document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208446>Question 1208446</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor the given expression. Express your answer so that only positive exponents occur.\r<BR>\n" );
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document.write( "(x^2 + 4)^(4/3) + x • (4/3)(x^2 + 4)^(1/3) • 2x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846865 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52802)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=846865\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "Edwin, thank you for sharing your thoughts with me.\r<BR>\n" );
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document.write( "My thoughts on these subjects are slightly different.\r<BR>\n" );
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document.write( "         P A R T    1\r\n" );
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document.write( "The population of US is about 320 millions.\r\n" );
document.write( "If we assume, that this population is distributed uniformly along/among the age in 80 sections according the years \r\n" );
document.write( "from 1 to 80, it gives us ~ 4 millions in each year of their age.\r\n" );
document.write( "Of them, 5 successive years, from 5th grade to 9th grade have some relation to algebra in their schools.\r\n" );
document.write( "It gives 4 x 5 millions = 20 millions.\r\n" );
document.write( "Only 20% continue their education in colleges and universities.\r\n" );
document.write( "It gives 0.2*20 millions = 4 millions.\r\n" );
document.write( "The rest 20 millions - 4 millions = 16 millions will never need Math in their everyday life.\r\n" );
document.write( "Of these 4 millions, only one of 10 has some inclination to Math. It gives 400,000 of those, who may be, or probably, \r\n" );
document.write( "will touch Math in their life.\r\n" );
document.write( "But in reality, the coefficient is not 0.1 for real using Math, or Math skills, or Math thinking.\r\n" );
document.write( "I think that this real coefficient is 0.01, instead, giving 40,000 of 5 years, who need to have accurate and progressive Math thinking per year.\r\n" );
document.write( "Of these 40,000, really only 0.1 of them (or 4,000) are able to be really creative and create new ideas providing real progress \r\n" );
document.write( "in different branches of Science and industry: Physics, Engineering, Chemistry, Biology, Economics, Medicine. \r\n" );
document.write( "(In this my count, I did not calculate those who will become teachers, professors etc.)\r\n" );
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document.write( "So, 4,000 per year are needed to have real mathematically creative thinkers, and 40,000 to accompany these 4,000.\r\n" );
document.write( "Saying \"creative thinkers\", I mean people having firm mathematical basics knowledge \r\n" );
document.write( "PLUS an ability to think on their own PLUS an ability to make a jump to another/new level - into the unknown.\r\n" );
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document.write( "So, 4,000 per year are the CORE, and 40,000 around them are to support and accompany.\r\n" );
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document.write( "These 4,000 and 40,000 are my focus. They are my focus.\r\n" );
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document.write( "Standard contemporary Math education in US can not provide neither these 4,000 of the CORE, nor 40,000 to accompany.\r\n" );
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document.write( "To provide and develop them, another system of Mathematical education is needed - the system NOT FOR ALL and NOT FOR EVERYBODY.\r\n" );
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document.write( "It is the system of Math education for those who are INCLINED to Math.\r\n" );
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document.write( "Actually, in the word and contemporary in the US such a system (or a sub-system) of Math education just EXISTs. It is the system of Math schools (RSM = Russian Math Schools).\r\n" );
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document.write( "First such schools in US were created by immigrants of 90-s from Russia (of huge wave of immigrants of this time), where the tradition of Mathematical schools and mathematical circles \r\n" );
document.write( "were created and were supported starting from 50-s and 60-s of the last century.\r\n" );
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document.write( "Probably, it was the greatest value, which this wave of immigrants brought to the US - the culture of the advanced mathematical education for children/young students, inclined to Math.\r\n" );
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document.write( "Yes, in US, equally as in the former USSR, just existed a popular literature of the school and elementary Math level, but in US it was oriented mostly to those students, who wished \r\n" );
document.write( "to become professional mathematicians and participate in Math Olympiads of the international level.\r\n" );
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document.write( "But the number of such students in not 4,000 per year and is not 40,000 per year - it is about 100 per year, and it is totally different category.\r\n" );
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document.write( "The Russian system of Math schools was targeted to create much wider circle of all of those, who will work in Science and Engineering, in general.\r\n" );
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document.write( "In Russia of that years, two tendencies fight one with the other - the tendency to teach for people and the tendency to teach for the state.\r\n" );
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document.write( "This fight was dramatic, and only those who observed it from inside, could understand the processes. Only one tendency could to win and to survive.\r\n" );
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document.write( "What is the difference between teaching for people and teaching for the state ? - Teaching for the state becomes bureaucratic; becomes formal and cuts off the enthusiasts. \r\n" );
document.write( "It becomes non-interesting. Also, the tone is changed: it becomes formal.\r\n" );
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document.write( "In 60-s, teaching for people was on increasing branch; but it was stopped by their government in 70-s and turned into teaching for state later. \r\n" );
document.write( "With the time (with the years) this education system, which was very progressive at its origin, heavily degenerated later.\r\n" );
document.write( "After their crisis in 90s, their educational system stopped existing anymore. // No, it still does exist formally, but does not produce any useful.\r\n" );
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document.write( "But some enthusiasts of the teaching Math for people survived and brought their ideas with this emigration wave.\r\n" );
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document.write( "Now in the US, parallel to RSM, do exist similar Math schools among Chinese diaspora and India diaspora.\r\n" );
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document.write( "Only such form of Math education - the system of Math schools, - is able to provide 4,000 creative minds in the US per year plus 40,000 accompanied minds per year.\r\n" );
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document.write( "  | They are those for whom I work here at this forum - they are my focus. |\r\n" );
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document.write( "It does not mean that only advanced Math should be presented there - NO. Everything from start to advanced topics should be covered uniformly \r\n" );
