What’s So Hard About Fractions? Fractions as we know them today, the symbols and the algorithms for performing operations, have developed over thousands of years, beginning with ancient Egyptians.
Through research of the origins, the development of fractions to appearing symbolically as we know them today, and of the developments of how we operate with them today and then connecting that knowledge with the observations of contemporary math education experts and personal interviews and observation of fifth grade students learning fractions It has become evident that the development of reactions took thousands of year and suffered many complications.
It Is most evident that knowledge of that which birthed and stunted the development of fractions is similar to what stunts instruction and confuses many students in their understanding of fractions and their algorithms as we know them today. Before delving into what is so hard about fractions it seems more appropriate to being to answer the following two questions; Why were fractions invented and what purpose do they serve? Ancient Egyptians, as long ago as BBC (collation knew that fraction were essential when needing more precise or even exact measurements, or agreements that fell between two whole numbers, without switching units.
Fractions therefore serve the purpose of measurement and for finding a value between two whole numbers, for more precise numerical value. That is, it has been deducted by historians that fractions were first deemed necessary in ancient civilization because of their need In precision of measurement, and not In terms of division. It wasn’t until the eleven century when the definition of fractions was determined to be the division between two numbers ( PDF citation). This is in contrast the way fractions are introduced in schools today.
To solve problems associated with numbers falling between two numbers, Ancient Egyptians developed unit fractions. They were breaking a whole number into parts and created unit fractions which they could then use. Unit fractions In today’s notation of fractions look like l/n for some positive number n. Except the Egyptian notation would not allow them to write 2/n or 3,n, so Instead, they used the sum of the largest unit fractions. That Is % would be written as h+h . The fractions would be written in hieroglyphics, which was the notation of ancient Egypt.
As might be apparent from the number sentence above, he Egyptian notation for fractions made adding fractions very difficult, as well as more complex operations. Perhaps they notated it this way because of how easy it was to compare fractions or because they could Just take the largest chunk at a time until there pieces got close enough to the precise measurement they were trying to get to. Because of this complexity in operating with them, the Egyptians had put on the Rhine Papyrus t a 2/n table for all mathematicians to consult when adding these unit fractions.
Since this list could never be exhaustive there was need for reform. Also for ancient Egypt, multiplying fractions was a tedious process that involved successive doubling. This was also noted on the Rind Papyrus as well as a problem about delving two loafs Into seven people. There were eighty seven problems In the fractions were to Egyptians and it also reveals how difficult they were to operate with, since mathematicians had to consult the rind papyrus for solutions. As fractions developed in different part of the world their notation became more sensible but still more problems regarding operations arose.
The Babylonians used them in their base 0 system as more of a number less than one but this was only understandable in context so it was hard to determine the place value if it were Just written without a context. The other problem here was that there needed to be a zero to show place of the missing units and also fractions needed a decimal place to know that they are not whole numbers. This highlights a historical problem with understanding fractions, fractions needed to be associated with in a clear context.
This will later be connected with contemporary problems faced by students who are trying to understand fractions. There were other civilizations, like the Romans, who attempted their own fraction notations and eventually the Arabic system developed the notation similar to the one we use today, with the numerator and the denominator separated by a horizontal line and sometimes a slanted line, but the problems with the most efficient way of performing operations with fractions were far from over.
For instance common denominators were initially found by multiplying the denominators and it wasn’t until later that it was commonly found by discovering the least common multiple. Also, the problem of dividing fractions and multiplying by to reciprocal is a elatedly new development. Up until recent years, this was done by finding a common denomination then Just dividing the numerator by the denominator. This goes to show that the algorithm we use today way by no means obvious to the most expert mathematicians even with the notation of fractions that we still use today.
Another problem faced in developing fractions, was developing decimal fractions that corresponded with the common fraction notation that was developed. In Development of Arabic Mathematics: Between Arithmetic and Algebra the author talked about those who were developing the decimal fraction notation and that they mound decimal fractions to play a pivotal role in understanding and conceptualizing fractions and their contextual meaning (peg 114) This is yet another example that highlights the fact that fractions are more fully understood in a developed context.
