Case Study: Algebra (Part 2): Applying Learning Strategies to Intermediate Algebra

Introduction

Algebra is a branch of mathematics that uses symbols and operations to solve problems. As they solve these problems, students are required to think abstractly and use multiple representations (e.g., symbols, equations, graphs, diagrams) to represent the math concepts. They are also required to generalize mathematical concepts that they previously learned, using those concepts as they transition from completing problems with numbers to solving problems with symbols (e.g., 6 + 3x + x2). However, many students have not obtained a strong foundation in these mathematical concepts. For this reason, these students are unprepared for algebra.

The National Assessment of Education Progress (NAEP) administers mathematics achievement assessments to fourth- and eighth-grade students each year and to 12th-grade students every four years. Student performance indicates the degree to which they have acquired the knowledge and skills expected at their grade level. The results are categorized into one of four levels: Below Basic (little mastery), Basic (partial mastery), Proficient (mastery), and Advanced (beyond mastery). NAEP data indicate the following for different student groups:

Source: National Assessment of Education Progress. (2024). NAEP mathematics achievement test results. The Nation’s Report Card. https://www.nationsreportcard.gov/ndecore/xplore/NDE

The National Mathematics Advisory Panel (NMAP) identified three mathematical concepts needed for students to be successful in algebra.

• Fluent with whole numbers—In addition to having a strong number sense, students must understand and be able to use place value and properties, compose and decompose numbers, and estimate. They must also be fluent in basic operations (i.e., addition, subtraction, multiplication, division) and be able to apply those operations to problem-solving.
• Fluent with rational numbers—Students must understand and be able to compute positive and negative fractions and understand the relationships among fractions, decimals, and percents. They should also understand rates, proportionality, and probability.
• Understanding of geometrical measurement—To solve algebra problems, students should understand similar triangles and be able to analyze two- and three-dimensional shapes. In addition, students are required to identify and use formulas (e.g., perimeter, area, volume) as well as identify unknown lengths, angles, and areas.

Students who have difficulties in any of these three concepts— as well as deficits in long-term memory, working memory, and organizational skills—will struggle with algebra due to the high cognitive load (i.e., the amount of working memory resources being used for a task) needed to solve problems.

*Note: As students graduate to more complex algebraic procedures, such as solving problems containing exponents and factoring polynomials, they can still use the same learning strategies that they used to understand the basics of algebra. This case study unit contains three of the four strategies introduced in Algebra (Part 1): Applying Learning Strategies to Beginning Algebra—using precise mathematical language, the Concrete-Representational-Abstract (CRA) method, and metacognitive strategies. As such, the basic information on the STAR sheets is the same. However, this case study unit shows how to apply them to more advanced algebraic concepts and offers examples of more advanced problems.

NMAP also states that the curriculum must concurrently develop their conceptual understanding, computational fluency, and problem-solving skills for them to be successful in algebra. Six essential instructional strategies, five of which will be discussed in one or more STAR Sheets, have been identified to assist students in developing their knowledge and skills in algebra:

• Explicit instruction—Using a structured instructional method, educators first provide students a rationale and clear expectations for learning skills or concepts, then model, scaffold, provide opportunities for practice and engagement, and offer feedback until students have independently mastered the skill or concept.
• Multiple representations—Educators use manipulatives, as well as pictorial and symbolic representations, to demonstrate concepts; students use multiple representations to organize their thinking to demonstrate conceptual understanding.
• Sequence or range of examples—Educators specify a sequence or pattern of examples or a variation in the range of examples.
• Visual representations—Students use visuals such as diagrams or charts to organize their work, reflect their understanding of a problem, and correctly solve it.
• Student verbalization—Students verbalize self-instructions or solution steps, and teachers encourage students to think aloud when solving problems.
• Use of metacognitive strategies—Students use these strategies for organizing information and for solving problems with self-reflection and self-questioning.

Each case study includes multiple STAR Sheets and cases.

