“5 is the same as 50, just with a 0.”

The kid also answered the “How do you know?” question:

“Because 5 is half of 10, and 50 is the same number as 5, just with a 0, and 10 is the same number as 100 just with another 0.”

What does this mistake say about the way kids see numbers and multiplicative relationships?

hillby

I really hope the kid didn’t see numbers this way, I can’t figure out anything other than verbal vomit because they didn’t know what else to write.
- mpershan
  
  How would you help this student? After all, the kid’s given a fairly clear argument.
Chris Robinson

Here’s my working hypothesis (because I’ll never really figure any of this stuff out): Students, through their mathematical experiences with algorithms and processes, treat numbers as objects to operate on rather than symbols or representations of quantity. Long story short, an underdeveloped number sense. Disclosure: I’ve been pushing this idea pretty hard in my classes this week, so I’m currently a little biased
- Michael Pershan (@mpershan)
  
  Interesting hypothesis, and you’re right that we’ll never really know. (That is, unless we have some more careful follow-up tools.)
  
  Here’s my superficial objection to your comment: of course it’s number sense. It has to be the number sense, since this is a mistake that shows a misconception about numbers. Any mistake involving numbers can be explained by saying that the student has an underdeveloped number sense, no? If that’s the case, then what do we gain by labeling this sort of mistake a “number sense” issue?
  
  This leads me to think that you’re talking about something more narrow than just “doesn’t properly understand numbers” when you talk about number sense. From your comment, I’m lead to believe that by “number sense” you mean something like “a constant awareness of the quantities that are being tossed around”? In other words, you want a student to be aware of the quantities that they’re operating on while they’re operating on them?
  - Chris Robinson
    
    Yeah, I think my definition of “number sense” is more narrow than just “involving numbers.” Rather, I see weak number sense as not being able to “sense” numbers as symbols for quantities, and being flexible in operating on those quantities. I’ve been seeing lately that a lot of my students have a very shallow conceptual understanding of what operations are actually doing to quantities. Instead, they apply the usual algorithms that should get them the “right answer.” I’ve been doing a lot of Brad Fulton and Bill Lombard’s Take Your Places number sense activities, and this hypothesis became very apparent when we were discussing strategies to form the largest and smallest products when given 4 integers (two digit multiplication). They were struggling to understand the effect of multiplication on the place values, instead falling back to the standard, two digit multiplication algorithm.
Michael Paul Goldenberg

Well, from one point of view, the statement is absolutely true. Unfortunately, it’s not one that in and of itself would likely lead to a career in STEM. But if the student really believes this, the teacher is obligated to make that belief problematic for her. How old is this student? Probably old enough to be disturbed if you ask, ‘since 5 is the same as 50, just with a zero,” then if I owe you $50, would you take $5 and consider us even”? Any number of variations on that sort of question will reveal just how married to this notion the student is, or if real world contexts brings things back to a more functional, practical perspective. If so, that doesn’t mean the student is just messing about. Plenty of students by folks like Jean Lave indicate the enormous gap many kids have between “school math” and everyday use of mathematics in their lives. The latter is almost always more highly developed in kids who actually use math outside of school. It’s when they are forced to abstract the math from their lives that things go kerflooey, because instruction has not succeeded in helping them see any connection between the domains.
Michael Pershan (@mpershan)

Guys, I don’t think that you should take this kid to literally mean that he sees no difference between 5 and 50. That’s an awfully ungenerous reading. What seems far more likely is that the kid is just saying that the relationship between 10 and 5 is the same as the relationship between 100 and 50, because of the resemblance between the numbers. This kid knows how to count.
Beth S

I think the misconception might be related to the child’s experience with multiplication of multiples of ten. Consider, for example, if they were taught “to multiply 50 x 5 all that they need to do is recall the fact 5 x 5 and add a 0” I suspect the child was taught this trick or shortcut before they had a firm grasp of place value. That is why taking the time for student to develop conceptual knowledge is crucial.
- Donna
  
  I agree that many educators and parents approach it this way. The best way to get them to stop saying “just add a zero” is to show how this does NOT work with decimals. 10 x 2.3 does not equal 2.30. We have to think about how our verbiage today will lead to misconceptions in the future.
  - Jeff Grigg
    
    But I can see how knowledge of “just add zero(s)” can translate directly into “just move the decimal point” (…adding visible zeros as needed, as all decimal numbers can be assumed to have an infinite number of leading and trailing zeros).
    
    I can’t criticize the student, as I’ve been reading that this is how schools are teaching them to do math these days. It’s a controversial change. But I can see the justification, as it’s kind of silly to do long multiplication or long division when computerized calculators are as accessible as pencils these days.
    - Beth S
      
      But what good is the calculator with no conceptual understanding? I fear if this child punched something into the calculator incorrectly they would not have the number sense to realize the resulting answer in the display was not reasonable.
Jeff Grigg

Certainly, the traditional approach is to use division.

Or to just count it out, as there aren’t that many. It took me about 15 seconds to do it that way.

