It is common for students to come unstuck by there inability to manipulate fractions (eg calculating gradients, algebraic fractions, etc) yet these same students can handle decimal numbers competently. How is this possible?
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12$\begingroup$ To me the skills are quite different, so I'm surprised that you seem to be surprised. Maybe you're viewing all this from a position of conceptual understanding that's too high to see the student's problems. For example, adding decimal expressions is essentially the same as adding integers and comparing sizes of decimal-expressed numbers is essentially an alphabetical ordering task, while adding and comparing sizes of fractional expressions typically requires relatively convoluted manipulations/changes with a pair of numbers (numerator and denominator). $\endgroup$– Dave L RenfroOct 20, 2021 at 11:27
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2$\begingroup$ @DaveLRenfro: Yes, and to add to that, students often try to memorize procedures rather than concepts. When you teach them to, for example, cross-multiply an equation with two fractions, find common denominators, etc., they are not thinking in terms of "why does this work?" They are thinking in terms of "how do I remember this so I can pass the test?" $\endgroup$– KevinOct 20, 2021 at 19:13
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3$\begingroup$ My theory is that the increase in comfort with decimals vs. fractions has to do with the way our students have used calculators in their classes in the past. Perhaps they have been using calculators that give answers in decimals, or were allowed to use calculators to work with decimals more often than to work with fractions, or simply found it easier to enter decimals due to order of operations nonsense. This sort of comfort in either case really only comes with many years of practice, which has been reduced by calculator usage. $\endgroup$– TomKernOct 20, 2021 at 23:10
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$\begingroup$ Your title question should be rewritten to "Is it possible to calculate with decimals without understanding how to calculate with fractions?" Understanding decimals and fractions are different than calculating with them. Many students understand fractions better than decimals, but calculate with decimals more easily. $\endgroup$– Amy BOct 21, 2021 at 17:07
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$\begingroup$ @TomKern Calculators likely are a big part, but decimals are legitimately easier in at least one respect: It is much easier to compare magnitudes of decimals. $\endgroup$– AdamOct 22, 2021 at 2:15
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3 Answers
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Normally I avoid semantics, but in this case, they're quite important.
Is it possible to understand decimals without understanding fractions?
No. Such an understanding is literally impossible.
With a strict and literal sense of the word "understanding", no. If you do not understand fractions - and tenths in particular - then the conventional meaning of $3.7$ as $3\frac{7}{10}$ cannot make sense. Without this basic factual knowledge, a precise understanding of $10.1\div3.7$ is literally impossible.
And make no mistake: By the hundreds of millions, students have no idea WTF is going on with fractions due to the way they were taught. So their understanding of decimals such as $3.7$ will range from non-existent to reasonable, but never strong nor precise.
Numbers Written in Decimal Format Activate Prior Knowledge of Whole Numbers & Money
When calculating $10.1\div3.7=$ ____ , it is possible for fractionally weak students to have reasonable
- intuitions
- estimates ("It's kinda like $\$10\div\$4$")
- written procedures and
- quotients in simplest decimal form
One reason that fractionally deficient students find $10.1\div3.7=$ ____ somewhat intuitive is that it looks and acts much more like whole number knowledge and money. Reasoning that $3.7$ is like $\$3.70$ which is nearly $\$4$ is basically correct, even if students have no understanding of tenths or hundredths or how those correspond to dimes and pennies. The algorithm for dividing $10.1$ by $3.7$ is just a couple of dots different from the algorithm for dividing $101$ by $37$.
Numbers Written in Fraction Format Activate Dread & Confusion
In contrast, a student could look at an equivalent expression with fractions, $\frac{101}{10}\div\frac{37}{10}$, and think: "Can I cancel the zeroes? Can I cancel the tens? What do I divide first? There's so much division! Do I multiply? Do I flip something? Then do I multiply after? What are the rules for denominators? Oh, I know, maybe I cross-multiply and divide after? Can I just multiply through by 10 and have no more fractions? That would make this question way easier. Wait... You want me to estimate this? How the heck do I do that? Oh, you want me to draw it? You can't draw $\frac{101}{10}$. $101$ out of $10$ is impossible. I can't show you $101$ out of my $10$ fingers. God, I f^%@&ing hate fractions. I hate them. I hate them so much."
As linked above, many students actually are that confused when it comes to fractions.
Conclusion
Numbers written in decimal format tend to activate prior knowledge of whole numbers while the exact same numbers written in fractional format will often activate dreadful confused memories of weird procedures. It's no wonder why students instinctively switch towards decimal numbers - those are familiar.
And students, by the millions, will, of course, soon desire the impossible conversion of $\frac{x+3}{x^2-9}$ to decimal format at which point weeder math classes will
- Yank them from any academic or professional future involving algebra or higher-level math and
- Leave a terrible impression of math and their own abilities forever.
