Teaching Primary Mathematics: Algebra and Algebraic Thinking

Defining Algebraic Thinking

• Conceptual Definition: Algebraic thinking involves moving beyond computational fluency to attend to the deeper underlying structure of mathematics. According to Cai and Knuth ($2011$, p. ix), it requires specific ways of thinking:

  • Analyzing relationships between quantities.

  • Studying change.

  • Generalizing problem solving.

  • Modeling, justifying, proving, and predicting.

• Habits of Mind: It develops not only new mathematical tools but also new habits of mind to understand relationships.

• Progressive Development: Ideas emerge from the use of materials, models, tables, and patterns of objects. These lead to:

  • Verbal descriptions of what is seen.

  • Mathematical descriptions using additive reasoning.

  • Advancement to multiplicative reasoning.

  • Formal notation (introduced when students see a need for succinct expressions).

• Global Integration: Algebraic thinking should not be seen as a new "strand" or isolated topic, but as an overarching conception that assists in growth toward abstract and sophisticated approaches to the world (Carraher, Schliemann, & Schwartz, $2008$).

Historical Origins of Algebra

• Leonhard Euler's Definition ($1767$): "Algebra is the science that teaches how to determine unknown quantities by means of those that are known."

• Historical Evolution Phases:

  1. Rhetorical Phase (Beginnings to $250\,\text{CE}$): Problems and solutions were expressed solely in words. Babylonian and Egyptian works (like the Rhind Papyrus, c. $1650\,\text{BC}$) used words like "heap" for unknown quantities.

  2. Syncopated Phase (Beginning c. $250\,\text{CE}$): Introduced by Diophantus, this involved shortened notation and abbreviated forms to rewrite mathematical problems in curtailed forms similar to equations.

  3. Symbolic Phase: The use of literal symbols to represent and manipulate ideas through specific rules (Bashamakova & Smirnova, $2000$). This was furthered by Viète and formalized by Descartes.

• Etymology: The term "algebra" comes from al-Khwarizmi’s book Hisab al-Jabr w’al-Muqabala, referring to the solution of equations.

• The Rule of False Position: A universal method used before symbolic equations where an initial guess is made and then adjusted based on the error. Examples include:

  • The Rhind Papyrus: "A heap whose seventh part is added to it becomes $19$." If the heap is assumed to be $7$, it becomes $8$. To adjust, multiply the initial $7$ by $\frac{19}{8}$.

  • Calandri’s Fish ($1491$): The head weighs $\frac{1}{3}$ of the fish, the tail $\frac{1}{4}$, and the body $300\,\text{g}$. Assuming a total weight of $120\,\text{g}$ leads to a body of only $50\,\text{g}$. Multiplying the initial guess by $6$ (since $300/50 = 6$) gives the correct weight of $720\,\text{g}$.

• Geometric Origins: Katz ($2007$) noted that early algebra in Mesopotamia, Greece, and India used geometric thinking, as literal symbolism required a level of abstraction not yet reached.

• Development of Negative Numbers: Arising from the need for linear equations to have answers. If "$6$ and a heap is $4$," the heap is $-2$. This was modeled using a number line as distances from zero, though the symbol $-2$ wasn't commonly accepted until after the $17\text{th}$ century.

Patterning and Generalization

• Types of Patterns:

  • Repeating Patterns: Involve a smaller part generating a whole pattern (e.g., shape sequences).

  • Growing Patterns: Each part increases by a consistent amount (e.g., the toothpick perimeter problem).

• Toothpick Perimeter Activity:

  • A sequence of shapes where each new shape adds a column of toothpicks.

  • Additive View: Shape $1$ (perimeter $4$), Shape $2$ (add $4$ bits to get $8$), Shape $3$ (add $4$ to get $12$).

  • Multiplicative View: Perimeter equals the shape number multiplied by $4$ ($P = 4n$).

• Arithmetic Sequences: Additive sequences like $2, 11, 20, 29$ (add $9$) or $63, 56, 49$ (subtract $7$).

• Geometric Sequences: Multiplicative sequences like $2, 8, 32, 128$ (multiply by $4$).

• Fibonacci Sequence: Named after Leonardo of Pisa ($1\, 2\, 3\, 5\, 8\, 13\, 21\, 34\, 55\dots$). Each number is the sum of the preceding two. This pattern is linked to the "golden ratio" found in the Nautilus shell and the Parthenon.

