1. Supporting Students Who Struggle with Math
Grades 3-5
Day2

2. Day 2: Agenda Building fluency An example with one set of facts
Strategies for fact fluency
Allegheny Intermediate Unit

3. Building Fluency
Allegheny Intermediate Unit

4. Memorizing Mathematics
Conceptual gaps force children to memorize mathematics, often producing the following difficulties:
Incomplete or inaccurate memorization of information
Lack of retention
Lack of transfer
Students may memorize the information partially or incorrectly, for example, memorizing part of the traditional subtraction algorithm
Students may memorize information for a test, then forget most of what they learned, which necessitates constant review.
Because children do not understand what they memorized, they cannot use it to assimilate new, even closely related material. For example, after memorizing the subtraction procedure for two-digit numbers, children often misapply it (12 – 8) or cannot apply it to three-digit numbers. It does not guarantee that they can apply it to a slightly different case such as 304 – 128.
Allegheny Intermediate Unit

5. Mastery of Facts
Number sense (meaningful connections) may affect how basic number combination knowledge is mentally represented in long-term memory and processed
Facts are a network of interconnecting relations
For example, an understanding of commutatively may allow us to store 5 x 8 = 40 and 8 x 5 = 40 as a single triplet: 5, 8, 40.
Allegheny Intermediate Unit

6. Conceptual Understanding
Understanding operations
Symbolic representations
Relationship between parts and whole
Understanding is gained through:
Problem posing
Hands-on exploration
Classroom discussions
Real-world examples
Allegheny Intermediate Unit

7. Possible Teaching Sequence
X 2
X10 X5 x1 X0 X3 X4 X6 X9 X8 x7
Allegheny Intermediate Unit

8. Understanding Multiplication and Division
Address the big ideas
Guide the types of questions that are posed
Explore symbolic representations
Use models to represent addition and subtraction
Number lines, manipulatives, ten frames, number charts
Explore concepts through problems and literature
Before math fact practice begins, understanding fo the operations is essential. Students who understand the concepts of addition and subtraction are able to understand the connections between math facts and real situations are better equipped to effectively solve problems by choosing the operation that makes sense. They are better able to make reasonable judgments about sums and differences and are better prepared to begin to remember the math facts because they understand what they are being asked to recall/memorize.
Allegheny Intermediate Unit

9. Classroom Environment
Discussion
Partner Work
Interactive bulletin boards
Word walls
Centers
Allegheny Intermediate Unit

10. 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.
Allegheny Intermediate Unit

11. 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?
Allegheny Intermediate Unit

12. 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
Allegheny Intermediate Unit

13. Doubling Task
Place 1 to 10 counters in plastic bags and give each pair of students 3 or 4 bags containing different quantities. Have additional counters available for students who need to explore the problems by actually creating the doubles.
Ask students to determine how many coins are in the bag now and how many would be in the bag after it fell into the pot.
Have them record an addition and multiplication equation that would solve the problem and to explain their answer.
Allegheny Intermediate Unit

14. 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.
Allegheny Intermediate Unit

15. 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?
Allegheny Intermediate Unit

16. 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?
Allegheny Intermediate Unit

17. 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?
Allegheny Intermediate Unit

18. Writing Doubles Word Problems
Work with the class to write a doubles word problem together..
Record the story on chart paper and label it with the appropriate number sentence.
Have students work with a partner to write a doubles word problem.
Allegheny Intermediate Unit

19. Building Automaticity
Short practice – daily routine
Games
Rolling for Doubles
Double Up
Fact Card Jumps
Doubles Match Up
Connect to Division
Allegheny Intermediate Unit

20. Games Play the games with a partner.
How is this targeted game practice the same or different than other multiplication games?
Allegheny Intermediate Unit

21. Mastery of Facts Slow Process: Three Phases
Relatively slow counting strategies
Relatively slow reasoning strategies
Mastery – relatively fast fact retrieval
Allowing children to use counting or reasoning strategies:
Using informal strategies can lead to discovering patterns and relations that underlie an efficient mental representation and processing of basic combinations
Practice using counting and reasoning strategies will, over time, lead to their efficient execution
Allegheny Intermediate Unit

22. Fluency and Automaticity
Learning Progression stages
Understanding – manipulatives and pictorial representations
Relationship – making connections within and across
Fluency – strategy development for accuracy
Automaticity – practice to facilitate automaticity
Fluency and Automaticity
Basic Facts
Computation
Allegheny Intermediate Unit

23. Fluency and Automaticity
Mass practice or Drill-and-Kill is NOT an efficient or targeted technique for many struggling students
Most practice efforts are NOF FLUENCY based – more accuracy based
Allegheny Intermediate Unit

24. Steps to Fluency and Automaticity
Specific criterion for introducing new facts
Intensive practice on newly introduced facts (more than 2 x)
Systematic practice on previously introduced facts
Adequate allotted time (5-10 min/day)
Record keeping
Motivational procedures
Allegheny Intermediate Unit

25. Steps to Fluency and Automaticity
3-4 facts
Specific criterion for introducing new facts
Intensive practice on newly introduced facts (more than 1x) each fact on 4 cards
Systematic practice on previously introduced facts
Adequate allotted time (5-10 min/day)
Record keeping
Motivational procedures
Continuous cycle
Regular schedule
Allegheny Intermediate Unit