Until fractions are taught as numbers and decimals are taught as fractions with base 10 denominators, the phenomena described in the original post will continue.
answered Oct 22, 2021 at 1:09
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$\begingroup$ Amusingly, if you perform long division on $\frac{x+3}{x^2-9}$ you will recover the laurent expansion of this function around the point at infinity. So, in a sense, one could consider this expansion as analogous to the "decimal expansion" of the rational function. $\endgroup$ Oct 22, 2021 at 16:08
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4$\begingroup$ Why do you use the dreaded worksheet notation, "When calculating 10.1÷3.7= ____"? What the equal sign and the underscore for? Can't you write just "When calculating 10.1÷3.7"? $\endgroup$ Oct 22, 2021 at 22:56
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1$\begingroup$ $3\frac{1}{7}$ should be written $3+ \frac{1}{7}$ for the sake of not ruining algebra later. $\endgroup$ Oct 28, 2021 at 7:20
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$\begingroup$ As a formalist, I'm not particularly inclined to say that $3.7$ is fundamentally $3\frac{7}{10}$, just as I'm not inclined to say $\frac{2}{5}$ is fundamentally $0.4$. Understanding decimals in terms of fractions this way is just one way of connecting decimals to something students might already know, but students' understanding of decimals might be robust enough on its own that they don't need to make this connection to be fluent. $\endgroup$– TomKernOct 28, 2021 at 20:20
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1$\begingroup$ @TomKern How else do you read 3.7 than "three and seven tenths"? When you draw a number line and put 3.7 on it, do you not break the unit amount into 10 equal parts? How then can we interpret 3.7 in any other way? $\endgroup$ Oct 28, 2021 at 20:30
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So much depends upon what you mean by "understand" decimals. Think about this in terms of Searle's Chinese Room. Also see this BBC Studios video with Marcus du Sautoy. In this thought experiment, we imagine writing a computer program that gives person A, locked in a room and who knows nothing about Chinese characters, enough instructions to mindlessly manipulate characters to trick person B outside the room and who understands Chinese, into thinking that person A "understands" Chinese.
Replace Chinese characters in the Searle Chinese Room thought experiment with decimals, and then with rational numbers (fractions). My sense is that it would be much easier to write the AI instructions for manipulation of decimals than it would be for fractions. So in this sense, I think the answer to your question is "Yes, it is possible to understand decimals without understanding fractions."
I do agree that reliance on digital calculators inhibits a person's ability to understand fractions. Think about the wild discussions about whether $0.9\overline{9}$ equals 1, or if it is a tiny bit less than 1. Clearly, people have a difficult time with this concept and this points to a subtle relationship between decimals and rational numbers. Yet if you ask people if $0.3\overline{3}$ is equal to $\frac13$ or slightly less, I think there is no misunderstanding: $99.\overline9\%$ of people will easily agree that $0.3\overline{3}$ equals $\frac13$. Why is this? I hypothesize that it is because the input $1\div3$ on a digital calculator results in $0.3333333333$. The brain parses the finite string as $0.3\overline3$ as so of course this equals $\frac13$. I have often thought how diabolically fun it would be to design a calculator that upon entering 1 from the keypad, the displayed result is $0.9999999999$.
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1$\begingroup$ Some interesting thoughts here, but: Disagree that 100% of people know that $\frac 1 3$ is $0.333...$. Note sure what the proportion is, maybe a majority, but I bet you'd be surprised at how many don't know that. $\endgroup$ Oct 23, 2021 at 1:44
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3$\begingroup$ @DanielR.Collins A graduate student in computer science could not decide which of 1/3 and 0.3 was greater in conversation with me last week without the use of a calculator. $\endgroup$ Oct 23, 2021 at 21:55
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If some student has grown up in a country where the metric system is used in everyday life (such as continental Europe), I would definitely say that this is possible:
A student not being familiar with fractions will not know how much $\frac{1}{12}h$ is; however, he is aware how much 5 minutes is.
Such a student will have no idea how much $\frac{7}{360}h$ is; however, he will have an idea how much 70 seconds is.
In the metric system, most "larger units" are $10^n$ times a "smaller unit".
Example: $1\text{km}=10^3\text{m}$, $1\text{m}=10^2\text{cm}$, $1\text{cm}=10\text{mm}$.
A child growing up in a country where the metric system is used in everyday life learns quite early that nearly every unit (kilometers, kilograms ...) works like this.
If you tell the student that $x=123.456789$, he will not be aware that $123.456789=123+\frac{456789}{1000000}$.
However, he might think that numbers work the same way as units and think of a pseudo-unit ($\triangle$) which is defined as: $\triangle:=\frac{1}{1000}$. However, not knowning about fractions, the student would not be aware that $\triangle$ is defined that way.
Now the student would think of $123.45678905$ as $123\text{k}\triangle + 456\triangle + 789\text{m}\triangle + 50\mu\triangle$.
I see no reason why it should not be possible that some students think about decimal numbers this way.
By the way
If you were told: "The time required is 0.345 days", most people would also have no idea how much $\frac{69}{200}\text{d}$ is, but they would think about $8\text{h} + 16\text{min} + 48\text{s}$.
