2.6+Future+of+Fractions

The readings reflections have two main purposes: 1) to hold you accountable for careful reading of and reflection on the readings assigned in class; and 2) to provide you with a record of what you've learned and thought about as a result of the readings.
 * Readings Reflections**

The readings reflections will be evaluated using the following criteria: Submit your readings reflection **before** reading anyone's on the Wiki page and then paste it into the existing reflection page for that current reading.
 * completeness and timeliness of the entries;
 * comprehension of the main ideas of the readings; and
 * depth and quality of integration of the ideas with your own thinking.

Provide a one page summary of "The Future of Fractions". Paste your reflection followed by your name. This is due **Monday**, Feb 27 by 2:00am. Notice MONDAY, not TUESDAY before class.

We Need Fractions!
I find fractions essential in mathematics and use them to represent information daily, and when I began reading the exert from 1979 predicting the extinction of fractions in mathematics as decimals take power, I was surprised and confused. How can a decimal replace fractions when in our daily communications we even refer to fractions-- there's 1/3 of the cake left, we wouldn't say there's about .3333 of the cake left, would we? The author of this article articulated exactly the same argument I battled after reading this mistaken prediction.

Usiskin (2007) points out that a fraction is a "convenient way to write numbers"(366-4), which I agree with because it does show a relationship between numbers. I was surprised that Usiskin did not include, however, that a fraction is the most accurate way to describe numbers oftentimes-- many decimals repeat to the point that writing out the accurate number would be tedious and oftentimes impossible. Usiskin then discusses just how many uses fractions have.

Like we have discussed in class when working with story problems, fractions represent "splitting up", or making divisions like dividing pie among people (367-2). Because it is difficult to think in terms of decimals for this type of problem, fractions play a fundamental role here. The author then argues that rates are always represented in fraction form-- like km/hr (367-4) where we are dividing distance over a quantity of time. It is also argued that many formulas use fractions-- and although negative exponents could yield an equivalent equation, we feel more comfortable without these (367-8).

Then comes in the 1979 argument that calculators compute decimals so quickly that fractions will not be necessary. The author, however, counters that many calculators today can compute fractions and that without a conceptual understanding of fractions, students cannot use calculators correctly (267-13-- after all, fractions represent division and isn't inputting 1 divided by 2 the same as 1/2? Thus, students are using fractions while using calculators, so why wouldn't we teach them how to keep their work from getting messy by writing 1/2 instead of 1 divided by two?

The author also points out the problems based on application are oftentimes best expressed as fractions. The probability question, for example, shows that fractions are the convenient way to express probability (368-4). I also especially liked the simplicity of the problem on asking how many kings are in a deck of cards of the total-- we think 4/52, not the decimal expansion of this number (368-6). Since scholars and standards agree that our students should build their conceptual knowledge, fractions need to remain as a way to represent numerical expressions. --- Katie

The article, “The Future of Fractions”, responds to speculation in the 1970’s that the advent of the metric system and calculators would render the use of fractions obsolete. The author disagrees with this assessment and proceeds to discuss the components, uses, and needs for fractions in every day life and as a foundation skill for Algebra.

The author initially reminds readers that a fraction is a representation of the division operation (1-2). Since a fraction represents division, every use of division leads to the use of fractions (1-5). Common uses for division in every day life are measurement, splitting up, rates, proportions, geometry, formulas, and sentence solving (2-1 through 7). It becomes clear to me, especially in measurement (English system) and probability applications, that the use of fractions in these situations is the only viable option that makes sense. The use of proportions only makes sense when utilizing fractions. It is important in many cases to see the part/whole or part/part in order to make sense of a situation or solution.

When looking at the use of calculators, most graphing calculators seem geared toward the input/output of values in decimal form. Scientific calculators make the input of fractions relatively easy and the output of fractions available. The developmental courses I have taught, which allow the use of calculators, usually recommends the use of the scientific calculator to enable the student to utilize technology without giving up the use of fractions for decimals.

