1. Building Fluency
Geri, the following is the feedback link for your session that we ask you provide to participants! 

2. IF we want to impact achievement for ALL students…
IF we want to impact achievement for ALL students… Geri Lorway Thinking101.ca

3. 40 years of research…

4. It’s not the curriculum…

5. It’s the teaching!

6. EVOLVE or go EXTINCT

7. Focus on number “sense” before number facts…….
Build a foundation for recall and transfer.

8. BUILD
EXPLAIN
REPRESENT
COMPARE
SELF ASSESS SYNTHESIS

9. Number Sense: Learners who have a well-developed Number Sense are able to succeed in early math (and beyond), while children who don’t are at much greater risk of falling increasingly further behind.  They also demonstrated that virtually any child could develop Number Sense (Case, Griffin, and Siegler,1994; Griffin, 2004).

10. Components of Number Sense:
EXPRESS Number
Identify and name multiple ways Trust the Count
additive composition vs equal units
MAGNITUDE & DENSITY
Components of Number Sense:
EQUAL
compare expressions using the equal sign
more, less , equal, not equal number properties emerge
INVERSE RELATIONSHIPS

11. Grade one: Number 6 to 10 Number 11 to 20 Equal Equal parts (multiplicative reasoning)
Sort one attribute Trust and repeat core forward and backward: predict
Examine, explore, replicate 3 D objects Direct compare measures
Grade 2: Number facts 6 to Number sense 11 to Number Sense to 100
Equality Inequality commutative and associative properties Zero & One
Sort 2 attributes Growth patterns (move pattern to number)
Match 2D to 3D: build vocabulary for attributes and properties
Iterate the unit to measure Compare size of units (multiplicative reasoning)
Grade 3: Number facts to 20 Inverse fluency to Number sense to 1000
Reasoning about missing parts (symbol) Sort 2 or more attributes
increasing & decreasing patterns: link to number
Multiplicative reasoning: fraction sense, multiplication sense (division) place value sense
standard units of measure 1 to to 1000
Reason about 3D objects Reason about polygons

12. Number Sense is not addition and subtraction facts
Number Sense is not addition and subtraction facts. It begins with automatic recognition of quantity and the ability to express a given number multiple ways. 
Subitizing forms the foundation, not the counting sequence.

13. Your brain knows small collection without the need to count:
2, 3, 5 (if arranged to see 2 and 3) 4 because 2 twos.
With practice and attention students learn to trust and use these.
Seven becomes 5 plus 2 or 4 plus 3 both of which are related to if you know number properties.

14. Push and circle 2, 3 and 5 to link fingers to brain in a positive way.
Make it automatic.

15. Use the subitize quantities to recognize and name collections to and including ten.
Additive Composition comes before equations.
Build tens to introduce the 10 frame, then use the ten frame as one way to explore magnitude & inverses as you practice “facts”.

16. READY? What do you see? 2s, 3s, 5s or a total. Discuss both.

17. Students come up and describe the parts.

19. Students come up and describe the parts.

21.
Here are the ones I found. Compare to your list.

22. Build and add to a pocket chart of expressions
Build and add to a pocket chart of expressions .As you work through lessons, add to the chart.

23. Use the list to find and circle parts in collections

24. Does it need to be this eight. No it does not
Does it need to be this eight? No it does not. However, if we all make the same 8 to start it will be easier to quickly see who is on track. However a student who makes a different arrangement is not wrong. (Be watching for the student who uses the chips to try to model the symbol 8 he or she is on the wrong track.
If we all make this 8 arrangement we can all talk about it and show our thinking.
Build 8 with your chips.

25. 5 + 3 Find this expression in your collection.
Slide the 3 in one move.

26. 5 + 3
3 + 5
Read each expression as you point to the chips. Gesture helps build recall.
Read this as or
The order does not change there is still 8.

27. Label the parts
Take this opportunity to practice a few 3s and 5s in the air before students print them. Be sure to model on the board.

28. 5 + 3 = 3 + 5
Because order does not matter we can record this equation.
Have a student read aloud. As student reads, point to the part.
Have students practice reading and pointing. Reading aloud is an important part of building recall.

