tag:blogger.com,1999:blog-65159627979048799612017-06-21T21:33:48.655-07:00Complex RootsA blog about various topics in math, including things I have learned, videos I have made for students, and thoughts about mathematics education. Ask Alephnoreply@blogger.comBlogger8125tag:blogger.com,1999:blog-6515962797904879961.post-83101829137206579062015-11-10T11:58:00.001-08:002015-11-10T11:59:02.961-08:00New Videos!Inspired by a tutoring session, I have some new vidoes.<br /><br />First is a two part series about how we can represent a function in different ways.<br /><a href="https://www.youtube.com/watch?v=I4mNy9ktVWo"><br /></a><a href="https://www.youtube.com/watch?v=I4mNy9ktVWo"> https://www.youtube.com/watch?v=I4mNy9ktVWo</a><br /><a href="https://www.youtube.com/watch?v=c7hb8UscaTM"> https://www.youtube.com/watch?v=c7hb8UscaTM</a><br /><br />One of the things I see when I tutor Algebra is that there is a disconnect between the graph of a function and the equation of a function. What I aim to do is show that we can use different methods to present the same function. In the second video, I go on to show how we can extract information about the function, like the slope and the intercepts from the different forms of the function.<br /><br />The other video I put up is just running through a way to solve for x and y intercepts of a line, if all we are given is a chart of values.<br /><br /><a href="https://www.youtube.com/watch?v=7ym7XhiHkGg">https://www.youtube.com/watch?v=7ym7XhiHkGg</a><br /><br />This will probably be all of the videos I post this week, as I am preparing for my in-person interview to try to be a member of Ada Developers Academy winter cohort!<br /><br /><br />Ask Alephhttps://plus.google.com/115373432537185140953noreply@blogger.com0tag:blogger.com,1999:blog-6515962797904879961.post-6349338938744908082015-10-22T19:06:00.001-07:002015-10-22T19:14:33.466-07:00Ask Aleph!As I was working on a new math video today, I decided it was probably a good idea to have a separate account for my math videos. Thus <a href="https://www.youtube.com/channel/UC3BQQybu6Hfdxza-AbX-q9Q?view_as=public">Ask Aleph</a> was born! Ask Aleph is my new YouTube channel where people can submit questions about difficulties they have with mathematics, and I try to come up with a simple way to explain the answer. I have additionally transferred this blog over to this account as well!<br /><br />The first video "<a href="https://www.youtube.com/watch?v=YyJ39shvFGY">Differentiation is Linear</a> " comes from one of my own personal questions I had when I took Calculus I. I never quite got the answer until I got to Linear Algebra, so I introduce the Linear Algebra definition of Linear to make sense of this statement for someone who has just taken Calculus I.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/YyJ39shvFGY/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/YyJ39shvFGY?feature=player_embedded" width="320"></iframe></div>If you have a question that has confused you about mathematics, go ahead and <a href="mailto:askaleph0@gmail.com">email me</a>, and I'll try to get a video up about it!<br /><br /> Also, if you saw my earlier videos, you might have noticed that the quality of my videos has improved some. Instead of my incredibly janky setup where I stream to Twitch and export to YouTube, I use the record setting on Open Broadcast Software, and only use it to record video. I record and edit the audio through Audacity. To edit the sound and video together, I was going to try to Blender, but I was running into some troubles, so I stuck with just using Microsoft Movie Maker, until I can find some more time to tackle a more complex video editing program. Unfortunately, I lost a lot a video quality during some process. I'll have to figure out where that happened and improve it for next time!<br /><br /><br /><br /><br />Sandra Granhttps://plus.google.com/108044886653901427095noreply@blogger.com0tag:blogger.com,1999:blog-6515962797904879961.post-92173319661323665812015-10-16T15:45:00.002-07:002015-10-16T15:45:50.609-07:00What am I doing with this Math Degree?This month I have started actively seeking out new students to tutor. With this, I have been trying a new tutoring style where I am trying to engage with my students on a more macro level. Instead of just helping them week to week, homework to homework, I am attempting to learn more about their math schedule and study habits when they aren't there. I am looking to find out when their tests and quizzes are to make sure they start preparing adequately ahead of time.