Math Facts through Conceptual Understanding

1. Math Facts through Conceptual Understanding Dawn Sparks Ann Sipe

2. Math Assessment Probe • Starting from what you know… • Fold assessment in half • Respond to one side only

3. Making the Case…Focus on number sense! • Research indicates that early number sense predicts school success more than other measures of cognition like verbal, spatial or memory skills or reading ability. Jordoan, Kaplan, Locuniak, and Ramineni, 2007

4. Washington State Report Card5th Grade Migrant Math Data

5. Migrant Math Data 5th Grade Math

6. Wa KIDS Kindergarten Readiness ESD 105

7. What is Mastery of Basic Facts: • Addition and subtraction by the end of 2nd grade • Multiplication and division by 3rdgrade • Fact fluency leads to other math fluent skills…

8. Guided Invention: • Fact mastery will not magically happen. • “Time is a poor intervention.” • Teacher’s job is to design tasks and problems that will promote invention of effective strategies, that these are shared in the classroom, thinking aloud.

9. Developmental Nature of Basic Fact Mastery

10. Phase 1: • Counting strategies- • using concrete objects • verbal counting Example includes: • 5 + 1 = count on starting at 5, 6 • 5 + 2 = 5, 6, 7 • 4 +7 = 7 then 8, 9, 10, 11

11. Phase 2: • Reasoning Strategy: • using known information to logically determine an unknown combination • 4 + 7, student knows 3 +7 = 10 so just adds on 1 more. • 10 x 8 is 80 half of that is 5 x 8 is 40

12. Phase 3: • Producing answers efficiently • just knows it • 4 + 7 = 11 • Retrieval within 3 seconds • Grade appropriate strategy • Efficient • Scalable

13. Knowing Facts from Memory-“Passive Storage View” Thinking that students will learn their facts if they just practice enough • What does this mean- • 100 isolated addition facts • 100 isolated multiplication facts • Also having to memorize subtraction and division- well over 300 pieces of information to remember • Not only having to memorize, but to keep practicing as well.

14. Effects of not knowing facts fluently… • Lack of basic fact automaticity has been shown: • Limit participation in math class discussions • Impede successful problem solving • Severely impair the development of the standard algorithms for multiple-digit addition and subtraction, long division and fractions • Misapply facts and not seeing reasonableness of answers • Inflexible thinking skills

15. Struggling Learners and Students with Disabilities: • Have difficulty memorizing so many isolated facts (but can be successful with strategies) • Drill creates, in a majority of students, unnecessary anxiety • Undermine student interest and confidence in mathematics

16. ELL and Migrant Supports • Scaffold the Language, but not the mathematic strategies • Use Realia and Models • Use Graphic Organizers • Use Student Interactions for additional practice • Use gestures for together, take apart, groups etc.

17. Explicit Strategy Instruction: • Strategies can be effective to learning math facts. • Supports students thinking rather than give the students something new to remember. • Key: Help students see possibilities and let them choose strategies that help them get to a solution without counting.

18. Resources • Origo The Box of Facts • Addition • Subtraction • Multiplication • Division • John Van de Walle • Research

19. Learning Facts: Addition

20. How many strategies?

21. Building on three strategies…

22. Operation ~ Strategy Clusters ~ Sub Strategies ~ Teaching Strategies

23. Addition Strategies • Count On • Count On 1 • Count On Turnarounds • Count On 2 • Count On 3 • Count On 0 • Use Doubles • Double • Double Plus 1 • Double Plus 2 • Bridge to 10 • All Facts

24. Count On: Addition Strategy Week One: Introduce Concept Through Real Objects Transitioning to Model Week Two: Reinforce Through Models Week Three: Practice through Games Week Four: Extend or Reteach

25. Prepare…Before teaching strategies students need to know… • Subitizing • Subitizing • Subitizing • Subitizing • The ability to instantly recognize the total quantity of objects in a group without counting

26. Week One: Introduce Concept Through Real Objects Transitioning to Model • Counting bottles • Cubes in a cup • Addition stories • Count on 1 Cards

27. Week Two: Reinforce Through Models • Count on Cards • Number Tracks • Cube Trains • Count on Cards with Numeral Cards • Does This Make Sense? • Moving from physical models to semi-abstract models to symbolic models (with numbers)

28. Week Three: Practice through Games • Spin, Count On 1, Record Fact • Reinforce with Count On Flash Cards • Count On 1 Bingo • Total and Expression