29. 5 + 3 = 3 + 5 commutative property
We name this property of numbers the commutative property.
I like to take a minute and have students write and read it with me.
Comm u ta tive. Cover read in mind. Look and check.

30. Slide your chips back together.

31. 6 + 2
We said was in this collection. Can you see it? Slide the 2 apart.

32. 6 + 2
2 + 6
Again you can read the collection as or Practice reading and pointing.

33. 6 + 2 = 2 + 6
Label the parts and record the equality.

34. commutative property 6 + 2 = 2 + 6
This is another example of the commutative property.

35. Slide your collection together.

36. 4 + 4
Can you see 4 + 4? Put your palm over 4 and slide apart in one group

37. 4 + 4

38. 4 + 4
Label and read either direction

39. 4 + 4 = 4 + 4
Record as an equality. Practice pointing and reading. It seems funny but you can read in either direction.
If we had 4 cows and 4 pigs we could change the order. In combinations and permutations students will need to think this flexibly.

40. 4 + 4 = 4 + 4
Commutative property

41. Slide together

42. 1 + 7
Slide one apart

43. 7 + 1
1 + 7
Label and read

44. 7 + 1 =
Record and discuss the commutative property

45. 6 + 2
5 + 3
7 + 1
4 + 4
These expressions are all related. Let’s see…

46. 6 + 2 is the same amount as 5 + 3, watch.

47. Slide one from the 6 to the 2, now you have
5 + 3.

48. 6 + 2 = 5 + 3
This is an equality. It is true because all you did was change the position of a dot.

49. 4 + 4 is the same amount as

50. 4 + 4 is the same amount as

51. Slide one from the 4 to have 5 and 3

52. 4 + 4 = 5 + 3

53. 6 + 2 is the same amount as

54. Slide one from the 6 to have 7 + 1

55. 6 + 2 = 7 + 1

56. 6 + 2 = 7 + 1

57.
7 + 1
4 + 4
Here is another way they are related…..

58. = 6 + 2

59. (3 + 3) + (1 + 1) = 6 + 2
Associative Property

60. = 3 + 5

61. 3 + (3 + 2) = 3 + 5
Associative Property

62. = 6 + 2
Associative Property

63. (3 + 3) + 2 = 6 + 2
Associative Property

64. I can build new ones
Start with 6 + 2
Associative Property

65. 6 + 2 = (2 + 4) + 2
Associative Property

66. Start with 7 + 1
Associative Property

67. 7 + 1 =
= 3 + 5
Have students place brackets.

68. associative property 2 + 6 = 2 + (4 + 2) 2 + 4 + 2 = 2 + (2 + 2) + 2
(2 + 2) + (2 + 2) = 4 + 4
Let’s look at the next one.

69. READY?

76. Describe what you saw.

77. Did you find these?.

78. Which ones can you relate?

79. commutative property
6 + 4 = 4 + 6
7 + 3 = 3 + 7

80. Can you show the commutative property for each of them?

81. equalities
6 + 4 = 5 + 5
7 + 3 = 8 + 2

82. Can you show more equalities?

83. 3 + 7 = 3 + (3 + 4) (3 + 3) + 4 = 6 + 4 3 + (2 + 3) + 2 = 3 + 5 + 2
associative property
3 + 7 = 3 + (3 + 4)
(3 + 3) + 4 = 6 + 4
3 + (2 + 3) + 2 =
3 + (5 + 2) = 3 + 7

84. Organized into arrangements that focus on 2, 3, 4
Limited to 10
Colour distractors
Simple to keep focus on number
See the whole as a whole

85. The ability to subitize and its impact on counting undergoes a long lasting development until the age of 17 years. A great percentage of students struggling with acquiring basic arithmetic skills exhibit developmental deficits in the correctness and speed of this special visual capacity.
Research demonstrates that subitizing and visual counting can be improved by daily practice. There was a significant improvement in basic arithmetic skills when students were given daily practice for 21 days.
Burkhart Fischer, Dipl. Phys., Andrea Köngeter, Dipl. Biol., and Klaus Hartnegg, Dipl. Phys. 2008

87. Subitizing sets the foundation of number. Not the counting sequence.
Additive composition Equal parts: unitizing
One to one correspondence vs many to one correspondence.
Pirie & Kieren: First model we learn seems to stick and override future models.