The question of whether the fraction or decimal comes first is an interesting idea to ponder. I agree with the author that the use and understanding of fractions must be a foundation for the use of decimals (3-9 through 11). How many children ask for 0.5 cookies? How many young children state their age as 3.5 years old (3-9)? Utilizing decimals mandates that a child first understand place values such as one-tenth and one-hundredth (3-11).

In summary, the thought that decimal representations of values will replace fractional representations is based on incorrect logic from faulty logic (4-4). Fractions are quite common in applications, but the mathematical curriculum of the 1970’s was not focusing on the applied application of mathematics (4-3). With the renewed focus on teaching mathematics in an applied setting, the importance of fraction manipulation is coming to the forefront of mathematical content concentration. //**(Susan Copeland) **//

Usiskin (2007) is talking about why fractions will still be in the future of mathematics. A fraction allows us to write numbers in a different way, and these numbers do not always need to be rational (1-2). This is important because there are many irrational numbers, and having different ways to write them is important. It also gives us a good way to look at multiplication, addition, subtraction, and division for any number less than one (1-3). This is important, because decimals are used quite often, and being able to know the fraction, or using fractions to get to decimals is a skill that is useful in everyday life. This skill would not be usable without fractions. Through all of page 2, Usiskin shows different uses for fractions, many of which are used outside of a classroom. This is evidence that fractions and mathematics in general is usable outside of the school. Calculators are making fractions less important, because as calculators evolve, they use more decimals, and do it more efficiently than a person (2-7 &2-8). This is important because I have said in relation to the Principles and Standards that technology can eliminate some of the challenge to math, and this shows that it can also eliminate some of the important elements of math. The example shown in relation to probability on (3-6) shows that fractions can make examples easier to visualize, whereas the decimal can make things harder to understand. So using decimals makes it harder to connect ideas, which is one of the standards for NCTM, thus fractions are a good way to go in following the Principles and Standards. Usiskin mentions looking at European curricula along with other curricula that have tried using both fractions and decimals at the same time, to see what the best way would be (4-1). This is a good idea, because if a method works well, then we would be smart to use it, compared to reinventing the wheel by making a new method. --**Josh Kaylor**

We may not realize it, but we use fractions quite often in our everyday lives. The article “The Future of Fractions” was written by Zalman Usiskin and intended to inform readers of how important fractions really are, and why they will continue to be used twenty five years from now. From the time we are little young kids, we use fractions even if we don’t realize it. Usiskin uses the example of splitting 2 pies evenly among 3 people, which means each person would get 2/3 of a pie (367-2). Thus, there is a real-life situation in which fractions are used. We may not realize when we use them, too, which is why it is important to define what a fraction is in order to get a better understanding of mathematics. They are directly correlated with division and also decimals, so by learning how fractions work, one can grasp concepts in mathematics a lot easier (366-5). One may argue, though, that fractions will be a thing of the past since new technology has been invented. The handheld calculator has contributed to math teachers putting more emphasis on decimals rather than fractions, since calculators use decimals (367-8). Money is another thing that is important to us that uses decimals, but fractions are still vital when discussing that, too. If two friends are splitting a bill at a restaurant and want to pay even amounts, they are taking //half// of the total bill. Without the background knowledge of fractions, the two friends would not realize that division is the process to use and that you are dividing by 2. Zalman Usiskin expresses his opinion in this article of why he thinks fractions are still important to learn in school. He states that they will be around and useful still, even years down the road so it is vital to learn about them and how they are correlated with decimals, percents and division. **--Shanna Thorn**

“The Future of Fractions” is an article written by Zalman Usiskin in 1979, the article’s main purpose was to provide a case against a growing theory in the 1970’s that common fractions would no longer be used in 25 years. It is now thirty three years later and, short of the world ending in the foreseeable future, I’d say that Usiskin was right on the money with this one. First let’s look at the reasoning behind the theory Usiskin is arguing against; in the 1970’s the combination of a movement towards the metric system and the advent of hand held calculators led many to predict that fractions would become obsolete (1-1). Usiskin provides several different reasons that fractions will continue to be used, some common uses being…

Splitting Up: Dividing 2 pies evenly among 3 people (2-1) Rate: Rate of speed if you travel 80km in 2 hours (2-2) Proportions: 5 grams for 6 people equals X grams for 8 people (2-4) Probability: Probability of drawing a king out of a deck of cards (3-4)