Fractions instruction today bombards students with many algorithms for computing and operating with fractions in a way that doesn’t allow much time for context. For example, the Harcourt curriculum and textbooks series devotes one unit to fractions. Titled, “Fraction and their Operations” the unit begins with writing decimals as reactions or mixed numbers and then moves on to adding and subtracting like fractions. Then before it moves into adding unlike fractions it shows 1/2 + 5/8 and then has a model of h with five 1/g’s attached to it.
There are two more problems with these models and then there are 14 problems that require that students find the sum. After this the curriculum moves into subtracting fractions in the same way, with 3 with similar models and then 14 asking them to find the difference. That’s all they have for adding subtracting and writing decimals as fractions. This sections is a total of 6 pages long. Then the curriculum moves onto common denominators, comparing, and estimating. Then they throw in mixed numbers and fractions into all of these operations and students are required to simplify.
There is also some strange most fifth graders, having already learned so much of fractions out of context, would be able to make the connections. The instruction of fractions that is used by the textbook and the many like it is problematic because it doesn’t allow students to conceptualize what they are doing when they perform these operations and therefore students don’t have a context for these complex ways of computing reactions. The models given are not explained and don’t show students manipulating fractions. It’s Just a picture that requires immense explanation.
The book unit doesn’t begin with an pictures of fractions or what they are and refreshers of why they use them or discussion of how they work, nothing to active the context of fractions, it simply moves straight into teaching to method for how to write decimals as fractions and mixed numbers and it is followed by many practice problems for implementing this method. This is not to say that practice problems don’t have a place in thematic instruction, but if it is only requiring rote knowledge and memorization, then the mathematics is lost and the students will more than likely forget the method.
The Harcourt instruction of fractions is also problematic because at no time does it require students to use fractions for measurement. As was mentioned earlier, historians believe that measurement was the primary reasons why fractions first originated and it wasn’t until hundreds of years later that there line between them was developed and it was even later that it was determined that the line between a and b meant to divide a into b.
This is consistent with the current instructions lack of attention to the intuitive and natural ways of learning that need to be considered when teaching fractions because of their complexity and many operations. Fraction is too often taught initially in terms of division and not as measurement, as is more logical. As mentioned, ancient Egyptians first started using fractions for more precision with measurement (citation). It seems the origins of fractions began in the context of measurements was a natural and necessary development.
Therefore teachers should learn from this historical reality and incorporate knowledge of it into heir instruction. Since fraction in measurement is a natural development and it more intuitive than an explanation of the division sign which developed hundreds of years later, teacher will likely be more successful introducing fractions in terms of measurement. My fifth grade class spent 3 months struggling through learning fractions and my mentor teacher was fed up.
The students would get one part of the test right one time they took it and then when she would retest the class, because all but a small percentage of the student were failing, she would notice new errors. They ad to compare, simplify, multiply, subtract and add fractions. This requires knowledge of LLC, GIF, and many other non-intuitive algorithms. There were so many operations that the instruction had to specifically gear toward how to do those operations, promoting memorization and rote knowledge over context and mathematical thinking.
After all, these students would see these types of problems on a standardized test and if they don’t simplify the whole problem they would fill in the wrong bubble and then get that whole problem wrong. I saw the biggest errors with comparing fractions. After the second class wide test failure, I grouped students according to their errors would work with them in a conference room on the problem areas. I would write on the board “The number one rule in comparing fractions is: repeat it back to me.
We would do a lot of problems and every time they would begin a comparing fractions problem I would have them say the rule aloud. Still, by the end of two weeks when it came time to take the test, most of them forgot this. I was inspired through various articles written by experts in math education, to teach this concept of comparing fractions in context. I would do this with an open ended question about comparing fractions that students would have to work together on and show me with pictures and verbal explanation how they know that one fraction is bigger than the other.
I did this by presenting a sub sandwich problem where 5 kids were given 4 subs, 4 kids were given 3 subs, and 6 kids were given 5 subs. They shared them fairly within their groups. I told the class that the children got upset because they heard they thought it was unfair. I ask the students to work in groups and show me if this was fair or not with pictures and verbal explanation. To my reprise none of the kids Jumped to the comparing fractions algorithms.
