Algebra

Assessing understanding of graphs of functions

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Problems about graphs of functions can be grouped into interpretation or construction tasks. The tasks may involve interpreting individual points, an interval, or the entire graph. The same may be said about construction tasks. It may involve point-plotting, a part of the graph, or constructing the whole graph.

Tasks involving constructing graphs are considered more difficult than interpreting graphs tasks but with the available graphing technology, constructing graphs is now easy. But not when you have to construct a relationship, not just graphs! In fact, I would consider it as an indicator of students deep understanding of graphs and functions when he or she can interpret and reason in terms of relationship shown in the graphs and from these be able to construct a new relationship, a new function. Here is a task you can use to assess this level of understanding. Note that in this task the graphs are not on grids to encourage holistic analysis of the graph rather than point-by-point. Interpreting graphs not on grids encourages algebraic thinking.

Below is a a sample a Year 8 student solution to the task above. This answer indicates that the student understands graphs and the function it is representing but he/she could still not reason in terms of relationship so resorted to interpreting individual points in x vs y and y vs z in order to relate x and z.

graph1.1 — solutions by point-by-point analysis

The figure below shows a solution of a Year 10 student who could reason in terms of the relationships of the variables represented by the graphs.

graph1.2 — reasoning in terms of relationship

A similar solution to this would be “x is directly proportional y but y is inversely proportional to z hence x would also be inversely proportional to z”.

Both solutions are correct and both solved the problem completely. Note that initially students will use the first solution just like the Year 8 student. The Year 10 however should be expected and encouraged to reason in terms of relationship.

A good assessment task not only assesses students’ mathematical knowledge and skills but also assesses the level of thinking and reasoning students are operating on. See posts on features of good problem solving tasks.

Evolution of the definition of function

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How do you define function? Do you teach relation first before teaching function? Does knowing about relation a pre-requisite to function understanding?

The concept of function “was born as a result of a long search after a mathematical model for physical phenomena involving variable quantities” (Sfard, 1991, p. 14). In 1755, Euler (1707-1783) elaborated on this conception of function as a dependence relation. He proposed that, “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (p. 15). Seventy-five years later, Dirichlet (1805-1859) introduced the notion of function as an arbitrary correspondence between real numbers. About a hundred years later in 1932, with the rise of abstract algebra, the Bourbaki generalised Dirichlet’s definition. Thus, function came to be defined as a correspondence between two sets (Kieran, 1992). This formal set-theoretic definition is very different from its original definition. Function is no longer associated with numbers only and the notion of dependence between two varying quantities is now only implied (Markovits, Eylon, & Bruckheimer, 1986). The Direchlet-Bourbaki definition allows function to be conceived as a mathematical object, which is the weakness of the early definition. However, the set-theoretic definition is too abstract for an initial introduction to students and is inconsistent with their experiences in the real world (Freudenthal, 1973; Leinhardt, Zaslavsky, & Stein, 1990; Sfard, 1992).

Textbooks, which often define function as a set of ordered pairs usually start the discussion with relation and introduce function as a special kind of relation. But relation is more abstract than function. Thus the supposed pedagogical value of having to learn relation first before one understands function is, in the opinion of Thorpe (1989), wrong. Freudenthal (1973) also expressed strongly that “to introduce function, relations can be dismissed” (p. 392). Thorpe went on to say that the use of the set-theoretic definition which defines function as a set of ordered pairs “was certainly one of the errors of the sixties and it is time that it were laid to rest” (p. 13). Amen to that but only until a certain grade level.

My references:

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht-Holland: Reidel.

Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning and teaching. Review of Educational Research, 60(1), 1-64.

Markovits, Z., Eylon B. A., & Bruckheimer, M. (1986). Function today and yesterday. For the Learning of Mathematics,6(2) 18-28.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59-84). (MAA Notes no. 25) Washington DC: Mathematical Association of America.

