1. Day 3 Number Sense Grades 3-5
Math & Science Collaborative at the Allegheny Intermediate Unit

2. Sharing
Share the mental math activity and game you did with your class.
What successes did your students have?
What challenges did they face?
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3. Read and Discuss
Discuss: “Why Children Have Difficulties Mastering the Basic Number Combinations and how to Help Them.” from TCM, August 2006
What are some reasons why students have difficulty mastering basic combinations?
What new ideas did you discuss in your
group about the relationship between
number sense and fluency after reading
the article?
Corinne’s Notes from the Article:
Three phases of mastering basic number combinations
Counting strategies (object counting or verbal counting) – starting from 1 or counting on
Reasoning Strategies (using known facts to derive new ones)
Mastery that is efficient
Adding it Up says attaining computational fluency (efficient, appropriate and flexible) is essential aspect of math proficiency
Vignettes (there are 4 in this article) can confront some of the things teachers might now emphasize or do in their classrooms
HOW CHILDREN LEARN BASIC COMBINATIONS:
Conventional wisdom says they grow out of rote memorization of facts through practice and reinforcement (e.g flash cards and tables)
Number sense view says that mastery that UNDERLIES COMPUTATIONAL FLUENCY grows out of discovering the patterns and relationships that interconnect the basic computations (part of phases 1 and 2 above)
People more easily learn a body of knowledge by focusing on its structure
Well-connected factual knowledge is easier to retain in memory and to transfer (ex. Kids who understand composition, are helped in understanding decomposition)
Children with a rich grasp of no. and arith.patterns are more likely to get to phase 2
Research shows that mental-arithmetic experts have a web of interconnected ideas
Phases 1 and 2 are NEEDED for laying conceptual groundwork and providing reasoning strategies that underlie computational fluency
REASONS FOR DIFFICULTIES – Defects in learners or inadequate instruction?
Conventionally, many kids labeled as LD have a heavy reliance on counting strategies; do not invent reasoning strategies; do not retain basic no. combos greater than a sum of 5; make mistakes recalling facts
Conventional view makes learning facts unduly difficult and anxiety provoking; discourages looking for patterns and relationships; deflects reasoning things out; undermines a productive disposition
Inefficiency promoted by memorization of discrete facts because it is much more challenging learning interrelated ideas. If sts do not understand teacher-imposed rules, then prone to associative confusion
Inappropriate application when memorizing instead of connecting or making sense – no effort or way to check. (e.g 2+5 is 3 or 7*0=7
Inflexibility because less likely to invent their own strategies and rely on counting
SO IN A NUTSHELL, KIDS HAVE DIFFICULTIES BECAUSE OF ROTE MEMORIZATION TECHNIQUES FOR DISCRETE FACTS
HELPING KIDS MASTER BASIS COMBINATIONS
Short-term vs long-term view
Conventional wisdom says lots and lots of practice and drill with timed tests, etc. Updated version of this gives set amount of time per fact and limits sets of facts to be learned, but this still does away with phases 1 and 2
Number sense view says that purposeful, meaningful, inquiry-based instruction is the way to go.
Adding it Up says that computational fluecy should be intertwined with conceptual understanding, strategic thinking and productive disposition. (Note that CC is explicit about conceptual and procedural being balanced – using conceptual to build to procedural)
Encourage the invention of , refinement of and sharing of informal strategies – these will get more efficient as number sense increases
Promote meaningful memorization by looking at patterns and relationships (e.g. fact families, build on known ideas, use different strategies)
Use practice wisely – to discover patterns, to make reasoning strategies more automatic
Put forth idea that Number combination proficiency should be broadly defined as any efficient strategies, not just fact retrieval
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4. “Mastery of a basic fact means that a student can give a quick response (in about 3 seconds) without resorting to inefficient means, such as counting by ones” (Van de Walle, Karp, Bay-Williams, 2013, page 171).
