Classroom Misconception: How Calculator’s Contributed to our Perspective of Public Education

I record every last detail and enter those details into a storage location. The space has an excellent way of sorting and determining facts based on the limited information provided. Once all the details are submitted, the tool spits out the exact answer that I was originally looking for. I have now successfully operated a calculator. The calculator was created as a tool to make the basic mathematics that we were already familiar with easier. As the technology developed, we saw an increase in their capabilities, including graphing functions. Those functions brought up are submitted through patents including the patents for enabling and re-enabling functions for graphing calculators from Texas Instruments. This innovation in the handheld calculator allows “apps, programs and additional information that may be embedded within the apps and programs in a calculator or other handheld calculating device” (Miller). These functions are great for society, but with every invention there are tradeoffs. The handheld electronic calculator may have accelerated the process of solving math problems with larger numbers, but it has also inhibited the ability to teach elementary-age students basic mathematics and contributed to the idea that a person’s sole purpose in life is to eventually join the workforce. The calculator may be starting to turn in this direction, but we can’t truly analyze its effects until we look at what the world looked like before the calculator entered the realm of mathematics.

Before the calculator was brought forward to our society, we were stuck doing problems the old fashioned way…by hand. Usually this math is basic in nature and develops into some harder topics. For example, Joe Moyal, a publisher of 36 works in the field of quantum mechanics, explained how Heisenberg’s Uncertainty principle made its way into the normal teachings because they were able to use previous proofs to come up with it. It almost didn’t come along because people “thought classical-looking mathematical descriptions were ruled out forever” (Moyal 108). Just by looking at some of the most basic of math that stood for hundreds of years allowed Werner Heisenberg to make the discovery that Moyal looked into deeply over his years of publications.  The methods of solving before Moyal’s time were relatively similar to how basic mathematics are taught to elementary school students. Students are taught times tables and all they are required to do is trace their finger across a page to find out a product of two numbers (Recorde 67). This account was written in 1699 and we still use times tables to this date to teach basic mathematics to children. This process of how basic math problems is taught has stood the test of time (over 300 years), but we continue to make developments in that field as more brilliant minds share their works and research with the world. In addition, using the times tables was a great way of getting students interested because it is somewhat hands-on as students have to trace their finger across the page to determine the product. A report from a meeting with the Maury County Teachers’ Institute asserts that objects incorporate the five senses to “exercise in abstraction, comparison and identification to the general notion of number in the abstract” (Figuers 3). I believe children learn through these hands-on activities because they are interesting to a relatively distracted mind. I remember when I was a kid, I wanted to do all of the interactive games or art projects because they didn’t make me write for long periods of time. The works that are shared by mathematicians have had an impact over the years, and we see it when students are initially taught how to do analytical problems.

The process of doing calculations changed significantly in the mid-1960s to the early 1970s. In 1965, desktop calculators were very popular in society and they “had to be plugged in, were the size of typewriters, and cost as much as an automobile” (Hamrick 633). The idea that access to a calculator cost as much as a car during these years is absolutely ridiculous to me. I can see how the initial calculators may have been targeted for a specific group of people that may have seen its uniqueness to society, but it seems like an outlandish expenditure at that point. Eventually, calculators were beginning to go mobile instead of just sticking to desktop designs. The initial handheld calculators were introduced by Texas Instruments in 1970 and were called “Pocketronics”. These options were cheaper than the desktop calculators as it cost $400 for a handheld calculator and $2000 for a desktop calculator at the time (Hamrick 636). I did a brief conversion on an inflation calculator that gives an estimation of how much $400 would be worth in 1970. I found the value to be right around $2500. This means that if the calculator was invented today, it would cost you $2500 to buy one of your own. Remember, this isn’t a graphing calculator, this is your basic calculator with simple addition, subtraction, multiplication and division functions. This shows how a new idea can cost a pretty penny when it first hits the shelves. Eventually, the prices dropped over time and calculators were beginning to be incorporated into the classroom setting.

