1. Teacher Understanding Unit 3
Deborah Ball
Prior to beginning this unit. Discuss discussion questions from Unit 2.
As you know now from Unit 2, our implementation choices are critical to student success. It is also important that we understand the nature of our students, the nature of learning and the nature of the mathematics itself. Deborah Ball (University of Michigan) has been at the forefront of articulating this issue and on investigating it through research studies.
The pre-test that you took in Unit 1 was designed by Ball and her research group to better understand what are the critical understandings about mathematics understanding and mathematics instruction that have an impact on student learning. What she has found is that there is a correlation between teachers strength in what she calls mathematical knowledge for teaching (MKT) and student achievement.
This is critical to us as we work to understand what we can do to improve our students’ success. What it means is that one of the most important things for us to do is to learn and understand the fundamental math that we teach better. This is not a quick cure. This is a journey. Because of how we were all taught mathematics, we have many conceptual holes in our understanding of fundamental mathematics. Again, the research is clear that these holes in our understanding affect our students. Our understanding itself changes the flow of our lessons, changes the way we respond to students inquiry and changes our emphasis in our discussions with students.
Units 3,4,5,6,7 and 8 of this foundation training will focus on beginning this journey. We think that you will see, over the course of the next year, that the work we do here will begin to affect your thinking, your instruction and, in turn, your students’ success.
Click on the link for further information and resources.

2. Defining Issue in Implementation
Teacher’s Own Understanding of Mathematics.
Liping Ma
You have read some articles by Deborah Ball already, but unit 3 is actually focuses around one of Deborah Ball’s doctoral students and her work. Liping Ma was born during the time of the Chinese cultural revolution and she was a young city kid who was moved away from the city to learn the ways of the land as part of the cultural revolution. Although she was still school aged the local farmers she now lived with noticed that she was bright and asked her if she would become a teacher. In China, elementary school teachers have the equivalent of an 11th grade education.
Eventually Ma came to the United States to continue her studies as a teacher and she began to notice that teachers in the United States did not seem to understand the mathematics they taught as well as her peers in China. This became the starting point of her work and Ma’s dissertation is now the book Knowing and Teaching Elementary Mathematics. It contains an impressive amount of information to start us on our journey.
Ma organizes her book around the questions that Ball had developed to probe teacher understanding. Ma interviewed teachers in China asking them the same questions and then compared the responses.
This lends itself nicely to a book that not only explores the mathematics, but explores the way teachers think about the math they are teaching both at the same time.

3. The Defining Issue in Implementation
---is the teacher’s own understanding of mathematics.
Liping Ma
The research continues to converge on the implementation choices.
Ma’s was qualitative documentation that the implementation choices of the teachers in countries that produce strong performing students is qualitatively different from those in the US.

4. Understanding and Instruction
We can only instruct our students as well as we understand the mathematics:
The better we understand the math, the better decisions we will make regarding what the student needs to achieve and how to instruct the student!
Have a participant read this slide.

5. Knowing and Teaching Elementary Mathematics Liping Ma
Compared and contrasted the pedagogy of Chinese and American teachers
Found that American teachers were much weaker in content and conceptual knowledge
Found American teachers teach procedurally rather than being driven by the logic of the mathematics (implementation)
Ma presented information through teacher responses to elementary math questions
These are some of the main points that you may have noticed from the reading.
We are now going to go through the mathematics and the thinking behind the four questions posed in this work

6. Expert Groups Chapters 1, 2, 3 and 4 Approach of the American teachers
Approach of the Chinese teachers
Largest pedagogical shift in thinking
Have each expert group prepare a short presentation highlighting
The approach of U.S teachers to topic
The approach of Chinese teachers to topic
Biggest pedagogical shift you may need to take regarding this topic and/or most interesting learning i.e. Your big “Aha” moment.

7. Problem #1 Subtraction 72 -15
How would you approach this type of problem if you were teaching second grade?
Here is the organizing question around chapter 1.
Let participants read then click.

8. Problem #1 Subtraction with Regrouping
American Teachers—Procedural approach
The “pedagogic insight "of the teachers
Once the student can take a ten from the tens place and turn it into 10 ones, then they can address the problem correctly. Problem solved.
Manipulatives suggested to explain this step only.
Manipulatives sometimes used in a way that they did not actually demonstrate process of regrouping.
Have expert group report out on the U.S. teachers approach.
Review this slide and then click to next slide until title shows.
Ask expert group--Why do we have steps for this slide?

