The IES practice guide Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students (Star, et al., 2015) provides three broad recommendations for improving the algebra skills of students. The three strategies include using solved prolems to engage learners, teach students to use the structure of equations, and teach students to intentionally choose specific strategies to solve problems. The following is a brief outline of the concepts and ways to implement them in your classroom.
Recommendation 1: Use Solved Problems to Engage Learners
The IES practice guide (Star, et al., 2015) suggests teachers should encourage the use of solved problems to engage learners in understanding algebraic logic and approaches. The practice guide provides evidence from four studies with adequate methodological quality to base the recommendation on. The rating of minimal evidence is based on the inability to generalize the findings to larger populations due to small sample sizes, and one of the studies finding negative outcomes when compared to the strategy in recommendation 2. Essentially, that it is better than normal activities, but not as great as teaching students to utilize the structure of equations. Obviously, utilizing all three recommendations in conjunction would be preferable.
Within this recommendation, teachers should have students discuss solved problems and how those solved problems are structured in whole group, small group, and individually. To that end, teachers should choose solved problems that directly reflect the lesson for the day or unit. The selection of non-examples would also be beneficial to illustrate common mistakes made in solving the specific problem type. The following is a sample solved problem.
Solve for x:
Discuss Solved Problems and Their Structure
The following are questions to facilitate discussion of solved problems.
What were the steps involved in solving the problem? Why do they work in this order? Would they work in a different order?
Could the problem have been solved with fewer steps?
Can anyone think of a different way to solve this problem?
Will this strategy always work? Why?
What are other problems for which this strategy will work?
How can you change the given problem so that this strategy does not work?
How can you modify the solution to make it clearer to others?
What other mathematical ideas connect to this solution? (Star, et al., 2015, p. 5).
These questions will allow discussion of the structure of the problems:
What quantities—including numbers and variables—are present in this problem?
Are these quantities discrete or continuous?
What operations and relationships among quantities does the problem involve? Are there multiplicative or additive relationships? Does the problem include equality or inequality?
How are parentheses used in the problem to indicate the problem’s structure? (Star, et al., 2015, p. 6).
Pick Problems That Reflect the Lesson Goal
The IES practice guide suggests using
problems that mirror the goal of the current lesson.
Select problems with varying levels of difficulty and arrange them from simplest to most complex applications of the same concept.
Display the multiple examples simultaneously to encourage students to recognize problems.
Alternatively, show the problems individually, one after the other, to facilitate more detailed discussion on each problem (Star, et al., 2015, p. 6).
The following is a description of introducing and discussing incorrect and correct problem solving:
Correct solved problem: x^2-4x-45 = (x-9)(x+5)
Incorrect #1: Student did not factor correctly: x^2-4x-45 = (x - 40)(x + 5)
Incorrect #2: Student did not factor correctly: x^2-4x-45 = (x + 9)(x - 5)
Questions to lead discussion
How can you show that the answers from students B and C are incorrect?
What advice would you give to students B and C to help them avoid factoring this type of problem incorrectly in the future?
How can you check that student A factored this expression correctly?
What strategy would you use to factor this expression and why did you choose that strategy? (Star, et al., 2015, p. 10).
Common issues and solutions
Issue 1. I already use solved problems, but students aren’t engaged.
Suggestion. Keep doing it! Modeling solving problems during whole-class instruction with think-alouds.
Issue 2. I don’t know how to find solved problems and am too lazy to make my own.
Suggestion. Curriculum materials and textbooks often have these. You could also use student work on homework.
Issue 3. Won’t incorrect problems confuse them?
Suggestion. No. Using correct and incorrect problems will help students understand the common errors made when solving problems.
Recommendation #2: Use the Structure of Equations
According to the WWC’s practice guide (Star, et al., 2015), the structure of the equations refers to the number, type, and position of quantities, including variables, operations, existence of equality or inequality, and simpler expressions nested inside more complex ones. For example, the structure of the following three equations is the same:
The underlying structure is 5 times an unknown number (x) or (x+1) or (3x-22), plus 19 equals 59. In their review of this process, the WWC reviewers once again found minimal evidence for the strategy, with four studies meeting standards without reservations and two met standards with reservations. Once again, though, the finding of minimal evidence should be viewed in light of the fact that this is not suggesting it does not work, only that there arent enough quality studies out there to allow us to generalize to a larger population.
One of the more common ineffective practices for teachers and parents alike is the use of imprecise language. Indeed, providing effective commands (defined as explicit and specific commands) is an evidence-based practice for improving student compliance (Losinski, Sanders, Katsiyannis, & Wiseman, in press). For example, Mr. Zeller saying, “everyone get your materials out”, is not considered an effective command. In this case, he should say, “students, please place your math textbook and a pencil on your desk”. The specificity of the command reduces any chance of miscommunication. The same is true for providing precise language in mathematics instruction. The following describes the use of precise language.
