Category: Assessment

Posted in Algebra, Assessment

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.

graphs
Relating graphs

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.

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.

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.

Posted in Assessment, Curriculum Reform

Features of good problem solving tasks for learning mathematics

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To develop higher-order thinking skills (HOTS) the mind needs to engage in higher-order learning task (HOLT). A good task for developing higher-order thinking skills is a problem solving task. But not all problems are created equal. Some problems are best suited for evaluating learning while others are best suited for assessing learning that would inform teaching. This post is about the second set of problems.The difference between these two sets of problems is not the content and skills needed to solve them but the way they are constructed.

What are the features of a good problem solving task for learning mathematics?
  1. It uses contexts familiar to the students
  2. What is problematic is the mathematics rather than the aspect of the situation
  3. It encourages students to use intuitive solutions as well as knowledge and skills they already possess
  4. The task can have several solutions
  5. It challenges students to use the strategy that would highlight the depth of their understanding of the concept involved
  6. It allows students to show the connections they have made between the concepts they have learned

It is this kind of problem solving task that is used in the strategy Teaching through Problem Solving (TtPS) which I described in the previous post. Here is a sample task:

Students solutions to the task can be used to teach area of polygons, kinds of polygons, preserving area, and meaning of algebraic expression. You can use the task to facilitate students construction of knowledge about adding, subtracting, multiplying and dividing algebraic expressions. Yes, you read it right. This is a good problem solving task for introducing operations with algebraic expression through problem solving! The problem above is also an example of a mathematical tasks that links algebra and geometry. Good mathematics teaching always links concepts.

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.