This article talks about the art of communicating mathematical ideas to novice mathematicians. Historically, those with the "privilege of knowing" mathematics, often teachers, conveyed these ideas to an audience by tailoring and organising their thoughts in the form of the following sequence of instructional techniques: the use of an instructional discourse, a logical sequencing of mathematical problems and exercises, and employment of visual aids. The author states that current pedagogical practices in mathematics are based on these historical techniques.
I really like the following quote from the article: "an initially passive observer who is gradually drawn into the train of instructional thought and hopefully led to the active realization or discovery of the mathematical concepts in question." It is interesting to note how a passive observer is assumed to be lacking in mathematical knowledge and that a teacher will be necessary to lead the observer to discover (if there is such a thing) mathematical concepts. This instantly creates a binary power structure between the teacher and the observer (clever/stupid, leader/follower, sophisticated/unsophisticated). The mere thought of leading someone to discover some new mathematical concepts is scary, especially in a classroom setting. Also, the phrase, "initially passive observer," implies that the observers are mentally lazy. What if the teacher fails to lead to the desired 'destination'? Or, more importantly, what happens if the students are unable to or don't want to realize or discover the euphoria of understanding mathematical concepts?
Yes, I like it when the art of educating our children hinges on 'hope'. Can we, as educators, rely on 'hope' in order to communicate mathematical knowledge to our students?
I really like the following quote from the article: "an initially passive observer who is gradually drawn into the train of instructional thought and hopefully led to the active realization or discovery of the mathematical concepts in question." It is interesting to note how a passive observer is assumed to be lacking in mathematical knowledge and that a teacher will be necessary to lead the observer to discover (if there is such a thing) mathematical concepts. This instantly creates a binary power structure between the teacher and the observer (clever/stupid, leader/follower, sophisticated/unsophisticated). The mere thought of leading someone to discover some new mathematical concepts is scary, especially in a classroom setting. Also, the phrase, "initially passive observer," implies that the observers are mentally lazy. What if the teacher fails to lead to the desired 'destination'? Or, more importantly, what happens if the students are unable to or don't want to realize or discover the euphoria of understanding mathematical concepts?
Yes, I like it when the art of educating our children hinges on 'hope'. Can we, as educators, rely on 'hope' in order to communicate mathematical knowledge to our students?
It's interesting to look at history to gain a better understanding of why we teach mathematics the way we do. The sequence of instructional techniques are definitely used quite extensively, but not necessarily always in that order I believe. I think that we often incorporate visual aids while we instruct, and go through a sequence of problems either solely by the teacher, together as a class, with a peer, or individually.
ReplyDeleteI am curious to know if Swetz offered any opinion on whether or not these techniques are still the most recommended, or if there are any other strategies that have proven to be more successful. For many subjects, it is recommended to start with a problem, and then take a step back and develop a way of solving it collaboratively. Rather than having one's hand held by the teacher step by step, the students create the steps, albeit led by the teacher. By engaging the students in the development of the problem-solving process, perhaps we can avoid the "passive observer".
It is interesting that the author comments on the current pedagogy reflecting the historical development of math. I have noticed it across the high school math curricula. This is the order in which the number system is taught: whole numbers in elementary school, integers in Math 8, rational numbers in Math 9, radicals in Pre-Calculus 10, and maybe imaginary numbers in Pre-Calculus 11 (when some teachers discuss imaginary roots in their lessons about the quadratic formula). Like the number system, students are taught how to solve linear equations before learning how to solve quadratic and higher-order polynomial equations. I can certainly see the chronological order here from a historical perspective. Some of my students seem to be comfortable with this order of learning. However, some of others can manipulate radicals or quadratic equations better than they do integers or linear equations. I am wondering if the order was reversed in the math curriculum, could the students who were good at doing radicals work with integers better?
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