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When I was a child, I learned basic mathematics (addition, subtraction) through practice and memorization of basics (6 + 7 = 13). However, while I was visiting a child (5 yrs. old) over New Year's, I found that she was unable to perform even simple addition (11+7) without the aid of a number line. Is there any way to prevent these aids from turning into crutches?

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If you were speaking of an eight year old, there would be need for concern, but five is young to be past needing these aids. Generally, a five year old has only just begun to learn the concept of addition and will likely grow out of the need for it in time and with "joyful" and encouraged practice. –  balanced mama Jan 5 '13 at 16:48
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It is my belief that facts are only memorized when there is a motivation to do so, and that by far the best motivation is relevance to something the individual cares about. Thus memorizing facts on their own is hard, but doing interesting problems which require those facts will lead to natural memorization through repetition.

Therefore I would recommend activities that require math but are not rote practice. For instance, dice games such as backgammon, yahtzee, and board games require the numbers up to 6 to be repeatedly added. Under this approach the child should progress from using the number line, to mentally visualizing a number line, to simply memorizing the sums and differences through natural repetition.

As a final comment, I would rather that a child laboriously compute solutions from first principles rather than progress too rapidly to memorization of results without understanding how they were obtained. In the first case they are much better equipped to handle problems that go beyond what they have previously encountered.

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Wish I could vote for this one more! So many good kernels, but especially, "I would rather a child laboriously compute solutions from first principles than progress too rapidly. . ." I'll add that we've started playing cribbage with our six year old to help her with speeding her in figuring facts with numbers that add to 15. –  balanced mama Jan 5 '13 at 16:52
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I have never met a child of older age (say, 7 or older) who had learned to use a number line, and who required it to do any sort of addition. Besides chance meetings with children over the years, for a while I tutored grade-schoolers in math. I've never heard of over-reliance on number lines being an issue, either. Thus I don't think there is much risk at all of that particular aid turning into a requirement long-term. Number lines are useful aids in developing number sense, and are not to be feared. (Calculators may be another matter when used improperly; for instance, promotion of calculator use is an oft-criticized feature of the Everyday Math curriculum.)

Many math curriculums for young children promote heavy use of manipulatives, with a good example being Cuisenaire and similar rods. Miquon Math is a well-regarded curriculum based heavily on use of such tools, and there is no indication that it leads to dependence on them later on. Singapore Math is an excellent primary math curriculum that features plenty of work with number lines across multiple years. Common Core math standards even show a guideline requirement that children must be able to represent addition on a number line-- by second grade.

Taking a step back, I wonder why you are worried over the math learning of a child who's so young, and whom you were merely visiting and presumably see only rarely. I think that you may not understand the wide variance in learning milestones among young children, based on physical cognitive development, family coaching/stimulation of learning in various areas and attitude towards learning in general, and prior learning environments outside the home.

For example, Waldorf teaching often results in delaying mathematics instruction significantly, and in some developed countries it is not abnormal to start kindergarten until age six or even seven. Here in the U.S., "redshirting", or holding children back in the hopes of fostering more maturity and perceived academic excellence compared to grade peers, results in many children entering kindergarten at age six.

PBS developmental milestones show that the child you mentioned is not doing badly when generally compared against age peers, and would not be out of place in a room with six-year-old children. At such ages, it is not unusual for simple math to be counting-based for many children, where a number line makes a lot more sense, and is certainly better suited as an aid than counting on fingers. (Elimination of counting on fingers is one of the potential benefits of using other aids such as rods, but it's not the end of the world when a kindergartner does teens arithmetic in this way, especially as many kindergartners cannot.)

In summary, there is no significant chance that any child will over-rely on number lines to do basic math as they get older, and there are significant benefits to using them in developing number sense. The particular five-year-old you visited is doing just fine.

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There are loads of tools which act as aids to memorisation, but that is exactly what they are - aids to memorisation

  • Wall charts of multiplication tables are an excellent way to get kids to memorise the times tables up to 12, but to help instil in them memorisation, we got ours to learn the 13 times table using the same concepts, but without any chart in front of them
  • Number lines are good, but you need to encourage the child to work through in their head what a number line is doing
  • Making a game, as Dima says, is a great idea - maths needs to be exciting. My eldest two love Numberphile videos, which make various number concepts very interesting.
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When I was 3 year old, my dad gave me a calculator and taught me to use it for addition and multiplication. I started playing with it and memorising the results, so at 4 I was already able not to use the calculator. So the calculation aids are not necessarily evil. Sometimes they help. Why don't you just play with the kid, so that the kid asks you question like what is 6 + 7, you answer, say, 15, and the kid corrects you? This way, they memorise the answers joyfully. My 4-year-old daughter is now using a wall calendar as a calculation aid, also it helps her to memorise the order of numbers up to 31. And I am trying not to hide the aid from her but to exploit her curiosity on the calendar. Whatever tool kids use, the goal, as I see it, is to encourage them to do math! Good luck!

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