Verification, solution strategy, multiple solutions, flexibility, efficiency
6.EE.2, 6.EE.3, 6.EE.7, 7.RP.2a, 7.RP.2b, 8.EE.7a, 8.EE.7b, A-RE I.2, A-RE I.3
Students need to know a variety of procedures for accurately and efficiently solving equations. Equally important, but often overlooked, is the need to understand how these solution strategies are related, and the need to select the proper strategy for a given context. These needs are characteristic of conceptual knowledge.
“Procedural knowledge refers to computational skills and knowledge of procedures for identifying mathematical components, algorithms and definitions. Procedural knowledge of mathematics has two parts: (a) knowledge of the format and syntax of the symbol representation system and (b) knowledge of rules and algorithms useful in mathematical tasks. Conceptual knowledge refers to knowledge of the underlying structure of mathematics. It is characterised as knowledge rich in relationships and includes the understanding of mathematical concepts, definitions and fact knowledge. Both procedural and conceptual knowledge are considered as necessary aspects of mathematical understanding” (Attorps, 2003, p. 1-2, italics added).
|1. Solve the equation 2(3x+4) = 6x−5
2. After solving, could you solve it in another way?
3. What would be the relationship between the graphs of each side of the equation?*
|The correct answer is empty set, meaning that there is no answer. This equation is a false equation (meaning that it leads to a contradiction).
Graphically each side of the equation represents a line and these lines are parallel to each other. 59% of high school student-pairs answered this question correctly. 57% of the participants used symbolic manipulation as their first choice of strategy
(I: Interviewer, S: Student)
|I:||Could you have solved these in another way?|
|S:||Student 1-Pair 1: Guess and check.
Student 2-Pair 1: I mean you could try to get . . . all the x values . . . on one side and then move all the others on the one side, pl—plug it into a graph.
|S:||Student 1-pair 2: Well in my head or on the calculator?|
|I:||On the calculator, how would you solve it on the calculator?|
|S:||Student 1-Pair 2: Well, let’s see . . .. I can do part of it in my head and part of it on the calculator. You subtract 2x from both sides and you’ve got 3x minus 9. I could solve it in my head easier than the calculator; I don’t know if I could do it on the calculator. There’s no specific way to set it up on the calculator.
S2-pair 2: At least not that we know of.*
What is intellectually needed to solve a problem is different from the set of math skills needed to solve a problem. Students may know the properties, like the Distributive Property, but not think to use it to solve a problem. Students in middle grades (even those who are classified as “advanced”) do not apply problem solving strategies (like “solve a simpler problem”) willingly. They prefer to be told what to do and follow the instructions.
When solving equations, students respond in different ways:
1. Guess and check—perhaps because initial equations can be solved easily using this strategy.
2. In tasks like 3x+2x=12, some will set the "first x" equal to something and then solve for the "second x." For example, they might set x=2 and then solve for the second x, getting 3.
3. Some use alternating methods--changing strategies, correctly or incorrectly, whenever the task changes.
4. Some will choose one procedure and apply it (incorrectly) to every equation regardless of the task.
For example, if 2x+4x = 12, then x*x=12-4-2. So, x^2=6 and x=Sqrt(6).
When a rule-based approach is used in algebra it causes confusion for students. The reinforcement of a strategy by solving a large set of exercises may help students to pass tests; however, this does not mean that they are learning algebra with understanding. When teaching students to solve equations, it can be helpful to have them compare multiple methods side by side. This process strengthens conceptual and procedural knowledge because:
1. It draws their attention to important features of the solution
2. It helps them to allow for the possibility of multiple solution strategies
When comparing methods, it is critical for the teacher to draw their attention to the important features--this makes them actively compare the methods. "Show-and-tell" is not enough. Developing flexibility in solving equations requires students to know which strategies are more efficient than others under particular conditions.
Instead of focusing on the difficulty of a question that students can solve, teachers can assess the sophistication level of the solution strategy. One guide for how to assess sophistication of these strategies is below.
Suggestions for Scoring Rubric for Solving Equations:
0. Unable to answer question
1a. Known basic facts
1b. Counting techniques
1c. Inverse operation
2. Guess and check
3a. Cover up
3b. Working backwards
3c. Working backwards, then known facts
3d. Working backwards, then guess and check
4. Formal operations/equation as object
5. Using a diagram (Linsell, 2009).
Teachers need to understand that students can solve one-step equations correctly but this does not mean that these students fully grasp inverse operations strategy for solving equations or that these students are ready to move to two-step equations. No matter how "simple" the equation is (one-step versus two-steps, one unknown versus two) students have difficulty solving equations with division in them. Teachers need to be aware of the fact that even if students can solve two-step equations correctly, they might not understand the strategy they use or alternative strategies.
Attorps, I. (2003). Teachers' images of the 'equation' concept. Paper presented at the Third Conference of the European society for Research in Mathematics Education, Bellaria, Italy.
Davis, R. B. (1975). Cognitive processes involved in solving simple algebraic equations. Journal of Children's Mathematical Behavior, 1(3), 7-35.
Filloy, B. & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. International Journal of Mathematics Education, 9(2), d1925.
Huntley, M. A., Marcus, R., Kahan, J., & Miller, J. L. (2007). Investigating high-school students' reasoning strategies when they solve linear equations. The Journal of Mathematical Behavior, 26(2), 115-139.
Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 36(4), 297-312.
Linsell, C. (2009). A hierarchy of strategies for solving linear equations. Paper presented at the Crossing divides: The 32nd annual conference of the Mathematics Education Research Group of Australasia, Palmerston North, NZ.
Linsell, C. (2008). Solving equations: Students' algebraic thinking. In Ministry of Education (Ed.), Findings from the New Zealand Secondary Numeracy Project 2007, 39-44. Wellington, New Zealand: Learning Media.
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An Experimental Study on Learning to Solve Equations. Journal of Educational Psychology, 99(3), 561-574.
Sleeman, D. H. (1984). An attempt to understand students' understanding of basic algebra. Cognitive Science, 8, 387-412.
Verikios, Petros and Farmski, Vassiliki; From equation to inequality using a function-based approach. International Journal of Mathematical Education in Science and Technology, 41(4), 515-530.
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?