Solved Problems & Innovative Teaching For Algebra Success: How It Works

Key Takeaways

• Using solved problems in algebra instruction helps struggling students analyze mathematical reasoning without overwhelming their cognitive resources.
• Teaching students to recognize algebraic structure improves their ability to make connections between different problems and solution strategies.
• Introducing multiple solution strategies enhances procedural flexibility, but it works best after students have developed foundational algebraic knowledge.
• Using precise mathematical language and reflective questioning techniques helps students deepen their understanding of algebra concepts.
• Modern solutions like Algebra Ace 9's structured approach to algebra instruction integrates these evidence-based practices to transform struggling students into confident problem-solvers.

Solved Problems: The Foundation for Transforming Struggling Algebra Students

Algebra demands abstract thinking, unlike anything students encounter in earlier mathematics. This cognitive leap from arithmetic to symbolic representation often creates barriers for struggling students.

What makes algebra particularly challenging is the mental burden it places on students. Algebra requires processing multiple pieces of complex information simultaneously, creating a high cognitive load that can overwhelm students' working memory and interfere with learning.

Solved problems offer a powerful remedy to this challenge. By showing both the problem and the complete solution steps at once, solved problems minimize the cognitive burden on students. Unlike traditional practice, where students must determine each step independently, solving problems allows them to see the entire solution path, helping them learn more efficiently.

When students analyze solved problems, they develop a deeper understanding of the logical processes used in solving algebra problems. This approach is particularly beneficial for students in remedial, regular, and honors algebra classes.

The most effective implementation involves guiding students to discuss solved problem structures with questions like:

• What were the steps involved in solving this problem?
• Why do they work in this order?
• Would they work in a different order?
• Could the problem have been solved with fewer steps?
• What other mathematical ideas connect to this solution?

These questions encourage active engagement rather than passive observation, prompting students to think critically about each step in the solution process.

Surprisingly, incorrectly solved problems also play a valuable role. Research shows that analyzing strategic, reasoning, and procedural errors helps students develop a better understanding of correct solution strategies. For example, when students compare a correct solution alongside an incorrect one for solving a system of linear equations, they develop stronger insights into why certain approaches work and others don't.

Teachers should incorporate solved problems across different instructional contexts. During whole-class instruction, solved problems can introduce new solution strategies. In small groups, students can critically analyze multiple solved problems and discuss their observations. For independent practice, teachers can provide incomplete solved problems where students fill in missing steps, or alternate solved examples with similar unsolved problems.

The Challenge of Algebraic Structure

Many struggling algebra students fail to recognize the underlying mathematical structure of expressions and equations. According to research reviewed by the National Center for Educational Evaluation, understanding structure—the mathematical features and relationships within algebraic representations—is fundamental to algebra success.

Structure in algebra includes:

• The number, type, and position of quantities and variables
• The number, type, and position of operations
• The presence of equality or inequality
• The relationships between quantities, operations, and equalities or inequalities

Why Students Struggle with Structure Recognition

Many students view algebra as disconnected procedures rather than a cohesive system built on logical structure. This fragmented understanding creates significant barriers to progress.

Consider the following equations:

• 2x + 8 = 14
• 2(x + 1) + 8 = 14
• 2(3x + 4) + 8 = 14

Though these equations appear different at first glance, they share a similar structure: in all three, 2 multiplied by a quantity, plus 8, equals 14. Students who recognize this structural similarity can apply consistent solution strategies, while those who don't see the pattern must approach each equation as an entirely new problem.

3 Strategies for Teaching Structure Effectively

1. Model Precise Mathematical Language

Precise mathematical language helps students understand the logical meaning behind algebraic structure. When teachers consistently use accurate terminology, students develop a stronger foundation for mathematical understanding.

For example, instead of saying:

• "Move the 5 over" → "Subtract 5 from both sides of the equation"
• "Flip it and multiply" → "Multiply both sides by the reciprocal"
• "Cancel out the numbers" → "The numbers add to zero" or "The numbers divide to one"

This precision in language helps students better understand quantities, operations, and the relationships between them.

2. Teach Reflective Questioning Techniques

Studies examining reflective questioning have found positive effects on both procedural and conceptual knowledge. When students ask themselves strategic questions while solving problems, they develop a more deliberate approach to algebraic problem-solving.

A student using reflective questioning might analyze an expression like (x² - 9)/(x - 3) by asking:

• What can I say about the form of this expression?
• What do I notice about the denominator of this expression?
• What has happened in similar problems I've solved before?
• Do I see any common factors in the numerator and denominator?

Through this structured self-questioning, the student might recognize that x² - 9 can be factored as (x + 3)(x - 3), revealing a common factor with the denominator.

3. Compare Multiple Representations of the Same Problem

Using multiple representations improves students' conceptual knowledge. By examining how the same algebraic relationship can be represented in different forms, students develop deeper structural understanding.

For instance, the equation y = (x - 2)² + 3 can be represented as:

• A symbolic equation in vertex form
• A graph showing a parabola with a vertex at (2, 3)
• A table of values
• An expanded form: y = x² - 4x + 7

Each representation highlights different aspects of the parabola's structure. The vertex form clearly shows the minimum point, while the expanded form makes it easier to find the y-intercept.