88. A key component of number sense is REASONiNG about relationships.

90. Overwhelming agreement:
We must attend to reasoning and arithmetic. Attention to one cannot outweigh attention to the other.

91. Key Relationship to reason about: Inverse Operations.
Fact Families dumbs down a foundational relationship that applies right up to algebra…. Use the term so that I do not have to re teach the concept in the higher grades.

92. My 7

93. 1.) Use a stickie note to cover 2 dots.

95. Record the subtraction

96. I had 7, I covered 2 so
7 – 2 = 5

97. What is the related addition?

98. I see 5, I lift to show 2 and say 5 + 2 = 7

99. Students should talk and lift several times.

100. ? 7 I call these What’s Covered cards. It says 7 but some are hidden.
I see 3 but the square is covering some.
The 7 means there are 7 dots…. How many are covered?

101. Build a collection.
I see 8

102. I lifted 5.
I record 8 – 5 = 3

103. I record 3 and drop the flap….

104. I record = 8.

107. Foundations of mathematical reasoning are built on visual-temporal-spatial reasoning.

108. Nearly a century of research confirms the close connection between spatialreasoning and mathematics performance. The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related.
(Mix & Cheng, 2012).
The connection does not appear to be limited to any one strand of mathematics. It plays a role in arithmetic, word problems, measurement, geometry, algebra and calculus. Research mathematics education, psychology and even neuroscience is attempting to map these relationships.

109. Nora Newcombe
Brent Davis
WW Sawyer
Grayson Wheatley

110. Ready?

117. Ready?

128. Mix & Cheng (2012)

130. 8 tens, 4 ones (10) + 4 (1)
5 tens, 34 ones (10) + 34 (1)
40 tens, 44 ones (10) + 44 (1)
2 tens, 644 ones (10) + 24 (1)
7 tens, 14 ones (10) + 14 (1)

131. These ways are all equal
2 tens, 64 ones = )
= 7 (10) + 14 (1)
8 tens, 4 ones =
5 tens, 34 ones = 5 (10) + 34 (1)
= =

132. Place 84 on this number line. What is near it?
Is it closer to 50 or closer to 100

133.
It is between 50 and 100.
It is closer to 100 than to 50

134. What is the closest decade? 0 50 100
Practice “facts” whenever you can. See why 4 and 6 are the

135. I put in the decades between 50 and 100.
Practice “facts” whenever you can. See why 4 and 6 are the
It is between 80 and 90.
It is 4 more than 80 and 6 less than 90.

136. What is one more, one less? Two more, two less?
Narrow in to the ones.
What is one more, one less? Two more, two less?
Practice “facts” whenever you can. See why 4 and 6 are the

137. I stretched the number line to have room.
Practice “facts” whenever you can. See why 4 and 6 are the
It is 1 more than 83. It is one less than 85.
It is 2 more than 82 and 2 less than 86.

138. Count by tens, forward and backward.
Practice “facts” whenever you can. See why 4 and 6 are the
What is 10 more? Ten less?

139. Count by tens, forward and backward.
Practice “facts” whenever you can. See why 4 and 6 are the
What is 10 more? Ten less?

140. We say what’s the difference between 84 and 100
How close to 100?
We say what’s the difference between
84 and 100
Practice “facts” whenever you can. See why 4 and 6 are the
I see one to 85, 10 to 95, 5 to 100 to 100.
84 +( )

141. We say what’s the difference between 84 and 100
How close to 100?
We say what’s the difference between
84 and 100
Practice “facts” whenever you can. See why 4 and 6 are the

142.
Practice “facts” whenever you can. See why 4 and 6 are the
I see one to 85, then 15 to 100.
84 +(1 + 15)

143.
Practice “facts” whenever you can. See why 4 and 6 are the
I see 10 to 94, 1 more to 95, 5 more to 100.
84 + ( ). Or I could say

144. No matter how I explain, the difference is 16 because
= 100 therefore
100 – 16 = 84
Practice “facts” whenever you can. See why 4 and 6 are the
I could go back 10 to 90 then 6 to 94.
100 – 10 – 6 or 100 – 16.

