Strands of Japanese curriculum in
mathematics are differently structured from the U.S. (I think Canada should be
close to the U.S. compared to Japan). In
the textbooks of my country, usually
students learn the way to proof in mathematics for their first time when they
are grade 8. After they learn three ways of triangle congruence
theorems/postulates, we can see many questions regarding mathematical proof
which contain the above theorems.
I posted one of example from the Japanese
textbook as below, with English translation. (I intentionally did a literal word-for-word
translation for the most sentences)
Here is the question.
右の図で、l // m として、l上の点Aとm上の点Bを結ぶ線分ABの中点をOとします。
点Oを通る直線nがl、mと交わる点を、それぞれP,Qとするとき、AP=BQとなることを示しなさい。
In the right figure, as l || m, let O be a midpoint
of the segment AB which is joined by point A on the line l and point B on the line m.
When taking the
point P for the point of intersection of the line l and the line n which goes
through the point O, and taking the point Q for the point of intersection of the
line m and the line n, show AP=BQ.
Here is the answer in the textbook (actually
the workbook).
<証明>
△OAPと△OBQで、
仮定より、OはABの中点だから、
AO
= BO ・・・①
対頂角は等しいから、
l//mから、平行線の錯角は等しいので、
∠OAP = ∠OBQ・・・③
①、②、③から、1組の辺とその両端の角が、それぞれ等しいので、
△OAP≡△OBQ
合同な図形では、対応する辺の長さは等しいので、
AP
= BQ
<Proof> In △OAP and △OBQ,
from the assumption, because O is a
midpoint of AB,
AO
= BO ・・・①
because opposite
angles are equal、
∠AOP = ∠BOQ・・・②
from l || m、because alternate-interior angles
of parallel lines are equal,
∠OAP = ∠OBQ・・・③
from ①, ②, and ③, because one pair of sides and two angles of both edges of the side
are respectively equal,
△OAP ≅ △OBQ
Because lengths of corresponding side in congruent
figures are equal,
AP
= BQ
When I was a junior high, I was taught to accurately
write the same way of mathematical proof as the model answer, in addition to memorizing
each phrase of triangle congruence theorems. Actually, I could not deeply
understand the meaning/concept of “assumption” in mathematics context but I did
not have doubt for writing it in the proof because my answer was correct. The
unit of proof in Japanese textbook and workbook usually starts with the
activity of filling blanks of model answers. And I suppose it will be tough for
Japanese language learners to write the proof as same as the model answer. Although
they might be accepted by the teacher and other peers even if they do not know
or misuse terminology in mathematics in discussions, writing proof requires
them to write down the answer with high accurate grammar and appropriate
terminology what they have learned before.
I am not sure about the way of proof in
Canadian textbooks so far, but I guess there are some differences between Japan
and Canada. (By the way, I may have a chance to get Canadian mathematics textbooks free of
charge, yay!)