9. Problem #1 Subtraction with Regrouping
Chinese Teachers—Decomposing and Composing a Higher Value Unit
Pedagogical insight: implemented the package of critical information embedded in subtraction
Saw this problem as connected to addition through composing and decomposing units
Demonstrated multiple ways of regrouping
Found the opportunity to explore the basics of our base ten system
Have expert group report out on how the Chinese teachers approached this problem.
Discuss and then show the rest of this slide as review.
Is an abacus a good graphic for this slide?
Also make sure that importance of Facts to 10 is discussed even if the expert group does not mention it.
Facts to ten is an important pre-skill for developing fluency within 20. Page 19

10. Math “Facts” vs. Subtraction within 20
U.S. deals with problems like 12-5, 15-7, etc., as FACTS to be memorized.
It does help to have these memorized, however, PEDAGOGICALLY, there’s more to it.
Chinese teach these “facts within 20” as the entry point for understanding our number system (develop “number sense”, emphasize base ten system)
Ask participants to report on question # 3--Main learning from chapter.
So let’s explore this further. Perhaps the whole key to understanding this chapter and this topic is to look at the essence of the difference between the way we in the U.S. approach these “facts within 20.”
Click slides.

11. 15
- 8 7
0 15 15
- 8 7
It is critical to see here, culturally, how we have learned to think about subtraction within 20. We see 15-8 as a fact to be memorized, as a fluency issue. Once you can memorize this, then we can get you to do harder subtraction problems. And, when we get you to do harder subtraction problems we will teach you the procedure to get things done. There is nothing wrong with efficient procedures BUT we must be aware of the mathematics they obscure. The real mathematics is happening underneath these procedures.
Click again.
If we look at this problem with the Procedure of crossing out and “borrowing” you see what? You haven’t “gotten” anywhere, right? This is how we see this problem.
What do the Chinese teachers see here?
•Different forms of the same value (Forms of a Value)
•Equivalent exchange (Equivalence)
•Exchange at the rate of ten (base-ten)
They are exploring the most important parts of our number system at this elementary level.

12. Equal Exchange, Base Ten and Form of a Value
Develop and ingrain the idea of equal exchange and the base ten system
1’s cubes and 10’s rods to enforce rate of exchange
Solve “problems within 20” emphasizing this base ten system and rate of exchange
Consider the difference in the type of Practice these students are getting. We all know that practice is critical for learning and expert development. The question is WHAT are we practicing?
Are we asking students to merely memorize ‘facts’ or have we taught them the pre-skills (facts to 10) and connected how this fits with the mathematical system (composing and decomposing, equal exchanges, exchanging at the rate of 10, forms of a number).
See page 19.

13. Pop Cubes and Tens “Composing and Decomposing”
Compose tens Decompose tens
It is important to practice the difference in language as you adjust your teaching to reflect your deeper understanding of the subject.
This looks simple enough, right? But it is not! What we do with students is very sophisticated, and this sophistication increases as we understand the mathematics more deeply.
For every ten ones we have, we can compose those ones into and equal value of 1 ten.
For every ten we have, we can decompose that ten into and equal value of 10 ones.

14. Subtraction within 20 Language Tips
Standard Form and Ones Form
Equal refers to value
A ten rod is not the same as ten ones
Chinese language: “1 ten 2 ones”
Mathematicians evaluate the form—
Is this the form I want my value in?
Remember to emphasize the following:
Standard Form versus Ones Form-- Mathematicians evaluate their values and the form they are in.
Equality is not the same as “the same thing”. Watch your language here.
Consider telling your students that in China they say the number 11 as 1 ten 1 one, and 12 as 1 ten and 2 ones, etc. Have students practice saying the numbers that way to develop comprehension in base ten.

15. Develop an Understanding of Base 10 and Equal Exchange
Have your students follow a 3-step procedure to think about the math. 1) set-up the problem with base ten blocks 2) evaluate the form 3) solve
Have participants practice their language. Set them up in pairs to practice with one being student and one being teacher and then changing places.
5-10 minutes in pairs.
Afterwards discuss how hard it is to make some of these changes even after you understand the logic of the changes.
Emphasize that This is the beginning of the journey.
The important thing is that you continue to pay attention to your implementation choices even at this most basic level doing this most basic math!

16. Know what you are teaching!
“Number Fact” (Automaticity)
15-8 Unlocking the Number System
It is not that automaticity is not important but automaticity itself does not build conceptual understanding.

17. Structures of Subtraction
Comparison (Difference between? Who has more?)
Deficit/Missing amount (What’s missing?)
The Classic “Take away” (How many are left?)
You’ve got some amount and “take away” from that amount. How many are left? ?
You compare to see who has more or less? ?
You need some more to get where you want to be. What is the missing amount?
One last note on subtraction. As you do word problems and discuss the math remember that there are 3 basic types of things that can happen in life that we can model and solve utilizing the operation of subtraction.
Ask yourself if a problem is a subtraction problem—Does it fit one of these three types?:
This is NOT a key word strategy. It is designed to help students understand the underlying structure of the mathematics. As you will learn through foundations training, key words don’t work. Teaching underlying structure and teaching students how to represent these structures to make sense of the mathematics does work and develops students sense-making skills. ?

