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Roots of a number and complex numbers
Posted by holy-cow on September 4, 2016 at 2:07 am[USER=7674]@MathTeacher[/USER]
I had some fun last evening while attending an open house at Mrs. Cow’s school. I was in the high school chatting with the upper level English teacher and the upper level math teacher as they stood by the side by side doorways leading into their classrooms. A young fellow who is beginning his senior year came by with his parents to chat with both teachers. He was discussing looking forward to the challenge of the pre-calculus class with the teacher who is in about his 49th year of teaching. I told him it gets easier as you move into the higher level classes because sometimes you can give imaginary answers and get the answers right. He looked at me like I was nuts. The math teacher grinned and said something about the square root of negative one. That’s when I added some discussion about roots of numbers and how many roots are imaginary. I reminded him that there are two square roots of a number but there are three third roots and five fifth roots, etc. The math teacher was amazed that I had remembered that trivia for so many years. Meanwhile the kid knew I must be on some sort of mind altering drug because that made absolutely no sense to him. But, since he said he intended to major in Computer Engineering, he needs to get all the irrational/complex/imaginary/un-real math he can find. Those guys eat, drink and sleep that stuff.
mathteacher replied 8 years, 1 month ago 8 Members · 26 Replies -
26 Replies
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Right you are, Mr. Cow. Sometimes I would take part of a period to discuss college majors. The assignment following the discussion was for students to find the first two years of course work for their chosen major at their preferred college. Computer science requires some heavy math courses and that was an eye-opener for those who thought that being good at playing games meant that developing games must be easy. Check it out at K-State or your preferred school.
I had a great young teaching comrade who was a recent computer science graduate from, ahem, THE University of North Carolina. We often debated whether the repeating decimal 0.999999… is really equal to one. We would send students with problems to stump each other. Kids loved it and we uncovered more than one math nerd in the bunch. Truth is, he was way smarter than me.
From what I’ve seen of Kansas schools, you folks have an excellent approach to education and some excellent and involved teachers. I’m sure the math teacher you chatted with is an excellent example.
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When I was in high school I sometimes played with numbers. I remember being so excited when I discovered a relationship between neighboring integers raised to a power and the factorial of that power. I started with a number line, raised the integers to a given power, then started subtracting left from right proceeding down. The common number at the bottom of each series is the factorial of the power to which the integers are raised.
In later years some mathematician types explained the phenomenon to me — it’s apparently a well-known relationship — but it went way over my head.
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It is unfortunate that most guidance counselors who are assigned to assist the young students find their way towards useful potential careers have so little real knowledge of what basic interests and skills are necessary. Odds are high that the parents have even less ability to guide their young in suitable directions. All sorts of tests are given but little benefit ever comes directly to the student. Telling them something like, “You have an aptitude for statistics.” means nothing if they have no concept of what exactly is “statistics”, what occupations hire statistics majors, what those occupations pay and where would someone need to live to be employed doing such work.
One very obvious source of help would be access to the schedule of classes required to acquire a bachelors degree in a specific curriculum, e.g., chemical engineering. But, additional help would be needed to explain what knowledge and pre-requisite skills are needed for each of those classes along with some guidance on what the individual class consists of in terms younger students can comprehend, i.e., organic chemistry as opposed to inorganic chemistry.
Many degree programs require far more math and science than anyone would guess. We had a couple of degree opportunities at K-State in Milling Science and Feed Science. To the unknowing it would appear that only a minor knowledge of mathematics would be required. Wrong. They took pretty much all the same science and math classes as engineers and physics and chemistry majors. Music majors had a special version of physics class aimed directly at their need to understand the science of music. Upper level psychology majors were required to take a class or two in statistics. Upper level economics majors took at least one calculus class.
University study is not for everyone. Too many learn this the hard and expensive way. Better guidance during their formative years would be a start towards correcting this problem.
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Jim, that’s a neat table and a great example of discovery through exploration.
The final factorial relationship is a consequence of the binomial theorem and the relationship between the degree of an expression and the number of differences required to get a constant. Proving it is tedious, but there are some good pieces on it on the internet.
dr. math is the go-to site for math explanations. Here’s his take on constant differences:
http://mathforum.org/library/drmath/view/56953.html
This one’s not so good on the factorial, but it shows how tedious proving it is.
https://mathlesstraveled.com/2012/02/16/differences-of-powers-of-consecutive-integers-part-ii/
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[USER=677]@GeeOddMike[/USER]
Love the shirt. Are these only available in Cambridge?
Did not know until a couple of minutes ago that the school mascot is named Tim the Beaver. A bit palindromic, if you ask me.
