Category: Teaching mathematics

Posted in Teaching mathematics

The Four Freedoms in the Classroom

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You will find that by providing the following freedoms in your classroom an improved learning environment will be created.

The Freedom to Make Mistakes

Help your students to approach the acquisition of knowledge with confidence. We all learn  through our mistakes. Listen to and observe your students and encourage them to explain or demonstrate why they THINK what they do. Support them whenever they genuinely participate in the learning process. If your class is afraid to make mistakes they will never reach their potential.

The Freedom to Ask Questions

Remember that the questions students ask not only help us to assess where they are, but assist us to evaluate our own ability to foster learning. A student, having made an honest effort, must be encouraged to seek help. (There is no value in each of us re-inventing the wheel!). The strategy we adopt then should depend upon the student and the question but should never make the child feel that the question should never have been asked.
classroom quote

The Freedom to Think for Oneself

Encourage your class to reach their own solutions. Do not stifle thought by providing polished algorithms before allowing each student the opportunity of experiencing the rewarding satisfaction of achieving a solution, unaided. Once, we know that we can achieve, we may also appreciate seeing how others reached the same goal. SET THE CHILDREN FREE TO THINK.

The Freedom to Choose their Own Method of Solution

Allow each student to select his own path and you will be helping her to realize the importance of thinking about the subject rather than trying to remember.

These freedoms help develop students skills and habits of mind.

Posted in Math investigations, Teaching mathematics

What is cognitive conflict approach to teaching?

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According to Piaget, knowledge is constructed when a learner encounters input from the environment and incorporates the new experiences to his/her existing schemes and mental structures (assimilation). When this new assimilated information conflicts with previously formed mental structures, the result is called disequilibrium – a cognitive conflict. This state of disequilibrium motivates the learner to seek equilibrium. The cognitive conflict approach to teaching is based on this assumption – that learners will seek this equilibrium.

Cognitive conflict approach to teaching exposes students in  situations where some of their existing understandings about an idea or a topic no longer hold. A famous example on this is the Chords and Regions activity:

Find a way of predicting the maximum number of regions created by chords connecting n points.

This activity is usually used to challenge students thinking that patterns observed will always hold true and that patterns can be used as proof. The pattern observed will not hold true for n > 5. You can read the result of this activity in this paper Chords and Regions.

The assumption that learners will seek equilibrium when they are put in a situation of disequilibrium, when they experience cogntive conflict isn’t often the case. In fact, a common challenge faced by the cognitive conflict approach is that students often possess ‘contradictory understandings’ (from a mathematical point of view) but they don’t feel the need to address the inconsistencies in their understandings. This is the reason why it is very hard to correct a misconception. Also, students often do not see the importance (or necessity) to engage in a process of modifying their understandings to resolve the contradictions and they tend to treat the contradictions as exceptions (Stylianides & Stylianides, ICME-11). In the above activity for example, instead of being challenged, students can just accept the fact that the pattern stops after n=5 and not try to think of a more general rule to cover all cases. It is also possible that students can just say Next time I’ll try up to 10 cases before generalizing. This is now the challenge to the teacher. As a teaching approach, the use of cognitive conflict has a lot of potential but it needs more than simply using the appropriate task to create the conflict. Our students can be very resilient.

Mary Pardoe via LinkedIn discussion suggests that a strategy that encourages students to confront, rather than avoid, a cognitive conflict is to challenge small groups of students to reach a group conclusion (about the situation) with which everyone in the group agrees. Students who individually might respond differently to the ‘conflict’ will usually then try to persuade each other that their own points of view are correct, and so they are motivated to explain and discuss their thinking. Sample teaching using this approach is described in Using cognitive conflict to teach solving inequalities.

Common misconceptions are also rich sources of tasks for creating cognitive conflict. Click the link Mistakes and Misconceptions and Top 10 Errors in Algebra for sample of tasks.

You may want to check the book below to get more ideas on teaching mathematics.

Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education)

 

Posted in Teaching mathematics

Should the historical evolution of math concepts inform teaching?

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Should the history of a math concept inform the way we should teach it? Some camps, especially those that strongly object to the usual axiomatic-deductive style of teaching, advocates the use of a “genetic” teaching model that takes seriously into account the historical roots of mathematical knowledge. Here are some studies that support this approach.

Harper (2007) compared the historical analysis with students’ empirical data and found a parallelism between the evolution of algebraic symbolism and the way students understand the use of letters in school algebra, concluding that “… the sequencing of conceptual acquisition appears to parallel that which is to be detected through the study of the history of mathematics.”

Moreno and Waldegg (1991) found that “… in situations involving the concept of infinity, the student response schemes are similar to the different response schemes given by mathematicians throughout the history of mathematics,…, when faced with the same kind of questions”

However, there are also those who contradicts this conclusions: For example, on solving linear equations, Arcavi argues that,

….solution methods generated throughout history are quite different from the usual methods generated by students. Consequently, we cannot assert that a reason for the study of linear equations is based on or inspired by parallels between history and psychology – these parallels do not seem to exist (Arcavi 2004, p. 26).

Herscovics acknowledges that while obstacles in the nature and evolution of knowledge are in parallel with some of those met by the learner and are associated with his/her cognitive evolution, she also warns that this parallelism should not be taken too literally, since learning environments in the past are significantly different from those of our learners now (Herscovics 1989, p. 82).

In their investigation of the parallelism between historical evolution and students’ conceptions of order in the number line, Thomaidis & Tzanakis (2007) has this to say:

If room is left for genuine problems to help the emergence of the new concepts, motivate students to appreciate their necessity, or formulate their own alternative ideas (as it happened historically), teaching will not be restricted to the presentation of formal constructs in their polished final form, as it is often the case under the additional pressure of factors peculiar to the modern educational system itself, but will help students conceive mathematics as a creative, adventurous human activity.

Like in most issues related to teaching and learning, there is no clear cut answer here,  but it will always pay to know for teachers to have a sense of how specific math concepts evolved in history. It could provide valuable information both in the design of instruction, in anticipating cognitive obstacles and, for making sense of students difficulties in learning the concept. Teachers must also always remember that the evolution of a math concept is always tending towards abstraction. And because definitions of math concepts are already abstractions of those concepts, starting with definition in teaching is a no-no. Read why I think it is bad practice to teach a math concept via its definition.