Let’s be honest, getting a kid interested in mathematics is not an easy task. The stigma that math isn’t fun or exciting gets into students’ heads. The calculator brought forth a tool that could quickly and efficiently make calculations. At the time of the invention, not many of them were being brought into classrooms and when they were brought in, there was likely only one calculator for the entire class. George Immerzeel, author of Attack Math and other publications on calculators in the education system, states that the calculator should be used less as a tool in early education, and more as an item of inspiration of students by placing it in an “interest center of your classroom” (Immerzeel 230). This strategy makes the calculator accessible for the students in the classroom, so that they know they can try it out. On the other hand, students should still learn the process of solving problems mentally or by hand. By that, I mean that the calculator shouldn’t be used to just plug and chug numbers when they are unfamiliar with the process of solving the problems by hand. After talking about the placement of the calculator, Immerzeel talks about strategies to make the students use the calculator. He speaks about giving a different student the calculator each day to use in the class (Immerzeel 230). This is a great way of involving the student because it keeps them engaged in the class discussion in a subject that tends to not be that popular in schools. James V. Bruni and Helene J. Silverman, of the Herbert H. Lehman College in the City University of New York, write how we should be taking advantage of the calculator. Bruni and Silverman say that students are “anxious to have a chance to use the calculator” (Bruni 494). They then assert that students want to learn about the technology because they know when something is relatively new to our society. This is because it is likely that the teacher was excited to try it out as well. When the teacher is truly excited about something, it can spark interest in the students’ minds. Once students have come to master basic mathematical process, they should start incorporating the calculator into their lessons as a form of confirming the student’s handwritten answers, like how Immerzeel suggested. This is what a study by Brian P. Beaudrie and Barbara Boschmans did. Students used calculators’ geometry and transformation functions, but did hands-on work before that. Students originally used the peg boards to do the problem, and then used the calculators function to confirm their original answer (Beaudrie 444). The students learned through the use of the physical peg board and eventually applied it to the use of the calculator. This may only be one study, but it shows how calculators can be used as a confirmation test instead of a complete substitution.  On the other hand, they analyze how students could build up a dependence on it. The calculator’s initial response was full of excitement, but now it something that we simply take for granted. In addition, it can potentially give students anxiety when they lose access to it.

I’ve seen it plenty of times that students become so accustomed to solving problems on their calculator that when they lose access to it on a test, they stress themselves out and bomb the test. Most of the students are familiar with the process of solving these problems, they have just made it a habit to use their calculator as a way of substituting for the math done by hand. David Corfield, current philosophy professor of Mathematics for the University of Kent, states it best that “there is a fine line between clarifying the boundaries of an old domain and extending beyond them into a new domain, but there are cases which are clearly on one side or the other” (Corfield 205). This means that it is okay to have uses for both the old and the new technology. Once we fail to see the positives on one side or another and throw that option out the window, we have made a complete substitution of the possible technique or technology. We as a society expect that because we have the tool provided to solve all of our problems for us that we don’t have to learn the hand done steps, which is horribly wrong. How are you going to know what to enter into the calculator if you don’t know how to approach the problem in the first place? The calculator can give an exact answer, but will only give an answer relative to the information provided to it. Why do people resort to the calculator to solve their problems? The answer is simple; we as a society choose the easiest route to get a job done, which is why the crutch of a calculator has expanded to phones and mobile calculation.

We as a society have grown quite fond of our technologies and have failed to realize how privileged we truly are. Tony Barnstone, an English professor at Whittier College, wrote of his experiences with “The Great Blackout of 1996”. When the power went out and he lost access to both his computer and his typewriter to write, he was so accustomed to having those luxuries that he couldn’t even bring himself to use “the lowly technology of the fountain pen” to do his writings (Barnstone 191). This is very reminiscent of people being afraid of change. Change is necessary to make progress in any field and it is how we advance as a society. Similar to Barnstone’s experience with his routine, workers don’t like to break routine as they fear losing steam in their efforts. Carol Somoano, a certified financial planner with Asset Planning Inc. in Cypress, shared her experiences using more ‘basic’ technology when the newer technology has been made available to her. She is stuck in routine of using an old HP calculator and said “I feel naked when I don’t have it and have to use something like Microsoft Excel instead” (White). Sometimes the simplistic technology is something that people are so accustomed to and they feel they perform better with it. On Barnstone’s side, he became so accustomed to using the newer technology, that he couldn’t bring himself to using the older devices. To me, this is similar to how the students that regularly use the calculator may panic when they lose access to it. Sure, it doesn’t seem like a huge deal when using it for basic addition and subtraction that almost everyone is familiar with, but once students start taking the higher level calculus courses where they are required to solve integrals without graphing capabilities, an issue arises. Integrals, determination of area under a curve, aren’t an easy process to solve and it takes practice to truly understand it, so why use a graphing calculator’s graphing functions to simply spit out the answer and you have no idea what it means. We live in a society so dependent on the handheld luxuries provided that we forget what life was like before we had them.

