The problem with fractions. Short history of fractions Controversy Conceptual understanding

1. The Invention of FractionsBy Jessica GoodfellowGod created the whole numbers:the first born, the seventh seal, Ten Commandments etched in stone,the Twelve Tribes of Israel - Ten we've already lost - forty days and forty nights,Saul's ten thousand and David's ten thousand.'Be of one heart and one mind' - the whole numbers, the counting numbers. It took humankind to need less than this;to invent fractions, percentages, decimals.Only humankind could need the conceptsof splintering and dividing,of things lost or broken,of settling for the part instead of the whole. Only humankind could find the whole numbers,infinite as they are, to be wanting;though given a limitless supply,we still had no wayto measure what we keepin our many-chambered hearts.

2. The problem with fractions. Short history of fractions Controversy Conceptual understanding Specific challenges for teachers Laying the groundwork Recommendations from experts Some activities Implications for us

3. 1 2 1 3 1 10 = = = “part” = Egyptian Fractions = “arm”

4. It will happen at the harvests, that you shall give a fifth to Pharaoh, and four parts will be your own. (Genesis, 42.24)

6. “To get into trouble is to get among fractions.” Old German Saying

7. Dennis DeTurck, Penn State Univ. http://www.youtube.com/watch?v=5d5RG9nx_7w&feature=related

8. Video interview with Denise Mewborn, Univ of Georgia: “Fractions are everywhere in the real world” http://wm.nmmstream.net/genasx/learningpt/mewbornwmv56842.asx

9. Did you know that 5 out of every 4 people have a problem with fractions?

10. “Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra.” Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008)

11. Children are told that a fraction is simultaneously at least five different objects: • Parts of a whole • The size of a portion • A quotient • A ratio • An “operator” - such as “2/3 of” Clearly, even those children endowed with an overabundance of faith would find it hard to believe that a concept could be so versatile as to fit all of these descriptions.

12. The curriculum should allow for sufficient time to ensure acquisition of conceptual and procedural knowledge of fractions (including decimals and percents) and of proportional reasoning. The curriculum should include representational supports that have been shown to be effective, such as number line representations, and should encompass instruction in tasks that tap the full gamut of conceptual and procedural knowledge, including ordering fractions on a number line, judging equivalence and relative magnitudes of fractions with unlike numerators and denominators, and solving problems involving ratios and proportion. The curriculum also should make explicit connections between intuitive understanding and formal problem solving involving fractions. Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008)

13. Critical Foundations of Algebra Fluency with Fractions By end of grade 4: proficiency with identifying and representing fractions and decimals, and comparing them on a number line or with other common representations of fractions and decimals. By end of grade 5: proficiency with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals. By end of grade 6: proficiency with multiplication and division of fractions and decimals, and with all operations involving positive and negative integers By end of grade 7: proficiency with all operations involving positive and negative fractions, and with problems involving percent, ratio, and rate including the extension of this work to proportionality Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008)

14. Fractions: Montessori Plan of Work

16. One day during math class, the professor was demonstrating how to simplify a fraction. "Take the fraction 16/64. Now, canceling the six in both the denominator and numerator, we get that 16/64 = 1/4." "Wait a second!" piped up one student. "You can't just cancel the six!" At this the professor turned around and stared at him incredulously, saying, "Oh, so you're telling us 16/64 is not equal to 1/4, are you?"

17. A Missouri farmer passed away and left 17 mules to his three sons. The instructions left in the will said that the oldest boy was to get one-half, the second oldest one-third, and the youngest one-ninth. The three sons, recognizing the difficulty of dividing 17 mules into these fractions, began to argue. Their uncle heard about the argument, hitched up his mule and drove out to settle the matter. He added his mule to the 17, making 18. The oldest therefore got one-half, or nine, the second oldest got one-third, or six, and the youngest son got one-ninth, or two. Adding up 9, 6 and 2 equals 17. The uncle, having settled the argument, hitched up his mule and drove home.

18. Teacher: How much is half of 8? Pupil: Up and down or across? Teacher: What do you mean? Pupil: Well, up and down makes it makes 3, but across the middle leaves 0.

19. Hardest part of fractions to master (poll) 17% fractions in relation to ratios, decimals, percents 7% fractions as numbers 9% improper and mixed fractions 49% deep conceptual understanding of fractions 7% problem solving with fractions 9% different representations of fractions 2% other

20. Vygotsky “Everyday” concept + “Mathematical” concept “Sublated” concept • Everyday: Fractions are partitions. • Mathematical: Fractional value depends on the unit-whole. • Sublated: Both everyday and mathematical concepts are lifted and preserved as a unified concept.

