11.3 Mathematical Processes: Problem Solving, Reasoning & Communication

Overview & Exam Relevance

Competency 006 (Mathematical Processes) is about how mathematicians think. It covers formal and informal reasoning, inductive and deductive arguments, problem solving, evaluating reasonableness, connections among representations, estimation, communicating with mathematical language, and tools and technology. This section covers those processes. The history of mathematics and financial literacy, also part of Competency 006, follow in the next section.

Mathematical Processes & Problem-Solving Pedagogy (Competency 006)

Competency 006 emphasizes mathematical inquiry, problem-solving heuristics, reasoning, and representation.

George Pólya's Four-Step Problem-Solving Model

Formulated in 1945 by mathematician George Pólya, this model serves as the foundational architecture for inquiry-based mathematics instruction:

1. Step 1: Understand the Problem: Identify what is known, what is unknown, and what constraints exist. Students restate the problem in their own words and state the problem's ultimate question.
2. Step 2: Devise a Plan: Select an appropriate heuristic strategy that connects the given data to the target unknown.
3. Step 3: Carry Out the Plan: Implement the selected heuristic, maintaining organized computational records and verifying each operational step.
4. Step 4: Look Back (Reflect and Check): Examine the solution for mathematical reasonableness, confirm proper units, explore alternative solution paths, and evaluate whether the method generalizes to other problem types.

Elementary Problem-Solving Heuristics

• Draw a Picture / Strip Diagram: Using rectangular visual bars (Singapore bar modeling) to represent part-whole or additive/multiplicative comparison relationships.
• Work Backward: Reversing operations sequentially from a known end result back to an unknown starting quantity.
• Make an Organized Table / Look for a Pattern: Structuring numerical trials into organized rows and columns to detect functional relationships.
• Guess, Check, and Revise: Making an informed mathematical conjecture, evaluating the magnitude and direction of the resulting error, and systematically revising the next estimate.
• Solve a Simpler / Related Problem: Substituting cumbersome multi-digit or fractional numbers with friendly single digits to identify the correct operational architecture.
• Act it Out / Use Concrete Manipulatives: Physically enacting the problem scenario with counters, cubes, or student actors.

The Lesh Translation Model (Multiple Representations)

Conceptual understanding in elementary mathematics requires translating fluently across five distinct modes of representation:

1. Real-World Contexts: Authentic everyday scenarios and situational problems.
2. Manipulative Models (Concrete): Tactile physical objects (base-ten blocks, fraction strips, counters).
3. Pictures and Diagrams (Semi-Concrete / Representational): Strip diagrams, coordinate graphs, number lines, area models.
4. Spoken Language: Verbal justifications, collaborative peer explanations, and academic discourse.
5. Written Symbols (Abstract): Formal numeric equations, algebraic variables, and operational symbols.

Students achieve deep conceptual mastery when they can translate a problem within and between all five representational modes.

Inductive and Deductive Reasoning

• Inductive reasoning looks at examples and makes a conjecture. A student notices that $3 + 5 = 8$, $7 + 9 = 16$, and $11 + 13 = 24$, and conjectures that "the sum of two odd numbers is even."
• Deductive reasoning proves a statement from definitions and accepted facts. Any odd number can be written $2a + 1$, so $(2a + 1) + (2b + 1) = 2(a + b + 1)$, which is a multiple of 2 and therefore even.
• A single counterexample disproves a general claim. "All prime numbers are odd" is false because 2 is prime and even.
• Valid conclusions from premises: "All squares are rectangles; figure P is a square; therefore P is a rectangle" is valid. "All squares are rectangles; figure Q is a rectangle; therefore Q is a square" is not.

Estimation and Reasonableness

Encourage students to estimate first and check last. Compatible numbers make mental estimates easy ($398 \times 21 \approx 400 \times 20 = 8{,}000$). If an exact answer is far from the estimate, students look for an error. Estimation is also appropriate when an exact answer is not needed, such as deciding whether $20 is enough for three $6.89 items.

Communicating and Connecting Mathematics

Students should express ideas with grade-appropriate language, standard English, precise mathematical vocabulary, and symbols. They should move among numeric, verbal, graphic, pictorial, symbolic, and concrete representations. Visual media such as graphs, tables, diagrams, and animations communicate quantitative information efficiently. Recognizing that a problem can be solved in more than one way, for example with a table, an equation, or a graph, deepens understanding of the connections between algebra and geometry.

Tools and Technology

Manipulatives (base-ten blocks, fraction tiles, geoboards), calculators, spreadsheets, dynamic geometry software, and virtual manipulatives let students explore ideas and test conjectures. Choose tools that develop understanding rather than replace it. For example, a calculator can let students explore patterns in multiplying by 10 after they understand place value.

Classroom Scenario Application

Classroom Context: Mr. Washington is facilitating a 6th-grade mathematics problem-solving workshop. He presents the following task:

"Sophia baked a batch of cookies. She gave $\frac{1}{3}$ of the cookies to her neighbor and then ate 4 cookies herself. Afterward, she had 16 cookies remaining. How many cookies did Sophia bake originally?"

Observed Challenge: Several students attempt to set up a complex symbolic algebraic equation ($\frac{2}{3}x - 4 = 16$), but become confused by fractional algebraic manipulation and make sign errors.

Diagnostic Analysis: Students are attempting abstract symbolic computation without grounding their thinking in concrete or visual heuristics. They need a systematic heuristic that aligns with Pólya's planning phase.

Targeted Instructional Plan:

1. Select the Heuristic: Mr. Washington guides students to deploy the Work Backward heuristic paired with a Strip Diagram.
2. Execute Working Backward:
   • The final state is 16 cookies.
   • Before eating 4 cookies, she had: $16 + 4 = 20$ cookies.
   • If she gave away $\frac{1}{3}$ of the cookies, the remaining 20 cookies must represent $\frac{2}{3}$ of the original batch.
   • If two-thirds equals 20 cookies, then one-third equals $20 \div 2 = 10$ cookies.
   • The complete batch (three-thirds) is $10 \times 3 = 30$ cookies.
3. Pólya's Step 4 (Look Back): Students check the result in the forward direction: Starting with 30 cookies, one-third is 10 cookies ($30 - 10 = 20$). Subtracting 4 cookies leaves 16 cookies ($20 - 4 = 16$). The solution is mathematically validated.