Tag: functions

Posted in Algebra, Curriculum Reform

Algebraic thinking in algebra

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Algebraic thinking is an approach to thinking about quantitative situations in general and relational manner. This kind of thinking is optimized by a considerable understanding of the objects of algebra, a disposition to think in generality, and engagement in high-level tasks which provide contexts for applying and investigating mathematics and the real-world.

big ideas in algebra
Ingredients in Algebraic Thinking
Objects of Algebra

The objects are the content of algebra which I classify into three overlapping categories. The first category and the most basic are those for representing changing and unchanging quantities and relationships. These include the idea of variables, numbers, graphs, equations, matrices, etc. The second category are ideas for working with unknown quantities which involve solving equations and inequalities under which are linear equations and inequalities in one variable, systems of linear equations and inequalities, exponential equations, quadratic, trigonometric equations, etc. The third and last category involves the ideas for investigating relationships between changing quantities which include directly and inversely proportional relationships; relationships with constant rate of change; relationships with changing rate of change; relationships involving exponential growth and decay; periodic relationships, etc.

Thinking dispositions

Knowledge of algebraic content do not necessarily translate in algebraic thinking. Computational fluency in simplifying, transforming, and generating expression for example, while important, do not necessarily involve a person in algebraic thinking if one is doing it for its own sake. Thinking processes that contribute to the development of algebraic thinking are those that require purposeful representations of quantities and relationships, multiple interpretations of representations, finding structures, and generalization of patterns, operations and procedures. These should become part of students’ thinking disposition.

High level tasks

The higher-order tasks in mathematics  include problem solving, mathematical investigations (sometimes referred to also as open-ended problem solving tasks), and modeling.

Posted in Algebra, Assessment, High school mathematics

Levels of understanding of function in equation form

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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.