<div><br /></div><div>One of the other things I am doing is I'm making <a href="https://www.youtube.com/playlist?list=PLfM4DQTQo2upR5wF1b48NEBANIyPxA8GU">videos</a> on topics I don't feel like I adequately covered, or just for questions that stumped me while I was tutoring. (I don't charge for the videos). What I hope to communicate through this method is, math isn't a knowledge base you just perfect memorize you go on. Even math majors can become temporarily stumped by a question, but with time and thinking, we can achieve understanding. (The videos are of questionable quality. I'm working on trying to get the static out of the background, and perhaps have a better system for recording videos than streaming on Twitch then exporting to YouTube). </div><div><br /></div><div>My videos right now, are very tailored to what we were discussing in the tutoring session. As in, I don't go into great detail about what the student already knew, but briefly cover it and then jump straight to the part which was more difficult to work through .But watching these videos now, I feel like I should perhaps do a more general video that describes the topic, and then do a second video talking about the specific (using a related problem, so I don't take away the experience of the student being able to engage with the material they were assigned). </div><div><br /></div><div>Aside from tutoring, I have also decided to apply to be a part of <a href="http://adadevelopersacademy.org/">Ada Developers Academy</a> Winter cohort. This program is very appealing to me. It is a year long, tuition-free, intensive program for women interested interested transitioning to software development. I think this could be a great opportunity to transition my love for problem solving that I learned into mathematics into a career. When I was in junior high and high school, I loved spending my night looking through the source code on webpages, trying to teach myself html. When I reached college, for some reason I can't fathom, I never even considered how to transition this into a career. </div><div><br /></div><div>Another reason I am considering this program is, although I like tutoring and I have several years of experience doing it, I think it's going to be really difficult to make a salary in which I could support myself doing it. I think also, one of the things that bothers me about private tutoring is that I'm essentially only working with students whose parents make enough money to afford them. I could volunteer, but all of the hours that students generally can be tutored (4-8pm on weekdays) I need to use to private tutoring to make enough money. I think if I could get a solid career in web development, I could invest a little time each week to volunteering for a tutoring program for students who can't otherwise afford such a service. </div><div><br /></div><div>Also, I feel like once I learn more about web development, perhaps I can get inspired to link my expertise in both areas to create something that could really help out a lot of students. </div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div>Sandra Granhttps://plus.google.com/108044886653901427095noreply@blogger.com0tag:blogger.com,1999:blog-6515962797904879961.post-86515863565483321392014-06-22T11:21:00.001-07:002014-06-22T11:21:18.090-07:00<div class="separator" style="clear: both; text-align: center;"><object width="320" height="266" class="BLOGGER-youtube-video" classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,40,0" data-thumbnail-src="https://i1.ytimg.com/vi/EjdPiTbXDqk/0.jpg"><param name="movie" value="https://www.youtube.com/v/EjdPiTbXDqk?version=3&f=user_uploads&c=google-webdrive-0&app=youtube_gdata" /><param name="bgcolor" value="#FFFFFF" /><param name="allowFullScreen" value="true" /><embed width="320" height="266" src="https://www.youtube.com/v/EjdPiTbXDqk?version=3&f=user_uploads&c=google-webdrive-0&app=youtube_gdata" type="application/x-shockwave-flash" allowfullscreen="true"></embed></object></div> With the week and half off I've had off between spring and summer quarter, I've been tweaking a setup that would allow me to stream and record math videos. To test my setup, I discuss why I feel it is a mistake to teach Algebra I (Pre-Algbera?) students FOIL when teaching them to multiply binomials. <br /><br /><br />Sandra Granhttps://plus.google.com/108044886653901427095noreply@blogger.com0tag:blogger.com,1999:blog-6515962797904879961.post-76662141974331037672013-01-28T17:27:00.001-08:002013-01-28T17:48:31.225-08:00Put your Mouth where your Money isToday I learned that the 10 Deutsche Mark banknote depicts (prolific) German mathematician, Carl Friedrich Gauss.