Naïve Conceptions

Marian Small

There is a lot of data on common naïve conceptions, and while we will address some common naïve conceptions in this piece, the real question is this: What can teachers do to minimize, as much as possible, the likelihood of misunderstandings?

Mistakes vs Naïve Conceptions

Naturally, students make errors when performing mathematical actions. We are human; we all make mistakes.

For example, a student might add 29 + 34 and record 64 as the result, incorrectly adding 9 + 4.

But sometimes it is not what we would call a mistake, but a conceptual misunderstanding, often what we might call a naïve conception, or what some people call a misconception, that produces the incorrect result.

For example, a student might think 1.234 is more than 4.2 since it has more digits. What the student has done is taken an idea that made sense in one situation (with whole numbers) and applied that idea when it no longer makes sense (with decimals). It is a natural thing for students to do, that is, generalize from what they already know, not realizing that the idea no longer applies.

One of the things we can certainly do is to teach for understanding, rather than simply supplying rules with minimal context. Another thing we can do is to be more “careful” in the language we use when we help students draw conclusions; for example, we can avoid saying addition makes things bigger because sometimes it doesn’t or saying that division is always about sharing since sometimes it’s not.

Let’s see how that might work in some specific situations.

Kindergarten to Grade 2

A student sees these two situations and says there are more stars than hearts.

This is not an unreasonable conclusion for a student to draw. There is something more about the stars; what is more is the area. A student may realistically not distinguish between more in area versus more in number since that distinction was never explicitly discussed.

Perhaps we help avoid this naïve conception by explicitly talking about the distinction with young learners.

A student sees the equation 3 + = 8 and adds 3 and 8 to say that 11 is the missing number.

This, too, is a reasonable conclusion for students to draw. They may know what an addition symbol means and assume that if you see an addition symbol, you add the numbers you see near it. They don’t worry about the ; the numbers they see are 3 and 8.

Here, too, it might be well worth taking the time to explicitly discuss the difference between

3 + = 8 and
3 + 8 = .

A student thinks you should write seventeen as 71.

What a student hears is the word “seven” followed by the word “teen.” The student knows that we write 7 for seven and may have learned that teen numbers mean there is a one, not focusing on where the 1 is in the number.

It may be worth taking the time to talk to students about the fact that with most numbers we write what we hear in the order we hear it, but this is not always the case. Twenty-seven is 27 and forty-two is 42, but we could take the time to point out how different the teen numbers are in that regard.

A student adds 14 + 9 by adding 14 + 10 and then adding 1.

This student is using a strategy they either learned or invented where they use a nearby number to simplify the calculation and then they compensate. Although it turns out that compensation here involves subtracting 1, they are focused on the fact that this is an addition question and assume the right thing to do is to add 1.

It may be important to point out to students that seeing an addition symbol does not always mean you add the numbers you see; for example, 3 + = 8 is solved by subtracting. But it also might be worthwhile bringing out the notion that if you add too much, you fix the problem by taking the too much away.

A student adds 45 + 36 and suggests that 711 is the sum.

This student is using place value appropriately, adding the tens and ones separately, but not taking into account that if there is no place-value chart that says that the 7 are tens and the 11 are ones, they might think that the 7 is hundreds.

We don’t want to say that it’s wrong to say that there are 7 tens and 11 ones since that is actually correct. So, what can we do?

Although there is great value in starting addition work in place-value charts, it is helpful to point out to students how we can misunderstand the meaning of numbers if those charts are not there, and we see only the digits.

You might, for example, contrast 70 ones, which we write as 70, with 70 tens, which is much greater.

Grades 3 to 5

A student thinks that each section is \(\frac{1}{3}\) of the whole.

Assuming that the whole is the entire circle area (which is a reasonable assumption, but still an assumption), we need students to know that since we are talking about area, each piece of area must be equal to each other piece to use fractional notation.

If we were talking about numbers of pieces, though, it is true that each section is \(\frac{1}{3}\) of the set of three pieces.

A student thinks that 5 × 1 must be 6, not 5.

Some teachers say that multiplication makes numbers bigger, but the problem is that this is not true all the time. However, if the student hears someone suggest that it is true, they will have trouble calling 5 × 1 “five” since then the answer is not bigger than the 5 you started with. They might think that since it has to be bigger, maybe it’s 6.

It is important that we are careful, as educators, to qualify generalizations we make if they are not always true. It is fair to say that multiplication often increases the size of a number but incorrect to suggest that it always does.

A student suggests that there is no fraction between \(\frac{3}{5}\) and \(\frac{4}{5}\).

It is natural for a student to suggest that there is nothing between \(\frac{3}{5}\) and \(\frac{4}{5}\), since we teach students to count by fifths, and \(\frac{4}{5}\) comes after \(\frac{3}{5}\).

There will be some students who might suggest \(\frac{3\frac{1}{2}}{5}\), and often we turn that back and say that you can’t write that, but, in fact, the student is making sense.

Rather than turning that back, ask them what \(\frac{3\frac{1}{2}}{5}\) looks like.

They might draw something like this:

This is actually excellent thinking.

We might point out that we often don’t see fractions written like \(\frac{3\frac{1}{2}}{5}\) and ask how else that same fraction might be written, leading students to \(\frac{7}{10}\), then pointing out that thinking of \(\frac{3}{5}\) and \(\frac{4}{5}\) as \(\frac{6}{10}\) and \(\frac{8}{10}\) leads to coming up with the same fraction a different way.

What is important, though, is that we ask the question as to whether there is such a fraction so that this discussion happens.

A student suggests that 0.5 is an odd number.