18. Structures of Addition
Join and Part-Part Whole
There is something, and you get more of it?
There are two kinds, how many all together?
Start Unknown
Some are given away, some are left, how many were there to start?
Compare--total unknown
- I know one amount and I have some amount more than that. How many do I have?
How many altogether? ?
What did I start with? ?
Taken Left
How many do I have? ?
Addition types Adapted from Carpenter, Fennema, Franke, Levi and Empson, 1999, p. 12 in Adding it Up, NRC 2001.

19. Problem #2 Multiplication
123
645
615
492
738
1845
What’s happening here?
How would you correct this mistake?
Here is the organizing question around chapter 2.
Let participants read then click.

20. Problem #2 Multiplication
American Teachers Approach—
Lining Up correctly
American teachers saw this as a problem of alignment and thus addressed with systems or “tricks” for alignment
Teacher’s understanding reflects the way they were taught multiplication
Have expert group report out on the U.S. teachers approach.
Click to this slide with just the title.
Then review this entire slide
Ask expert group--Why do we have an elephant for this slide?

21. Problem #2 Multiplication
Chinese Teachers Approach:
Elaborating the Concept of Place Value and Powers of 10
Reinforce concept of Base 10 system
Develop concept of Distributive Property
Place Value a logical system (not so much a “place”)
Developing foundation and connections for higher thinking in mathematics
Have expert group report out on how the Chinese teachers approach this problem.
Discuss and then show the rest of this slide as review.
Are numbers a good graphic for this slide?--why?
Ask participants about their response to their main learning from chapter.
Make certain that participants know that all teachers saw this as lining up the partial products incorrectly, but the Chinese teachers gave emphasis to the place value system and the values of the digits rather than just a place for the digits.

22. 1 2 23
x 47 16 1 9 2
Consider what we are saying, communicating and thus practicing with students as we go through the procedure of multiplication.
3 x 7 equals 21 we will carry the 2.
And 2X7 equals 14 plus the “2” that I carried and that gives me 16.
Then we “add a 0” to “move it over”
And 3 times 4 equals 12, and we “carry the 1” I don’t really know why I’m carrying a one but that is what I’m told to do so I will do it.
Then, finally, 2 times 4 equals 8 and I add that to my 1 and I get 9.
Then I add those two numbers together to get “my answer”.
Where is the math here? It is buried underneath the procedure.
1081

23. 23
x 47 21
140
120
800
1081
Consider what we are saying, communicating and thus practicing with students as we go through the procedure of multiplication.
Let’s see how just a shift to the partial products method and the language we use changes drastically the lesson we are teaching and the practice the students are getting with number.
With time students can learn a more efficient system if needed, but they will do so with a deeper understanding of the math.
Model:
Okay, let’s see. We have 47 times 23. That means we have a pile of 23 things and we need 47 of those piles! That would take us a long time to do with manipulatives or drawing it out. But if we think about what we know about the math, we can do this without manipulatives.
We can chunk this problem up and then add up all of the partial products to get our total answer. We know we can do this from our work with the distributive property, right?
Okay, 7 x 3 is 21. Let’s put that down to remember it.
7 x 2 -now is that a 2? No, that is a 20--right? So this is 7 x 20. We know that 7 x 2 is 14 and this is one more power of 10, so 7 x 20 is Put that down. (and etc.)
Note: As you continue to model the method, make sure your modeling reflects the number system (powers of ten in particular) and not just a procedure…

24. Problem #2 Multiplication
Develop understanding of the numbers:
123= ( 1 ) x ( 2 ) x 10 + ( 3 ) x 1
645= ( 6 ) x ( 4 ) x 10 + ( 5 ) x 1
Also we could ask:
What does 123 x 5 represent?
What does 123 x 4 represent?
What does 123 x 6 represent?
Let’s reflect again on the methods used by the Chinese teachers.
Notice how these teachers tended to go back to a simpler problem not because they felt the students did not have a procedure internalized but because they wanted to go back and explain some important things about the math. Particularly, the ideas of the distributive property (partial products) and the idea of powers of ten and how that works in our place value system.

25. Multiplication and Properties
Distributive Property
Reinforce the concept of 10s
12 * 5 = (10 + 2) * 5
60 =
60 = 60
Reinforce willingness to “play” with numbers
12 * 5 = ( 6 + 6) * 5
60 =
Teaching students the distributive property with theses numbers (before a variable is introduced) affords them the opportunity to understand the Equality involved here and gives them practice with base ten and developing flexibility with problem solving. It reinforces the idea of composing and decomposing number, partial products and sets them up to understand the distributive property in algebra.