document.write( "and carefully explained.\r\n" );
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document.write( "But one thing should be EXCLUDED: teaching inside the box, which is a standard way to teach in average school. \r\n" );
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document.write( "So, my principle of teaching Math is to give students a view at least one step out the box. \r\n" );
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document.write( "There is another important issue in Math literature for children and young students -\r\n" );
document.write( "   - it is taking a right TONE.\r\n" );
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document.write( "The tone should not be formal - it should be normal tone conversation with child or young teenagers.\r\n" );
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document.write( "It should contain emotions, humor, explaining what is right, what is wrong, what is excellent and why.\r\n" );
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document.write( "One wise scientist (physicist, of course - who else can create such a profound thought) and famous teacher/professor said once \r\n" );
document.write( "that collections of problems in Math and/or in Physics are similar to fair tales for children - they teach the students to life.\r\n" );
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document.write( "Ok. These are my thoughts about mathematical education.\r\n" );
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document.write( "         P A R T    2\r\n" );
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document.write( "Now about what happens at this forum, as my view.\r\n" );
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document.write( "I am at the forum from the middle of 2011 (after my retiring).\r\n" );
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document.write( "Till 2015, among the visitors, there was significant part of real students with their problems - I think, about 60%.\r\n" );
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document.write( "About 20% were half-mad people or totally illiterate in Math, which I will not count, for clarity.\r\n" );
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document.write( "In the interval 2015-2019 (pre-covid era) I observed the increasing number of teachers, who wanted to improve their skills, \r\n" );
document.write( "asking to explain them, how to present the solutions of the problems to students.\r\n" );
document.write( "Such requests are easy to identify by the words \"let x be something and y be another something\" after a normal word problem.\r\n" );
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document.write( "I think that among these people, was also a significant amount of those who came to the forum in order for to use our solutions\r\n" );
document.write( "to fill other web-sites and re-distribute them among other web-sites.\r\n" );
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document.write( "OK. But at the covid-era, many things changed significantly.\r\n" );
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document.write( "The number of those, who don't know Math AT ALL and are UNABLE TO TEACH, but WANT TO TEACH became astronomic.\r\n" );
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document.write( "I call them quasi-tutors. \r\n" );
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document.write( "In the same proportion, the number of those who transported our solutions to other web-sites, also became astronomic.\r\n" );
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document.write( "As always, the flow of non-sensical problems from semi-mad persons and totally illiterate in Math persons was kept the same 10-20%.\r\n" );
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document.write( "As always, the number of routine normal Math problems was about 15-20%.\r\n" );
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document.write( "The percent of good and very interesting math problem was/is about 5% - 8%.\r\n" );
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document.write( "         P A R T    3\r\n" );
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document.write( "After the covid era, new times came - the Artificial Intelligence era.\r\n" );
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document.write( "Now we, the tutors of this forum, work 50% to feed this AI - we produce the basic solutions for it just one or two years.\r\n" );
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document.write( "Do you like it or not - but it is so. It is the reality, independently of your wishes and your thinking.\r\n" );
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document.write( "Still 5% - 10% of semi-mad posts do exist at the forum.\r\n" );
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document.write( "50% of posts are for AI.\r\n" );
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document.write( "10% - 20% are for real/human visitors.\r\n" );
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document.write( "10%-20% are regular/routine school assignments.\r\n" );
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document.write( "The rest 5% are for upper level problems.\r\n" );
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document.write( "        Is this AI good or bad in the school Math education ?\r\n" );
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document.write( "Really, I don't know the answer to this question for to be sure.\r\n" );
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document.write( "From one side, AI can replace the most majority of the average school Math teachers.\r\n" );
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document.write( "For 95% of school students it will be almost the same, with no difference.\r\n" );
document.write( "As they did not know Math in the past and don't know Math in the present - exactly in the same way/proportion \r\n" );
document.write( "they will not learn it from AI.\r\n" );
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document.write( "A real difference is/(will be) for 1% (or less) of advanced students. \r\n" );
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document.write( "What can change differently, is THIS: now 95% of school students do not know Math, but many of them are neutral to Math.\r\n" );
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document.write( "         After AI teaching, I afraid that many of these 95% will HATE Math.\r\n" );
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document.write( "I believe (it is only my subjective belief) that real knowledge of Math can be passed only by human and from human to human \r\n" );
document.write( "from hands to hands, from mind to mind and from head to head - and not via computer or via artificial intelligence.\r\n" );
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document.write( "From the very smart and professional people (even from those thinkers who work in AI industry), \r\n" );
document.write( "I heard their opinion that computerized education can not replace a traditional education.\r\n" );
document.write( "One can complement another, but can not replace, and I firmly believe in it.\r\n" );
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document.write( "And THEREFORE, in the era of AI, the Math school mathematical education may become the only real alternative \r\n" );
document.write( "to all other forms of mathematical education for those school students who need it.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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