presenter’s guide to multiple representations in the teaching of mathematics – part 1 by...

Presenter’s Guide to Multiple

Representations in the Teaching of Mathematics

– Part 1By Guillermo Mendieta

Author of Pictorial Mathematics

www.pictorialmath.com

Pictorial Mathematics: Helping Teachers Build a

BridgeBetween the Concrete and

The Abstract

Mathematics Is a Field of Representations

2 groups of 3 2 x 3

Six

3 repeated 2 times

3 + 3

The creation, interpretation, translation and transformation of

these representations defines much of the work done in

mathematics

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

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

“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

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;

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.

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;

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

• 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.

• 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

Illustrating Multiple Representations

Within the Concept of Multiplication of Mixed Fractions

Symbolic Representation

1 12 x 12 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.

Most teachers were taught (and are teaching) a symbolic, procedural procedural approach to multiplying mixed fractions similar to the following:

Step 1

Change the

1 12 x 12 2

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 5

2

122

to an improper fraction

Step 2

Change the to an improper fraction

1 12 x 12 2


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 3

2

112

Step 3

1 12 x 12 2


Multiply the numerators, then multiply the denominators.

Your new fraction is

5 3 x 2 2

154

Step 4

1 12 x 12 2


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

Step 5

1 12 x 12 2


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,

31 12 x 1 = 32 2 4

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.

The Pictorial Representation:

Let’s take a look at the pictorial representation of

repeated times.

1 12 x 12 2

Can be read as groups of

Or

as repeated times

122

112

112

122

112

122

1 12 x 12 2


112

122

repeated times

We first draw what will be repeated, 112

1 12 x 12 2


112

122

112

repeated times

This picture shows 1 x

112or

repeated only once.

1 12 x 12 2


112

122

112

repeated times

This picture shows 2 x or

Repeated 2 times.

112

1 12 x 12 2


112

122

112

repeated times

This is repeated 2 times

We are supposed to repeat 11

2

We need to repeat half more times.112

two and a half times.

1 12 x 12 2


112

122

repeated times

112

This is repeated 2 times

1 12 x 12 2


112

122

repeated times

112

This is Repeated times122

112

Repeated 2 times112

Repeated ½ times

1 12 x 12 2


112

122

repeated times

To get the total of 1 1

2 x 12 2

We combine all the wholes and parts together.

1 12 x 12 2


112

122

repeated times

The picture now shows that

334

1 12 x 12 2

is equal to

So far, we have seen three different types of representations for the multiplication of mixed fractions:

1. Symbolic/numeric:

2. Descriptive, written: groups of

1 12 x 12 2

122

112

3. Pictorial:

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.

Top 10 Reasons

To use Multiple Representations

In the Teaching of Mathematics

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

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

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

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.

Reason number 6Increases 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.

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.

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.

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.

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.

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.

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

presenter’s guide to multiple representations in the teaching of mathematics – part 1 by...

Documents