Posted in Assessment, Curriculum Reform, Elementary School Math, High school mathematics, Number Sense

Assessing conceptual understanding of integers

Assessing students’ understanding of operations involving integers should not just include assessing their skill in adding, subtracting, multiplying and dividing integers. Equally important is their conceptual understanding of the process itself and thus need assessing as well. Even more important is to make the assessment process  a context where students are given opportunity to connect previously learned concepts (this is the essence of assessment for learning). Because the study of integers is a pre-algebra topic, the tasks should also give opportunity to engage students in reasoning, number sense  and algebraic thinking. The tasks below meet these criteria. These tasks can also be used to teach mathematics through problem solving.

The purpose of Task 1 is to encourage students to reason in more general way. That is why the cells are not visible. Of course students can solve this problem by making a table first but that is not the most ideal solution.

adding integers
Task 1 – gridless addition table of integers

A standard way of assessing operations involving integers is to ask the students to perform the operation. Task 2 is different. it is more interested in engaging students in reasoning and in developing their number and operation sense.

subtracting integers
Task 2 – algebraic thinking and reasoning in numbers

Task 3 is an example of a task with many possible solutions.  Asking students to find a relation between the values in Box A and Box B links operations with integers to the study of varying quantities or quantitative relationship which are fundamental concepts in algebra.

Task 3 – Integers and Variables

More readings about algebraic thinking:

If you find this article helpful, please share. Thanks.

Posted in Algebra, High school mathematics

Evolution of the definition of function

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.

Posted in Algebra, Assessment, High school mathematics

Levels of understanding of function in equation form

There are at least three representational systems used to study function: graphs, tables and equations. But unlike graphs and tables that are used to visually show the relationships between two varying quantities, students first experience with equation is not as a representation of function but a statement which state the condition on a single unknown quantity.  Also, the equal sign in the early grades is taught to mean “do the operation” and not to denote equality between the quantities of both sides of the sign. To complicate the matter, when equation is used as a representation of function, it takes an additional meaning, that of a representation of two varying quantities! For example, let y be number of t-shirts and x be the number of t-shirts to be printed. If the cost of printing a t-shirt is 2.50 then the function that defines y in terms of x is y = 2.50x. Understanding function equation form is not an easy concept for many students.

I believe that if mathematics teachers are aware of the differing level of abstraction in students’ thinking and reasoning  when they work with function in equation form then the teachers would be better equipped to design appropriate instruction to lead students towards a deeper understanding of this concept.Failure to do so would deprive students the opportunity to understand other advanced algebra and calculus topics.I would like to share a framework for assessing students’ developing understanding of function in equation form. This framework is research-based. You can download the full paper here or you can view the slides in my post Learning Research Study Module for Understanding Function.

The framework is in terms of levels of understanding. You can use it to design tasks or assess your students understanding of function in equation form. Each of these levels are “big ideas” or schemas in the understanding of function.
Level 1 – Equations are procedures for generating values.
Students at this level can find x given y or vice versa. Some can generate a number of pairs but not really see the equation as a rule for all pairs of values in a situation.
Level 2 – Equations are representations of relationships.
Students at this level understands domain and range, can generate pairs of values and graph it. They also know that that relationship is unique and true for the values in the domain and range.
Level 3 – Equations describe properties of relationships.
Students at this level can interpret the properties of the function like rate of change and intercepts form the equation but can do it by generating values.
Level 4 – Functions are objects that can be manipulated and transformed
This is the highest level. At this point students see the equation as a math object. They can do composition of function, can find its inverse by algebraic manipulation and can interpret the meaning or effect of the parameters to the graph of the function. 

You can reference the above framework:

Ronda, E. (2009). Growth points in students’ developing understanding of function in equation form. Mathematics Education Research Journal, 21, 31-53.