This is what John Van de Walle has to say about basic fact mastery.
Let’s see what the CCSSM have to say about it.
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5. Modules are built on the strands
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6. foundational development occurs PreK- Grade 2
25%
K-2=49%
16%
30%
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7. Mental Math
Encourages students to build on number relationships to solve problems instead of memorized procedures
Using number relationships helps students develop efficient, flexible strategies with accuracy
Causes students to be efficient to avoid holding numerous quantities in their heads
Strengthens students’ understanding of place value
4 x 7, 2 x 14
Does this strategy always work?
How might we show one of these strategies with a concrete model?
Array. Area model?
Number line?
Story problem?
Writing the problem horizontally encourages a student to think about and utilize the value fo the whole number.
Number relationships provide the foundation for strategies that help students remember basic facts.
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8. Video: Doubling and Halving
The teacher chooses to use a context as students consider the relationship between 4 x 7 and 2 x 14. How does this context help support student thinking?
How is the commutative property addressed in the discussion?
How do the teacher’s questions help students begin to build an understanding of doubling and halving?
The teacher is continually assessing her students during the number talk. What student understandings and misconceptions will she use to guide her next instructional steps?
8 x 25
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9. Array Discussion
The class had shared strategies for solving 8 x 25 before the teacher introduced the array model. Why do you think this instructional decision was made? How did she link previous strategies to the array?
Noel incorrectly refers to columns as rows in the array. The teacher does not correct her. Why do you think she choose to ignore this error?
How does the teacher connect the students’ additive thinking to multiplication?
How does the array support student understanding of multiplication? The commutative property? The distributive property?
32 x 15
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10. 32 X 15
What examples support the strong learning community that exists in the classroom?
What strategy closely resembles the standard algorithm? How are they similar? Different?
The distributive property is interwoven in most of the students’ strategies. What examples do you notice of this property being used?
The topic of efficiency surfaces in this discussion. Which strategies would you deem ot be most efficient and why?
Choose one of the strategies to model with an open array.
496 ÷ 8
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11. 496 ÷ 8
How does the succession of problems provide a scaffold for students to solve 496 ÷ 8?
How does the scaffold provide multiple ways for students to access the problem,?
In what other ways could 496 ÷ 8 be solved using the prior problems?
Jillian had a lovely strategy for solving this problem; however, she struggled to state what her answer was. Where is Jillian's answer in the recording on the board? What mathematical concepts are embraced in her strategy?
Students struggle to follow Jackson’s strategy. What components of division and multiplication does Jackson understand?
The teacher repeatedly asks students to explain where their answer is in their strategy. Why is that an important focal point throughout the discussion?
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12. Place Value Standards
PA Core Standards talk about “strategies based on place-value, properties of operations, and/or relationships between operations.”
Look at the standards. What strategies are they focusing on? What are “strategies based on place-value, properties of operations, and/or relationships between operations.?
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13. PA Core
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14. 2.OA.2
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15. 1.OA.6
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16. Precursors to Fluency
The sequence of number names, both starting at 1 and not starting at 1
How to count a set, keeping track of the items they counted
Understanding relationships of more, less , and same
Skip counting starting from 1 and from other numbers
Cardinality
Conservation
1-to-1 correspondence
Making tens – link to understanding place value
Subitizing
Decomposing and composing numbers
Understanding part-part-whole
Number sense
Take a few minutes and talk with your group about each one of these. Pass out concepts cards, 1 or 2 to a table, and tell participants their table will be responsible for reporting out on that concept.
What does it look like?
How will you know if students have understanding of the concept?
How can you help students move beyond counting by ones to other chunks?
What does understanding part-part-whole mean?
This activity is meant as more of a formative assessment activity to get information for us.
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17. Fluency demands…
An understanding of the meaning of operations and their relationship to each other.
The knowledge of a large repertoire of number relationships. (The “patterns” of our number system.)