About the Strategy

Using precise mathematical language requires the use of formal math terms (e.g., negative five rather than minus five for –5). For algebra, mathematical language includes both mathematical vocabulary terms and symbols. Using precise mathematical language:

• Helps to ensure clear communication among educators and students
• Facilitates better understanding of concepts for students
• Allows students to explain their mathematical reasoning

What the Research and Resources Say

• Mathematics vocabulary is necessary for the demonstration of mathematics proficiency (Hughes et al., 2016; Powell et al., 2018).
• Mathematical language should be consistently taught during mathematical instruction (Peeples et al., 2023; Hughes et al., 2016; Powell et al., 2018).
• Teachers must model clear and concise mathematical language through lessons, and they should expect students to use clear and concise mathematical language when explaining their solutions, providing support as needed (Hughes et al., 2016; Powell et al., 2018).
• The ability to solve word problems is dependent on understanding the language in the problems (Hughes et al., 2016; Powell et al., 2018).

Types of Activities to Implement

Educators should teach students to use precise mathematical language and provide good models when presenting content and providing feedback. Following are several instructional practices that educators can use to help students learn mathematical language.

Explicit Instruction for Pre-Teaching Mathematical Language

Educators can increase students’ mathematical language by explicitly pre-teaching novel terms and symbols to students before they encounter them in their work. When using explicit instruction to pre-teach mathematical language, educators should:

• Select language needed to understand the concept
• Provide a clear and precise definition
• Provide examples and non-examples
• Check for understanding
• Provide feedback
• Review
• Provide opportunities for independent practice

The examples below demonstrate how to implement the seven steps of explicit instruction. The first is an example of how to pre-teach a vocabulary term and the second is an example of how to introduce a symbol.

Frayer Model

The Frayer Model is a graphic organizer used to build students’ understanding of key vocabulary terms across all subject areas. In mathematics, it also can be used to increase students’ understanding of symbols. This type of graphic organizer includes:

• The term—The vocabulary term or symbol is placed in the center circle.
• The definition—A clear and precise definition is provided in the upper-left quadrant and, if applicable, it can include an algebraic formula.
• Characteristics—Features that help the student recognize, identify, or distinguish the term or symbol are recorded in the upper-right quadrant.
• Examples—Synonyms, concrete applications, or relevant illustrations of the characteristics of the term or symbol are placed in the bottom-left quadrant. In mathematics, this might include tables, graphs, or diagrams.
• Non-examples—Antonyms, inappropriate applications, or relevant illustrations that do not fit the characteristics of the term or symbol are recorded in the bottom-right quadrant and might include tables, graphs, or diagrams.

Educators should provide explicit instruction on how to develop and use the Frayer Model. After asking students to create a Frayer Model on their own, teachers should review the completed models to ensure the information is correct. Below are two examples of a Frayer Model, the first for the term equation and the next for the absolute value symbol.

Self-Correcting Activities

After students have been explicitly taught new vocabulary terms, they can use self-correcting word cards to continue practicing the words. This consists of two sets of cards: one set contains the vocabulary terms, and the other set contains the definitions. Each matching term and definition has the same symbol in the upper-right corner. After students match the term with the correct definition, they can check their work by making sure the word and the definition have the same symbol. This activity can be completed independently, with a peer, or in small groups.

Students in intermediate algebra courses should be familiar with common algebra terms and symbols. Below are lists of the most common.

About the Strategy

The Concrete-Representational-Abstract (CRA) approach is a strategy for teaching mathematical concepts by systematically connecting concrete objects or student-drawn representations to the abstract equation. It allows students to understand a concept before memorizing the algorithms. The components of the framework are:

• Concrete—Students interact and manipulate three-dimensional objects (e.g., algebra tiles, integer chips). This interaction helps students understand the concept, instead of simply having them solve the algorithm.
• Representational—Students use two-dimensional drawings to represent problems. These pictures may be presented to them by the teacher, through the curriculum used in class, or students may draw their own representation of the problem.
• Abstract—Students complete the algorithm without any concrete or representational assistance.

What the Research and Resources Say

• CRA is effective for all age levels and can assist students in learning basic concepts, operations, and applications (Bundock et al., 2021; Flores et al., 2024).
• By using CRA, students gain a conceptual understanding of the problem and process instead of completing the problem using a memorized algorithm (Bouck et al., 2019; Bone et al., 2023; Namkung & Bricko, 2021).
• Virtual manipulatives are effective in helping students understand algebra concepts, and students prefer them over traditional concrete manipulatives (Bone et al., 2022; Bouck et al., 2018; Satsangi et al., 2018).

