Tag: math exercises

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 Math investigations

Exercises, Problems, and Math Investigations

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The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in.

Mathematical activities/tasks can be categorized into three types: exercises, problem solving, and math investigations.

Standard exercises

These are activities with clearly defined procedure/strategy and goal. Standard exercises are used for mastery of a newly learned skill – computational, use of an instrument, and even new terms or vocabulary. These are important learning activities but must be used in moderation. If our teaching is dominated by these activities, students will begin to think mathematics is about learning facts and procedures only. This is very dangerous.

Problem solving activity

These are activities involving clearly defined goals but the solutions or strategies are not readily apparent. The student makes decision on the latter. If the students already know how to solve the problem then it is no longer a problem. It is an exercise. Click here for features of good problem solving tasks. It is said that problem solving is at the heart of mathematics. Can you imagine mathematics without problem solving?

Math investigations

These are activities that involve exploration of open-ended mathematical situation. The student is free to choose what aspects of the situation he or she would like to do and how to do it. The students pose their own problem to solve and extend it to a directions they want to pursue. In this activity, students experience how mathematicians work and how to conduct a mathematical research. I know there are some parents and teachers who don’t like math investigation. Here are some few reason why we need to let our students to go through it.

  1. Students develop questions, approaches, and results, that are, at least for them, original products
  2. Students use the same general methods used by research mathematicians. They work through cycles of data-gathering, visualization, abstraction, conjecturing and proof.
  3. It gives students the opportunity communicate mathematically: describing their thinking, writing definitions and conjectures, using symbols, justifying their conclusions, and writing and reading mathematics.
  4. When the research involves a class or group, it becomes a ‘community of mathematicians’ sharing and building on each other’s questions, conjectures and theorems.

Students need to be exposed to all these type of mathematical activities. It is unfortunate that  textbooks and  many mathematics classes are dominated by exercises rather than problem solving and investigations tasks, creating the misconception that mathematics is about mastering skills and following procedures and not a way of thinking and communicating.

Samples of these tasks are shown in the picture below:

Click here and here for a sample teaching using math investigation.