The use of fractions is ingrained in everyone at an early age for instance most people would look at the example for dividing 2 pies evenly among 3 people and say that each person would get two-thirds of a pie, on the off chance a person used a calculator and got the answer of .6666 they would still recognize the number and call it 2/3’s. Another point that Usiskin brings up is that in some cases the use of a decimal can take away the meaning of the answer (3-3). In the probability example above if you use fractions it can easily be followed that there are 4 kings out of a 52 card deck (4/52) which simplifies to 1/13 signifying the one king out of each suite of 13 cards, but without the use of fractions in the answer you get 0.076923 which is correct but the number loses its meaning compared to 1/13.

The last thing Usiskin talks about is that if decimals are becoming more prevalent in a curriculum which should be taught first, decimals or fractions (3-9)? To me it seems obvious, fractions should be taught first. In the beginning of the article Usiskin talks about how fractions first and foremost are a representation of division (1-2), which means that if someone is typing 2 divided by 3 they have to input it into most calculators as 2/3 which is a fraction. Not only this but when a person is to learn about decimals they learn that .1 and .01 have the meaning of one-tenth and one-hundredth (3-10).

In short fractions are an integral part of mathematics and our everyday lives and will always be used regardless of what the future brings and Usiskin does a good job of bringing this fact to light. **- Hutch**

In “//Future of Fractions//”, Usiskin bring up the relevance of fractions, how people still use fractions today, and that he does not believe fractions will ever become obsolete. Osiskin shows us how even though 1/16 = 0.0625, fractions are much easier to multiply by hand and with other fractions than it is to multiply decimals with decimals (1-8).

First there is //splitting up//. Here we can look at dividing up 2 pies among 3 people, everyone gets 2/3 of a pie (2-2). Even with SI units we can take 1.3 square kilometers and divide it among //n// people, leading us to the conclusion that everyone gets 1.3///n// square kilometers (2-2).

“Every rate starts as a fraction”, Osiskin states. If you traveled any distance in a certain amount of time, such as 80 kilometers in 2 hours, then you could set it up as [80 kilometers]/[2 hours] = 80/2 kilometers per hour or 40 kilometers per hour (2-3). Fractions are interwoven in rates and they are necessary for solving any problem with rates.

//Proportions// are said to be the most common form of applied mathematics by Osiskin (2-7). Proportions are used in geometry, algebra, when using similar figures, and other areas of applied mathematics (2-8). Here is an example of how to use proportions: If 6 tablets of sanitizer are needed for every 5 gallons of water, then how many tablets are needed for 8 gallons of water? Setting up fractions, we can see this better: 6/5 = x/8. Using fractions with proportions seems natural; using decimals with proportions seems unnecessary.

Another trivial way of using fractions is with //formulas//. A few examples that the author gave were finding the mean or average, slope, and volume of a sphere (2-10). Most formulas in physics and mathematics are given in fraction-form, where all that we have to do is find the value for our variables and plug them into our formulas.

//Sentence-solving// is the last main use that the author gives. Whenever we have an expression written out in “sentence form”, or as //ax = b//, and we know what our //a// and //b// are, then we are looking to find what the value of our //x// is. Dividing both sides by a we find that x = b/a. Other was that you can write this are b ÷  a or ba-1 ( 2-12). It always seems easier to keep a fraction as a fraction and never to write it out in decimal form, especially for fractions that give long strings of repeating numbers.

The author asks “Which comes first? Fractions or decimals?” and seems to be left in the air. Reading through his discussion about whether to teach factions or decimals first, I found that there is no importance whether one is taught before the other, but that teaching them at the same time is more important than not (3-18).

The reasons to which people think decimals are conquering the existence of fractions seem skewed and just wrongly thought. Calculators do make working with decimals easier, but the rise of calculators does not mean that fractions will become obsolete. We are simply able to use decimals more and easier than before, but fractions, I believe, will always be around and interwoven with our mathematics. We are not going to see fractions removed from curriculum. **(Ryan Sherman)**