Thorpe, J. (1989). Algebra: What should we teach and how should we teach it? In S. Wagner & C. Kieran (Eds.) Research issues in the learning and teaching of algebra (pp. 11-24). Reston, VA: NCTM.

Algebra, Math investigations

Solving systems of equations by elimination – why it works

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Mathematical knowledge is only powerful to the extent to which it is understood conceptually, not just procedurally. For example, students are taught the three ways of solving a system of linear equation: by graphing, by substitution and by elimination. Of these three methods, graphing is the one that would easily make sense to many students. Substitution, which involves expressing the equations in terms of one of the variables and then equating them is based on the principle of transitive property: if a = c and b = c then a = b. But, what about the elimination method, what is the idea behind it? Why does it work?

While the elimination method seems to be the most efficient of the three methods especially for linear equations of the form ax + by = c, the principle behind it is not easily accessible to most students.

Example: Solve the system (1) 3x + y = 12 , (2) x – 2y = -2.

To solve the system by the method of elimination by eliminating y we multiply equation (1) by 2. This gives the equation (3), 6x + 2y = 24. Thus we have the resulting system,

6x + 2y = 24

x – 2y = -2.

The procedure for elimination tells us that we should add the two equations. This gives us a fourth equation (4), 7x = 22. We can then solve for x and then for y. But we have actually introduced 2 more equations, (3) and (4) in this process. Why is it ok to ‘mix’ these equations with the original equations in the system?

Equation (3) is easy to explain. Just graph 3x + y = 12 and 6x + 2y = 24. The graph of these two equations coincide which means they are equal. But what about equation (4), why is it correct to add to any of the equations? The figure below shows that equation (4) will intersect(1) and (2) at the same point.

Is this always the case? Think of any two linear equations A and B and then graph them. Take the sum or difference of A and B and graph the resulting equation C. What do you notice? This is the principle behind the procedure for the elimination method. But before students can do this investigation, they need to have some fluency on creating equation passing through a given point. The following problem can thus be given before introducing them to elimination method.

Is there a systematic way of generating other equations passing through (3,1)? This will lead to the discovery that when two linear equations A and B intersect at (p,q), A+B will also pass through (p,q). With little help, students can even discover the elimination method for solving systems of linear equations themselves from this. This problem is again another example of a task that can be used for teaching mathematics through problem solving . The task also links algebra and geometry. Click this link for a proposed introductory activity for teaching systems of equation by elimination method.

Algebra

What is algebraic thinking?

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In my post Arithmetic and Algebra, I wrote that it’s how you solve a problem that tells whether you are doing algebra or arithmetic, not the problem itself. Here’s a description of algebraic thinking that I think teachers in elementary school mathematics might find useful especially when they are teaching about numbers and number operations:

Algebraic thinking is about generalising arithmetic operations and operating on unknown quantities. It involves recognising and analysing patterns and developing generalisations about these patterns.(NZCER)

I find the description clear, concise, can easily be committed to memory and can form part of teachers everyday discussion with a little effort. The keywords that should be remembered are patterns and generalizations. You can’t actually separate one word from the other. If you see a pattern, you can’t but make some generalizations. It’s human nature. If you make generalizations it must from the patterns that you recognize.

Patterns about what?

In the description of algebraic thinking above it says patterns about arithmetic operations (add, subtract, etc) or relationships between numbers for example in equations. Of course it could also involve patterns in shapes, colors, positions in sequences. In short, you also use algebraic thinking in geometry.

Here’s my other favorite description of algebraic thinking:

Algebraic thinking involves the construction and representation of patterns and regularities, deliberate generalization, and most important, active exploration and conjecture. (Kaput, NCTM, 1993).

It is similar to the first but added representing patterns and regularities observed and active exploration as important processes. Without these, generating cases needed for making conjectures/generalizations and verifying them would be difficult.

Click algebraic thinking to see the collection of articles and lessons in this blog about this topic. You may also want to check on these book for other lessons..

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