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[USER=50]@Holy Cow[/USER]
Your K-State example touches on another important point, choosing the right school. Schools with an engineering focus are likely to require significant STEM classes, even for non-technical majors. On the other hand, liberal arts colleges are likely to require more humanities courses even for technical majors. Many employers know this and act accordingly when recruiting recent college graduates.
Preparing students for college and/or work is very much neglected by schools and parents alike. I expected that when I taught in the inner-city high school, but I was shocked by the degree of neglect in the magnet high school. Anyone who can find an effective way to address that problem will earn great honor. Because it’s education, however, there won’t be any money associated with the achievement. 🙁
Kids mature at different rates, but most of them are forced through a one-size-fits-all pipe called public education. What works for one is detrimental to another. Overall, I think that we do a pretty fair job, but we lose far too many smart and potentially productive students.
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[USER=7674]@MathTeacher[/USER]
The latest swing in educational goals for the public schools in Kansas is called College and Career Ready. The huge catch in this idea is that the individual student is to be identified very early as to which of those two paths is more suitable. Another term for that is “tracking”. Then something very simple to an individual education plan (IEP) is to be developed and followed, as is currently done for special education students. IEP’s are developed by administration, faculty, parents and the student all together to set forth precisely how this specific student is to be serviced. So, for standard students, this would mean determining in about Third Grade that little Johnny needs to plan to spend his working life asking people if they want fries with their order. Meanwhile, little Jonathon needs to plan to be a surgeon. Little Johnny will be in the underwater basketweaving courses and little Jonathon will be in the academically challenging courses along the path to high school graduation. Fifty years ago there was a nation being condemned for such thinking. Most called it Russia but some called it the USSR. -
[USER=50]@Holy Cow[/USER] and [USER=7674]@MathTeacher[/USER]
How can someone who spends their whole life in academics prepare some one for life outside academics? The education of our Children falls on all of us. We in the private sector need to reach out to the school systems and offer our thoughts on what the schools need to be teaching to help ready kids for the ever changing job market. There should be job and career fairs twice a year from middle school on to help connect students to those in society who are actually doing what the students are proposing. Those in academics need to listen to those business owners who are trying to help.
When my younger brother was in High School they asked my parents to be on an advisory board. This board was made up of Principles, teachers, and parents. They all met once a week to help “improve” the school. My folks went to every meeting, and come out feeling belittled, every idea was overlooked, and the only talked about pushing a college education. See my parents only have a high school education and where not good enough to give solid advice on education. Even though they run a successful buisness and make about three times what any teacher in the room makes. -
Third grade? Wow. Kids who don’t read well by third grade are at a distinct disadvantage, but some of them will come around.
The gap between the haves and have nots is affecting future generations more than ever. Technology among pre-schoolers is astounding, but pre-schoolers without access to the tech are rapidly going to the end of the line. It used to be access to books, now it’s access to computers.
I taught a Vietnamese immigrant whose father was an educated upper middle class citizen in Vietnam Nam. Here, he worked as a janitor. His son said that his parents were sacrificing their own well-being so that he and his sister could have a better life. And that son was a smart and hard-working student, more than thankful for his parent’s sacrifice.
I’ll bet Mrs. Cow has similar stories about parents sacrificing (and not) for their kids. Schools, I think, are often more serious about educating kids than some parents are about raising them. It surely takes both parties full efforts.
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[USER=10458]@Jones[/USER]
Excellent comments. What you have described is part of why I, at age 63, am serving on our local school board and serve as their representative on another board that runs a quasi-governmental education service center with over 500 employees. The family and community must be involved in operating our public schools. Simply paying one’s taxes does not count as being involved.
About 25 years ago the state came up with the idea of site councils for each school building. They were very similar to the advisory board you mention. They collapsed within a few years for much of the same reason you mention. The so-called “professionals” refused to listen to anyone who thought change was needed. They wanted all members to be cheerleaders for whatever they were providing at the time. Due to their training, they were able to manipulate the parents and community members who were donating their valuable time to the effort.
We are in process of finally starting new site councils within our school district. Our Board is taking this very seriously and setting it up such that the “professionals” are there to provide information only. The councils are to be led by the volunteers. Meanwhile, the “professionals” are kicking and screaming. “Tough cookies”, I say.
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[USER=677]@GeeOddMike[/USER]
Love the Angry Nerds and Wicked Smaaht tee shirts as well. Not sure I’m willing to spend $34 plus shipping and handling to acquire the very nice shirt you highlighted above. Less than one percent of people who would see me wearing it would understand the M part or the I part or the T part, let alone all three.
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[USER=10458]@Jones[/USER]
You are so right. I started teaching in my mid- fifties after careers in insurance and clinical trials came to abrupt halts. As it turned out, I taught with several “lateral-entry” teachers. Some were just looking for a paycheck while others really wanted to teach. I count myself in the latter group, of course. One of the things that we all brought, however, was an outside perspective.I think that community involvement failures are often due to miscommunication and control issues. School administrators are not going relinquish control of their schools, but community members who are sacrificing their time want to be taken seriously. It’s a difficult climate for real give and take. Schools cry out for more involvement but often don’t want what they get. And there’s more than one community member with a personal agenda to push.
