The
topics for this week’s reading are somehow out of my expectation, although they
are interesting as well. I have thought some relevant topics about mathematical
proof may be to explore why the reasoning process is a relatively boring
subject for students, but Hanna and Barbeau’s article discusses the potential
achievements students can make beyond the wordy mathematical proofs, while Zack
showed his understanding and improved teaching skills through a series of
classes dealing students’ counterarguments of a given plausible but wrong
solution.
Indeed,
proof is a powerful and helpful tool for students to understand mathematical
concepts. Zack’s improvement and understanding of children’s math thoughts
strongly supports Hanna and Barbeau’s argument that teachers and students can
improve their understandings through proofs. But proof itself may be scary for
first-time learners if his/her instructor focuses more on the format of writing
a mathematical proof. The situation may be more challenge if a student has a
below average math skills because there is (almost) no formula for finding a
proof. My own experience tells me that math mainly contains two parts:
reasoning (proofs) and application. Things (do math problems) can be easy if my
job is to simply use some rules; but it is always a challenge task to
understand a proof because of its mathematical writing styles and the
relationships between each step. Things can be out of control if I am asked to
write a mathematical proof. That means I have to construct a persuasive reason
and convey it into mathematical language.
I believe
this is the challenge part for some students, as conveying the reason into
proper mathematical language always involve a certain level understanding of
necessary mathematical knowledge. This requirement is hard to meet because (a) an
average students may not master all the previous knowledge well, and (b) there
are (is) limited (not as open as proofs should be in a class/test setting) ways
for students to choose when conducting a mathematical proof because of their
understandings and knowledge background. And thus a proof is somehow boring if
students cannot figure out solutions by themselves.
Anyway,
what interest me in proof are the different-than-usual proofs of some familiar
proofs. I am looking forward to explore more on this area.
Ps: I wonder where are the additional readings for
this week as I am absent from the off-campus meeting.
Hi Shan.
回复删除You make two very interesting points in your comments: (i) many students know the reason why something is true, but cannot formulate it into proper mathematical language. (ii) There are no steps that a teacher can outline for proving a result. There are techniques for proving results, such as breaking the proof up into smaller proofs, starting from the end instead of the beginning, but at the end of the day I can't tell my students are sure way of going about proofs. I think this frustrates students, and is one of the reasons proofs are avoided before university.
I think the key to going about proving a theorem is to take "intuition" and translate it into formal mathematical language. This relates to the issue of moving between "representations" (e.g. visual vs formal) and is something that has been researched extensively in mathematics education.
For me, the key as educators is to first help students develop some intuition for the subject, and move to the formal ways of presenting proofs later.
I am glad the way you pointed out the necessary mathematical knowledge required in order to conduct proofs efficiently. I also concur that proofs test one's ability to convey/translate one's reasoning efficiently into mathematical language. Often students struggle with mastering the fundamental mathematical concepts that there is not enough foundation for the students to build their proofs on. The most common mistakes I observe in "proving" trigonometry identities is related to algebraic deficiencies. At math 12 level, students still struggle to perform correct algebraic procedures. Often these rudimentary errors become the stumbling block for students to enjoy proofs. This is why I argue that proofs are typically reserved for the mathematically-able and not incorporated in the everyday teaching pedagogy for general student population. What do you think? Are proofs suitable for students of all mathematical abilities?
回复删除Hi Shan!
回复删除Although I read the same article as you, I feel as if you brought in an additional point that I over looked. That point is writing in mathematics. You mentioned that there is no "rule" for writing proofs, which is the opposite of how I think most student view mathematics. I think most (definitely not all) children just memorize steps or rules to follow without any conceptual understanding. Therefore, when a student is asked to write a proof, it may be very difficult for them because they cannot just proof something by saying "because that's just the way the rule works" or "that's just the steps my teacher taught me".
I am taking an independent study this summer and my focus is writing in mathematics. After seeing your thoughts on this article, I might end up researching a good bit on just writing in math in general, but also writing proofs in math.
Hope your wisdom teeth stuff is going well. See you Wednesday!
Thank you Keri. So glad you like it.
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