26. Concrete Multiplication
Finally--consider how this teaching feeds into the type of support that is very concrete. While you will show this concept to all students as you introduce multiplication, the struggling learner may need more repetitions at this concrete level to internalize these concepts about number and about multiplications.
The concrete multiplication mat is showing 1 ten and 2 ones for a total of three time. ( 12 x 3 ) or

27. Concrete Multiplication12
x 3 36
Notice how this concrete multiplication reinforces the distributive property and the partial products concepts.
We have here 12 x 3, but the teacher can also show the student that we have 3 tens and 3 2s as well. We then add these two partial products together to get our total number of 36.
3 tens (30 ones)
6 ones

28. Concrete Multiplication with Regrouping14
x 3 12 30 42
3 tens (30 ones)
12 ones

29. Develop the Distributive Property and Reinforce Base Ten14
x 3
We can then build this understanding as we go to a situation where a ten needs to be composed or “carried” to the next digit.
Click once and 14 x 3 = (10 + 4) x 3 shows the use of the distributive property.
Click a second time and 10 ones are circle to show how they compose 1 ten. Since 12 ones = 1 ten and 2 ones, when we add the 3 tens and 1 ten and 2 ones we have the solution of 42.
14x 3 = 3(10+4)=
(12 ones = 1 ten and 2 ones)=
= 42

30. STRUCTURE: 3 Types of Multiplication: 4 x 3
Repeated Addition
Array/row-column
Counting Principle
Repeated Addition: I am going to get 3 dollars each day I dog sit for my neighbors and they will be out of town for 4 days.
Array/Row-Column: Area Questions such as “My garden is 4 feet by 3 feet, how many square feet is it?”
Or questions such as, “I need to set up the chairs in 4 rows of 3 chairs, how many chairs do I need?”
Counting Principle: I have 3 types of cones and 4 flavors of ice cream, how many unique one-scoop cones can I make?
Also works for others with same structure like 3 tops and four pairs of shorts, how many different outfits, etc.
These are the three types of things that happen in the world that we can model and then solve using multiplication.
Students should be explicitly taught these structures and then use them to understand word problems and to model the problems at the representational level.

31. Know what you are teaching?
Multiplication --Place value as a place Multiplication as an algorithm
Multiplication --Unlocking the System
Open the floor for discussion.
If time allows have small groups reflect on their learning.
What is the difference between place value as a place with multiplication as an algorithm and unlocking the system? What does it take to unlock the system?
Possible answers include: Place value as a place and algorithms do not facilitate understanding the system. Emphasis on the role of the value of the digit would provide conceptual understanding for students which will facilitate the math connections across the components of number sense.

32. Problem #3 Division of Fractions
1 ¾ divided by ½
Give a Story Problem to show what is happening with this expression.
Here is the guiding question for this chapter. Even though this question is more difficult then where we would begin with students, division by 1/2 has proved to be a very difficult concept for the U.S. teachers in Ma’s study.
Before the expert group presents, allow participants to come up with the beginning step of the Griffin prototype. Let them wrestle with how to write a story problem for this question.

33. Sharon Griffin Core Image of Mathematics2 1
X =
“one”
“two”
“three”
What Sharon Griffin (Robbie Case) found in their work with young students was that students must understand quantity and the earliest form of a number line—i.e. counting independently.
Recall Griffin’s Core Image of Mathematics. By writing a story problem you are essentially accessing the concrete/quantity of the mathematics.
Quantity
Counting Numbers
Symbols

34. Prototype for lesson construction2 1
Touchable
visual
Discussion:
Makes sense
Of concept
Learn to Record these ideas
We will use this prototype for lesson construction to see if we can make sense of this very difficult concept. But remember, it is only difficult for us because when we were young, we were not taught this concept conceptually. It is actually a late elementary or early middle school concept. It is in the SCoS in NC in the 6th grade.
Quantity
Concrete display of concept
Mathematical Structure
Discussion of the concrete
Symbols
Simply record keeping!
V. Faulkner and DPI Task Force adapted from Griffin