• Octagon Pattern (NAPLAN-based):

  • $1$ octagon = $8$ toothpicks; $2$ octagons = $15$ toothpicks; $3$ octagons = $22$ toothpicks.

  • General rule: $7n + 1$, where $n$ is the number of octagons.

• Tower of Hanoi: A growing pattern where the minimum moves for $n$ disks is investigated. This puzzle was invented by the mathematician Lucas in the late $19\text{th}$ century.

Problem Solving Models and Counter Activities

• Tug of War Activity: Five oxen equal eight donkeys. An elephant equals one ox and four donkeys. This represents algebraic equivalence using animals as variables.

• Balancing Fruit: Using colored counters to show how addition/subtraction on both sides maintains balance (e.g., $3$ bananas and $4$ kiwifruit balance $13$ kiwifruit).

• Bread Buying Analysis:

  • Larry: $6$ rolls + $1$ loaf = $\$5.70$.

  • Mehmet: $4$ rolls + $2$ loaves = $\$8.60$.

  • Method: Doubling Larry's purchase reveals $12$ rolls + $2$ loaves = $\$11.40$. Comparing to Mehmet shows $8$ rolls cost $\$2.80$.

• Age Problems (Relational Thinking):

  • Caitlin's sister is $7$ years older; total age is $25$.

  • Strategy: Subtract the difference ($25 - 7 = 18$), halve the result ($18 / 2 = 9$). Caitlin is $9$, sister is $16$.

• Bus Conductor Problem: Passengers board based on stop number ($1$ at stop $1, 2$ at stop $2$, etc.).

  • Standard formula discovered by a Year $6$ student: Multiply the stop number by the stop number plus one, then divide by two ($\frac{n(n+1)}{2}$).

• Nadia’s Birthday Savings: Saving $\$1$ on June $1$, $\$2$ on June $2$, etc.

  • Visual strategy: Moving the first row to the last creates rectangular arrays.

  • For June ($30$ days): $15 \times 31 = \$465$.

Transitioning to Symbolic Algebra

• Barriers to Comprehension:

  • The Equals Sign: Often misunderstood as "find the answer" rather than a sign of equivalence ($3 + 2 = 4 + 1$).

  • Directionality: Students may accept $2 + 3 = 5$ but struggle with $5 = 2 + 3$.

  • Notation Confusion: In arithmetic, $36$ uses place value concepts, but in algebra, $3x$ indicates multiplication.

  • Additive vs. Multiplicative Mindsets: High school mathematics relies on multiplicative ideas, which are essential for factoring and ratios.

• Number Puzzles and the "Vanishing Variable":

  • Puzzle: Choose a number, add $6$, add the original number, subtract $4$, subtract the original number, subtract $1$, the result is always $1$.

  • Symbolic representation: $n \rightarrow n + 6 \rightarrow 2n + 6 \rightarrow 2n + 2 \rightarrow n + 1 \rightarrow 1$.

• Succinct Expressions (Activity 7.8): Translating verbal rules to symbols:

  • Result is $5$ less than original: $n \rightarrow n - 5$.

  • Result is triple the number plus one: $n \rightarrow 3 \times n + 1$.

  • Result is $100$ divided by the number: $n \rightarrow 100 \div n$.

Practical Applications and Resources

• Academic Passport: Schoenfeld ($1995$) describes algebra as the gateway to the job market and higher schooling.

• Children’s Literature for Algebra:

  • Each Peach Pear Plum (Ahlberg): Early years patterning.

  • Anno's Mysterious Multiplying Jar (Anno): Number patterns.

  • Spaghetti and Meatballs for All! (Burns): Patterns and reasoning.

  • Dinner at the Panda Palace (Calmenson): Growing patterns.

  • Drummer Hoff (Emberley): Repeating patterns.

• Geometric Generalizations:

  • Cutting Squares: A pattern where a square is cut into smaller squares. Rule: $n = 3c + 1$, where $c$ is the number of cuts.

  • Folding Paper: Investigating the maximum number of regions created by $n$ folds ($1 \text{ fold} = 2 \text{ regions}, 2 = 4 \text{ regions}, 3 = 8 \text{ regions}$), which follows the pattern $2^n$.