The younger generations today are dramatic about the loss of technology. They can go forever without learning real-life analytical skills, but flip out the second they lose access to their phones. How often is it that when someone goes to the grocery store and resorts to their mobile phone’s calculator application to determine which of two products is a better deal? It shouldn’t be that hard to know how to determine a unit price, or even make a close approximation. You’re in a grocery store, not NASA. I was in the grocery store a couple days ago with some of my friends to pick up some groceries and saw a group of guys trying to determine the better deal between two products. To find out, they pulled out their phones for a 2 for $5 or a $2.75 each sale. This is a relatively easy calculation and they took the easy way out of it by resorting to the use of the calculator, especially when there is a high likelihood that they were college students. On the other hand, it may not be the children that are the problem. Maybe they are reacting similarly to how some adults do when their phones are dying. I’ve seen adults frantically searching for an outlet at the fear of losing access. Maybe children are just seeing the example their parents set forth and they follow in suit because kids are usually instructed to follow the etiquette and actions that their parents do. There’s a reason that when you go online and search phone addiction, you’ll see plenty of results like “10 Signs That You’re Addicted to Your Phone” or “How to Beat Your Phone Addiction”. In addition, you could potentially find some cartoons explaining how reliant we are on these devices. I gave a quick search on YouTube and found a dark-humor view of smartphones called “Very funny cartoon animation about smartphone addiction”. All of the characters in the video are complete oblivious to everything that is going on around them because their phones are their top priority (Suncheez79). Even though this is exaggerated, this is a message that mobile technology can dominate our interests if we don’t have a way of controlling it. The calculator by definition is an example of mobile technology that is overused due to its availability today. When they were first introduced, not many could afford to have them so they only were owned by a select group of people. Now you can buy a basic calculator or even a scientific calculator for around $10. With their widespread availability, more people are getting acquainted with them and their positive effects in terms of quick and reliant calculations. Now we need to focus more on those people knowing how to properly use a calculator. In addition, they need a reminder of how important hand-done calculations are for the future of our workplace. But, is being a worker a person’s sole purpose in life?

Yes, we want to live in a world that has competent workers so that we can be assured that we can live the lives that our parents gave for us before, but is that the purpose of the human race? Our schools are almost force-feeding education to students and what good is that if they make students disinterested by doing so? The calculator may be helpful in society, but it is another tool that we are putting in the hand of every student to solve problems. By giving these students these problems, is there the slightest thought in the back of your head that maybe we are taking away students’ individualities by making all of them learn the exact same things as if we as a society are thinking of them as robots? I think that we need to offer more variety in how we are teaching students so that students know similar processes, but are given the opportunity to focus on something they are interested in. We focus so much on making sure that students can get a right answer, but what’s the point if they don’t know the process. Even if they make a mistake in the process, you can track down the mistake and address it and see if they continued to follow the right steps to get to an answer that makes sense from their original calculations. Also, students are losing out on the philosophical thought of mathematics when they are instructed in this manner. In Charles Seife’s book Zero: The Biography of a Dangerous Idea, he makes claims to how the concepts of zero and infinity “destroyed the Aristotelian philosophy” (105). Before that, Seife goes into how a set of calculations determine our output on life. Seife uses an example of having a 50% chance of earning $100, we have an expectation of getting $50 out of the experience (as 0.5 multiplied by 100 is 50). Later, Seife talks of how we incorporate zero and infinity to this discussion and it causes us to make extremist claims, his claim being that there is a God (Seife 103). I believe his claims are radical, but there is some merit to chance that he brings forward. We live in a world that looks at opportunity cost and other factors to decide whether we will follow through on a decision. We now teach these concepts to students in comparison strategies, but we also teach the concepts of infinity and zero (even though we focus more on zero). We teach students the concept of dividing by zero and how it is undefined, but we look at the limits before students take an intro-calculus class? They don’t recognize dividing a number minutely close to zero becomes infinity as the number gets closer and closer to zero. In an interview with Andrew Hargreaves, Dean of the School of Education at Boston College, he asserted that “it is inspiring visions rather than imposed system targets that are more likely to move a whole system forward successfully and sustainably” (Gillies 201). This is a direct shot at standardized testing where teachers are supposed to get students up to a certain competency for these exams. These imposed targets can cause confusion in students, and when they become confused, they become disengaged with the class. Sometimes during the process, students that get the process get less help from the teacher because the teacher has to do more to assist the struggling students. Those students that are not getting help, may see more confusion, then they become disengaged. This endless loop goes on and on if we can’t recognize that not every kid is the same, so their educational experience should be as well. Once students find that thing they have a high level of interest in, they will find school to be more interesting. If school is more interesting to them, they will be able to find some sort of an opportunity that allows them to follow whatever passion they may have. No person should be forced into a job that they don’t have a high level of interest in. That’s the purpose of an education: to give students the opportunity to find what they want to do later in life. They may have to work extra to do other extravagant events, but at least they will know what it takes to get to that point. Students need to be given the opportunity to choose what they study, even if it is giving them a limited number of choices, give them the option to choose the topic that interests them the most. By giving papers and tools (such as calculators) for students to mindlessly solve problems only cripples their ability to learn.