35. Division of Fractions Lesson Construction2 1
We are now asking you to develop a lesson for fractions to ensure that YOU communicate the three areas. In particular Ma asked that we create a story problem – so that we are communicate the quantity of division of fractions. Note: Participants will probably only have time to come up with a story problem around which they could develop the quantity component of this prototype for lesson construction.
Now that you can tell a story problem, you can use the story problem to communicate quantity to the students. From there we can explain to the students the mathematical structures that they are learning with this mathematical skill.
Use a subtraction with renaming as an example to follow. i.e (What would we do with the quantity? (Story problem, base ten blocks, etc.) What would we talk about and how would we represent the problem? (decomposing a higher unit value, form of the same value, equality, etc.) Use a copy of the Prototype for Lesson Construction to discuss the “how to” of this task.
Go for it! write a story problem and fill in your lesson plan for the division of fractions problem –1 ¾ divided by ½ .
NOTE: Participants will generally struggle with this task. Let them know that they are not alone, this is a difficult task for American teachers because we were not taught to think about quantity and mathematical structures. We were taught tricks to manipulate symbols: Flip and Multiply.
Call on a brave soul to share their story. NOTE: If participant gets it right, congratulate them. Ask if anyone had a different solution. Get someone to say out loud their mistaken story. X1/2 or divided by 2. Participants will be confused as to why that’s not divided by ½. Again, support them for being typical because they are a product of the instruction they received. Teachers did not “lay down the synapses” for you to think about quantity as you solved problems of division of fractions.
Teachers told you that when you divide things get smaller. This is erroneous. Here is the importance of the language piece again. A 4th grade teacher says this to their students when introducing division that’s a problem. What could you say instead?
Discuss second oval as well. What mathematical structures would you want to convey as you teach division of fractions?
Why does the number “get bigger” proportional thinking/reasoning. 2) Division as repeated subtraction.
Questions will likely persist. Assure people that this will become clearer when we have a Bean Party!
Quantity
Concrete display of concept
Mathematical Structure
Discussion of the concrete
Symbols
Simply record keeping!
V. Faulkner and NCDPI Task Force adapted from Griffin

36. Division of Fractions U.S. teacher’s approach Flip and multiply
Answers don’t match
Confused division by 2 with division by 1/2
Have expert group report out on the U.S. teachers approach.
Review this slide and then click to next slide until title shows.
Ask expert group--Why do we have a flip for this one?

37. Division of Fractions Chinese teacher’s approach
Gave a mathematically accurate story problem
Explained the mathematics behind the operation
Gave multiple mathematical constructs for division of fractions
Have expert group report out on the Chinese teachers approach.
Review this slide and then click to next slide until title shows.
Ask expert group--Why do we have a chef for this one? (because we can use 1/2 cup portions to explain division of fractions)

38. Division of Fractions—Ah Ha!
What does it mean to divide by a fraction?
Have expert group report out on their AHA moment in this chapter.
They will likely report that it was very confusing to them. Do not prolong any discussion of division of fractions because you will explain it in detail in the next few slides.
Instead emphasize getting participants to talk about how confusing the concept was for them.

39. Problem #3 Division of Fractions
1 ¾ divided by ½
Give a Story Problem to show what is happening with this expression.
Using this guiding question of 1 ¾ divide by ½, we want you to work in a group of 2 or 3 to chart out this concept using the lesson plan outline and making sure to utilize a story problem to help you explain the concrete/quantity aspect of the concept.

40. Division as Repeated Subtraction
Bean Party!
Ok there seems to be some confusion about how we explain this to children. Division of fractions is a 6th grade concept.
Start out modeling 2 x 3. I am starting where I am confident that I have everyone in this room. Demonstrate. Watch the language.
2 cup servings for 3 people. Two cups, two cups, two cups. Connects kids to whole number multiplication.
What if I have six cups of beans and I want to give each person 2 cup serving. Okay now I am going to start with 6 cups and divide by 2 cups. Be very Explicit. Make sure they are with you.
So consider that we are having a bean party and I’ve 6 cups of beans. Everyone loves beans! So I am going to give all of my guests 2 cups. So that is 6 divided by 2 yields 3 portions.
You know what I have more friends than that so I am going to give one cup portions. So now what will happen. MODEL 6 divided by 1 with beans.
Okay—but you know I have a lot of friends, and I am hoping that they won’t notice if I’m sort of cheap with the beans. How can I serve more people. You know I am just going to give them a ½ a cup a piece. Let’s see what happens… Wow ---I can serve 12 people. 6 divided by ½ is what?. GET AUDIENCE to tell you that there will be 12 servings.
Why did the number get larger? (As the unit that we are dividing by becomes smaller, we can access more portions.) This is proportional reasoning and can be illustrated through a diagram. See next slide.

41. For each Cup of Beans— I get two portions if I divide by 1/2
So what about ½ cup servings. Out of each cup---I would get 2 servings---6 cups x 2 servings is 12, so I can serve 12 people.
What if I had 1/3 cup servings? How would our diagram look? See next slide.

