Mathematical Proof in Japan

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、

              ∠AOP = ∠BOQ・・・②

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!)