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I talked with my friend on December 29 2020. Then I talked with him again on January 03, 2021.

Q: What was the year when you last talked with your friend?

A: 2021.

Q: And what was the year the previous time you talked?

Α: 2020.

Comment: So dutifully doing the arithmetic, $2021\, (-) \,2020 = 1$. This means that 1 year has passed right?

This is an almost universal joke around the change of the year ("it's been a year since we last talked", etc). But how to explain this to a little child?

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    $\begingroup$ How old is the child? What does the child understand about being a year older than the day before on his/her birthday. $\endgroup$
    – Amy B
    Jan 7 at 15:35
  • $\begingroup$ @AmyB There is no specific child here. And essentially what you ask is exactly my question: how to help understanding that the one-unit change in the "year-counter" does not mean that "one year has passed". Both are valid, but they don't mean the same thing. $\endgroup$ Jan 7 at 15:47
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    $\begingroup$ My approach (as a parent) was to let the child explain it and try to understand their explanation (through discussion). $\endgroup$
    – Raciquel
    Jan 7 at 16:04
  • $\begingroup$ Being a year older on birthdays is a more familiar context to a younger child than the change in year. When trying to explain things to anyone it is helpful to have a more familiar context). $\endgroup$
    – Amy B
    Jan 7 at 17:14
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    $\begingroup$ Even if there is no specific child - you should still specify an age range. A 3 year old, 6 year old, and 9 year old would all be dealt with differently. $\endgroup$
    – Amy B
    Jan 7 at 17:15
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I would use a number line. This is the most straight forward way to explain being "in between" integers while giving some intuition with a visual. It is possible that a school-age child would be familiar with it, too.

The number line is particularly useful because you can demonstrate that the distance between a point near the end of 2020 and a point near the beginning of 2021 has length less than one.

You might want to start with examples of smaller numbers first, or introduce the line before moving all the way to 2020. You could even do this in context, saying "think about a change from the year 5 to the year 6". Be sure to talk about what it means to be "in" a year as being between the starting point of each year.

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My approach might be to try to use smaller units of time to draw comparisons. I would look at a clock make sure the child knew about seconds or minutes. We could talk about how long one second is, and then how long one minute is, and how on a standard digital clock, the time is usually displayed up to the minute. We could even watch the time change as it goes from one minute to the next.

Then I would probably steer the conversation towards how long ago something was. Whether it was one second ago, 10 seconds ago, or 60 seconds ago, and what the displayed clock might look like. Then advancing the conversations from seconds and minutes, to minutes and hours, and increase the time units on up from there.

This is all hypothetical, but since your question seems to be only hypothetical, this may satisfy your question. If you do try it, let me know how it turns out!

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