Rafaela Borasi's Controversial Statements About (Mathematics) Education (Exit Slip)

Sixth Controversial Statement: “A good teacher should not confuse the students”. I disagree that mathematics is ambiguous. What is ambiguous are not the concepts of mathematics, but rather, the ways in which they are taught and the ways the learners acquire them. Students have different learning styles and level of cognition; therefore, a good teacher should provide multiple teaching strategies to address this issue. It is said that there are “many paths to the top of the mountain”; hence, the teacher should provide students a variety of solutions or problem-solving techniques to solve a math problem (for example) rather than conform to one standard solution. It is also plausible to encourage the students to explore and generate their own ways in solving a problem as long as they do it the right way. In this way, the students would be able to discover their learning strengths and enhance self-evaluation. Giving students multiple tips, ideas, or examples is not intended to introduce confusion but to offer a wide range of options which could aid them grasp the concepts effectively and meaningfully.

I always thought of effective and efficient way of teaching as being able to deliver a lesson in a limited amount of time and basically, being able carry out the objectives of the curriculum in the classroom religiously. However, today’s discussion left me the following worth-noting insights:

• we should let students think and develop learning strategies independently and not always adapt the preferred procedures or solutions provided by the teacher or the curriculum material. In this way, we will be able to stimulate their creative thinking skills.
• Time-constraint in both teaching and learning has a detrimental effect to the teaching-learning process. Time-management is an important part of classroom management but better things are achieved and done in the absence of time pressure.  
• For humans or any well-thinking animal species, learning from mistakes is essential for self-improvement. We learn most from trial and error, discovery learning, and “learning by doing” which involve learning from mistakes. We even can learn from the mistakes of others like our peers as well. 
• The value of formal mathematics or fundamental and theoretical principles behind formulas must be inculcated in the minds of students. This may be impractical at the first thought but understanding them may help minimize the stress of memorizing formulas. (the iceberg and remote control analogy were good examples)
• The importance of history and philosophy as being the backbone of knowledge must be acknowledged in teaching as well. The fact that the genesis of mathematics can be attributed to ancient Babylonians and Egyptians could make us appreciate the efforts of the ancient civilizations, of our ancestors (perhaps). Values formation and appreciation should be incorporated into teaching and thus, teachers shouldn’t emphasize the cognitive and psychomotor (skills) learning domains only. 

The discussion left me wondering, what is the true measurement or criteria of efficiency and effectiveness in teaching and learning? Is it good to say that your teaching is efficient if the students were able to solve math problems with the use of conventional procedure in short-given time? Or is it better if they’re given the freedom to explore on other possible solutions which they feel more comfortable of using though it requires an ample amount of time?