1. Presenter’s Guide to Multiple Representations in the Teaching of Mathematics – Part 1 By Guillermo Mendieta Author of Pictorial Mathematics www.pictorialmath.com

2. Pictorial Mathematics: Helping Teachers Build a Bridge Between the Concrete and The Abstract

3. Mathematics Is a Field of Representations 2 groups of 32 x 3 Six 3 repeated 2 times 3 + 3

4. The creation, interpretation, translation and transformation of these representations defines much of the work done in mathematics

5. How we choose to represent a mathematical concept or skill will greatly impact: 1. Students’ understanding of the concept 2. Students’ attitude towards the concept 3. The types of connections students make with the concept 4. The level of access students have to learning the concept 5. The type of prior knowledge we tap from our students

6. While there are many definitions of mathematics, all mathematical activity involves one or more of the following six processes: Representing ideas and concepts Transforming these ideas within a given representational system Translating these ideas across representational systems Abstracting Generalizing Establishing relationships between concepts, structures and representations

7. “The depth of conceptual understanding one has about a particular mathematical concept is directly proportional to one’s ability to translate and transform the representations of the concept across and within a wide variety of representational systems.” - Guillermo Mendieta, Pictorial Mathematics

8. There are eight widely used representational systems used in the teaching and learning of mathematics: 1. Written mathematical symbols (Symbolic) – these can include numbers, mathematical expressions, i.e. x + 2, <, etc. 2. Descriptive written words: For example, instead of writing 2 x 3, we might write “two groups of three” or “three repeated two times” 3. Pictures or diagrams – figures that may represent a mathematical concept or a specific manipulative model, such as the ones used throughout Pictorial Mathematics;

9. 4. Concrete models/Manipulatives – like Base-10 blocks, counters, etc., where the built-in relationships within and between the models serve to represent mathematical ideas; 6. Spoken languages / Oral representations – i.e. the teacher saying the number one hundred thirty-two is quite different from the teacher writing the number 132 on the board for students to see; 5. Concrete / Realia: where the objects represent themselves; for example, candies that are being used to count or to graph. The candies themselves are not representing anything other than candies.

10. 8. School word problems: “If Mary is three years older than Carl, and Mary will be 34 next year, how old is Carl now?” 7. Experience-based – or real world problems, drawn from life experiences, where their context facilitates the solution;

11. In school mathematics, which of the eight types of representations are most often used? Which are neglected? Why? 7. Experience-based 6. Oral representations 3. Pictorial Representations 4. Concrete/Manipulatives 2. Descriptive written words 5. Concrete/Realia1. Written Math Symbols 8. School word problems

12. Most concepts in school mathematics can be represented using any of these eight representational systems. Important Observations about Multiple Representations Each different type of representation adds a new layer or a new dimension to the understanding of the concept being represented. Some students will find some representations easier to understand than others.

13. It is not practical, or efficient to use each of the eight types of representations to teach every math concept Given that most high stakes assessments rely heavily on the symbolic, pictorial, and written representations, we must help students make strong connections between these and other representations we might use in our teaching Most of us will teach using the representations we feel comfortable with, and these may not be the ones our students need the most. Important Observations about Multiple Representations

14. Illustrating Multiple Representations Within the Concept of Multiplication of Mixed Fractions Symbolic Representation Try to recall the instructions you were given to carry out this multiplication. If you can’t recall the exact words, think about what you would tell a student to do to carry out this operation. Share your thoughts with a partner.

15. Most teachers were taught (and are teaching) a symbolic, procedural procedural approach to multiplying mixed fractions similar to the following: Step 1 Change the The Symbolic, standard procedure used in schools: To do so, multiply the whole number (2) by the denominator (2) and add it to its numerator (1). In our example, this gives us 2 x 2 + 1 = 5. Thus, (5) is the new numerator of the your first fraction. Keep the same denominator (2). The new improper fraction is to an improper fraction

16. Step 2 Change the to an improper fraction The Symbolic, standard procedure used in schools: To do so, multiply the whole number (1) by the denominator (2) and add it to its numerator (1). In our example, this gives us 1 x 2 + 1 = 3. Thus, (3) is the new numerator of the your second fraction. Keep the same denominator (2). The new improper fraction is

17. Step 3 The Symbolic, standard procedure used in schools: Multiply the numerators, then multiply the denominators. Your new fraction is

18. Step 4 The Symbolic, standard procedure used in schools: If the numerator of your new fraction is larger than its denominator, divide. In our example, 15>4, so we divide. 4 15 -12 3 3

19. Step 5 The Symbolic, standard procedure used in schools: Based on the results of your division, your answer will have The quotient as the whole number of your mixed fraction, the remainder as its numerator, and the divisor as its denominator. 4 15 -12 3 3 Quotient Remainder Divisor Thus,

20. Note about this Symbolic Procedure Even when students are able to remember all the steps, in the right order, this symbolic procedure does not lead most students to a conceptual understanding about multiplying mixed fractions.

