Canadians are depressingly ignorant of all Canadian history, and military history in particular. This, I believe, is a legacy of: a) poor history teaching; and b) general 1960s-70s Trudeau/liberal anti-military bias.
An excellent, if controversial book on the subject was written by Jack Granatstein, currently the head of the Canadian War Museum in Ottawa. Who Killed Canadian History? made some waves, but at least it got people debating the subject: no easy task in Canada! http://www.canoe.ca/JamBooksReviewsW/whokilled_granastein.html
Fortunately, in the last 5 years, there has been something of a (minor) renaissance in the observance of Canadian military history, sparked mainly by the “Canada Remembers” program of 1995, which commemorated the 50th anniversary of the end of WWII, http://www.umanitoba.ca/cm/vol2/no25/canada.html, the efforts of the War Amps “Never Again” series, http://www.waramps.ca/video/nvra.html, and the belated recognition of just how important the Battle of Vimy Ridge (in France) was to maturing us as a nation
http://www.vac-acc.gc.ca/general/sub.cfm?source=history/firstwar/vimy
Just over one week ago, Canada finally got an Unknown Soldier, laid to rest in Ottawa with full honours. http://www.vac-acc.gc.ca/general/sub.cfm?source=Memorials/tomb Interestingly, an undertaking was signed as part of the arrangements for the exhumation that no effort will ever be made to identify the soldier.
Now, to answer the OP question: very little is taught in Canadian schools about the War of 1812. Many, but probably less than say, a third, of Canadians might be able to tell you that “we” burned the White House, but most likely have no idea as to whether it happened in 1814 or 1914.
To say that “Canada defeated America,” is of course, stretching things beyond the breaking point. There certainly were Canadian Militia involved in a number of battles and skirmishes, but the bulk of the fighting was undertaken by British Army regiments in Canada, and the Royal Navy. The only major role played in the war by “Canadians” was that of the First Nations (“indians” to you), who might not have considered themselves “Canadians.” The word at the time really referred to les Canadiens, the Francophones in Lower Canada.
Without the alliance (made between sovereign nations) of the British forces and the First Nations under Tecumseh, probably the most able military leader of the war of 1812, the British would quite possibly have lost to the American forces. General Sir Isaac Brock (who with Tecumseh should be considered the saviours of Canada), said of him, “A more sagacious or a gallant warrior does not, I believe, exist”.
Sadly, few Canadians remember the vital role played by these men in the history of our nation.