The development of the handheld calculator and its prominence in our society has shaped how we see mathematics today. Major industries are able to apply their vast knowledge of mathematics and perform calculations with the assistance of calculators at a relatively rapid pace compared to how it was before the calculator had become so readily available. In addition, the calculator sparked interest in mathematics for the students, but also caused some students to use it as a substitution for the basic mathematical operations that we often take for granted. The students that used the calculator as a crutch may have taught us that we shouldn’t be forcing mathematics upon students. Instead, we should give the students a set of options that allows for a more interactive experience in their learning. The educational system has a belief that all students have to be taught the same thing and we are to prepare them for “the real world”, but what good is that if we take away the student’s individuality in the process? The calculator’s influence of convenience brought up a lot of realizations that likely wouldn’t have come into play had we stuck to the hand-done calculations, both good and bad. The fact that something so small that we use every day has such a large impact in our daily lives makes us appreciate the technologies we have available, but also causes us to think about our reliance upon them.

 

Works Cited

Barnstone, Tony. “Technology as Addiction.” Technology and Culture 41.1 (2000): 190-193. JSTOR. Web. 7 Apr. 2017.

Beaudrie, Brian P., and Barbara Boschmans. “Transformations and Handheld Technology.” Mathematics Teaching in the Middle School, vol. 18, no. 7, 2013, pp. 444-450. JSTOR, doi:10.5951/mathteacmiddscho.18.7.0444. Accessed 4 Apr. 2017.

Bruni, James V., et al. “Taking Advantage of the Hand Calculator.” The Arithmetic Teacher, vol. 23, no. 7, 1 Nov. 1976, pp. 494-501. JSTOR, http://www.jstor.org/stable/10.2307/41189069?ref=search-gateway:83e7c5746794748e8596699b34f28e1b. Accessed 4 Apr. 2017.

Corfield, David. Towards a Philosophy of Real Mathematics. Cambridge, Cambridge University Press, 2006.

Figuers, T.N. “Relative Educational Value of Mathematics.” News about Chronicling America RSS, The Columbia Herald, 21 Apr. 1899, chroniclingamerica.loc.gov/lccn/sn96091104/1899-04-21/ed-1/seq-3/#. Accessed 11 Apr. 2017.

Gillies, Robyn M. “Education Reform: Learning from Past Experience and Overseas Successes.” Change! – Combining Analytic Approaches with Street Wisdom, ANU Press, 2015, pp. 193-204. JSTOR, http://www.jstor.org/stable/j.ctt16wd0cc.17. Accessed 13 Apr. 2017.

Hamrick, Kathy B. “The History of the Hand-Held Electronic Calculator.” The American Mathematical Monthly, vol. 103, no. 8. Oct. 1996. Pp. 633-639. Accessed 2 Apr. 2017.