21. Deep conceptual understanding of fractions: Groundwork is important • Variety of models for fractions, relating/interchanging models • Regions – not just circles (square, triangle insets, rectangles) • Constructive Triangles • Sets • Linear – “Where do fractions live?” • link with measurement • “Fraction Tower” • Problem Solving • Creating problems • Rate problems • Solving problems • Fraction language in everyday life • 2% milk • “Give 110%”

22. The emphasis of instruction should also shift from the development of algorithms for performing operations on fractions to the development of a quantitative understanding of fractions. -Bezuk & Cramer • Linger over operations and how they make sense • Why doesn’t division by a fraction result in a smaller number? • Lots of equivalence experience and ordering of fractions experience. • Use geometry sticks as fractional parts of the “ten” stick • Caution: geometry sticks are measured from hole-to-hole. Cuisenaire rods or Fraction Towers may work better. • Paper folding • Use words for denominators, instead of numbers • Vary the unit-whole (make the half be the unit-whole)

23. The emphasis of instruction should also shift from the development of algorithms for performing operations on fractions to the development of a quantitative understanding of fractions. -Bezuk & Cramer • Linger over operations and how they make sense • Does this result make sense? (estimate) • 12/13 + 7/8 = about 2 • 1/2 + 1/3 ≠ 2/5 because 2/5 < 1/2 and must be bigger than either addend • 3/7 < 5/9 because 3/7 < 1/2 and 5/9 > 1/2

24. By postponing most operations with fractions at the symbolic level until grade 6 and using instructional time in grades 4 and 5 to develop fraction concepts and the ideas of order and equivalence, teachers will find that their students will be more successful with all aspects of operations with fractions and will have a stronger quantitative understanding of them. Bezuk & Cramer Teaching about Fractions: What, When, and How

25. Developing fraction concepts and the ideas of order and equivalence. Extend the students’ concept of the unit. Extend the students’ physical models to include number lines and length rods. Extend the students’ concept to a new interpretation: the quotient model (3 pizzas shared by four people) Generate equivalent fractions with manipulatives, then with diagrams.

26. Equivalent fraction paper folding

27. Equivalence Chart

28. Developing fraction concepts and the ideas of order and equivalence. Compare fractions with like denominators (2/7 and 3/7) and like numerators (2/4 and 2/8) with manipulatives, eventually to verbalize a rule for ordering fractions with like numerators that does not rely on changing them to equivalent fractions with like denominators.

29. Developing fraction concepts and the ideas of order and equivalence. Order pairs of fractions by comparing them to 1/2 or 1. For example, 3/ 10 is less than 2/3 because 3/10 is less than 1/2 and 2/3 is greater than 1/2. (Notice that although this pair of fractions has unlike numerators and unlike denominators, an ordering decision can be made without finding equivalent fractions with the same denominator).

30. Developing fraction concepts and the ideas of order and equivalence. Model addition and subtraction with manipulatives and diagrams, emphasizing estimation and judging the reasonableness of answers.

31. Developing fraction concepts and the ideas of order and equivalence. (At grade 6) Addition and subtraction of fractions at the symbolic level. Multiplication and division of fractions also introduced, but the goal should not be to calculate the answer, since it can easily be obtained using whole number ideas, with no conceptual understanding. Children should demonstrate an understanding of multiplication and division by modeling a problem with manipulatives or by naming the problem for the manipulative model. Students should be able to create story problems for a multiplication or division sentence or write a multiplication or division sentence for a story problem.

32. The ability of the child to reconstruct the unit when given a part, is an indicator of concept stability of a part-whole understanding of fractions. (Cramer, Behr, Post, & Lesh, 1997; Post, Cramer, Behr, Lesh, & Harel, 1993). Whole number procedures are often used when teaching fractions. Many children are introduced to fractions by being shown a set of elements, for example a rectangle divided into three equal pieces, some of which are shaded and told to count. The cardinal number of the count of shaded objects is the numerator, while the cardinal number of the count of shaded and unshaded elements becomes the denominator. And thus fractions become tied to counting and matching. It is when children are asked to find the unit that they, drilled to look for the number of parts first, show up the weakness of procedural understanding.

33. Concept-of-Unit Activities Use fraction circles to solve these problems: The black circle is 1. What is the value of each of these pieces? 1 blue 3 grays 1 pink 3 yellows

34. Concept-of-Unit Activities Use fraction circles to solve these problems: Change the unit: The yellow piece is 1. Now what is the value of those pieces? 1 blue 3 grays 1 pink 3 yellows

35. Concept-of-Unit Activities Use counters to solve these problems: Eight counters equal 1. What is the value of each of these sets of counters? 1 counter 2 counters 4 counters 6 counters

36. Concept-of-Unit Activities Use counters to solve these problems: Change the unit: Four counters equal 1. Now what is the value of those sets of counters? 1 counter 2 counters 4 counters 6 counters

37. Concept-of-Unit Activities Use geometry sticks to solve these problems: The orange stick equals 1. What is the value of these rods? purple green brown pink

38. Concept-of-Unit Activities Use geometry sticks to solve these problems: Change the unit: The pink stick now equals 1. What is the value of these rods? purple green brown pink

39. Ordering Activities • Compare: Pairs of fractions with like denominators: 1/4 and 3/4 • 3/5 and 4/5

40. Ordering Activities • Compare: Pairs of fractions with like numerators: 1/3 and 1/2 • 2/5 and 2/3

41. Ordering Activities • Compare: Pairs of fractions that are on opposite sides of 1/2 or 1: 3/7 and 5/9 • 3/11 and 11/3

42. Ordering Activities • Compare: Pairs of fractions that have the same number ofpiecesless than one whole: 2/3 and 3/4 • 3/5 and 6/8

44. Child: Dad, can you help me find the lowest common denominator in this problem please? Father: Don't tell me that they haven't found it yet, I remember looking for it when I was a boy!

45. How do we know that the following fractions are in Europe? A/C, X/C and W/C ? Their numerators are all over C's.

46. Do you know who invented fractions? Henry the 1/8th!