Representing Mathematically: Foundations, Models, and Learning Trajectories

Big Ideas in Mathematical Representation

• Mathematical Variability: Mathematical concepts can be expressed through a wide variety of representational methods. A profound understanding of a concept is achieved only when one can navigate and translate between these different representations.

• Concrete Foundations: Students using concrete materials and physical models tend to develop more comprehensive and precise conceptual understandings. These tools increase student motivation and improve the ability to apply mathematical knowledge to real-life contexts.

Diagnostic Case Study: Conceptions of a Half

Conceptual misunderstandings often arise from the specific types of representations used during instruction. In an interview with a 10-year-old student, the following progression occurred:

• Initial Definition: The student defined a half as "When you divide something into 2."

• Symmetry Misconception: When asked to show a half on a square, the student justified the result by saying the parts were "the same shape and size." However, when presented with a parallelogram, the student initially claimed it could not be halved because it lacked lines of symmetry. This revealed a belief that "half" is inherently linked to symmetry.

• Diagonal Partitioning: When the parallelogram was cut along a diagonal, the student accepted the resulting pieces as halves after flipping and physical verification, concluding that halves must at least be "the same shape."

• Area vs. Shape: The teacher challenged the student further with rectangles cut into non-identical shapes (e.g., different types of triangles or quadrilaterals that maintain equal area). This forced the student to reconsider if "half" requires identical shapes or if it pertains to sharing a common mathematical property, such as length (in a string) or area (in a shape).

Defining Mathematical Representations: Internal and External

Representations are tools used to build and communicate mathematical information. They are categorized into two environments: within the mind and in the external world.

Internal Representations

These are mental models or images constructed by the individual. They include:

• Verbal/Syntactic: Language-based internal structures.

• Imagistic: Mental imagery.

• Formal Notational: Internal processing of formal symbols.

• Affective: The emotional response or state associated with the mathematics.

• Cognitive Representations: These are constructed by students as they attempt to make sense of a concept or solve a problem.

External Representations

These are physical, observable configurations in the environment. They include:

• Traditional Representations: Graphs, number lines, algebraic equations, and tables of values.

• Concrete Materials: Physical manipulatives used in teaching.

• Instructional Representations: Definitions, examples, and models provided by teachers to impart knowledge.

Taxonomy of Concrete Representations in the Classroom

Traditional External Representations: Number Lines

Number lines facilitate connections between different number types (fractions, decimals, real numbers) and develop relative size awareness.

Properties and Usage

• Primary Education ($F-4$): Supports addition, subtraction, and general number sense.

• Middle Years ($5-9$): Used as measuring scales, Cartesian axes, and to represent real number properties like density (the principle that between any two unequal numbers, there exists another number).

• Shift in Perspective: The number line marks a transition from viewing numbers as "the count" of discrete objects to viewing numbers as a distance from zero. In this model, all numbers (counting, rational, and real) exist on a continuum. For example, $2.75$ is the distance $2.75$ from zero.

Classification of Number Lines

• Structured: All numerical markings are present.

• Semi-structured: Only some numerical markings are present.

• Empty: No markings are pre-given. This requires students to understand number as length and fosters active strategy development, though it requires a prerequisite understanding of the line's structure.

Conventions of Orientation

• Horizontal Number Line: Precursor to the $x$-axis; numbers increase as they move to the right. Symmetry point is $0$.

• Vertical Number Line: Precursor to the $y$-axis; numbers increase as they move upward. Symmetry point is $0$.

Representations in Problem Solving: Tables of Values

Tables of values organize information in an ordered fashion, relating two sets of numbers (variables). Two distinct contexts for these tables include:

1. Combinatorial Problems: e.g., mapping the number of people to the number of handshakes.

2. Equation-based Variables: e.g., mapping time to distance at a constant speed ($120\,\text{km/h}$).

Patterns of Mathematical Thinking

Students generally search for relationships in tables in three ways:

• Recursive (Looking down columns): Recognizing patterns in single variables, such as "handshakes increase by $2, 3, 4, 5...$".

• Linguistic/Narrative: Using language to describe change, such as "As time increases by 1 hour, distance increases by $120\,\text{km}$".

• Relational (Looking across rows): Finding a direct mathematical relationship between the two variables. For handshakes, this is expressed as: $\frac{n \times (n - 1)}{2}$, where $n$ is the number of people. This is the most powerful form of mathematical thinking and allows for calculation of large values (e.g., handshakes for $321$ people).

Historical and Cultural Contexts: Mayan Numeration

Analyzing varied representational systems, such as the Mayan base-$20$ system, fosters discussion on base-$10$ and the role of zero.