Immerzeel, George. “ONE POINT OF VIEW: The Hand-Held Calculator.” The Arithmetic Teacher, vol. 23, no. 4, 1 Apr. 1976, pp. 230-231. JSTOR, http://www.jstor.org/stable/10.2307/41188948?ref=search-gateway:f36c1822d0c5flflffa167aa9c4a652a. Accessed 3 Apr. 2017.

Miller, Michelle A., et al. “Functionality Disable and Re-Enable for Programmable Calculators.” US20130290389A1 – Functionality Disable and Re-Enable for Programmable Calculators – Google Patents, 27 June 2013, patents.google.com/patent/US20130290389A1/en?q=handheld%2Barithmetic%2Bcalculator&sort=new&page=1. Accessed 21 Mar. 2017.

Moyal, Ann. “The Reflective Years.” Maverick Mathematician: The Life and Science of J.E. Moyal, ANU E Press, 2006, pp. 103-115

Recorde, Robert, et al. Arithmetick, or, The Ground of Arts. 1699

Seife, Charles. Zero: The Biography of a Dangerous Idea. London: Souvenir Press, 2000. Print.

Suncheez79. “Very Funny Cartoon Animation about Smartphone Addiction.” Youtube, 22 Apr. 2015, http://www.youtube.com/watch?v=6Mwpmjf6cwE. Accessed 7 Apr. 2017.

White, Ronald D. “Dented and Duct-Taped, Old HP Calculators Still Beloved by Planners.” Los Angeles Times, 28 Aug. 2013, http://www.latimes.com/business/la-fi-mo-old-hp-calculators-20130826-story.html. Accessed 16 Apr. 2017.

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Presentation: Calculators and their Impact on Public Education

When going to the store and you are trying to find the better offer between two products, do you do the calculation in your head, or do you pull out your phone’s calculator application to do it for you? If you resort to your calculator application, you are not alone. The majority of our society has been quick to use the tools we have provided, and the calculator is one of the most universally available to us. The handheld electronic calculator may have accelerated the process of solving math problems with larger numbers, but it has also inhibited the ability to teach elementary-age students basic mathematics and contributed to the idea that a person’s sole purpose in life is to eventually join the workforce. To truly understand the impact that the calculator has had on our society, we have to look back to life before the calculator and when it was first brought forward.

The process of doing calculations changed significantly in the mid-1960s to the early 1970s. Before the 1960s, all calculations were done by hand, similar to the way we are taught basic mathematics from 1st through 6th grade. In 1965, desktop calculators were the most common calculating device and they had to be plugged in, were the size of typewriters, and cost as much as an automobile. The idea that access to a calculator cost as much as a car during these years sounds ridiculous, which is why it is likely that the initial calculators were only owned by a small portion of the population. The main selling tactic of Texas Instruments was to sell to male workers, because there was still a large difference in the number of male and female workers in the workplace. Eventually, calculators were beginning to go mobile instead of just sticking to desktop designs. The initial handheld calculators were introduced by Texas instruments in 1970 and were called “Pocketronics”. These options were cheaper than the desktop calculators as it cost approximately $400 for a handheld calculator and approximately $2000 for a desktop calculator at the time. I did a brief conversion on an inflation calculator that gives an estimation of how much $400 would have been worth in 1970. Through my search, I found the value to be right around $2500. This means that if the calculator was invented today, it would cost you $2500 to have one of your own. Remember, this isn’t a graphing calculator, this is your basic calculator with simple addition, subtraction, multiplication and division functions. This shows how a new idea can cost a pretty penny when it first hits the shelves. Eventually, the prices dropped over time and calculators were beginning to be incorporated into the classroom setting.

When something new is brought into a learning setting and is readily available to students, they are going to want to try it out. Teachers were also learning about the new devices, just like the students, and they recognized that letting the students use them would keep them engaged in their teachings. Most of the classrooms could only get one calculator, as they still had a relatively hefty cost for the amount of functions they had. With that limitation, some teachers would allow one student to use a calculator for the entire day. Throughout the day, the teacher would call on the student with the calculator to answer questions for calculations relating to the class discussion. By doing this, the student with the calculator became more engaged in the class discussion, and they also became more excited about the subject of mathematics. As time will tell, the calculator developed in both processing speed and the function capabilities that were programmed to them. They can now do functions for graphs, statistics, even determinants and other matrix functions. With these new functions, we have to maintain the grasp on the fact that the calculator should still be used as a confirmation test to the hand-done calculations instead of making it a complete substitution for the math that we have been doing for hundreds of years by hand.