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-VBIspYPP9p0/UQcjTTjv8BI/AAAAAAAAGbQ/79Aock5PrSE/s1600/10-deutschmark-german-mark-gauss.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-VBIspYPP9p0/UQcjTTjv8BI/AAAAAAAAGbQ/79Aock5PrSE/s1600/10-deutschmark-german-mark-gauss.jpg" height="161" width="320" /></a></div><br />Not only that, but the bank note also depicts one of his achievements: the formula and (detailed) graph of the Normal (or Gaussian) Distribution, one of the most important probability distributions.<br />(Click bank note to see the detail of the formula). <br /><br />I wonder if a case can be made for understanding what is important to a country by what they put on their money. Canada has hockey players on their 5 dollar bill. Hmmm..... <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-02yeR-XhS9o/UQclGPztv4I/AAAAAAAAGbg/gzjUzIuze1A/s1600/Canada+2003+BC-62a-i+Back.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-02yeR-XhS9o/UQclGPztv4I/AAAAAAAAGbg/gzjUzIuze1A/s1600/Canada+2003+BC-62a-i+Back.JPG" height="150" width="320" /></a></div><br /><br /><br /><br />Sandra Granhttps://plus.google.com/108044886653901427095noreply@blogger.com1tag:blogger.com,1999:blog-6515962797904879961.post-83502352620253573442013-01-07T17:00:00.000-08:002013-01-07T17:32:30.000-08:00Equal and Equivalent are not EquivalentOne of the things that I love most about upper level math is the precision of language. Words have very specific meanings and if you don't say something precisely right, then often it is wrong. This is not referring to the idea that "People like math because there is one right answer." This is literally referring to the words we use to write out mathematical statements.<br /><br />Although this is always true in math, I did not realize the depth of it until my Linear Algebra class, where by the end of the semester we ended up getting points knocked off for incorrect language. In Linear Algebra you learn matrix mathematics as a way to solve systems of equations. In Linear Algebra you can make a statement like "the columns of matrix A are linearly independent" but you can't make a statement like "matrix A is linearly independent". Linear independence does not have a definition with regards to matrices. <br /><br /><br />Much of precision in mathematical writing has its roots in logic. For instance <i>"A is true if B is true"</i> is very different than <i>"A is true if and only if B"</i> is true. In this circumstance, the second statement is biconditional, meaning that you could swap A and B and the statement is still true. That is not the case for the first statement.<br /><br /><b>Today I learned that equal and equivalent are not words that you can use interchangeably in mathematics. In general, numbers are <u>equal</u> and mathematical statements are <u>equivalent</u>.</b><br /><br />To better understand this statement, let's look at what a mathematical statement is. <br /><br />A <i><span style="color: #6aa84f;">mathematical statement </span></i>can be defined as any mathematical sentence that has a truth value (meaning, any mathematical sentence that is true or false).<br /><br /><u><span style="color: #674ea7;">Examples:</span></u><br /><br />a.<span style="color: #3d85c6;"> 5 is a prime number. </span><br /><br />This is a mathematical statement because it has a truth value. This sentence can be either true or false. In this situation it is true.<br /><br />b. <span style="color: #3d85c6;">12 and 13 are solutions to the equation x + 1 = 0</span><br /><br />Again, this is a mathematical statement because it has a truth value. In this situation, it is false. A statement does <u>not</u> have to be true for it to be a mathematical statement.<br /><br />c. <span style="color: #3d85c6;">20</span><br /><br />The number 20 is not a mathematical statement because it has no truth value. 20 is neither true or false, it just is. <br /><br />The Oxford-English dictionary defines <span style="color: #6aa84f;"><i>equal</i></span> as "being the same in quantity, size, degree or value" (as an adjective) or "to be the same in quantity or amount" (as a verb). Meaning, equal refers to things that have a measurable size, like numbers. Mathematical statements do not have a measurable size, they merely have a truth value.