42. What if I divide by 1/3? How would my diagram look?
How many portions would I have?
Why are my number of portions getting larger?
How does this idea of quantity tie into the math structure of proportional reasoning and repeated subtraction?
Gives new meaning to flip and multiply (multiplication by the reciprocal)

43. Division of Fractions: Measurement Model
How many ½ cups would go into a
1 ¾ cup container?
How many ½ cup scoops?
How many ½ cup servings?
½ of a ½
1 3/4
So what about our original problem?
Using the 6 cups divided by 2 quantity structure, we can model for kids that 1 ¾ cups divided by ½ cup units is 3 ½.
Use your beans and cups to model this problem.
Keep in mind that the unit that we are measuring to is “1/2” cup units. We can measure out 3 “1/2” cup units and then we have ½ of a ½ unit left. So---we have 3 and ½ (1/2 cup) servings from 1 ¾ of a cup of beans. Stress the unit here!!
1/2 1
1/2
1/2

44. Division of Fractions: Bean Party! Division as Repeated Subtraction2 1
“Why does this constant amount of beans “looks like it is getting bigger”?
Division as repeated subtraction
2 ½ / ½
Using 2 ½ divided by ½, have participants discuss their new learning in small groups and then report out. (If time allows)
Quantity
Concrete display of concept
Mathematical Structure
Discussion of the concrete
Symbols
Simply record keeping!
V. Faulkner and DPI Task Force adapted from Griffin

45. Division Structures Measurement/Repeated Subtraction
“How many 2s can I get out of 10?”
Partitive/Unitizing/Fair Shares
“How many would one person get? or “What would that mean in relation to 1?”
Product/Factor
“If I have an area of 10 and one side is 2, how long is the other side?” 2
If I have 10 cups of beans and I give out 2 cup portions, how many servings will that provide? 10 2 2 2 ? ? 2 10 ?
If 2 people find $10 how much will each person get ? 10 2 ?

46. Teaching Operations on Fractions
“Invert and Multiply”
vs.
Developing the concept of the procedure
Again we have this same issue of the instruction we received obscuring the mathematics. Learning the algorithm does NOT mean that you understand the mathematics.
On the other hand, understanding the concept can help you to learn and remember the algorithm!

47. Problem #3 Division of Fractions
Why DO we “Flip and Multiply” anyway???
Eloquent Explanation: Remember.
We can apply this to our problem by what the Chinese Teachers call “maintaining the value of a quotient”
Remember that anything times one is equal to its original identity. This is the identity property of multiplication. These properties tend to be taught as something to memorize and “know” rather than something that we UTILIZE as mathematicians. They also become obscured when we emphasize procedures over mathematical constructs.
I will digress here to make this point. For starters let’s remember how the identity property works. Do you agree that we are multiplying 10/2 by one when we multiply by 5/5? 5/5 is just one of the many FORMS of ONE that we can chose to maintain the value of the quotient. Correct?
Make sure audience is on board with this.

48. Rethinking Adding Fractionsx
See Unit 3 Slide 48 Dialogue. x +

49. Maintaining the Value of the Quotient
Click on the division of fraction problem ¾ divided by ½.
Click the equivalent form and make certain participants know why these are equivalent. (4 (division sign 2) is the same as 4/2, right?
On the third click—we can multiply the numerator and denominator by 1—we are going to use the form of the number of 2/2 as that will allow us to simplify the denominator to 1.
On the fourth click we again see an equivalent form of our division problem, since 2 divided by one is equivalent to 2.
In the fourth click--Our denominator now simplifies to 1 since 2 divided by 2 is 1.
Click 6 –we did show our “flip and multiply” equivalent.
Click 7—When we multiply the numerator—we have 6 and we multiply our denominator we have 4 which simplifies to 1 and 2/4 and then 1 ½.

50. Maintaining the Value of the Quotient
Watch This!!
Now, let’s apply this understanding to this mathematical proof for why we flip and multiply to our Division of fractions problem.
Let’s look at this very first step. We can see that 1 ¾ is an equivalent to 7/4.
Do you see in the second step that all we have done is multiply by 1? (2/2)
In the third step we have written out in vertical form, but look here that in the Numerator we are multiplying by 2 and in the Denominator we have multiplied by 2. Now in vertical form we call these the Dividend and the Divisor, but that is semantics.
Note: take your time here and make sure that the participants understand and really see here that you are just multiplying by 1.
What form of one have we chosen? Why?
Now the rest follows easily from that first bold step of just multiplying by 1.
As we simplify our expressions we get to 1 3/4 * 2 and eventually to the most simplified form of this expression--3 1/2
This is a simple symbolic proof for flipping and multiplying.

51. Teaching Operations on Fractions
“Invert and Multiply”
vs.
Developing concept of the operation
“Do the same thing to the top as you do to the bottom”
Multiplicative Identity of Multiplication or Maintaining the Value of the Quotient
Which of these strategies should we use with our struggling math students?