It’s no surprise that a large portion of the population doesn’t consider math to be one of their favorite subjects. Maybe it’s because it can be difficult at times, but maybe it is because we have become so accustomed to using the tools we have provided to do our math problems for us. People shouldn’t struggle to complete simple multiplication and division processes when they lose access to their technology. How will students be able to work in their desired jobs without knowing the processes of long division or multiplication with zero placeholders if they were to lose access to their calculators? These are processes that they should’ve learned in elementary school and mastered throughout the rest of their education. They should be getting these concepts, but maybe by forcing these concepts and expectations on them, we are losing the grasp on the purpose of an education.

When we have the idea that we are training workers when sending students to schools, we defeat the purpose of an education. Andrew Hargreaves, the Dean of the School of Education at Boston College, asserts that “it is inspiring visions rather than imposed system targets that are more likely to move a whole system forward successfully and sustainably”. Every student’s experience should be different from the other, as it takes away their individuality when we as a society are just programming our future wave of workers. The calculator is a key tool of this idea because of our standardized tests that we are giving to students. Students take standardized tests to see level of understanding by area and these tests permit the use of calculators. Instead of going through the process of solving problems that will likely be on the tests by hand, many teachers are just teaching their students about the calculator functions so they can just plug and chug with the numbers and commands. After this, students see an answer on their calculator’s screen and don’t know what it means. In addition, they don’t know if the answer obtained is correct or not, they just assume it is correct because there is the stigma that the calculator is never wrong, when in reality, the calculator can only determine an answer based on what you enter into it.

The calculator’s prominence in our society has created a quicker solution to the basic problems that we do on a regular basis. On the other hand, it’s prominence has possibly led to a dependence upon it. If there is a middle ground where we can use the calculator to speed up the process of solving problems while still making an effort to master the hand-done math we have done for years, the calculator can reobtain its original purpose.

Dented and Duct-Taped, Old HP Calculators Still Beloved by Planners

White, Ronald D. “Dented and Duct-Taped, Old HP Calculators Still Beloved by Planners.” Los Angeles Times, 28 Aug. 2013, http://www.latimes.com/business/la-fi-mo-old-hp-calculators-20130826-story.html. Accessed 16 Apr. 2017.

This article is how some of the more basic calculators that we thought were leaving the shelves are still fairly popular in some businesses. A lot of workers don’t have the need for the multi-function graphing calculators that can be confusing to use at times, so they keep it simple and use the basic function calculators (like the old HP ones). A big shock that I had about this article was asking users about their experiences with them. One of them even mentions taking a class to learn how to use it properly! Nowadays, we a higher amount of videos teaching how things work and instruction manuals are always available, so the process of learning to use calculators is much more individualized. These people also talk about how they don’t want to break from their routine of using the calculator. Carol Somoano, a certified and fee-only financial planner with Asset Planning Inc. in Cypress, said things like “I feel naked when I don’t have it and have to use something like Microsoft Excel instead”. Now we are seeing students learning how to use Microsoft Excel and its functions in their classes, similar to how there were classes to learn to use the calculator. This leaves me wondering whether Microsoft Excel will be replaced by something bigger and better like how the basic HP calculators were.

The Reflective Years

Moyal, Ann. “The Reflective Years.” Maverick Mathematician: The Life and Science of J.E. Moyal, ANU E Press, 2006, pp. 103-115

This chapter of Maverick Mathematician focuses on Joe Moyal’s accomplishments through his publications in the field of quantum mechanics. He made a total of 36 publications throughout his career, but also is known for making comments about the philosophy of mathematics and its developments. He speaks highly of those that revolutionized the subject and explains how simplicity actually allowed a deeper understanding of the subject. He goes into depth about how Heisenberg’s Uncertainty Principle was discovered due to Werner Heisenberg refusing to rule out “classical-looking mathematical descriptions”. Had he ruled them out, we may have lost a grand discovery in the field of quantum mechanics that has direct links to chemistry and physics as well. The field of quantum mechanics requires a lot of both complex and simple (mostly complex though) calculations that use a wide variety of mathematical principles. Most of Moyal’s work was done between 1950 and 1980, which means that later on in his career, he could have had access to a calculator. His impact may have been limited on that end because his more popular works were done on the philosophy of mathematics in the early 1950s.