<br /><br /><span style="color: #6aa84f;"><i>Equivalent</i></span> is defined as "equal in value, amount, function or meaning." In the Oxford-English dictionary we do see some overlap in these definitions. First of all the definition of equivalent has the word equal in it, which I think for the purposes of mathematics, should probably be switched to "being the same in". The definition also reuses the term of value. Nonetheless, we can see the general, albeit subtle, difference between these words. Mathematical statements are statements that have function and meaning (true or false), not a measurable quantity, therefore it is correct to use equivalence rather than equal. <br /><br />Sandra Granhttps://plus.google.com/108044886653901427095noreply@blogger.com0tag:blogger.com,1999:blog-6515962797904879961.post-50047615054660334562012-11-02T17:00:00.001-07:002012-11-02T17:06:12.893-07:00Complex Roots (Also quadratic factoring)<br />Although I am far past the initial idea of complex roots in my mathematical education, it seems correct to explain it here, since it is the name of my blog.<br /><br /> The first time I consciously remember a teacher saying the phrase, "What are the roots of this equation?" I was in a Trigonometry class my first semester at Seattle Central Community College. I strained my brain, but I seriously could not remember coming across this term earlier in my high school (which included Algebra 1 & 2, Geometry and Calculus). That night I had go teach the concept to myself (since I was too embarrassed to ask in class, which is a great thing to learn to overcome). <br /><br /><br /><b>Roots </b><br /><br /><br />The term <i style="color: #6aa84f;">root</i> is used in reference to a function.<br />The <i><span style="color: #6aa84f;">roots</span></i> of a function, f(x), are the values of x that will cause the function to be zero.<br /><br /><div style="color: #674ea7;"><u>Examples:</u></div><br />a. <i style="color: #3d85c6;">f(x) = 2x</i><br /><br />This equation has one root. That root is x = 0. No other value of x will cause f(x) to equal zero.<br /><br />b. <i style="color: #3d85c6;">g(x) = (x</i><span style="color: #3d85c6;">²</span><i style="color: #3d85c6;"> - 16)</i><br /><br />There are two ways you can think of solving this. The first thing that I see when I look at this is a difference of squares. Difference of squares can be factored every time using this method.<br /><br /><br /><div style="color: #3d85c6;"><i>(x²</i><i>- c²</i><i>) = (x + c)(x - c)</i></div><br /><br />Using that method in this equation we have<br /><br /><div style="color: #3d85c6;"><i>g(x) = (x</i>²<i> -16)</i></div><div style="color: #3d85c6;"><i> = (x + 4) (x - 4)</i></div><br /><br />At this point we can see that if either<i> <span style="color: #3d85c6;">(x + 4) = 0</span></i><span style="color: #3d85c6;"> </span>or <i style="color: #3d85c6;">(x - 4) = 0 </i>then the whole function will equal zero due to the Zero Product Property (any number multiplied by zero will equal zero).<br /><br /><br />The roots of this equation are therefore<i style="color: #3d85c6;"> x = -4</i> and <span style="color: #3d85c6;">x = 4</span>.<br /><br /><br />The other way we could have done this problem was not by factoring but just considering the problem <i style="color: #3d85c6;">(x² - 16)</i>. What values of <i>x</i> would give us 16? What numbers squared equal 16? 4 and -4.<br /><br /><br /><br /><b>Quadratic Formula</b><br /><br />Not all equations can be factored using simple algebraic methods, such as the difference of squares method. When it is not obvious what the factors are, or you cannot find a method to easy factor the equation, it becomes necessary to use the quadratic equation. <br /><br />The term <i style="color: #6aa84f;">quadratic equation</i>, refers to an equation with one variable (x, usually) with a degree of two (meaning, the largest exponent in the equation is 2). The basic form for a quadratic equation is<br /><br /><div style="color: #3d85c6;"><i>ax²+ bx + c = 0</i></div><br />A quadratic equation will always have two roots. <br /><br />Where x is the variable and a, b, and c all represent constants. To use the quadratic formula, you must put the equation in this form. Once the equation is in this form, you plug the values from that equation into the quadratic equation.<br /><br /><i><span style="color: #3d85c6;">(-b </span><span class="st" style="color: #3d85c6;">± </span><span style="color: #3d85c6;">√ b² - 4ac ) / 2a </span></i>.<br /><br />(Full disclosure: For some reason, the quadratic equation was one of the gaps in my education. I still have hard time recalling it from memory. I seriously had to double check it for this post. Don't be like me, just pound it into memory the first time you learn it.)<br /><br /><div style="color: #674ea7;"><u>Example:</u></div><br />c. Factor <span style="color: #3d85c6;"><i>f(x) = x² + 5x - 2</i> </span>using the quadratic equation.<br /><br />The value for a, b and c are a = 1, b = 5 and c = -2. We simply just plug these into the quadratic formula.