Imprecise vs. precise mathematical language
Precise mathematical language
Take out the x.
Factor x from the expression.
Divide both sides of the equation by x, with a caution about the possibility of dividing by 0.
Move the 5 over.
Subtract 5 from both sides of the equation.
Use the rainbow method.
Use the distributive property.
Solve an expression.
Solve an equation.
Rewrite an expression.
A is apples.
Let a represent the number of apples.
Let a represent the cost of the apples in dollars.
Let a represent the weight of the apples in pounds.
Plug in the 2.
Substitute 2 for x.
To simplify, flip it and multiply.
To simplify, multiply both sides by the reciprocal.
To divide a fraction, invert and multiply.
To divide fractions, multiply by the reciprocal.
Do the opposite to each side.
Use inverse operations.
Add the opposite to each side.
The numbers cancel out.
The numbers add to zero.
The numbers divide to one.
Plug it into the expression.
Evaluate the expression.
Use reflexive questioning
One of the key suggestions the authors use is having students utilize reflexive questioning. This involves asking themselves questions that uncovers the structure of the problem: The following are examples of reflexive questions:
What am I being asked to do in this problem?
How would I describe this problem using precise mathematical language?
Is this problem structured similarly to another problem I’ve seen before?
How many variables are there?
What am I trying to solve for?
What are the relationships between the quantities in this expression or equation?
How will the placement of the quantities and the operations impact what I do first? (Star, et al., 2015, p. 20)
Using diagrams to denote the underlying structure
The following is an example of using a diagram to identify the structure of a problem. Students are asked to compare each.
Question: Compare a diagram and an equation to represent Timmy’s total online gaming costs per month if Timmy has a fixed/starting cost (f) of $50 plus a game cost (g) of $4.50 for every game. Timmy used 5 games last month. What was his total gaming cost (T)?
Equation (where n = the number of games used).
T = f + ng
T = 50 + 5(4.50)
T = $72.50
Common issues and solutions
Issue 1. Teachers enjoy simplifying language, and students like it.
Suggestion. Imprecise language may cloud student understanding during standardized assessments. Precise language should not be treated as more complicated, but more mathematically accurate. Precise language promotes the use of common language across contexts.
Issue 2. Students rush through problems.
Suggestion. This could be due to two problems: First, problems may be too easy, and students can motor through them without much thought. If this is the case, offer problems that are similar but look different. Second, students may be using strategies they know well, by may not be correct. Assign students reflexive questions to develop understanding and use of varied strategies.
Issue 3. Students don’t use the diagrams
Suggestion. Some students will get to the answer without them, however using diagrams can bring the underlying structure to light. Thus, teachers should encourage the use of diagrams to help students learn the structure.
Recommendation #3: Intentionally Choose Specific Strategies
The WWC practice guide (Star, et al., 2015) suggests teaching students a variety of strategies, though it doesn’t stress that students need to be fluent in all of them. Six studies met WWC group design standards without reservations. Four of the six showed positive effects of teaching alternative strategies and two found negative or mixed effects. This resulted in the classification of this strategy as one with moderate evidence. Within this domain, it is recommended that teachers instruct students to recognize and choose strategies to solve specific problems. According to the Star and colleagues,
Provide students with examples that illustrate the use of multiple algebraic strategies. Include standard strategies that students commonly use, as well as alternative strategies that may be less obvious. Students can observe that strategies vary in their effectiveness and efficiency for solving a problem (Star, et al., 2015, p. 27).
The following is an example of using different strategies to solve problems.
Question 3a + 9b – 7a + 2b – 8a (if a = 6 and b = 8)
3a + 9b – 7a + 2b – 8a
3(6) + 9(8) – 7(6) + 2(8) – 8(6)
18 + 72 – 42 + 16 - 48
3a + 9b – 7a + 2b – 8a
–12a + 11b
–12(6) + 11(8)
-72 + 88
Levi’s restaurant bill, including tax, but before tip, was $23.00. If he wanted to leave a 12.5% tip, how much money should he leave in total?
23.00 * 1.125 = x
x = $25.86
10% of $23.00 is $2.30, and one quarter of $2.30 is $0.56, which totals $2.86, so the total bill with tip would be $23.00 + $2.86 or $25.86.
Issue 1. Whenever I teach multiple strategies, kids get confused.
Suggestion. You’re right, it gets confusing. Start with one until they have mastered it, then present a second to show a different way of solving the problem. Let them practice with it, then they will be able to choose the one they feel more comfortable with.
Issue 2. Our textbook only covers one strategy, what am I supposed to do?
Suggestion. Professional development? Google? What Works Clearinghouse?