29. Week Four: Extend or Reteach • ___ + 1 = ___ Any Number-What’s the Rule • Moving to the teen numbers • Investigate Number Patterns • 8 + 1 • 18 + 1 • 28 + 1 • 38 + 1 • Target Number

30. Supporting Migrant Students Through Family Support… • Take Games Home • Instructions • Materials • Send home strategies in newsletters • Language

31. Count On Turnarounds: Addition Strategy Week One: Introduce Concept Through Real Objects Transitioning to Model Week Two: Reinforce Through Models Week Three: Practice through Games Week Four: Extend or Reteach

32. Count On Turnarounds… Week One: Introduce Concept Through Real Objects Transitioning to Model • T-shirt • Count On Strategy Cards • Blending Realia and Strategy Cards

33. Count on Turnarounds… Week Two: Reinforce Through Models • Count on Strategy Cards • Using a Number Line • Graphic Organizers • Count on Flash Cards

34. Count on Turnarounds… Week Three: Practice Through Games • Flash Cards with Turn Around Strategy • Bingo • Total Version • Expression Version • Assessment and Monitoring

35. Count on Turnarounds… Week Four: Extend or Reinforce • Write 1 + ___ = ___ Develop a Rule • Use same models with larger numbers • Use Frames ___ + ___ = 18 ___ + ___ = (numbers between 10-30) 18 = ___ + ___ (after practice change position) • Using a number line to show strategy

36. Repeating with new strategies and operations • Same structure for subtraction-opposite of addition • Same format for multiplication and division

37. Multiplication Strategies • Use Tens • Five Facts • Doubling • Two’s Facts • Four’s Facts • Eight’s Facts • Use a Rule • One’s Facts • Zero’s Facts • Build Down and Build Up • Nine’s Facts • Six’s Facts • Last Facts

38. Use Tens: Prepare • Assumptions- • Students should already know tens facts • Be familiar with the turnaround strategy • Turnarounds are an application of the commutative property • Realia-cubes in a bag • Suggesting stories involving food • Sentences reflecting multiplication (groups) • Chocolate Bars and turnarounds-yum yum! • Connecting with language

39. Use Tens: Prepare • Assumptions- • Students should already know tens facts • Be familiar with the turnaround strategy • Turnarounds are an application of the commutative property • Making Models using Dice and Grid Paper • Everyday situations involving 10 • Unifix Cubes Modeling 10’s

40. Count On Turnarounds:Multiplication Strategy Week One: Introduce Concept Through Real Objects Transitioning to Model Week Two: Reinforce Through Models Week Three: Practice through Games Week Four: Extend or Reteach

41. Introduce: Five Facts • What are our experiences with fives? • Using students to model with hands • Clocks • Use Ten Strategy Cards

42. Reinforce: Fives Facts • Use 10 Strategy Cards • Number Lines • Finding Unknown Numbers

43. Practice: Five Facts • Multiplication Flash Cards • Spinning Five Facts • Tally Game • Assess and Monitor

44. Extend: Fives Facts • ___ x ___ = 40 • Using 5’s and 10’s write four number sentences that would be true. • Repeat using numbers 80, 90, 110, 120 and so on. Number Lines Tables

45. Multiplication Strategies • Use Tens • Five Facts • Doubling • Two’s Facts • Four’s Facts • Eight’s Facts • Use a Rule • One’s Facts • Zero’s Facts • Build Down and Build Up • Nine’s Facts • Six’s Facts • Last Facts

46. What to Do When Teaching Basic Math Facts • Ask students to self-monitor • Focus on self-improvement • Drill in short time segments • Work on facts over time • Involve families • Make drill enjoyable • Use technology • Emphasize the importance of quick recall

47. What Not to Do When Teaching Basic Facts • Don’t use lengthy timed test • Don’t use public comparisons of mastery • Don’t proceed through the facts in order from 0-9 • Don’t work on all the facts at once • Don’t move to memorization too soon • Don’t use facts as a barrier to good mathematics • Don’t use fact mastery as prerequisite for calculator use.

48. Math Assessment Probe • Starting from what you know… • Fold assessment in half • Respond to one side only

49. Resources • ESD Fact Fluency Courses • Migrant Fluency- • Origo: The Box of Facts • Elementary and Middle School Mathematics by John Van de Walle

50. Identify fact fluency is a developmental process. • Identify research based strategies that help develop fact fluency • Understand strategies that are ineffective for fact fluency development • Learn specific strategies that help develop fact fluency