Types of Activities to Implement

Below, you will learn about each stage of CRA. When using this strategy, educators should:

• Be very familiar with the concrete objects prior to teaching and having students interact with them
• Provide modeling at all three stages of the CRA approach
• Continuously monitor student work during the concrete and representational levels, asking them questions about their thinking and providing clarification as needed

Examples

Below are multiple examples of how educators can use CRA to teach students to multiply and factor polynomials. For the concrete stage, the concrete manipulatives are represented as follows.

Note: Other manipulatives can be used to represent y, y², y³, x² and x³.

Multiplying Polynomials

About the Strategy

Metacognitive strategies enable students to become aware of and understand their own thinking and how to transfer their learning to new contexts. Metacognition is the process of thinking about thinking or understanding one’s own learning patterns; this includes understanding the task, knowing what strategies work best for oneself, monitoring personal progress, and evaluating the results afterward. More specifically, metacognitive strategies help students learn to:

• Plan—Students decide how to approach the mathematical problem, first determining what the problem is asking and then selecting and implementing an appropriate strategy to solve it.
• Monitor—As students solve a mathematical problem, they check to see whether their problem-solving approach is working. After completing the problem, they consider whether the answer makes sense.
• Modify—If students determine that their problem-solving approach is not working or that their answer is incorrect, they can adjust their approach as they work to solve a mathematical problem.

What the Research and Resources Say

• When paired with cognitive strategies, metacognitive strategies have been shown to increase the understanding and ability of students with mathematics learning difficulties and disabilities to solve math problems (Pfannenstiel et al., 2015).
• Middle school students who received cognitive and metacognitive strategy instruction outperformed peers who received typical math instruction (Montague, Enders, & Dietz, 2011; Pfannenstiel, 2015).

Types of Activities to Implement

Below are examples of how three metacognitive strategies can be used in algebra.

Cue Cards

Cue cards, like those below, can help students remember important information such as formulas, definitions, or rules.

Mnemonics

In general, mnemonics are approaches that help students retain and recall information; therefore, they are sometimes referred to as memory-enhancing strategies. Some are simply words, sentences, or pictorial devices created to help students remember content (i.e., mnemonic devices), while others contain a more coordinated series of steps to help students perform a task (i.e., mnemonic strategies). Although there are different types of mnemonics, FOIL highlighted below is a first-letter strategy—wherein the first letters of the words in a list of items or steps are used to create another word (acronym) or sentence (acrostic).

FOIL

FOIL (First, Outside, Inside, Last) is a mnemonic device can help students remember the steps for multiplying two binomials. The figure below illustrates the guidelines students should follow to apply FOIL to algebra problems. This mnemonic is helpful when students complete the abstract phase of the CRA model.

First (F) — multiply the first terms in each binomial (x • x = x2)
Outside (O) — multiply the outside terms in each binomial (x • –3 = –3x)
Inside (I) — multiply the inside terms in each binomial (4 • x = 4x)
Last (L) — multiply the last terms in each binomial (4 • –3 = –12)

Graphic Organizers

Graphic organizers are visual aids designed to help students organize and comprehend substantial amounts of text and content information. As illustrated in the Using Precise Mathematic Language STAR Sheet, a type of graphic organizer known as the Frayer Model can be used to build students’ understanding of key vocabulary terms across all subject areas. In the area of algebra, graphic organizers can provide students with steps and examples needed to solve mathematical problems. The first graphic organizer below provides steps on how to solve an exponent equation. The second provides a sequence of steps for factoring polynomials.

Exponent Equation

Factoring Polynomials

Problem: f ² + 9f + 18
Degree is 2 (quadratic)
Number of terms is 3 (trinomial)
Coefficient of the squared term is 1

Steps for Factoring

1. Place variables in parentheses
   (f       ) (f       )
2. Find greatest common factor (GCF)
   

   9 = 1 x 9</br />3x3

   18 = 1 x 18 2 x 9 3 x 6

3. Divide
   

   9 ÷ 3 = 3

   18 ÷ 3 = 6

4. Identify signs and complete parentheses
   (f + 3) (f + 6)
5. Check by distributing Factoring polynomials

   

   Multiply f by 6 — (f • 6 = 6f )
   Multiply f by f — (f • f = f ²)
   Multiply 3 by f — (3 • f = 3f )
   Multiply 3 by 6 — (3 • 6 = 18)
   Combine like terms — (6f + 3f = 9f )

6. Practice problems
   x² – 3x – 9
   x³ – 3x + 8