• Symbols: A dot ($.$) represents $1$, a horizontal bar represents $5$, and a shell represents $0$.

• Vertical Positional Value: Numbers are written from bottom to top. The bottom row represents ones, the second row represents groups of $20$, and the third row represents groups of $400$ ($20 \times 20$).

Case Study: Olympic Medal Tallies and Representation

The 2014 Winter Olympics medal counts demonstrate how data can be organized differently to change rankings.

The Interplay of Language and Mathematics

Language acts as an integral representation in the construction of meaning. Students' utterances provide windows into their internal thinking.

• The Mathematical Register: This is a specialized genre of speech where everyday language is intertwined with precise mathematical terms.

• Ambiguous Language:

  • More/Less: In daily life, these compare two separate sets ("Jill has more than Jack"). In math, they can imply change over time ("If I give you two more") or quantify an exact difference.

  • Between: Daily usage often implies one object in the middle of two others. Mathematical usage often entails identifying an entire set of numbers (e.g., "What numbers are between $26$ and $35$?").

• Folding Back: If a student struggles with abstract language, teachers should "fold back" to concrete materials and the simpler language associated with physical actions.

Diversity in Language and Discourses

• ESL and Indigenous Students: Dominant oral traditions and dialects like Aboriginal English (AE) may mismatch with school-based Standard Australian English (SAE). This is not a cognitive deficit but a mismatch of discourses that requires bridges to specific numerical vocabulary (e.g., "next to", "how far", "one more than").

Building Abstract Thinking via Representational Trajectories

Abstraction involves a spiral process moving from concrete to symbolic recording.

The Abstraction Cycle

1. Concrete Stage: Physical materials (e.g., counting 8 physical counters in a ten-frame).

2. Semi-concrete Stage: Pictorial representations (e.g., images of objects).

3. Abstract Stage: Using symbols only (e.g., the numeral $8$).

Application to Generalizations

Spiraling through loops adds depth. For example, understanding that all numbers ending in $8$ are two away from the next ten ($b8 + 2 = (b + 1)0$). This abstraction can then be extended to decimals (e.g., $1.8$ is $0.2$ away from the whole number $2$).

Commutative Law Example

Using finger puppets (2 red, 3 blue) and then turning the hand (3 blue, 2 red) provides a kinaesthetic, concrete representation of the turnaround property. This leads to the abstract formula: $a + b = b + a$.

Euler's Formula

In 3D geometry, students can record the faces ($F$), edges ($E$), and vertices ($V$) of shapes in a table. By looking across rows, they derive the formula: $V - E + F = 2$. Returning to concrete modeling here is used to test the generality of the abstract concept.

Evaluating Quality in Models: Epistemic Fidelity and Accessibility

When choosing materials, two key criteria must be considered:

• Epistemic Fidelity: The accuracy of the mapping between the material and the knowledge domain. Bundling sticks have high epistemic fidelity for place value because they physically show groups of ten. Counting frames have lower fidelity as they rely on abstract threading positions.

• Accessibility: How easily a student can relate the material to the concept. MAB blocks have high fidelity for whole numbers but low accessibility for decimals because students are socialized to see the unit cube as "one whole" rather than "one hundredth" ($0.01$).

Challenges in Fractional and Geometric Understanding

• Prototypical Thinking: Students often fail to recognize shapes in different orientations if they only see "base-sitting" examples. For instance, a square may not be recognized as a rectangle or a parallelogram if the student has a rigid mental prototype.

• Fraction Misconceptions: Many students believe fractions must be symmetrical. Research shows only $58\%$ of 10.5-year-old students correctly identified halves in a trapezium, and only $49\%$ could correctly partition a hexagon into sixths. Misconceptions often arise because instruction relies too heavily on textbook diagrams where shapes are already perfectly partitioned.

Multi-Representational Learning Environments (MRLE)

MRLEs utilize various external representations simultaneously to provide information in multiple forms.

• Synergy: Linking representations creates a whole greater than the sum of its parts.

• Cognitive Translation: The ability to transfer information from one form (like an equation) to another (like a graph) is fundamental to success.

• Bird Data Case Study: Below is a table mapping variables to demonstrate data translation.

Digital and Virtual Manipulatives

Interactive technologies allow for virtual manipulatives—dynamic symbols that represent concrete materials.

• Selection Criteria: Tools must have adjustable difficulty, provide supportive feedback, have clear instructions, and require teacher scaffolding to help students "see" the embedded math.

• Pedagogical Benefit: They provide opportunities for guided exploration and can help bridge the gap between concrete interaction and abstract symbolic manipulation.