Career and technical education (CTE) classes offer specific job training and are usually taught by people recruited from specific fields. My last school began life as a CTE school but was forced to expand to include an arts curriculum due to declining enrollment. We offered paths in carpentry, diesel mechanics, computer science, drafting, nursing, and several other fields. I was an academic teacher and I sometimes felt that academics were secondary, but that truly was not the case. By the way, CTE teachers usually make significantly more money than their academic counterparts.
There will always be debate over what secondary education should be aimed toward. I personally don’t think that high schools should be workforce training institutions, except for CTE curricula. Instead, they should provide the platform that supports further learning to master a chosen field. We’re very effective at doing that for the top 10% of our students, but we certainly don’t reach everybody. The monetary cost of doing that would be staggering.
A high school graduate should know specific material and, maybe, some applications of that material. Most importantly, though, a high school graduate should be able to learn and appreciate the role of more learning in his future. Then they are as career-ready, and college-ready, as we can make them.
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I think Dr. Duane F. Marble identified a critical challenge facing educators when he said
“We must cease confusing mastery of software commands with attaining a grasp of critical intellectual concepts” For more on Dr. Marble see: http://ucgis.org/ucgis-awards/duane-marble).As MathTeacher says, learning to learn is critical. Anyone imagining that there is an end to education limits their prospects significantly.
DMM
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[USER=7674]@MathTeacher[/USER]
I agree, the CTE schools are a great path for a lot of students. The trades taught at these schools give students, who don’t want to go to four years of college to start out on the right foot; with a skill that they can, in most case make a great living at. Just a couple of weeks ago I talked to a recent high school graduate. I asked him if he was going to school this fall, he said no that he was going to work under is Uncle which is a contractor. But then he done something most graduates that are not going to college do, which is explain why college is not for him. The explanation is irrelevant, but we push college so hard that kids feel like they are lesser in some sense if they don’t go. When in reality if we are honest and upfront with the choices kids have they can choose for themselves the appropriate path in life.
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Those of us who live in small, rural communities have a special challenge with helping kids find potential careers. That is, the range of potential careers they might be able to see with their own eyes during their standard day to day life is much less than what exists in metropolitan locations. The jobs held by their parents and the parents of their classmates and neighbors are not nearly so diverse as in dense population areas. If one doesn’t know a specific job opportunity exists, why would they aspire to enter that line of employment.
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[USER=50]@Holy Cow[/USER] . That is so true. Kids today have to make some of the most important decisions of their lives before they’re 20 years old. Without some serious guidance, they have very little to base those decisions on. They’re no better equipped to make them than most of us were. The difference is that today the economy is far less forgiving.
[USER=10458]@Jones[/USER] . That college thing has been catastrophic for far too many kids. A few years of student loans culminated by the wrong degree or worse, no degree, and a young person is behind a financial eight ball for decades. And a bankruptcy filing does not eliminate student loans. All other debts can be gone, but that one stays.
Earlier this year, I was introduced to a non-profit organization started by a retired credit union president. Kids they work with become thoughtful decision makers. I was on the scholarship committee and that was one of the best experiences of my life.
Maybe we need more of these; here’s a link: http://www.cfsfc.org
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Not that I’m a math nerd by any stretch, but I noticed something one time surveying. I saw a direct correlation between the number of points occupied to the number of vetcors. I began to notice that I could sample an entire tract’s measurements, via GPS, to constrain to it, with very few points, but it generated near the requisite amount of vectors for a “sample”. 7 points made 21 measurements, 8 points made 28 measurements, et cetera. The formula I came up with was N ((N-1)/2) = number of vectors where N is number of points from original tract, or anything for that matter. The first formula was very long and I saw a way to condense it. I always shoot for 8 points because it’s close to the magic 30 required for a “sample”. Also works REALLY well for fitting an existing database, like say an O&G unit. Shows problem areas VERY well too. I’ve used this method to find old busted traverses from days gone by of others and isolate the error.
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[USER=29]@Kris Morgan[/USER] . That’s a fine piece of analysis. Your formula gives the sum of the first (n – 1) integers and seeing that 7 points create 6 unique lines and so on, and then finding a formula, may just qualify you as a math nerd.:)
We teach that as (n + 1)n/2 in order to get the sum of the first n integers, but we make the students derive it from a theoretical beginning. It’s interesting to do some Googling and see how Gauss found the sum of the first 100 integers as an elementary student.
I wish we could convince more people that math is fun.
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