6.EE.5, 6.EE.6, 6.EE.7, 7.EE.4a, 8.EE.7, A-CED.1
We never really grasp a concept until we internalize it and make it our own. Watching someone else work through an example will not help us internalize an idea nearly so well as working through some ourselves. At the same time, we usually need at least a little guidance for new ideas. It is important to strike a balance between these needs and to provide a rich enough variety of examples for students to work through that they can see patterns emerge and recognize common principles at play in seemingly different situations. Formal procedures, while useful and important, do not generally enhance understanding or intuition, and development of intuition should not be sacrificed for the sake of formal procedures.
|1/3(x + 4) = 2||
1/3(x + 4) = 2
3/1 * 1/3 (x + 4) = 2 * 3/1
(x + 4) = 6
x + 4 - 4 = 6 - 4
x = 2
|1/3(x + 4) = 2||One third of something is 2, so that "something" must be 6. Something plus 4 is six, so that "something" is 2. X = 2.|
(I: Interviewer, S: Student)
|I:||Can you solve 1/3(x + 4) = 2 for x?|
|S:||Yes, I can multiply both sides by 3 over 1, then subtract 4 from both sides and get x = 2.|
|I:||Can you solve for x without writing anything on your paper? Can you do this one in your head?|
|S:||That is a lot to keep track of in my head. I would probably make a mistake because I'm not good at mental math. I could guess and check but that always takes forever. I think it would be hard but I could probably do it.|
Students learn from constructing meaning from multiple examples. They will not learn a procedure by seeing one clear example (as is done in most textbooks). They need to try things over and over and test their procedures to see what works.
“Transmitting algebraic rules by the teacher, memorizing them by students and practising them in a mechanical way is worthless” (Demby, 2007, p. 68). Students’ misconceptions and incorrect ways of solving equations are documented multiple different ways. The underlying student difficulties indicate that students are unaware of the reasons for the manipulation they perform, and students do not grasp the meaning of an equal sign.
When students evaluate expressions, their procedures may fall into one of the categories below:
1. Automatization—students do it, but cannot explain how or why they did what they did (I don’t know why, it’s just what my teacher said to do)
2. Formulas—students apply a memorized formula (like difference of 2 squares)
3. Guessing-Substituting--students guess what the simplified version is, and then check by plugging in a number like 2 or 3 to check if they are correct
4. Preparatory Modification of the expression—students change something in the expression before evaluation (like changing subtracting a negative to addition or division by an integer into multiplying by a fraction
5. Concretization—students convert variables into a concrete idea (like 3x + 2x is 3 apples + 2 apples = 5 apples)
6. Rules—students identify a memorized rule (like when adding 3x and 2x you add the coefficients and leave the variable alone)
7. Quasi-rules—students apply rules, but do it inconsistently (sometimes raising numbers to an exponent, sometimes not)
8. Students will use their own invented procedures (correct or incorrect) rather than use what they have been taught in school.
Presenting mathematical concepts through stand-alone examples and repetitious practice does not foster understanding. We appear to teach algorithms too soon and assume once taught they are remembered. This approach does not help students overcome misconceptions. Instead, we should provide examples of problems that make students see the benefit of using informal procedures, like in the example above. Other examples help them to see the necessity for formal procedures as in a task like: 6/a = 15. Equations can be thought of as using metaphors as well. Four types are:
1. A story of something happening to x: For 4(x+2)=4, “There is a number. You add 2 to the number, and then multiply by 4 and the number becomes 4.” This helps with single variable equations with no variable on the other side. It can reinforce the misconception that you do a procedure to what is on the left to get the answer on the right of the equals sign.
2. The function machine: Students are given a set of inputs and outputs and need to find the rule. This works well for data provided in a table format.
3. The recipe: For y = 2a + b, “To make y, you need to double a and add it to b.” This connects known formulas like A= L x W to algebraic equations. It also shows students that procedures can happen on the right side of an equation.
4. The balance model: Two expressions are seen to be equal and if you operate on one, you must operate equally on the other. (This should be used with caution with students who are not at the formal-operational mode according to Piaget.)
A child’s errors are actually natural steps to understanding—they provide insight for the teacher into the pupil’s thinking. Students need to confront the conflict between their misconceptions and the principles they have learned or else the connections may not be made.
Brown, G., & Quinn, R. J. (2006). Algebra students' difficulty with fractions: An error analysis. Australian Mathematics Teacher, 62(4), 28-40.
Demby, A. (1997). Algebraic procedures used by 13-to-15-year-olds. Educational Studies in Mathematics, 33(1), 45-70.
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.
MacGregor , M. E. (1999). How Students Interpret Equations: Intuition Versus Taught Procedures. In H. Steinbring, M. G. Bartolini-Bussi & A. Sierpinska (Eds.), Language and Communication in the Mathematics Classroom (pp. 262-270). Reston, VA: NCTM.
Rossi, P. S. (2008). An uncommon approach to a common algebraic error. PRIMUS 18(6), 554-558.
Tirosh, D., & Stavy, R. (1999). Intuitive rules: a way to explain and predict students' reasoning. Educational Studies in Mathematics, 38(1-3), 51-66.
Vaiyavutjamai, P. & Clements, M.A. (2006). Effects of classroom instruction on students’ understanding of quadratic equations. Mathematics Education Research Journal, 18(1), 47–77.
Yannis T. & Constantinos T. (2007). The notion of historical “parallelism” revisited: Historical evolution and students’ conception of the order relation on the number line. Published online: 2 May 2007 # Springer Science + Business Media B
- 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?