A Mathematical Beginning

Today, my grade 5 son came to me with a subtraction problem


According to him, he has yet to learn how to subtract “big” numbers. He was quite confident in his ability to subtract 13 from 20 but faltered as soon as three digit numbers were presented. So I set the problem up for him (in the way that I learned to subtract “big”numbers) and I realized that I could not explain the “borrowing” thing in a meaningful way. And why would he understand that anyway? All too often we present algorithms to students without an adequate explanation of why it works and to the student who looks for meaning, we do a complete injustice.

Think about it,

“I can’t take 6 away from 2, so I borrow 10 from the three to make it twelve and the three becomes a 2 and so on….”

What sort of logic is at work here?? Sure it works but few kids at this age understand why? A “good” math student follows that procedure without thinking about it! The fact that they don’t think about it, concerns me! I will go out on a limb here and suggest that our “good” math student would have difficulty explaining why it works and will will struggle when asked to either communicate their understanding or apply such knowledge to an unfamiliar problem.

So I proceeded to explain an algorithm for subtraction based on his previous experiences with place value (and one which at least makes sense to me)! It went something like this…

Start with 932 and subtract 300, then subtract 50 and then 6. This is intuitive and we didn’t even have to “borrow” anything! He did this all in his head. I thought he would get hung up when subtracting 50 from 632 but to my delight (and surprise) he gave me the answer in a matter of seconds. When asked how he did it so quickly he responded, in writing,

“I took away the 2 off 32 and subtracted 30 equals 500, then subtract 20 equals 80, then add 2 again equals 82!”

Marvelous! Intuitive, and without a pencil and paper…

He then proceeded to complete the remaining subtraction questions 2005-189 and 40031-177 in exactly the same manner never needing to put pencil to paper.

I guess this serves as a reminder for math teachers as we launch into a new school year….young children, and IB students will look for meaning in everything you teach. Provide them with a context, an intuitive approach and link new topics to their previous learning experiences. Have students communicate their understanding – start a class blog, collaborate in Google Docs or simply have students write in an old-fashioned journal.

Successful students are those whose teachers have given them the opportunity to learn the necessary skills to face confidently, the challenges of the unfamiliar situation and the 21st century. Good luck!

  • Hilary Sacharewicz
    October 21, 2010

    I always explain in terms of money.If you want £6 from me and I have £52 but only 2 are as coins I need to break one of the £10 notes into 10 £1 coins in order to pay you. This seems to make sense to most pupils. I hate to see it called borrowing anyway, because there is no way you’re going to put it back, decomposition is a much more realistic term.

  • Mark Bethune
    October 26, 2010

    This is also a very nice way to look at it. I am glad there are others who do not like the term “borrowing”. Decomposing is a useful term not only for a situation like this but also for decomposing fractions which is an important algebraic skill in calculus and when investigating certain types of rational functions. Thanks for the comment.

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