<br /><i><br /></i><br /><div style="color: #3d85c6;"><i>[-5 <span class="st">± </span>√ 5² - 4(1)( -2)] / 2(1) </i></div><br /><div style="color: #3d85c6;"><i>[-5 <span class="st">± </span>√41] / 2</i></div><br />The roots for this equations are <i><span style="color: #3d85c6;">x = [-5 </span><span class="st" style="color: #3d85c6;">+ </span><span style="color: #3d85c6;">√41] / 2</span></i> and <i><span style="color: #3d85c6;">[-5 </span><span class="st" style="color: #3d85c6;">- </span><span style="color: #3d85c6;">√41] / 2.</span></i><br /><br />Although these numbers look "complex" they are not actually complex numbers. They are two real numbers, real roots to the equation.<br /><br />(Fun fact: There is also a cubic formula and a quartic formula, a general equation for solving for the roots of a third degree or fourth degree polynomial. The cubic formula is rather big, but still digestible, and the quartic formula is ridiculously huge. There actually can be no quintic or higher general function, however. This was proven in the 1800s. The proof for it is called Abel's Impossibility Theorem). <br /><br /><b>Complex Roots</b><br /><br />A <i><span style="color: #6aa84f;">complex number</span> </i>is a number that includes the imaginary unit. The imaginary unit is (√-1). The imaginary unit is represented by a lower case <i>i</i>. The imaginary unit is literally defined as <i>i² = - 1 </i>All complex numbers can be written in the form <i>a+bi</i> where a and b are constants. <i>a</i> is referred to as the real part of the number and <i>bi</i> is referred to as the imaginary part of the number. The idea of an imaginary number, intuitively, is difficult to grasp. I will not attempt to explain it here (mostly because I have problems grasping it myself). <br /><br />Some quadratic equations, when put into a quadratic formula, give complex roots instead of real roots. This happens when the <i>4ac </i>term of (√<i> b² - 4ac</i>) is larger than the <i>b² </i>term.<br /><br />( <i>b² - 4ac</i> is also referred to as the <i style="color: #6aa84f;">discriminate</i> of the quadratic equation. A quadratic equation can have either one real root, two real roots, or two complex roots. If the discriminate = 0, then the equation will have one real root, if the discriminate > 0 the equation will have two real roots, if the discriminate < 0 it will have two complex roots. This can be useful to get an idea of what you are working with before you star the problem).<br /><br /><div style="color: #674ea7;"><u>Example:</u></div><br />d. Factor <i style="color: #0b5394;">f(x) = x² + 2x + 10</i><i style="color: #0b5394;"> </i>using the quadratic equation.<br /><br />The quadratic equation is <i><span style="color: black;">(-b </span><span class="st" style="color: black;">± </span><span style="color: #3d85c6;"><span style="color: black;">√ b² - 4ac ) / 2a</span>.</span></i><br /><br /><i><span style="color: #3d85c6;">a = 1, b = 2, c = 10</span></i><br /><br /><i><span style="color: #3d85c6;">[-2 </span><span class="st" style="color: #3d85c6;">± </span><span style="color: #3d85c6;">√((2)² - 4(1)(10)] / 2(1)</span></i><br /><br /><i><span style="color: #3d85c6;">[-2 </span><span class="st" style="color: #3d85c6;">± </span><span style="color: #3d85c6;">√4- 40] / 2</span></i><br /><br /><div style="color: black;">You can tell that we will have complex roots based on the fact that the discriminate will be a negative number. </div><br /><i><span style="color: #3d85c6;">[-2 </span><span class="st" style="color: #3d85c6;">± </span><span style="color: #3d85c6;">√-36] / 2</span></i><br /><i><span style="color: #3d85c6;">[-2 </span><span class="st" style="color: #3d85c6;">± </span><span style="color: #3d85c6;">(</span></i><i><span style="color: #3d85c6;">√36)</span></i><i><span style="color: #3d85c6;">(√-1)] / 2</span></i><br /><i><span style="color: #3d85c6;">[-2 </span></i><i><span class="st" style="color: #3d85c6;">± 6</span></i><i><span style="color: #3d85c6;">(√-1) / 2</span></i><br /><i><span style="color: #3d85c6;">[-2 </span></i><i><span style="color: #3d85c6;"></span></i><i><span style="color: #3d85c6;"></span></i><i><span class="st" style="color: #3d85c6;">± 6i] / 2 </span></i><br /><br /><i><span style="color: #3d85c6;">-1 </span></i><i><span style="color: #3d85c6;"></span></i><i><span style="color: #3d85c6;"></span></i><i><span class="st" style="color: #3d85c6;">± 3i</span></i><i><span class="st" style="color: #3d85c6;"> </span></i><br /><br /><span class="st" style="color: black;">The roots of this equation are</span><i><span class="st" style="color: #3d85c6;"> x = </span></i><i><span style="color: #3d85c6;">-1 </span></i><i><span style="color: #3d85c6;"></span></i><i><span style="color: #3d85c6;"></span></i><i><span class="st" style="color: #3d85c6;">+ 3i </span></i><span class="st" style="color: black;">and </span><i><span style="color: #3d85c6;">x = -1 </span></i><i><span style="color: #3d85c6;"></span></i><i><span style="color: #3d85c6;"></span></i><i><span class="st" style="color: #3d85c6;">- 3i</span></i><span class="st" style="color: black;">.