reasoning process.

25. CLAIM, SUPPORT, QUESTION
OBJECTIVES
(1) Engage students to explore strategies for uncovering mathematical relationships. (2) Help students support their claims through logical reasoning. (3) Develop students’ self-directed learning capability through the raising of questions by students.

26. When making a claim, are students:
CLAIM, SUPPORT, QUESTION
ASSESSMENT
When making a claim, are students:
looking for generalisations that get to the truth?
When supporting a claim, are students
anchoring the claim with solid evidence?
recognising special cases?
from Making Thinking Visible

27. GEOMETRIC PROPERTIES OF CIRCLES
CLAIM, SUPPORT, QUESTION
GEOMETRIC PROPERTIES OF CIRCLES
Students are given a pseudo-real life scenario.
They need to use their knowledge of geometrical properties of circles to solve the problem of dividing food equitably during a shipwreck.
The Claim-Support-Question Strategy is used to engage students in making their thinking visible.

28. CLAIM, SUPPORT, QUESTION

29. CLAIM, SUPPORT, QUESTION
TYPES OF NUMBERS

30. SAMPLE OF STUDENT’S WORK
CLAIM, SUPPORT, QUESTION
SAMPLE OF STUDENT’S WORK

31. SIEVE OF ERATOSTHENES Claim Support Question
Your friend claims that the above steps just needs to be repeated as far as the number 7 for all primes smaller than 100 to be identified.
Support
Can you support your friend’s claim?
Question
How do we determine if a number is prime?
Suppose that we have found the multiples of all the prime numbers up to 7 and then we proceed to find the multiples of 11. This will be superfluous as 11 X 2 is the same as 2 X 11, which we did earlier. Therefore the process just needs to stop at number 7. To be more specific, we just need to find the multiples of the prime between 1 and the square root of 100.

32. QUESTION AND ANSWER
Thank You