145. We say what’s the difference between 84 and 50?
How close to 50?
We say what’s the difference between
84 and 50?
Practice “facts” whenever you can. See why 4 and 6 are the

146. I went back from 84 to 54, then back another 4.
Practice “facts” whenever you can. See why 4 and 6 are the
I went back from 84 to 54, then back another 4.
84 – 30 = – 4 = 4. I subtracted 34.

147. I started at 50 and added 30 to 80, then 4 to 84. 50 + (30 + 4) = 84.
Practice “facts” whenever you can. See why 4 and 6 are the
I started at 50 and added 30 to 80, then 4 to 84.
50 + (30 + 4) = 84.

148. We say what’s the difference between 84 and 50?
How close to 50?
We say what’s the difference between
84 and 50?
Practice “facts” whenever you can. See why 4 and 6 are the
I see 50 now add 30 to get to 80, then 4.
50 + (30 + 4)

149. I see 84, subtract 4 to 80, then – 10, - 10, -10 I subtracted 34.
Practice “facts” whenever you can. See why 4 and 6 are the
I see 84, subtract 4 to 80, then – 10, - 10, -10
I subtracted 34.

150. I see 84, subtract 10s so 74, 64, 54, then subtract 4
Practice “facts” whenever you can. See why 4 and 6 are the
I see 84, subtract 10s so 74, 64, 54, then subtract 4
I still subtracted 34.

151. No matter how I explain, the difference between 84 and 50 is 34 because
= 84 therefore
84 – 50 = 34
Practice “facts” whenever you can. See why 4 and 6 are the

152. Number Sense: Learners who have a well-developed Number Sense are able to succeed in early math (and beyond), while children who don’t are at much greater risk of falling increasingly further behind.  They also demonstrated that virtually any child could develop Number Sense (Case, Griffin, and Siegler,1994; Griffin, 2004).

153. Components of Number Sense:
EXPRESS Number
Identify and name multiple ways Trust the Count
additive composition vs equal units
MAGNITUDE & DENSITY
Components of Number Sense:
EQUAL
compare expressions using the equal sign
more, less , equal, not equal number properties emerge
INVERSE RELATIONSHIPS

154. Grade one: Number 6 to 10 Number 11 to 20 Equal Equal parts (multiplicative reasoning)
Sort one attribute Trust and repeat core forward and backward: predict
Examine, explore, replicate 3 D objects Direct compare measures
Grade 2: Number facts 6 to Number sense 11 to Number Sense to 100
Equality Inequality commutative and associative properties Zero & One
Sort 2 attributes Growth patterns (move pattern to number)
Match 2D to 3D: build vocabulary for attributes and properties
Iterate the unit to measure Compare size of units (multiplicative reasoning)
Grade 3: Number facts to 20 Inverse fluency to Number sense to 1000
Reasoning about missing parts (symbol) Sort 2 or more attributes
increasing & decreasing patterns: link to number
Multiplicative reasoning: fraction sense, multiplication sense (division) place value sense
standard units of measure 1 to to 1000
Reason about 3D objects Reason about polygons

167. Which Set Does Not Belong?

168. Interventions that benefit all students: Using visual spatial models to engage students in reasoning about number space and shape
STUDENTS Build Explain Represent Compare numbers before they “practice” facts.
BUILD AND DESCRIBE NUMBER: multiple expressions
COMPARE EXPRESSIONS USING THE EQUAL SIGN: number properties & number operations emerge as students explore equal & not equal
DEVELOP MAGNITUDE & DENSITY: how
INVERSE RELATIONSHIPS
Immerse students in vocabulary and encourage, guide, expect, it to be used.
Engage students for 10 minutes a day for 6 weeks, then ……

