C-R-A Approach to Mathematical Understanding instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, base-ten blocks, pattern blocks, fraction bars, geometric figures, etc)- the more real-life the better Concrete Level transforms the concrete model into a representational (semiconcrete) level, which may involve drawing pictures; using circles, dots, and tallies; or using stamps to imprint; constructing tables or graphs. Representational Level Resources: Hands On Standards,

Super Source, Teacher's Manual, The “doing” stage using concrete objects to model problems The “seeing” stage using representations of the objects to model problems Semi-Abstract/Abstract Level models the mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the number The “symbolic” stage using abstract symbols to model problems

Resources: Math literature, virtual manipulatives, singapore models 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. Using Multiple Representations Increases Student Achievement Reason number 1 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. Using Multiple Representations Increases The Dept of Students’ Understanding Reason number 2 By using a wide variety of representations with the key concepts, you are differentiating instruction and building on wider set of student’s strengths. Every representation taps a different bank of experiential knowledge and student aptitudes. It Facilitates the Delivery of Differentiated Instruction Reason number 4 Depending on the context, the audience and other factors, one approach may be more effective than another in any given situation. 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. It Values Different Approaches Reason number 5 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. Increases student engagement and motivation Reason number 6 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. The Real World is Multidimensional Reason number 7 Connections and Relationships 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. Reason number 8 Introduces a Change of Pace 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. Reason number 9 The Nature of Mathematics is about Representations 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. Reason number 10 To use Multiple Representations

In the Teaching of Mathematics Top 10 Reasons Transformations across representations. Translations across representations and 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. 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: 3. Pictorial: groups of 2. Descriptive, written: 1. Symbolic/numeric: So far, we have seen three different types of representations for the multiplication of mixed fractions: is equal to The picture now shows that repeated times The Pictorial Representation: This picture shows 2 x or

Repeated 2 times. repeated times The Pictorial Representation: or

repeated only once. This picture shows 1 x repeated times The Pictorial Representation: We first draw what will be repeated, repeated times The Pictorial Representation: this symbolic procedure does not lead most students to a conceptual understanding about multiplying mixed fractions. Even when students are able to remember all the steps, in the right order, Note about this Symbolic Procedure Thus, Divisor Remainder Quotient 3 3 -12 4 15 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. The Symbolic, standard procedure used in schools: Step 5 3 3 -12 4 15 If the numerator of your new fraction is larger than its denominator, divide. In our example, 15>4, so we divide. The Symbolic, standard procedure used in schools: Step 4 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 The Symbolic, standard procedure used in schools: Change the to an improper fraction Step 2 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. Symbolic Representation Illustrating Multiple Representations

Within the Concept of Multiplication of Mixed Fractions Important Observations

about Multiple Representations Most of us will teach using the representations we feel comfortable with, and these may not be the ones our students need the most. 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 It is not practical, or efficient to use each of the eight types of representations to teach every math concept 8. School word problems 1. Written Math Symbols 5. Concrete/Realia 2. Descriptive written words 4. Concrete/Manipulatives 3. Pictorial Representations 6. Oral representations 7. Experience-based In school mathematics, which of the eight types of representations are most often used? Which are neglected? Why? 7. Experience-based – or real world problems, drawn from life experiences, where their context facilitates the solution; 8. School word problems: “If Mary is three years older than Carl, and Mary will be 34 next year, how old is Carl now?” 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. 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; 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; “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 Establishing relationships between concepts, structures and

representations Generalizing Abstracting Translating these ideas across representational systems Transforming these ideas within a given representational system Representing ideas and concepts While there are many definitions of mathematics, all mathematical activity involves one or more of the following six processes: 5. The type of prior knowledge we tap from our students 4. The level of access students have to learning the concept 3. The types of connections students make with the concept 2. Students’ attitude towards the concept 1. Students’ understanding of the concept How we choose to represent a mathematical concept or skill will greatly impact: Pictorial Mathematics:

Helping Teachers Build a Bridge

Between the Concrete and

The Abstract 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. It Gives Students With Different Learning Styles Wider Access to the Same Content Reason number 3 repeated 2 times This is repeated times The Pictorial Representation: two and a half times. We need to repeat half more times. We are supposed to repeat This is repeated 2 times repeated times The Pictorial Representation: Multiply the numerators, then multiply the denominators.

Your new fraction is The Symbolic, standard procedure used in schools: Step 3 Some students will find some representations easier to understand than others. Each different type of representation adds a new layer or a new dimension to the understanding of the concept being represented. Important Observations

about Multiple Representations Most concepts in school mathematics can be represented using any of these eight representational systems. 3. Pictures or diagrams – figures that may represent a mathematical concept or a specific manipulative model, such as the ones used throughout Pictorial Mathematics; 2. Descriptive written words: For example, instead of writing 2 x 3, we might write “two groups of three” or “three repeated two times” 1. Written mathematical symbols (Symbolic) – these can include numbers, mathematical expressions, i.e. x + 2, <, etc. There are eight widely used representational systems used in the teaching and learning of mathematics: The creation, interpretation, translation and transformation of these representations defines much of the work done in mathematics We combine all the wholes and parts together. To get the total of repeated times The Pictorial Representation: Repeated ½ times Repeated 2 times Repeated times This is repeated times The Pictorial Representation: Can be read as groups of

Or

as repeated times Let’s take a look at the pictorial representation of

repeated times. The Pictorial Representation: to an improper fraction 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 The Symbolic, standard procedure used in schools: Change the Step 1 Most teachers were taught (and are teaching) a symbolic, procedural procedural approach to multiplying mixed fractions similar to the following: 3 + 3 3 repeated 2 times Six 2 x 3 2 groups of 3 Mathematics Is a Field of Representations

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# CRA Math Understanding

Describes the Concrete-Representational-Abstract Approach to teaching Mathematics

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