26. Practice Math Facts Peer to peer or individual or small group
Students must say the fact (“Four times five is twenty”)
Error Correction Procedure
The only correct response is the correct answer to the problem
All other responses should be corrected
For example, hesitating, saying the incorrect fact, using a strategy
Allegheny Intermediate Unit

27. Practice Math Facts Stop student and say correct answer
Say correct answer with student
Have student say correct answer
Partner says correct answer
Fact is placed three cards back to make sure student has opportunity to re-practice the fact while the correction is still in short-term memory
Allegheny Intermediate Unit

28. Facilitating Fluency and Automaticity
Instruction for conceptual understanding must occur first
Automaticity activities must be cumulative
Newly introduced facts receive intensive practice while previously introduced facts receive less intensive practice, but still systematically planned
Fluency/automaticity activities should occur for no less than 10 minutes
Automaticity practice must be purposeful and systematic as well as carefully controlled by the tecaher
Allegheny Intermediate Unit

29. Allegheny Intermediate Unit

30. Allegheny Intermediate Unit

31. Role of Teachers
Encourage children to use efficient strategies to retrieve facts, not just memorization
Help children conceptually understand operations before drilling facts
Encourage children to look for patterns and relations
Encourage children to build on what they know
Practice should focus on making reasoning strategies more automatic, not on drilling isolated facts
Allegheny Intermediate Unit

32. 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
Allegheny Intermediate Unit

33. Strategic Thinking Math fact strategies
Focuses attention on number sense, operations, patterns, properties, number concepts
Big ideas: concept of tens, knowing the order of addends will not affect the sum, various numbers can create the same product (e.g., 8 x 5 = 40 and 4 x 10 = 40)
Allegheny Intermediate Unit

34. Allegheny Intermediate Unit
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
Allegheny Intermediate Unit

35. Time to Practice
Use the index cards to create a set of math fact cards you might use with a particular student or a small group of students.
Practice with your partner.
Students must say the fact (“Four times five is twenty”)
Error Correction Procedure
The only correct response is the correct answer to the problem
All other responses should be corrected
For example, hesitating, saying the incorrect fact, using a strategy
Stop student and say correct answer
Say correct answer with student
Have student say correct answer
Partner says correct answer
Fact is placed three cards back to make sure student has opportunity to re-practice the fact while the correction is still in short-term memory
Allegheny Intermediate Unit

36. Read and Discuss
Read the article “Why Children Have Difficulties Mastering the Basic Number Combinations and how to Help Them.”
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?
Allegheny Intermediate Unit

37. Understanding Multiplication and Division
Students who understand these concepts recognize the connection between math facts and real situations
i.e., 2 vases of flowers with 9 flowers in each vase would be represented by 2 x 9)
Are better equipped to effectively solve math problems by choosing the operation that makes sense
Are better able to make reasonable judgments about products and quotients
E.g., 2 x 9 can’t be 11 because you have 2 groups of 9 flowers. That doesn’t make sense.
Allegheny Intermediate Unit

38. Standards for Mathematical Practice
Take time to read the Standards for Mathematical Practice.
What is something that stood out/interested you? Something you have a question about?
As a table group, discuss: To what extent do you think these practices are embedded in the daily work of teachers and students?
How do number talks support development of the Mathematical Practices?
Math & Science Collaborative a
t the Allegheny Intermediate Unit

39. Allegheny Intermediate Unit
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 =
83 – 56 =
How might we show one of these strategies with a concrete 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.
Allegheny Intermediate Unit

40. Number Talks: Supporting Fluency
Engage students in mental math through grappling with mathematical problems
A powerful tool for helping students develop computational fluency because the expectation is that they will use number relationships and the structures of numbers to add, subtract, multiply and divide.
7 x 7
Math & Science Collaborative at the Allegheny Intermediate Unit

41. 7 x 7 As you watch the video, consider the following questions:
What evidence in the video supports student understanding of multiplication?
How do students’ strategies exhibit number sense?
How does the teacher connect math ideas throughout the number talk?
How does fluency with smaller multiplication problems support the students’ strategies?
What examples of properties can be observed in the strategies and discussion?
Which strategies were most accessible to you? More challenging?
How could the strategies for 7 x 7 be connected using an array?
4 x 7
2 x 14
Allegheny Intermediate Unit

42. Doubling and Halving
Watch the video and consider the following questions:
How does using a context support student reasoning about the commutative property?
How is a context used to support student thinking about the doubling and halving strategy?
How does the context and the array bring student understandings and misconceptions to the forefront?
What questions does the teacher pose to help students confront their misconceptions?
8 x 25
Allegheny Intermediate Unit

43. Arrays: 8 x 25
As you watch the video, consider the following questions:
How does the array model support the student strategies?
How does breaking the factors into friendly numbers promote the goals of efficiency and flexibility?
How do the teacher’s questions foster an understanding of multiplication?
How could you represent other multiplication problems with an array?
How does the array help connect additive thinking to multiplicative reasoning?
Which math understandings and misconceptions are addressed in this model?
Allegheny Intermediate Unit