My question about whether or not to teach long division to an 11 year old evoked some interesting responses. However, none was more balanced and illuminating than that of Antony Caruso, a Grade 8 teacher with over twenty years experience in the York Region Catholic Board. His insights and passion for teaching are self evident in a recent e-mail he sent me. Here is a portion of what Antony wrote:
It's a question that still plagues me as an elementary teacher who teaches math. I think that there needs to be a balance in teaching the algorithms (the 'short-cuts' as I like to call them) and the understanding behind the concept (or modeling).
This 'new math' was ushered in with our board-wide purchase of the Nelson Math textbooks. With an emphasis on having students explain their thinking, these new texts were poorly received by teachers and parents alike. Although initially apprehensive about their new focus myself, the new texts did force me to rethink the way I taught math. My old way was me usually at the blackboard, giving a teacher-directed lesson, the students copying the board notes, then assigning homework questions. The next day would be spent correcting the homework then teaching the next lesson in the same way. I grew to dislike this method. We spent so much time correcting homework that it left me little time to go into great detail about certain topics. My favorite lessons were the ones I created that explained some of the reasoning behind the concepts. You mentioned the lesson about pi - I created an activity that had students do what you suggested, measure circles themselves and find out the relationship between the circumference and the diameter. I would then explain the centuries-old quest in finding the exact value of pi, starting with its mention in the bible. We recognize Pi Day (March 14 @ 1:59) and even have a contest to see who can memorize the most digits in pi. Appropriately, the winner gets a pie!
I have grown to really like the new Nelson textbook. I find there is a balance between having the students use models to explain -3 + 2 = -1 and learning the simple shortcut.
But I digress ...
The question that you posted was whether students should learn long division. I believe that they should learn it, through simple questions. Students have such a reliance on calculators that they take its answers as gospel, not realizing that many times they input errors that cause mistakes (many are using their calculators as they use their phones - they input with two fingers instead of one and it causes the most errors). I do feel that the students should know their basic facts, especially multiplication and division facts. When I look back to see who were my most successful math students, they were usually ones who knew their facts well. How can you teach factoring skills when students are stuck on trying to figure out 24 / 4?
I don't blame the students though. Elementary math teachers used to spend a great deal of time practicing multiplication and division questions with students until the number facts were ingrained in their DNA. It doesn't happen anymore and often division is one of the last topics taught, often lost in the mad rush that is June.
I have noticed a troubling trend though. In the past few year years, I had a few students arriving in Grade 8 not knowing the difference between an even and an odd number, and not being able to do even the most simple of division questions. We decided to do something about it as a division this year. We set aside two weeks in the fall to review and practice basic multiplication and division skills. We told them that this would not be graded as part of their report card mark. Rather, it was simply to help them improve skills that we felt would help them with this year's math. Many gave a sigh of relief - many students revealed to us that they felt embarrassed that they didn't know their simple facts and were glad that something was being done about it. They also were happy that it didn't count as part of their report card, so students could progress at their own rate without pressure. Students were not allowed to use their calculators for multiplication and division questions, but they were allowed to use the multiplication tables we provided them. The tables were a hit, and were especially useful later, when used in reducing fractions!
Knowing basic multiplication and division facts helps greatly with estimation. We play a game where I put up prices of basic grocery items up on an overhead in front of the class. It is in their shopping cart and they are in line at the cash. I tell the students that they have $20 on their person and no debit nor credit card. They have to tell me, without benefit of a calculator, if they have enough money to buy all items before they reach the cashier. I will throw a couple of items that are multiples (4 cans of tuna). I play another similar game called the Costco game. I tell the students that they are at Costco, where everything is sold in bulk. They are thinking of purchasing a package of 8 cans of tuna at a certain price. They know that the very same tuna can be purchased at No Frills for a certain price per can. Is the Costco package the better deal? Again, they are told that they have no calculators. These are real-life scenarios.
In conclusion, I feel that there should be a balance between modeling concepts and learning the shortcuts. I think the two go hand-in-hand. Long division, using simple questions, should be part of that. I recall a short story that I read while I was a student at Brebeuf College back in the early eighties. Written by Isaac Asimov, it takes place in the near future. One citizen becomes an instant celebrity as he displays a remarkable, almost magical skill that no one seems to possess - the ability to do arithmetic using pencil and paper in a world where everyone relies on a computer. Could that story be a warning or a prophecy?
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