But the student’s logic is almost right. They just lost the “half” and used “5” again instead. Here’s the correct logic:

“Because 5 is half of 10, that means that there are *TWO* fives in ten.
“10 is the same number as 100 just with another 0.
“So, likewise [proportionally] we should take *TWO* and add a zero.
“Answer: 20.”
- Beth S
  
  The student’s logic is “almost right”? Where is the number sense?
Michael Paul Goldenberg

The more I look at this, the more annoyed I get at all the trickery teachers give to kids in lieu of conceptual understanding, as if the kids couldn’t possibly understand the concepts that justify the shortcuts. Having worked with kids who really struggle to understand, I do get why teachers are tempted to give shortcuts that, if memorized, will work, at least up to a point (and as we see with Donna’s example of 2.3 x 10 does NOT yield 2.30, there is a definite, inevitable problem with saying that to multiply by 10, just append (or ‘add’) a zero to the right-most digit).

With that in mind, look at Beth’s example: “if they were taught ‘to multiply 50 x 5 all that they need to do is recall the fact 5 x 5 and add a 0′”. As I look at that, what I want to say is, “remember that 50 is 5 groups of 10” [or “five tens” or 5 x 10]. Now you’re multiplying that by 5, so you have 5×5 tens or 25 tens. And THAT is 250.”

Too wordy? Maybe. But I think it would hold up to the issue Donna raises, though not as slickly as “move the decimal point one place to the left for each power of 10 [or trailing zero, IF you’re dealing with whole numbers],” at least not at first. If I look at 2.3 x 10 the way I looked at 50 x 5, I might say, “2 x 10 = 20, and 0.3 x 10 = 3.0 (how the student arrives at that, of course, goes back to fundamental understanding of place value and decimal numbers. . . so maybe I’m caught in a loop here?) and that gives 20 + 3 = 23.

I suppose in the end, what we want kids to realize is what it means to multiply or divide by 10 or a power of 10. And to be able to ask themselves in advance what the outcome should be (at least approximately) when they multiply or divide by 10, 100, 1000, etc., and later, by 1/10, 1/100, 1/1000, etc.

And that means having a solid foundation in place value combined with a real understanding of arithmetic operations with rational numbers. So kids won’t claim that multiplying always “makes things bigger,” and that division “always makes things smaller.” The ideas here are subtle and intimately related to one another. I’m not averse to having kids reduce concepts they’ve GRASPED to some shortcuts, but my concerns are: 1) do they actually HAVE those concepts well-understood to begin with, and 2) will teachers (and students) recognize the need to ‘return to first principles’ (i.e., those underlying concepts) periodically to make sure that using the tricks/shortcuts hasn’t allowed the more important conceptual understanding to completely erode.

And obviously, this isn’t an issue restricted to this particular mathematical area. It just happens to be a rich one for seeing the strengths and weaknesses of shortcuts: without the underlying concepts, I don’t find them valuable – they’re a bit like handing someone a leaky life-preserver when they need to get a lot further than to the edge of the swimming pool. Hope that’s clear and useful.
Joshua Zucker

I think the student is confusing the proportionalities here. They’re answering the value of n that makes 5:10 as n:100, and quite sensibly getting 50, and not realizing that the question is 1:5 as n:100. They see that this is a question about proportionality and they deploy some proportional reasoning, which is great! But they don’t recognize which things are actually in proportion. (Maybe this is the same as what Jeff was saying above?)

I think the habit of doing a small case first is what they need: revisiting the fact that counting to 10 takes 2 people, so they can see that counting to 100 will proportionally take 20 people. The problem is that they leap toward using numbers they find and some vague sense of proportionality to end up with an answer, rather than taking that first step of thinking about what makes sense. Maybe they need to develop the habit of thinking about not only the small case but also an estimation/prediction of how big the answer should be …
Alexis Leader

I think that the student is having a hard time understanding place value. The student didn’t understand the difference between 50 and 5 or 10 and 100. The was confused about how the extra place value had changed the number. Since the student was confused about how 50 and 5 were different the student didn’t see how numbers could be made up of other numbers, the student didn’t see 10 as made of 2 fives and thus didn’t see 100 as made up of 25 fives.
- Jeff Grigg
  
  Ummm… Last time I checked, 100 was not made up of 25 fives. Not very much. ;->
  
  (25 x 4 = 100, and 20 * 5 = 100, using conventional approaches.)
  
  (And *DARN*, the original post image is now gone! :-/ )
Marissa

I can understand how the student got to this conclusion. When I tried to solve this question myself, I thought, how many times would the classmates speak before the number counted up to 100. Then I thought, how many times does 5 go into 100, and I realized the answer is 20. I am somewhat confused how the student though that the answer was half of 100, but I am more concerned about the students came about these multiplicative relationships. I know that the student is confused about place value in his response. The believed that the 0s no matter as long as the whole numbers are the same. So, 50=5 and 100=10. Yet, what the student needs to realize that if 10 is made up of 2 5s, then 100 is made up of 20 5s. This is the correct moment when the 0s could be exchanges among the place values.

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