52. Marilyn Burn’s Fraction Kit
It is important to show the real quantities behind fractions.
The top bar represents one whole, the bar below the top bar is ½ of the whole bar. (1/4, 1/8 and 1/16 follow these first two bars.)

53. Fraction Game
See Trainer Notes for Dialogue on Fraction Game—Slide 53.

54. Implementation
Teacher’s discussion of the Mathematical Structure is critical.
Deborah Ball has found that teacher knowledge affects student growth.
That was a powerful lesson---we can see how much the manipulatives helped the teacher to help the students to understand the concept.
What Deborah Ball has found is that the manipulatives alone will not teach the math—again-consider the TIMSS research about implementation, Ma’s work about teacher knowledge, or our adaptation of Griffins work about conveying mathematical structure.
This mathematical structure circle in Griffin’s model is critical for the teachers and student understanding.
Consider the following--- (Go to next slide.)

55. Manipulatives and “Magical Hopes”
Deborah Ball
How do you explain this to a student?
Deborah Ball in her article “Magical Hopes” poses the following student error.
If we represent 1/6 like so (click once) - Why then is the following not true? (click 4 more times).
How do we explain this concept to the student? Our whole is one whole circle (6/6). We are keeping this unit. We begin to think that the manipulatives are enough for student conceptual understanding. This is not true however; manipulatives alone are not enough.
It is important to have this conversation with the child but we must help the child with the conceptual understanding.
It is all about the concept of the UNIT. (Go to the next slide.)
1/6 + 1/6 = 2/6
1/6 + 1/6 = 2/12

56. Game from SRA Real Math I roll 5/6 What should I do with my 5/6ths?
4/6 + 1/6 = 5/6
Real Math from SRA/McGraw-Hill has a game that supports kids thinking with composing and decomposing fractions and helps us to see that we can represent our value different ways. The key is that the unit whole is maintained. One circle is a unit whole that does not change over the course of a given situation. A pair of students would have a sheet of paper with lots of unit circles on the paper, divided up into different fractional parts. From wholes, 1/2s, 1/3, 1/4 1/5 and 1/6ths. We will look now at what might happen on one particular roll.
If I roll two dice and the Red die is my numerator and the white die is my denominator, I can then use what I roll to “claim” a circle. You win a circle by claiming more than half of the circle. Let’s say I roll 5/6. What should I do? I have a color marker and am trying to win circles.
Animation will show empty circles. And “ I roll 5/6ths”. Let’s start dominating circles! I color in 1/6, 2/6, 3/6--HAVE I WON this circle yet? (click up to 3/6ths) No I am at half. Okay, 1/6 more (click) and now have I dominated this circle? Yes, I have more than half, my opponent cannot win this circle. So what should I do with my last 6th? Waste it on this circle? NO, I will put it on another circle so that I can start to get my color on that circle. (click).
Click again and show that it is 5/6ths but we have put it into 4/6 and 1/6. Is this 5/12ths? No. Do you see that it becomes clear that you are just decomposing your 5/6 into two parts and demonstrating your value in a different form. This will reinforce the idea of the UNIT WHOLE. As well as different forms of a value, composing and decomposing, and deeper understanding of what it means to add fractions.

57. Prototype for Lesson Construction2 1
The Idea of the
Unit
In the conceptual understanding of 1/6 + 1/6 , it is important that the we talk about the “unit”. We are combining like quantities within the unit of 6/6.
What we have found in our work, which is supported by the research, is that teachers need support in this implementation of the mathematics so that they are able to deeply convey mathematical structures.
Quantity
Concrete display of concept
Mathematical Structure
Discussion of the concrete
Symbols
Simply record keeping!
V. Faulkner and NCDPI Task Force adapted from Griffin

58. “Magical Hopes”
How is it that Sean thought 6 could be both odd and even?
Why were manipulatives not the answer for Sean?
What assumption do we make as teachers about the connections that students make with the mathematics?
How can we help students see the relationship between 4/8 and 4/4 and 3/3 and 5/5?
What do we need to keep in mind as we use manipulatives as a mathematical tool in developing conceptual understanding?
See Discussion Question possible responses in Trainer Notes.