I have even had teachers tell me that 0.5 is odd, so this is not an unreasonable thought for a student to have. Students have been focused on the fact that, with whole numbers, numbers that end in 5 are odd, so they simply overgeneralize that to a decimal situation.

So, what could we have done to try to avoid this?

Perhaps we could have been much more explicit that we only talk about odds and evens with integers (or whole numbers), not with decimals. But we want to give students reasons for this. We could help them understand, for example, that it is tricky since 0.5 is the same as 0.50, so it can’t switch from odd to even when it’s the same number.

Or we could have pointed out that “even” means divisible by 2, so students assume if you can divide a number by 2 and get the kind of number you started with, the number is even. But since every fraction or decimal can be divided by 2 to get a fraction or decimal answer, every fraction or decimal would be even, and then the classification is meaningless. For example, \(\frac{1}{5}\) ÷ 2 = \(\frac{1}{10}\), so students might call it even. Or 0.5 ÷ 2 = 0.25, so students might call it even.

If everything is even, there is nothing left not to be even.

A student believes that this diagram shows \(\frac{5}{8}\) and not 1\(\frac{1}{4}\).

What is not clear to the student, in this case, is what the whole is.

Is the whole the 4 little sections that are clumped together and separated from the other 4 little sections, or is it the whole 8 pieces?

One of the things we can do is ensure that whenever students are working with fractions, they are required to circle or in some other way identify the whole.

A student believes that \(\frac{3}{10}\) > \(\frac{3}{8}\) since 10 > 8.

This, too, is a logical conclusion on the surface. If 10 is more than 8, why wouldn’t tenths be more than eighths? As adults, we know why, but maybe we have a responsibility to very explicitly discuss this with students. Students need to see that tenths mean the whole is split into 10 sections and eighths means the whole is split into 8 sections, and it is because the tenths have more sections that each section is smaller.

Grades 6 to 8

A student thinks that \(\frac{2}{3}\) + \(\frac{3}{4}\) = \(\frac{5}{7}\).

Especially when fractions are stacked, it is natural for a student to see the addition symbol and add the tops and add the bottoms.

So, what can we do to minimize this naïve conception?

One thing we can do is to ask whether \(\frac{2}{3}\) + \(\frac{1}{3}\) = \(\frac{3}{6}\), that is, use fractions with common denominators where students are somewhat more likely to realize that we don’t just add the tops and the bottoms.

But some students think that maybe \(\frac{2}{3}\) + \(\frac{1}{3}\) is \(\frac{3}{6}\), so we need alternative strategies.

Perhaps we explicitly ask students why you can’t add the numerators of the two fractions and the denominators of the two fractions to get the numerator and denominator of the sum.

We might give them lots of examples, so they end up seeing that if they compare their result to the fractions they started with, instead of the answer being greater than the two fractions they are adding, as it should be, it turns out to be between them, so it doesn’t feel right.

For example, consider \(\frac{2}{5}\) and \(\frac{3}{4}\). \(\frac{5}{9}\) is more than \(\frac{2}{5}\) but less than \(\frac{3}{4}\), not more than both.

Consider \(\frac{1}{3}\) and \(\frac{7}{8}\). \(\frac{8}{11}\) is more than \(\frac{1}{3}\) but less than \(\frac{7}{8}\), not more than both.

A student thinks that the value of 3x + 6 when x = 6 is 42.

This, too, is a natural suggestion from a student. They are accustomed to seeing, for example, 3 to mean a number in the 30s, so it is natural to think of 3x when x is 6 as 36.

Again, it is important that we are really clear about what we mean when we write 3x, that is, it is x + x + x, so it could be 150 if x were 50. When x = 6, 3x + 6 = 24.

A student thinks that 373 must be prime since 3 and 7 are prime.

This particular naïve conception is a bit less likely than some of the others you’ve seen here, but it does occur. In fact, 373 is prime, as is 353, which just makes the assumption even more tempting.

But, of course, 333 is not prime since it is 3 × 3 × 37.

Then maybe a student thinks it works unless the digits repeat. So, we might need examples like 375, which is clearly not prime, even though each digit is prime.

Again, as with the other examples, talking about this is a way to dispel the naïve conception.

A student thinks that 0.5% of 400 is 200.

Because 0.5 means \(\frac{1}{2}\) and because 50% means \(\frac{1}{2}\) and students view 0.5 as 0.50, it is almost surprising when a student does not assume that 0.5% of 400 is half of it.

I propose that what we need is more conversation about what 0.5% means. Just as 1% is less than 2% since 1 is less than 2, 0.5% is less than 1% since 0.5 is less than 1.

A student thinks that −10 > −8 since 10 > 8.

Again, a natural assumption is that −10 > −8 and, in fact, the absolute value of −10 is greater than the absolute value of −8 since it is farther from 0 on the number line.

But students need to fully understand that being to the right on the number line always makes the number greater, and since −8 is closer to 0 than −10, it is to its right.

A student thinks that −4 + (−3) = +7 since two negatives make a positive.

Here is another example of where a student hears something (probably in the context of multiplying two negatives) and overgeneralizes to a situation where it does not apply.

One thing we can do is to be careful not to throw around phrases like “two negatives make a positive” since there are a lot of conditions on that statement we are not saying.

I feel that we can do better by having students use fewer terms like “plus” and more phrases like “putting together 4 negatives and 3 negatives means …”

In Summary

The examples presented in this article are just examples; there are many more. I hope that you see that these naïve conceptions are almost natural for students unless we, as the educators, take deliberate steps to try to minimize their occurrence.