This was good for my lesson but unsettling after having worked 3 months with these students on these operations and not one of them made a connect with that and the problem. Instead students too out 1 half from the subs that each got and then compared what was left over. All students were involved and they were given a context for how we are comparing when I had one of the students do the algorithm on the board and lead a discussion about what we must be doing. I gave the students two problems to use the algorithm on and roughly 20 our out 25 students of the class did it accurately.
Through these experiences teaching fractions I learned why in the historical development of fractions, context was so important. When I was teaching the kids rote memorization or the algorithms they were more prone to forgetting and had no conceptual understanding of what we were comparing because when the were given this real world problem, not one of them connected it to the algorithm. Now having learned comparing fractions in context, the students will be able to apply this algorithm and be able to simultaneously visualize how it’s actually manipulating the fraction.
I think this lesson was also good because it asks kids to use fractions in both division and in measurement. I hope this experience of connecting fractions to a real life, visual scenario, will transfer to their understanding of other fraction operations and algorithms. Leading experts in Mathematics educations suggest teaching fractions in ways very different from today general method and are influencing instruction of fractions to focus further on conceptualizing. One way they have done this is by researching the impact that representation has on learning fractions.
In an article in the April 2008 edition of Mathematics Teaching in the Middle School experts reveal their findings of eight years of research of what role representation plays for young students learning fractions. “Concrete models are critical forms of representation and are needed to support students’ understanding of, and operations with, fractions. Other important representations include pictures, contexts, students’ language, and symbols. Translating among all these representations makes ideas meaningful to students. The article later mentions that teaching fractions through representation is good for promoting a strong foundational understanding of fraction and that these understandings helps dents later understand why you find common denominators when performing Babylonians, today’s Arabic system fractions still need to be learned in context. Mack, another expert in research for math education, also discovered the role context has on learning in researching the use of demanding students to use informal knowledge when working with fractions.
In the article, Learning fractions with understanding. Building on informal knowledge, Mack stated that “Informal knowledge can provide a basis for developing understanding of mathematical symbols and procedures in complex content domains. ” By creating problems for dents that are in the context of their real life, tremendous gain we made in student learning. This is yet another connection to the historical reality of context being pivotal to successful understanding of fractions.
The algorithm for dividing fractions has a unique history and is there is little understanding in students today for how it works, yet researchers are now connecting instruction with its more natural unfolding, as was seen throughout history. In a 2007 article of Mathematics Teaching in the Middle School experts in math education focused their research on the algorithm for dividing fractions and how students understand it. They have discovered through extensive years of research that students most generally get a word problem for division after having the division algorithm taught to them and instantly invert and multiply.
When they are later asked why they did this they said “because it is a division problem”(citation). This reveals that students don’t know the reason why they are performing these operations. As mentioned about the historical development of the division algorithm as we know it today, fraction began with measurement, then once they were in the form you see them in today they were first vided by finding a common denominator and then dividing the numerators, it was much later that they were divided using the algorithm of inverting and multiplying that we use today.
In the article, experts are teaching division in exactly this sequence. They are given division problems where students are required to first use measurement in word problems to divide fraction. Then students are given word problems that lead them to the finding like denominators algorithm. Once students have fully developed these concepts that are easier to understand with dividing reactions and are seeing why you’re getting such a large outcome each time, then the invert and multiply algorithm is introduced with extensive discussion as to why this works (citation).
This connects the historical natural unfolding with the way students learn fractions. Also after all of this discussion, context, and ground work that is laid to fully conceptualize what is being done when you divide fractions, students are much less likely to forget how do to these types of problems further down the road.
When you consider all that has transpired throughout history to develop fractions as e know them today it is very evident why fractions and their many operations is hard for students to understand to. The reason why fractions were first invented was to be more precise in measurement and yet in the fractions operations unit we looked at, which is used in Lansing Public Schools, fractions used for measurement is never discussed.
Also throughout history we saw various times in development where expert mathematicians struggled to create fractions within an accurate context and then to operate with them most efficiently, within that context yet schools onto require that students know the context, they often Just require them to fractions is long overdue and leading math educators are doing what they can to make sure students are conceptualizing fractions in terms of measurement and division through open ended questions and problem solving.