59. Problem #4 Area and Perimeter
Imagine that one of your students comes to class very excited. She explains that she has discovered that as the perimeter of a closed figure increases the area also increases. She shows you a picture to explain what she is doing. P = 16cm A = 16 cm2 P= 24cm A = 32cm2 How would you respond to this student?
4 cm
4 cm
8 cm
4 cm
Guiding question for 4th chapter
Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you this picture to prove what she is doing.
(participants read slide)

60. Area and Perimeter U.S. Teacher’s Approach Praised student for answer
Looked it up in a book
Have expert group report out on the U.S. teachers approach.
Review this slide and then click to next slide until title shows.
Ask expert group--Why do we have a teacher proudly displaying a book for this one? Draw out the fact that teachers did not trust their own understanding of the math to explore the topic,

61. Area and Perimeter Chinese Teacher’s Approach
Understood student’s misunderstanding
Supported student in a deeper understanding of area and perimeter
Have expert group report out on the U.S. teachers approach.
Ask them about their biggest pedagogical shift in reviewing this chapter.
Review this slide and then click to next slide until title shows.
Ask expert group--Why do we have this graphic for this slide--(Chinese teachers tended to show the 3 conditions)

62. Similar portions of U. S. and Chinese teachers accepted the theory immediately.
From this point, paths diverged. Note the number of U.S. teachers who were not sure and provided no further exploration.

63. A Condition in which Student’s Claim is supported
Perimeter = 16 units
Area = 16 square units
Perimeter = 24 units
Area = 32 square units
Let’s review the learning from this chapter.
This is the student’s claim and the thinking behind her proof:
The increase in area is in green based on an increase of 4 units to one pair of opposite sides. The other pair of sides remains the same.
A Condition in which Student’s Claim is supported

64. Clarifying the Conditions
Three Cases
The length or width is increased
The length and width are increased
Either the length or width increases while the other dimension decreases
Teachers questioned when the conditions were favorable to meeting the claim and when they were not.
In the first two cases, area will always increase.
In the third case, the area may increase, decrease or remain the same.

65. Explaining the Conditions
In the first two conditions, an area is added.
Length Increases 4 4 8 4 8
Width Increases
Original Figure
Reinforce that in this condition the area will ALWAYS increases. This is what the student was picturing when she made her claim.
When the length or width is increased from 4 to 8, the area doubles from 16 to 32.
When both the length and the width double, the area quadruples from 16 to 64. 8 4
Length and Width Increase 8

66. Explaining the Conditions
In the third condition, the area can decrease, increase or stay the same.
Perimeter =28 units 4
Area = 40 sq units 10
Case 3
(P=44 A=40)
Case 2
(P =30 A=56)
Case 1
(P=30 A=36)
When perimeter increases, area can decrease, increase or stay the same. All bets are off. You case destroys the claim that “if my perimeter increases my area always increases”
When we understand the relationships better we can begin to see the claim as hinging on conditions. In the first two conditions the claim is correct. But not under these conditions.
Case 1 length increases, width decreases, area decreases
Case 2 Length decreases, width increases, area increases
Case 3 Width decreases, length increases, Area stays the same 8 2 3 20 12 7

67. Solution
When an increase in the perimeter is caused by the increase in either or both the length and the width of a rectangle, the area of the figure will increase accordingly.
When an increase in the perimeter is caused by an increase in one dimension (length or width) followed by a decrease in the other dimension, the area of the figure may or may not increase.
The claim in conditionally correct.

68. Distributive Property
Area = LW
Area = (L + 4) W (Length is increased by 4.)
Area = LW + 4W
The additional area is 4W.
Explanations why some conditions supported the student’s claim and others did not.
Diagrams of additional area added to the originals figures support this as well as the Distributive Property.
The area of the original rectangle is LW. When 4 is added to the length, the area of the new figure is increased by 4W. (The distributive property can be used to demonstrate this increase in perimeter of 8 as it relates to an increase in area of 4(W) or 4W.

69. Teacher Exploration Affected by Intention Affected by Strategy
Must have interest
Must have self-confidence
Knowledge and Attitude
Affected by Strategy
Knowledge of topic
Ways of thinking in mathematics
Basic principles of the discipline related to the approach
Most American teachers did not show any intention of exploration so they did not come up with a strategy.
Ways of thinking provides a jump from the basic knowledge level
Principle---conditionality of mathematical proposition (i.e. distributive property_

70. Key Factors in the Difference
Knowledge of Topic
Mathematical Attitude
Significant in the solution of the Area and Perimeter Problem
In order to foster exploration by our students, we must embrace mathematical ways and approach new situations independently.
Chinese teachers prepare small lectures –a complete presentation of the concept. Organized talk
The Chinese teachers tend to think rigorously, use mathematical terms in discussions and justify their opinions with mathematical arguments.

71. References
Royer, J. M. (Ed.) (2003). Mathematical Cognition. Greenwich, CT.: Infoage Publishing.
Ma, Liping (1999) Knowing and Teaching Elementary Mathematics. Edison, NJ, Lawrence Erlbaum Associates.
Ball, Deborah (1992) “Magical Hopes: Manipulatives and the Reform of Math Education”, American Educator, Summer 1992
Ball’s Website: