2.9+Reflection+9+Future+of+Fractions


 * Recall you'll post your discussion on this page for the "Thoughts on Fractions" paper by Usiskin.**

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:
 * 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.

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. This is due by Wed by 11:45 pm.

Will fractions one day be obsolete? That is the topic of “The Future of Fractions.” The author of the article believes that this is an absurd thought. He begins first with a definition of a fraction (366-3), which he calls “indicated division,” which is an easy way to represent many different numbers. When I first read the title I thought, why would fractions ever become useless?

He continues to the use of fractions, arguing that they are not simply for measurement, as many people think. Fractions, he says, also represent a variety of mathematical concepts such as dividing (“splitting up” (367-1)), rates, proportions, formulas and sentence-solving. I think fractions are the basis of many more mathematical concepts, ideas and rules than people would really think.

In the section, Calculators and Operations with Fractions, the author argues that due to the wide range of use of fractions they will be with us for a long time and there is no sign of them going anywhere anytime soon. Even with calculators, fractions are still important representations, especially for things such as rates. It helps the students to see why travel time and distance is written as miles per hour or kilometers per hour. The “per” represents the fraction; because even once you “divide” the fraction you still produce another fraction except the denominator is now 1.

The example he gives, (368-5) “In a random drawing from a deck of 52 cards, what is the probability of drawing a king?” I think the question he asks; “Does 0.076923 convey the same information” helps people to see why fractions are important. Although one could argue that the decimal answer tells them they have around an 8% chance of selecting a king. I do not think that is as concrete as saying I have a 1/52 chance.

Next, the author discusses which should come first, fractions or decimals? I have never experienced these two concepts being taught separately. I think they should be taught at the same time considering they go hand in hand with one another, including percents. Also in this section of the article, the use of calculators comes into play. As of now, I am against letting middle school students use calculators for simple addition, subtraction, multiplication and even division. I believe that it is important that they know how to do these in the heads instead of relying on a calculator that they might not always have handy in life. However, the author argues that, “with calculators, the crucial skill is knowing //when// to multiply, not how (368-13).” I can understand that logic but I still believe they should know both, when and how to multiply.

Ending his article, we see some reasons why some people think fractions will go (369-2) including, since decimals are increasing in importance, fractions must decrease, fractions are only used for measurement, fractions constitute a numeration system and operations with fractions are not particularly important. For me, I think these types of reasons are not based on logic but on what people see fractions as, and I think many of these people do not realize how prevalent fractions really are or how common they are. Just because people do not see eating a slice of pizza as the same thing as eating 1/8 of a pizza does not mean you are not eating a fraction of your pizza! Kaitlin Froehlke

This article explores whether or not fractions will become obsolete and attempts to convince the reader that this will not be the case (366-2). The author first describes fractions as indicating division (336-3). He clarifies that they are not components of a number system, nor do they have to represent rational numbers (336-4) (since rational numbers are defined with the numerator and the denominator being integers). Additionally, for many numbers fractions are merely a more convenient representation than decimals – they are simply easier to deal with (366-5).

Next, the author argues that fractions are used for more than just measurement (366-6). Therefore, this implies that switching to the metric system for measurement does not eliminate our use of them. In every instance of division, fractions are useful, including splitting up, rates, proportions, formulas, and sentence-solving, among others (367). All of these are methods to convey division, and it is likely that different students will view a single problem from multiple perspectives.

The author then brings up the idea that the widespread use of calculators does make fractions less important than they were before this time (366-6). In fact, for the purpose of understanding, converting fractions to decimals via calculator actually can cause more difficulty (366-5). As shown in the example of the cards (368-5), a decimal representation does not convey the same information as a fraction does, and so I would prefer to use fractions in certain situations and percentages in others.

I believe calculators can be very helpful tools for mathematics in general and fractions in particular, but they do not become practical unless there is a underlying mathematical understanding that the calculator enforces and enhances. The author makes the point in this article that, “with calculators, the crucial skill is knowing //when// to multiply, not how” (368-13). Students have the ability to punch numbers into a calculator, but knowing what these numbers mean or how they relate to one another is a matter of conceptual understanding, which a calculator does not necessarily provide. The calculator’s role is simply to calculate, not further one’s understanding. As teachers, we need to promote true knowledge of mathematical ideas, not emphasize routine computations. I think fractions represent more of a conceptual idea of this, whereas it is more appropriate in other problems to use decimals. We will have to decide what is appropriate for each, based on the individual tasks at hand.

What should we teach first in our classrooms, fractions or decimals? I believe we must consider that in order to understand the meaning of one tenth, one one-hundredth, in decimals and in the metric system, it is essential to have a basic knowledge of fractions (368-12). Thus, I think it would be wise to introduce fractions before decimals to ensure a more solid foundation of these concepts. Clearly, though, there are a large number of similarities between fractions and decimals, so we cannot completely separate the two. The way operations with fractions and decimals work are extremely related, and so are their applications. Furthermore, the author states that there is no simple solution to the dilemma of which order to use for teaching. He concludes by reaffirming that we will continue to use fractions in the future, and this may even occur more than in the past (369-7).

Mandi Mills

The subject matter that this article covers is absurd and I now wonder about the state of the world for it having to exist in the first place. Why would fractions ever become obsolete just because we can easily do decimal calculations? As the article points out, fractions and decimals are closely related where numbers like 1/16 are equivalent to the decimal 0.0625. (1-5) It is discussed how it is easier to multiply 1/16 and 1/8 than there decimal counterparts, but since some people believe that calculators will do all the work anyway, I do not believe this strengthens his argument.

The author shows us some of the basic uses of fractions. It is smart that he chooses to use simple real world examples to prove his point. His examples include how fractions can be used to split up items between people and it is a much easier concept to understand and work with in fractions than decimals. (2-2) Among his given uses of fraction comes formulas. The sheer quantities of formulas that use fractions are staggering. He gives several examples, and then writes a formula using a negative exponent which would confuse more people than having fractions in the first place. (2-10)

The most crucial points brought up in the article are those that have to do with understanding mathematics. Usiskin gives an example of a problem with probability which he continues to solve using fractions. He argues he could have solved the problem with decimals “but the decimals would disguise what is going on.” (3-5) With fractions and the story problem, it is easy to see that 1/7 means one out of seven days of the week. In decimal notation, that would be 0. 142857 repeating. Fractions give us a much simpler representation on both a symbolic level and representational level.

There is then a discussion on curriculum, whether or not decimals or fractions should be first. This is a more worthwhile section that the rest of the article seeing as the sheer value of fractions should not be understated. There is a point made about one-tenth, one-hundredth, and one-thousandth which is how we view decimals. (3-12) As Usiskin points out, this would require at least a rudimentary understanding of fractions to understand the workings behind decimals. I feel that fractions and decimals can be taught side by side for a while, but working with fractions has a much higher benefit than decimals.

It is also interesting looking at the arguments against fractions which is given near the end of the article. (4-3) This makes me wonder what exactly people learned in school. Their use of faulty logic and misconceptions are staggering. The most worrisome belief people had for thinking fractions would go away is “They think that operations with fractions are not particularly important.” (4-3) The sheer amount of applications of fractions is astounding and those who think this way must have had to work with fractions on many occasions. Obviously, the people who think that fractions are unimportant did not understand fractions well so that should make it even more important that we do an excellent job teaching fractions today.

-Marcus Edgette

“The Future of fractions” opens up as a topic of whether fractions will be obsolete one day. The article is to, “Convince the reader of the falseness of such a statement” 366-2. First the author describes what a fraction is by saying it’s a division 366-3. He says fractions provide a convenient way of representing numbers and are more easily to deal with than decimals 366-3. He also says that a fraction is not a component of a numeration system, and that a fraction does not necessarily represent a rational number 366-2.

The author also states that fractions are used for much more than measurement 366-4. He gives examples of fractions in splitting up, rates, proportions, formulas and sentence solving. 367-1. He says you can replace a fraction by writing the multiplicative inverse to the negative 1 power 367-9, but all this would do is stir up confusion when one wants to represent something simple. For example, when writing 1/7 you wouldn’t want to write 0.142857 repeating.

He also talks about the use of calculators. He believes calculators can be used to obtain the numerator and denominator of an answer 367-12, but does an answer in decimal form help you better understand a solution rather than using a fraction? In the article he gives an example of a problem dealing with card. The problem is, “In a random drawing from a deck of 52 cards, what is the probability of drawing a king” 369-6. The answer is 4/52 but if you were to put this in the calculator it would say 0.076923 repeating. Talking this answer out with a middle school student would make them understand that you have a 4/52 chance not a 0.76923 chance of getting a king. Fractions just make things easier to understand and to reason with.

In the article the author mentions whether the teaching of fractions or decimals should come first. He says, “We should teach a particular operation with fractions at nearly the same time as we teach the same operation with decimals, and teach the applications simultaneously” 368-15. I feel that they should both come simultaneously. It is with fractions where you come up with numbers that are not whole, and it is with decimals that you can write a number in fractions. The two go hand in hand. It is important to note the difference between the two but also we must show how they are related. The author also say that the use of fractions will continued to be used more often despise the fact that how calculators have made higher mathematics more widely used 369-5. It is my job as a future educator to show the importance of using fractions and that everything doesn’t need to be in decimal notation.

Fredrick Martin

When I read the opening paragraph to this article, I immediately agreed with the author before I even read any of his justification. The first fraction that came to mind was that of a repeating decimal. There is no way to write that repeating decimal in a nice way, to multiply it with other decimals, and so on. To take a fraction, whose decimal counterpart is repeating, and disregard it seems unnecessary. The way I look at that is taking the number pi. If we were to use only decimal representations, in the cases of repeating decimals, it would be like taking 3.14 as the value of pi. Usiskin makes this point as well. In addition, and primarily, he makes the point that fractions indicate division (1-2). I think this point is lost a lot of the time when teaching students. I know when I was in middle school, ½ was one half and .5, but not talked about as 1÷2. I know if this was focused on more, rather than the decimal form of the fraction, I would understand fractions better. Our society is formed around fractions. As Usiskin pointed out, rates are fractions. A primary example of this is, as the article said, kilometers per hour (2-3). How can fractions be considered on the way out when they are a central part of how we calculate how fast we are driving? Speed limits can never be simply 70; 70 what? Additionally, I also thought about the idea of averages in real life examples when I saw the first paragraph of this article. This work points out the idea of if we have 80 students in 2 classes; there is an average of 40 students per class (2-4). There is that word per again. Every time we say something per something else, we are referring to a fraction. I am someone who never has gotten along with calculators. Calculators decrease the amount of skill work there is to be done with decimals because they are so quick and efficient (2-10). As a middle school student, I did not do much work with decimals. In fact, today I have a hard time doing arithmetic with decimals without a calculator. Is this because my teachers forced me to use the technology I didn’t like? Or is it because I was taught fractions better? I feel comfortable with fractions and I feel this is because of the amount of practice I got with them. Finally, I agree with the idea of people needing an understanding of fractions to understand decimals. In a random deck of cards, the chance of pulling a king is one of every thirteen cards or 1/13, which makes more sense than 0.076923 cards(3-6). Also, money is a big issue where I think fraction knowledge needs to be understood first. If a student knows it takes 100-dollar bills to make a hundred dollar bill, they should also be able to see one dollar is 1/100. If they can only see .01, then they are not seeing the full meaning of what that dollar is (3-12). Fractions are not on their way out. We see them every day and there is no way to get rid of them. Those who are trying to say they are on their way out of fad like Roman Numerals are under false understanding (4-3) or are not thinking clearly in my opinion. They must not have been taught the real meaning of a fraction and this is why I, someone who has been taught fractions, know they are here to stay. Katey Cook

When I first read the idea that fractions would become obsolete due to the modern calculator, I initially saw how that might make sense. Decimals seem like such a more concise representation of a partial number. This article strongly convinced me otherwise though through the following points:

- The multiplication of two fractions is sometimes easier to use fractions, - Rate is calculated by a quotient represented by a fraction, - Splitting up equally: showing the division of two pies amongst three people, - Proportions and “similar figures” would be more difficult to study in any other form, - Many formulas also depend on fractions, and in - Sentence solving: what if 7x=1? 1/7 is more clear than 0.142857. Is drawing a king better represented by 1/13 or 0.076923? A good point.

I also agree that it makes sense that fractions be taught earlier in the curriculum than decimals, because in order to understand decimals, students should have a prior knowledge of what one-tenth or one-hundredth is (368-10). Maybe teaching simultaneously would suffice as well.

Also, just because the use of decimals goes up does not mean that the application of fractions must decrease. That leads me to the intriguing closing statement, about how he said the use of fractions will be more common in the future than it was back then… That is a very interesting conjecture, one I can’t definitely oppose or support. It sure would not be difficult to believe.

- Tim Hollenbeck

The Future of Fractions article claims that fractions are absolutely necessary even if there are calculators and we switch to the metric system. I feel as though this is correct because we cannot fully understand decimals without fractions. In its essence fractions are a method of dividing (1-4) and division is still necessary for everyday functions. It also makes the point that because most fractions deal with small numbers or can be reduced to smaller numbers which are easier to deal with. (1-6) Furthermore there are a number of different uses that cannot be changed such as rate, average, the volume of a sphere and more. (2-8) Usiskin makes another great point most of the logic the argument for decimals displacing fractions is extremely flawed (4-6). In summary Usiskin makes it very clear for many reasons that fractions will never truly cease to exist. We really only touched on the two uses of fractions in class however there are many different uses for fractions. I believe that when we leave the exploration of fractions for the students we allow them to even invent new uses for them. Using fractions is also making sense of division and applying it to real life scenarios such as proportions and speed as given in the article (2-2, 4). I wonder though how the students will make fractions meaningful for themselves so that they can remember it. In the article about increasing student confidence we learned about what makes a student learn math better and how confidence will affect the student’s learning. I think that the same is true for the teacher, when we as teachers can exude confidence about knowing our subject and our students and having confidence in them, then the student will catch on and be confident which will allow them to have a reason for making the task interesting for themselves.

Although the author of Future of Fractions has great point I also think I’ll have to disagree about the use of calculators making the use of fractions more important and the use of decimals less important (2-10). I feel that calculators are a way for the student to become lazy and not really learn unless it is an absolutely time consuming procedure. If that is the case then learning how to work with fractions is comparable to mere procedure which is what the calculator is good for. Other than these thought I feel like Usiskin has made a great case for why we will never stop using fractions. ~Denise Slate __

Fractions soon to be extinct?! Like dinosaurs or the cassette tape?! Say it ain’t so! Over forty years ago, some people believed that fractions were going to become outdated because of the popularity of the decimal. It is now 2011, and the fraction is still prevalent despite advances in technology. The assumption statement about fractions becoming obsolete is ridiculous (1-1). I agree with the standpoint of the author in this article, and the fact that people even speculated fraction extinction really grinds my gears.

In order to hypothesize that in the future a fraction will become obsolete next to the decimal, it is necessary to know what a fraction is. Based on the reasons, I feel confident in saying that those people did not truly understand fractions (4-3). Usiskin mentions that in certain situations fractions are easier to deal with than decimals (1-4). Also, it is true in some cases that fractions can be harder to work with, but this does not occur often enough to do away with them completely. As a counterargument, we could say that decimals could eventually become archaic! The examples in the article showed that fractions were easier to work with in some cases (1-4). So why keep decimals around? Of course, this is a silly argument, but the point is that these mathematical representations are situational and sometimes fractions will be easier to deal with and at other times, decimals.

I use fractions in cooking and in conversation. For example, I could say, “Hey give me half of your pizza!”. It would seem odd to say “give me 50%” or “give me .5”. Fractions are incorporated in everyday jargon, whether we realize it or not. Fractions can be used in splitting up things (such as a pizza), rate, proportions, formulas, and sentence-solving (2-1). Even with calculators being decimal friendly, fractions are a big part of life and make certain tasks easier, like cooking or rates.

If we teach our students what a fraction truly is, this calculator worry should not even be a problem. We need to take the reasons why people think fractions will go away and disprove them all with our curriculum (4-3). Students need to learn that fractions are not just used for measurements, fractions represent an operation, and that operations with fractions are important. By incorporating these into our curriculum, we can extinguish any notion of fractions becoming extinct.

Hailey McDonell

In the article entitled the “Future of Fractions” Zalman P. Usiskin goes through some reasons why fractions will not disappear both in the world of math and in our everyday lives. I thought his explanation of what a fraction is was very interesting. He talked about having fractions in base eight, base ten, and in Babylonian mathematics (1-5). I never really thought about fractions in any way other than using the base of 1. He mentions the convenience of using fractions to represent certain numbers (1-6). I can remember when I was younger that I always relied on my calculator that I preferred the decimals, even though those can be much more tedious to write out and can often lead students to inaccurate answers. In High School I can remember teachers emphasizing exact answers and forcing us to use fractions instead of decimals for some of our answers.

Usiskin goes on to talk about the uses of fraction and division leads us to use fractions (1-7). He goes on to share other uses for fractions outside of measurement, such as rates of change (2-3). When states that anytime the per is used, it shows a rate, and ultimately that rate can be found by use of division (2-4). I never viewed this as a fraction. It is very true and eye opening what he is pointing out. Many times I take fractions for granted and don’t always realize how essential fractions are to our everyday lives. Usiskin continues his article mentioning how some people argue that calculators will reduce the use of fractions. He goes through a problem on page three that deals with probability. He shows that we can solve the problem using fractions, and that the calculator just gives us the decimal form of the number (3-4). One thing he did do was show how we could use negative exponents instead of fractions (2-10). I agree with him that most people wouldn’t prefer that method of writing formulas and numbers.

He talks about the order in curriculum in a math classroom. He talks about what the sequence should be, fractions before decimals or decimals before fractions (3-13). He then goes on to give the arguments others have for fractions disappearing (4-3). The logic of some of these arguments seem pretty bad to me. For instance some think that since decimals are increasing in importance, the importance of fractions must decrease. He points out that means the “the sum total of the two must remain equal.” I think fractions will be here to stay for a long time. As educators we should emphasize both fractions and decimals. I think that I will place more emphasis on fractions, like many of my professors have done in my past math classes.

Mike Freeland

This article explores the idea of fractions and their place in our mathematics going forward. The author begins by firs explaining the idea of what a fraction actually is. A fraction is indicated division, says the author. (366-3) In other words a fraction is more than a representation of its decimal equivalent but it is the implication of a mathematical expression. Further in the next paragraph he also argues the point that in many cases it is easier to deal with fractions than its decimal counterpart such as 1/16 and 0.0625. I n many of my higher level math classes I dealt with decimals more often fractions. It wasn’t until my physics classes that my use of fractions really started to take hold. In beginning physics courses calculating distance and time traveled, rate of speed as well as rate of increase of speed. All of these variables were usually algorithmically derived from the classical mechanics equations. The author also gives a great example in the form of proportions. In particular his recipe example really showed how fractions can be easier than decimals.(377-4) Not only does this help in dealing with fractions but in everyday life recipes actually do need to be modified according to the number of guests. Any time a real world example can be used to show students how the topic they are learning relates to their real lives is a great way to get them engaged and interested.

Further on in the reading the topic of ‘what came first? The fraction or the decimal?’ is discussed. The best argument he could have made when he supposed that to accurately understand what a decimal represents one must first know the meaning of a tenth, hundredth and thousandth.(368-10) In my opinion the fraction must come first. The use and manipulation of fractions can lead to higher comprehension of numbers and a heightened confidence in mathematical manipulation in general.

Although he does give some half thought out reasons as to why fractions will go away at the end of his paper it is clear to see that these are not unbiased statements (369-1). Regardless, what really struck me about this article is that he was not denying the need and importance of calculator based decimal work. He simply is stating that it doesn’t have to be one or the other, both can share an equal part of the curriculum. Christopher Cardon

The interesting article “The Future of Fractions” by Zalman P. Usiskin revisits a movement in the United States that occurred in the 1970s: the slow disappearance of fractions due to technological advances. The author’s goal is to dismiss this absurd thought that is “based on incorrect reasoning from faulty assumptions” (366-1).

He begins by giving multiple clear definitions of what a fraction is. First, he mentions that “a fraction is an indicated division” (366-2). At first, this definition doesn’t make too much sense to me; it doesn’t give much of an explanation. An easier way to grasp what a fraction is, is to understand that a fraction is a much more convenient way of representing some numbers (336-3). I think this is especially true for repeated decimals like thirds or ninths.

He then goes on to describe many of the ways fractions are used. Most Americans associate fractions with measuring, and I caught a hint of condescension by the author because he hinted that “the layman” use fractions in this manner while mathematicians have multiple complex uses for fractions. When I think of fractions I don’t jump right to measurement; I don’t think I ever really have. Mathematicians use fractions to split up things, calculate rates, make proportions, formulate formulas, and solve sentences (367).

The author goes on to argue that fractions are used in so many ways that it’s highly unlikely that they will disappear. Some believe that decimals are slowly replacing fractions because calculators could only use decimals, but the authors has argued that calculators have become so powerful that they can now handle fractions just as easily as decimals.

To conclude, he tries to understand why some people have thought fractions will phase out (369-2) and he came up with four different assumptions: “1) decimals are increasing in importance, 2) fractions are only used for measurement, 3) fractions constitute a numeration system, 4) operations with fractions are not particularly important” (369-3). He finds fault in the logic of each assumption and believes fractions will be around for at least the next 25 years.

Tori Ward