</span><br /><i><span style="color: #3d85c6;"> </span></i>Sandra Granhttps://plus.google.com/108044886653901427095noreply@blogger.com0tag:blogger.com,1999:blog-6515962797904879961.post-9500871922790467612012-10-31T14:59:00.002-07:002012-10-31T15:00:21.302-07:00Fundamental Theorem of Math Blogs<br /><style><!-- /* Font Definitions */ @font-face {font-family:Times; panose-1:2 0 5 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:"ＭＳ 明朝"; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:128; mso-generic-font-family:roman; mso-font-format:other; mso-font-pitch:fixed; mso-font-signature:1 134676480 16 0 131072 0;} @font-face {font-family:"ＭＳ 明朝"; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:128; mso-generic-font-family:roman; mso-font-format:other; mso-font-pitch:fixed; mso-font-signature:1 134676480 16 0 131072 0;} @font-face {font-family:Cambria; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 0 0 0 1 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"ＭＳ 明朝"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} @page WordSection1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} </style><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">In the Internet age, there is a general demand that any question we have, there is an answer for. This convenience in some ways has corrupted our education. If there is a problem we don't know the answer to, it is extremely tempting to just Google the problem to find the solution. We have lost touch with the idea that discovery requires struggle, effort. This is a huge problem in mathematics. From arithmetic through things like the first calculus series, differential equations and linear algebra it is mostly possible to type verbatim a problem in a textbook and find an answer for it online. I myself have done this many times. My justification, and the justification I have heard for other people is.. I had no idea how to start and I just needed some direction. We feel like it is no different than going to office hours, or asking in class how it is done. The main difference here is, the teacher had no say about how much of the problem you saw, and if you did see the problem completed in completion, there is no possible way for the work to be your own anymore. I came to this realization during my Linear Algebra class, where half the class was accused of cheating by coping a written proof that used information that was not within the bounds of what we learned in this class. She did not say who in the class it was, but I think some people came to her office, possibly even ones that didn't cheat, with guilty consciences about to what extent they were using "outside resources." My professor's point of view? She viewed using any answer that she did not give cheating. She truly saw homework as a line of communication between herself and the students. That circumventing her and consulting the Internet with your problems was not admitting that you were having difficulty with the subject. She also felt it cheated you out of genuinely struggling to find the answer on your own. She felt that this struggle was paramount to actually understanding what was going on.</span><br /><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">This is the purpose of this blog. This blog is a journal of my struggle to understand mathematics in a truly meaningful way. This doesn't mean I need to understand the application of everything. I just want to be able to demonstrate true understanding of what I am learning about, and I want to start using the Internet in a new way. I want to share how I came to understand the things in math in my own language. By posting it on the Internet, I want to open up my work and my wording for the world to criticize. This blog may contain mathematically incorrect information for this reason. What this math blog does contain is mathematics to the best of my knowledge. </span></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">One of the roadblocks to starting this blog was actually quite silly. I wanted a clever name for it. Instead of struggling with this issue, I outsourced the problem to my overly clever friends, who came up with a fairly sizable list of good ideas, such as...