21. The Pictorial Representation: Let’s take a look at the pictorial representation of repeated times. Can be read as groups of Or as repeated times

22. The Pictorial Representation: repeated times We first draw what will be repeated,

23. The Pictorial Representation: repeated times This picture shows 1 x or repeated only once.

24. The Pictorial Representation: repeated times This picture shows 2 x or Repeated 2 times.

25. The Pictorial Representation: repeated times This is repeated 2 times We are supposed to repeat We need to repeat half more times. two and a half times.

26. The Pictorial Representation: repeated times This is repeated 2 times

27. The Pictorial Representation: repeated times This is Repeated times Repeated 2 times Repeated ½ times

28. The Pictorial Representation: repeated times To get the total of We combine all the wholes and parts together.

29. The Pictorial Representation: repeated times The picture now shows that is equal to

30. So far, we have seen three different types of representations for the multiplication of mixed fractions: 1. Symbolic/numeric: 2. Descriptive, written: groups of 3. Pictorial:

31. The second part of of this power point presentation (coming soon) will address the other five representational systems and it will address the most important representation-operations teachers and students need to focus on when they are working on developing conceptual understanding: For now, Part 1 will close with the ten top reasons mathematics educators should pay special attention to the types of representations they use and engage their students with. Translations across representations and Transformations across representations.

32. Top 10 Reasons To use Multiple Representations In the Teaching of Mathematics

33. Reason number 10 Mathematics is about representing ideas and relationships through symbols, graphs, charts, etc. Effective teaching involves the purposeful and effective selection of the representations we engage our students with. The Nature of Mathematics is about Representations

34. Reason number 9 Using multiple representations for a given concept introduces a change of pace in our instructional practice. Students who listen to a lecture, then work with physical models and create pictorial representations for their oral presentation, experience a much richer pace of instruction that we use only one representation. Introduces a Change of Pace

35. Reason number 8 Using multiple representations provides more opportunities for students to make meaningful connections and discover relationships between the concept being studied and their own prior knowledge.The representations themselves are doors to a whole set of different types of possible connections. Connections and Relationships

36. Reason number 7 The Real World is Multidimensional Real world problems do not come neatly packaged in one representation. Defining the questions and finding alternative solutions often involves reading text, searching on the internet, interpreting graphs, creating tables, solving equations, designing models, and working with others. Using Multiple Representations prepares students for the real world of problem solving.

37. Reason number 6 Increases student engagement and motivation Multiple representations increase the level of engagement and the level of motivation of your students. Some will be more motivated and more engaged when you use models and pictures, while others will connect better to the standard symbolic representations.

38. Reason number 5 It Values Different Approaches It conveys the idea that there is not one single way to solve problems; different people, with different perspectives and different strengths may offer a different way approach a problem. Depending on the context, the audience and other factors, one approach may be more effective than another in any given situation.

39. Reason number 4 It Facilitates the Delivery of Differentiated Instruction Every representation taps a different bank of experiential knowledge and student aptitudes. By using a wide variety of representations with the key concepts, you are differentiating instruction and building on wider set of student’s strengths.

40. Reason number 3 It Gives Students With Different Learning Styles Wider Access to the Same Content We all learn differently. Some students who “could not get it or see it” through the traditional symbolic representation will “see it” when you use a visual or pictorial representation.

41. Reason number 2 Using Multiple Representations Increases The Dept of Students’ Understanding Research on the role that representations play in the teaching and learning of mathematics strongly suggests that the depth of someone’s understanding of a mathematical concept is directly proportional to their ability to represent, translate and transform this concept within and across representations. Different representations of a concept add new layers of understanding for that concept.

42. Reason number 1 Using Multiple Representations Increases Student Achievement It prepares students for high stakes testing, which includes a large number of questions that focus on interpreting, translating and transforming mathematical relationships across and within representational systems.

43. This Concludes Part I of The Presenter’s Guide to Multiple Representations For Part 2, 3 and 4 of this series of powerpoint presentations on multiple representations will be available at www.PictorialMath.com