169. Improve all students: 10 minutes a day everyday devoted to practice with imagery.
If you are interested in accepting the challenge……. I will share 6 weeks of tasks… Every day
See the difference.
Good Grouws Wheatley

170. Finding Similarities and Differences
Are we teaching to reach all students?
Finding Similarities and Differences
The brain seeks patterns, connections, and relationships between and among prior and new learning
The ability to break a concept into its similar and dissimilar characteristics allows students to understand and often solve complex problems by analyzing them in a more simple way

173. Graphic Organizers for Similarities and Differences
Finding similarities and differences can increase student achievement by 45%
Graphic Organizers for Similarities and Differences
Compare
Classify
Create metaphors and analogies

174. Finding similarities and differences can increase student achievement by 45%
Guidance in identifying similarities and differences enhances students' understanding of and ability to use knowledge.
Independently identifying similarities and differences enhances students' understanding of and the ability to use knowledge.
Representing similarities and differences in graphic or symbolic form enhances students' understanding of and ability to use knowledge.

175. Summarizing and Note Taking increases student achievement by 34%
These skills promote greater comprehension by asking students to analyze a subject to expose what’s essential and then put it into their own words.
verbatim note taking is ineffective because it does not allow time for processing the information.

176. Summarizing and Note Taking increases student achievement by 34%
To effectively summarize, students must delete some information, substitute some information, and keep some information.
To effectively delete, substitute, and keep information, students must analyze the information thoroughly.
Being aware of the explicit structure of information is an aid to summarizing information.

177. Summarizing and Note Taking increases student achievement by 34%
Teach students how to process information for their own note taking.
Use a variety of organizers
Graphic Organizers for Similarities and Differences

178. 10-2 Strategy For every ten minutes of new learning provide students two minutes to process the new learning.
You can either pay attention or make meaning.
So time to process is essential to transfer learning to long term memory.

179. Generating Nonlinguistic Representations Increases student achievement by 27%
Research says that knowledge is stored in two forms: linguistic (in ways associated with words) and nonlinguistic (mental pictures or even physical sensations like smell, touch, kinesthetic association or sound)
The more we can use nonlinguistic representations while learning, the better we can think about and recall our knowledge

180. Generating Nonlinguistic Representations
Increases student achievement by 27%
Research says that knowledge is stored in two forms: linguistic (in ways associated with words) and nonlinguistic (mental pictures or even physical sensations like smell, touch, kinesthetic association or sound)
The more we can use nonlinguistic representations while learning, the better we can think about and recall our knowledge

181. Reinforcing Effort and Providing Recognition increases student achievement by 29%
Feedback should be timely.
The larger the delay in giving feedback, the less improvement one will see.
Feedback should be specific to a criterion, telling students where they stand relative to a specific target of knowledge or skill.

182. Students can effectively provide some of their own feedback.
In fact, non-authoritative feedback produces the most gain.
Feedback should be corrective in nature.
The best feedback shows students what is accurate and what is not.
Asking students to keep working on a task until they succeed appears to enhance student achievement.

183. Effective Praise Specifies the particulars of the accomplishment
Provides information to students about their competence or the value of their accomplishments
Is given in recognition of noteworthy effort or success at difficult tasks
Attributes success to effort and ability

184. Problem Solving
Problem solving is about finding the best solution, not just any solution.
Problem solving of unstructured problems - those that do not have clearly defined goals and usually have more than one solution- are the kinds of problems we find in everyday life.
Example: Ask students to build something using limited resources. This will generate questions and hypotheses about what may work or not work.

187. Teachers need specific support in understanding how to develop number sense in students, to guide their learning as they plan for and provide instruction (Ball & Cohen, 1996) and, ultimately, to ensure that they are spending time encouraging students to do the thinking that will improve number sense.

189. This model represents discussions and connections that are to be made in virtually every math lesson. This is not a progressive model wherein once one wedge is taught it is then seen as review material. Rather, each wedge is to be connected to each lesson throughout the curriculum