Towards a Philosophy of Real Mathematics

Corfield, David. Towards a Philosophy of Real Mathematics. Cambridge, Cambridge University Press, 2006.

This monograph by David Corfield gives insight on how mathematics developed over time and how proofs from older mathematicians were being used to prove new principles by newer mathematicians. I specifically turned my focus to the 9th chapter of the book on the values in mathematics. It speaks to how we have developed internally from the field of mathematics. We are a confident group because of it and it makes a stronger society from it. I found it interesting in the definition of development and progressiveness being split into 5 categories. The categories were that it “allows new calculations to be performed in an existing problem domain”, “a connection between already existing domains”, “a new way of organising results”, “opens up the prospect of new conceptually motivated domains” and “directly leads to successful applications outside of mathematics” (205). The last one interests me the most as it is a sign that your research can make connections to the subject without directly being about the subject. My paper is on the calculator, but I can make draws to the general scope of mathematics and the philosophy behind it.

Education Reform: Learning from Past Experience and Overseas Successes

Gillies, Robyn M. “Education Reform: Learning from Past Experience and Overseas Successes.” Change! – Combining Analytic Approaches with Street Wisdom, ANU Press, 2015, pp. 193-204. JSTOR, http://www.jstor.org/stable/j.ctt16wd0cc.17. Accessed 13 Apr. 2017.

This excerpt is from the book Change!, where it focuses on how the education systems overseas may be outperforming ours. It speaks loudly to how other nations also are looking for those with degrees to take jobs in their economy, but it also goes into a deeper thought on how that is done. The big idea I got from this article was  that “it is inspiring visions rather than imposed system targets that are more likely to move a whole system forward successfully and sustainably”. I interpret this as schools should be focused less on students meeting certain expectations (from standardized tests and other performance measuring techniques) and more on inspiring students to pursue something they are interested. If a student isn’t interested in the topic they are studying, why would they be motivated to find a job in that field or give their best work in that field. After this, it goes into strategies that teachers are following to make the education system better like seeking feedback from students, understanding students learn in multiple ways and developing positive relationships with students. No kid wants to go to the same boring lecture over and over again. A little variety keeps classes interesting and that’s exactly what nations like Australia are doing. We need to aspire to do similar things in the classroom to help students find a greater interest in their schooling instead of making it a requirement for them to meet the minimum competency that we expect now.

Presentation Start

When going to the store and you are trying to find the better offer between two products, do you do the calculation in your head, or do you pull out your phone’s calculator application to do it for you? If you resort to your calculator application, you are not alone. The majority of our society has been quick to use the tools we have provided, and the calculator is one of the most universally available to us.

The process of doing calculations changed significantly in the mid-1960s to the early 1970s. In 1965, desktop calculators were very popular in society and they had to be plugged in, were the size of typewriters, and cost as much as an automobile. The idea that access to a calculator costed as much as a car during these years is absolutely ridiculous to me. I can see how initial calculators may have been for a specific group of people that may have seen it uniqueness to society, but it seems like an outlandish expenditure at that point. Eventually, calculators were beginning to go mobile instead of just sticking to desktop designs. The initial handheld calculators were introduced by Texas instruments in 1970 and were called “Pocketronics”. These options were cheaper than the desktop calculators as it cost $400 for a handheld and $2000 for a desktop calculator at the time. I did a brief conversion on an inflation calculator that gives an estimation of how much $400 would be worth in 1970. I found the value to be right around $2500. This means that if the calculator was invented today, it would cost you $2500 to buy one of your own. Remember, this isn’t a graphing calculator, this is your basic calculator with simple addition, subtraction, multiplication and division functions. This shows how a new idea can cost a pretty penny when it first hits the shelves. Eventually, the prices dropped over time and calculators were beginning to be incorporated into the classroom setting.