This includes inverse relationships – means more than saying addition is the opposite of subtraction. Students need to know why this is so.
This includes how 4 x 5 is related to 4 x 50, how is related to
10 behaves differently in than it does in 24 x 10
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18. Fluency demands…
A thorough understanding of the base-ten number system, how numbers are structured in this system, and how this system behaves in different operations.
Knowing how a number can be composed and decomposed and using that information to be flexible and efficient with solving problems.
Involves three components.
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19. Fluency
Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of sub-problems and making use of intermediate results to solve the problem.
Accuracy depends on several aspects of the problem-solving process, among them, careful recording, the knowledge of basic number combinations and other important number relationships, and concern for double-checking results.
Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to be able to choose an appropriate strategy for the problem at hand and also to use one method to solve a problem and another method to double-check the results.
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20. How Can We Help Students with Facts?
Ongoing practice and engagement with math facts tasks
Hands-on activities and thoughtful discussions
Conceptual understanding of operations
Strategic thinking
mult anddiv
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21. Conceptual Understanding
Understanding operations
Symbolic representations
Relationship between parts and whole
Investigating the meaning of facts through hands-on activities and thoughtful discussions
Understanding is gained through:
Problem posing
Hands-on exploration
Classroom discussions
Real-world examples
p.2 Our goal in today’s math class has shifted from memorizing facts and procedures to increased understanding of math skills and concepts.
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22. Meaningful Practice
Builds on understanding of operations and using strategic reasoning to explore math facts
Practice 5 – 10 minutes daily throughout the school year
Vary the practice activities - ensures that students are motivated and engaged
Automaticity is achieved through brief, frequent, interactive activities
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23. Understanding Multiplication and Division
Address the big ideas
Guide the types of questions that are posed
Explore symbolic representations
Use models to represent multiplication and division
Number lines, manipulatives, area models, arrays
Explore concepts through problems and literature
Understanding of the operations is essential.
Initial understanding develops through a context
Deeper understanding can develop through additional contexts while fluency with the operation in play develops
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24. Understanding Multiplication and Division
What does it mean to understand multiplication?
Look at the standards for the operations. Read the standards dealing with multiplication and division.
Then talk with your group. Develop a chart, graphic, or model showing what students need to know to show an understanding of multiplication.
Note that many of these are in OA, beginning at grade 2 with foundational work for multiplication. However, some are in the NB standards as well.
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25. Building Understanding While Focusing on Fluency
Use models to represent multiplication and division
Arrays, set models, area models, number lines
Use problem contexts/real-life situations
Make sure all four categories of problems are addressed
Productive talk
Classroom environment
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26. Big Ideas Central to Understanding Multiplication and Division
Our number system is a system of patterns.
Numbers can count objects or groups.
The order of factors does not change the product.
Addition and multiplication are related operations.
Multiplication and division are inverse operations.
Numbers are flexible.
Patterns in facts can help students make sense of, and remember, the facts. For example, noticing that the products of x6 facts are twice the products of x3 facts or that x 5 facts have a 0 or 5 in the ones place.
From experience with counting, addition and subtraction, students have developed an understanding that 2 means 2 objects. In multiplication, a number can represent the size of each group and the other number represents the number of groups.
Five groups of two has the same product as 2 groups of five. The representation may look different, but the total is the same.
Both operations combine parts, but multiplication deals with equal groups and a more efficient way of combining the groups.
This means more than just saying the operations are related or inverse.
Numbers can be broken apart in different ways. The distributive property is the foundation for understanding that splitting large factors and then finding the sum of two products is a way to find unknown products.
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27. Questions Posed What do the numbers in the equation represent?
What patterns do you notice in the factors and products?
Does the order of the factors affect the product? Give examples to justify your thinking?
Do you notice a connection between this multiplication equation and this division equation? Explain.
Can you break apart one of the factors to help you find the product?