</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;"><br /></span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Sum Blog</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Off on a Tangent</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Different(ial) Strokes</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Road Sines</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- The Right (Angle) Stuff</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Integrated Education</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Linear Algeblog</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Fractions of a Thought</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Actue Math Blog</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- You Can Count on Me</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Parabloggic Curve</span></div><div class="MsoNormal"><span style="font-family: Times; font-size: 10pt;">- Complex Roots</span></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">I ended up choosing Complex Roots for its multiple meanings. </span></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">First of all for it's mathematical meaning (which will be given in the next post).</span></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">Secondly, I like the imagery that the word "roots" brings. Roots in a more poetic sense can refer to a foundation. I think many people have complex roots in mathematics. For some people, at some point in their life, they got convinced that they were "bad" at mathematics. Or weren't a "math person." These people may not have been bad at math their whole life, but the just hit a wall at some topic and never broke through it. Some other people want to enter into a science or engineering career and see upper level mathematics as this huge roadblock to getting there. People's emotional foundations with math can be very complex. </span></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">Third, in a more academic sense, mathematics is an extremely cumulative subject. Rarely do we learn something in mathematics that is not a precursor for something to come. At some point, a class may have moved to fast for us, or we skipped a class, or we got sick and there ended up just being a hole in our mathematical education. There is an assumption as we progress through mathematics that we have perfect understanding of all the mathematics that came before that class. There is an assumption we all have the same mathematical foundation, the same mathematical roots. But this is not true. Some of our roots may be more developed than others. Even if we never missed a class, got a 4.0, over time we forget things that we don't use that much. </span></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">As I've progressed through upper level math (I am in Linear Analysis right now, which is the last class in the Linear Algebra, Differential Equations series. This series usually comes after the first calculus series). I have found some of the questions that take the most time in class do with a gap in understanding from something much more basic. In my Linear Analysis class the teacher spent 20 minutes explaining how to factor a polynomial of degree 3, then how to divide polynomials, both topics covered in Algebra.</span></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><br /></div><div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: Times; font-size: 10.0pt; mso-bidi-font-family: "Times New Roman";">No matter what your mathematical roots are, I hope that you find this blog insightful in some way. I plan on most of my posts being about explaining a mathematical topic that I am currently working on. I also plan on returning to topics where my own roots are not well developed. Since one of my goals in life is to become a math teacher, I may also write my thoughts and research about mathematical education. I am also willing to write posts on roots you want to develop. In fact, I hope to be able to respond to any and all questions and criticism I receive. I want to live in a world where failure and mistakes in mathematics are completely fine. Where it is totally fine to not understand something at the same rate as someone else, or to understand math at the rate your school wants you. I will never insult *anybody's* question, no matter how basic they are. Nobody is ever stupid for not knowing a piece of mathematical knowledge. It is just simply where you are in mathematics. </span></div>Sandra Granhttps://plus.google.com/108044886653901427095noreply@blogger.com0