Cooperative Learning for Structure Recognition

To reinforce structure recognition, applying cooperative learning strategies like:

• Think, write, pair, share: Students think independently about a problem's structure before discussing with partners
• Partner coaching: One student solves a problem while explaining the structure, and the partner provides feedback
• Jigsaw: Different groups analyze different aspects of algebraic structure, then teach their insights to others

These collaborative approaches help students articulate their understanding of structure and learn from their peers' perspectives, creating a richer learning environment for struggling algebra students.

Developing Strategic Flexibility in Problem Solving

Strategic flexibility—the ability to choose appropriate solution methods for different problems—is perhaps the most transformative skill for struggling algebra students. When students learn to recognize, generate, and intentionally select from multiple solution strategies, their algebra performance improves dramatically.

A strategy is more than a series of steps; it's a general approach that requires students to make choices based on problem specifics. Teaching alternative strategies significantly improves students' procedural flexibility, though with an important caveat: this approach works best once students have developed some basic procedural knowledge.

When to Introduce Multiple Solution Methods

The timing of introducing multiple solution methods is crucial. Studies show negative effects when teaching multiple strategies to students with no prior algebra knowledge, suggesting a sequential approach is best. First, ensure students are comfortable with one solution method before introducing alternatives.

For instance, students should first master solving quadratic equations using the quadratic formula before learning factoring or completing the square as alternative approaches. This foundation gives them a reference point for comparing new strategies.

4 Ways to Build Students' Strategic Thinking

1. Present Alternative Solution Strategies Side-By-Side

When students see different solutions to the same problem simultaneously, they can more easily identify similarities and differences. Consider this system of linear equations:

5x + 10y = 60

x + y = 8

Students might solve this using:

• Substitution method
• Elimination method
• Graphing method

By displaying these solutions side-by-side, students see that while all approaches reach the same answer (4, 4), the substitution method might be more efficient for this particular system because one equation is already in a form that makes substitution straightforward.

2. Guide Students to Analyze Strategy Efficiency

Include reflective questions that help students evaluate different strategies:

• What strategies could I use to solve this problem?
• Of the strategies I know, which seem to best fit this particular problem? Why?
• Is there anything special about this problem that suggests a particular strategy?
• Why did I choose this strategy to solve this problem?
• Could I use another strategy to check my answer?

These questions develop students' metacognitive skills, helping them become more deliberate in their strategy selection.

3. Incorporate Incorrect Solutions for Analysis

Analyzing incorrect solutions develops critical thinking and deepens understanding of valid strategies. For example, when factoring quadratic expressions, teachers might present:

Correct: x² + 5x + 6 = (x + 2)(x + 3)

Incorrect: x² + 5x + 6 = (x + 1)(x + 6)

By comparing these solutions, students can identify why the incorrect factorization fails and reinforce their understanding of the distributive property.

4. Use Comparison Activities for Deeper Understanding

Small-group comparison activities are particularly effective for developing strategic flexibility. Implement a partner comparison activity where:

• Students are paired based on using different strategies to solve the same problem
• Partners discuss the strategies they used and their reasoning
• Each student then attempts to solve a new problem using their partner's strategy
• The class discusses the advantages and limitations of each approach

This collaborative approach helps students appreciate multiple perspectives and develop greater flexibility in their problem-solving approach.

Small-Group Strategy Comparison

For a more structured comparison, create activities where students work in small groups to evaluate different approaches. For example, when teaching systems of equations, provide groups with the same problem solved using different methods (substitution, elimination, and graphing). Ask students to:

• Identify which method was most efficient for this particular system and why
• Discuss when each method would be most appropriate
• Create examples of systems that would be most efficiently solved with each method

Comparison activities are more effective than studying individual strategies in isolation.

Overcoming Implementation Challenges

Teachers often worry that teaching multiple strategies will confuse struggling students or take too much instructional time. You can address these concerns by emphasizing that:

• The goal isn't mastery of all strategies, but rather developing the ability to compare and choose appropriate approaches
• Multiple strategies should be introduced gradually after foundational knowledge is established
• Comparison activities can be integrated into existing lesson plans rather than requiring additional instructional time
• Even students with learning difficulties benefit from strategic flexibility when it's taught with appropriate scaffolding

By helping students develop strategic flexibility, educators transform struggling algebra learners into confident problem-solvers who can adapt their approach based on the specific demands of each problem.

Fostering a Growth Mindset for Algebra Success

The strategic flexibility approach naturally promotes a growth mindset in mathematics. When students see that they can tackle the same problem in multiple ways, they begin to view algebra as a flexible thinking system rather than a rigid set of procedures to memorize.

This shift in perspective is often the turning point for struggling students, as they begin to see themselves as mathematical thinkers who can make strategic choices rather than passive followers of procedures.

For professional development and curriculum resources that integrate these research-based strategies for developing algebraic thinking, visit Algebra Ace 9.