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28. Classroom Environment
Discussion
Partner Work
Interactive bulletin boards
Word walls
Centers
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29. Introducing Concepts of Multiplication and Division
Compare methods for solving problems.
Provide multiple opportunities for students to represent and solve multiplication problems.
Set models
Arrays
Area Models
Number lines
Explore division
Show students a word problem, such as Bailey made favor bags for each of the 5 guests at her birthday party. She put 3 stickers in each favor bag. How many stickers were in the 5 bags altogether? Have students share strategies.
Students need to represent multiplication situations with concrete, pictorial, and symbolic representations.
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30. Looking at One Example: x2
What are the big ideas around multiplying by 2?
Multiplication by 2 is same as doubling.
Numbers stand for a variety of things. Operation symbols help us determine what the numbers represent.
Our number system is a system of patterns.
Order of factors does not change the product.
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31. Possible Questions to Support the Big Ideas X2
What does it mean to have twice as much? What does it mean to double a quantity?
What does it mean to have half as much?
What do the numbers in the equation mean?
What patterns do you notice in the products?
Does the order of the factors affect the products? Give examples to justify your thinking.
How are a sum and a product the same? How are they different?
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32. Literature Connection
Two of Everything– read and discuss.
After story, discuss what doubled. Ask:
Are doubling and twice as many the same? Explain.
Can you find twice as many by adding? How?
Can you find twice as many by multiplying? How? pp
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33. Word Problems
Students need to visualize the facts using a concrete model and move from concrete/visual experiences to symbolic representations. They need to use concrete items and draw pictures.
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34. Word Problems Pose problem such as the following:
Mrs. Short baked some chocolate brownies. She placed 6 plates on the table and put 2 brownies on each plate. How many brownies did she put on plates?
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35. Observe Patterns with Twos
Have students think about a series of brownie problems, and write a multiplication equation to solve each one.
1 plate with 2 brownies on each plate.
2 plates with 2 brownies on each plate. …..
10 plates with 2 brownies on each plate.
What patterns do you notice?
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36. Commutative Property
Provide students with manipulatives and paper. The paper can represent the baskets. Have them determine the answer to the following:
Colin had 2 baskets with 3 apples in each basket, how many apples did he have?
Colin had 3 baskets with 2 apples in each basket. How many apples did he have?
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37. Building Automaticity
Short practice – daily routine
Games
Rolling for Doubles
Double Up
Fact Card Jumps
Doubles Match Up
Connect to Division
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38. Other Groups of Facts X10 X5 x1 X0 x3 X4 X6 X9 X8 x7
For the group of facts assigned to your group, read the pages for your set of facts.
X10 X5 x1 X0 x3 X4 X6 X9 X8 x7
Refer to chart on page 10 for rationale for this sequence.
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39. Sharing Give a brief overview of you group of facts. Share:
An interesting activity for your fact
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40. Article Discussion
Read and discuss: “Fluency: Simply Fast and Accurate? I Think Not!” by NCTM Past President Linda M. Gojak
How do the ideas in the article resonate with your ideas?
What is the same? What is different?
What stood out for you?
Make sure that it comes out that fluency entails 3 main parts:
Efficiency
Accuracy
Flexibility
My takeaways (Corinne)
Entire 2nd and 3rd paragraph really stand out to me
Memorization is one of the least effective ways to reach fluency (4th paragraph)
Like the questions Linda G poses in the 7th paragraph (Do we give students the opportunity to think about what they know and understand and use it in ways that make sense to them? ….)
8th paragraph mentions “mastery of facts” I think that as soon as we see the work “master” we assume that means to memorize – WRONG
“As we think about fluency, we should realize that it is more than procedural.”
Last paragraph…I am not sure our students actually enter school thinking the goal is to do math fast and get it right. I agree that we encourage it, sometimes inadvertantly, sometimes purposefully.
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41. Fluency Look back at your ideas about fluency.
What ideas, if any, do you want to add to your ideas?
Put up their post-its from day 2.
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