Presenter’s Guide to Multiple Representations in the Teaching of Mathematics – Part 1

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Presenters Guide to Multiple Representations in
the Teaching of Mathematics Part 1

• By Guillermo Mendieta
• Author of Pictorial Mathematics
• www.pictorialmath.com

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

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Mathematics Is a Field of Representations
3 3
2 groups of 3
2 x 3
Six
3 repeated 2 times
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The creation, interpretation, translation and
transformation of these representations defines
much of the work done in mathematics
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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
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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

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The depth of conceptual understanding one has
about a particular mathematical concept is
directly proportional to ones ability to
translate and transform the representations of
the concept across and within a wide variety of
representational systems. - Guillermo Mendieta,
Pictorial Mathematics
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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, lt, 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
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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
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
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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?
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In school mathematics, which of the eight types
of representations are most often used? Which are
neglected? Why?
5. Concrete/Realia
1. Written Math Symbols
6. Oral representations
2. Descriptive written words
7. Experience-based
3. Pictorial Representations
8. School word problems
4. Concrete/Manipulatives
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Important Observations about Multiple
Representations

• Most concepts in school mathematics can be
  represented using any of these eight
  representational systems.

• 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.

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Important Observations about Multiple
Representations

• 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.

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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 cant
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.
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The Symbolic, standard procedure used in schools
Most teachers were taught (and are teaching) a
symbolic, procedural procedural approach to
multiplying mixed fractions similar to the
following
Step 1
Change the
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
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The Symbolic, standard procedure used in schools
Step 2
Change the to an improper fraction
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
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The Symbolic, standard procedure used in schools
Step 3
Multiply the numerators, then multiply the
denominators. Your new fraction is
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The Symbolic, standard procedure used in schools
Step 4
If the numerator of your new fraction is larger
than its denominator, divide. In our example,
15gt4, so we divide.
3
4 15
-12
3
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The Symbolic, standard procedure used in schools
Step 5
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.
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Quotient
4 15
Divisor
-12
Remainder
3
Thus,
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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.
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The Pictorial Representation
Can be read as groups of Or as
repeated times
Lets take a look at the pictorial representation
of repeated times.
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The Pictorial Representation
repeated times
We first draw what will be repeated,
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The Pictorial Representation
repeated times
This picture shows 1 x
or repeated only once.
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The Pictorial Representation
repeated times
This picture shows 2 x or Repeated 2 times.
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The Pictorial Representation
repeated times
This is repeated 2 times
We are supposed to repeat
two and a half times.
We need to repeat half more times.
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The Pictorial Representation
repeated times
repeated 2 times
This is
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The Pictorial Representation
repeated times
Repeated times
This is
Repeated 2 times
Repeated ½ times
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The Pictorial Representation
repeated times
To get the total of
We combine all the wholes and parts together.
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The Pictorial Representation
repeated times
The picture now shows that
is equal to
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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
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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
Translations across representations and
Transformations across representations.
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.
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Top 10 Reasons

• To use Multiple Representations
• In the Teaching of Mathematics

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Reason number 10
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.

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Reason number 9
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.

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Reason number 8
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.

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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.
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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.
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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.
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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
students strengths.
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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.
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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 someones
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.
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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.
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This Concludes Part I of The Presenters 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