tag:blogger.com,1999:blog-57519568322919252242024-03-12T20:32:51.225-07:00Building BlocksAnonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-5751956832291925224.post-76387581335859877752017-04-16T22:43:00.000-07:002017-04-16T23:04:43.997-07:00A Neat Diophantine Equation<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">I recently ran into the following interesting problem: Can you find distinct integers such that </span><br />
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"><br /></span>
$$<br />
\begin{eqnarray*}<br />
a^{2}+b^{2}+c^{2}+d^{2} & = & abcd?<br />
\end{eqnarray*}<br />
$$<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;"></span><br />
<br style="background-color: white; color: #1d2129; font-family: Helvetica, Arial, sans-serif; font-size: 14px;" />
<span style="background-color: white; color: #1d2129; font-family: "helvetica" , "arial" , sans-serif; font-size: 14px;">I like this problem because a friend recently told me there are no interesting problems accessible to high school students, and that all the interesting ones require advanced math degrees. This is one of those problems that crushes that argument. Not only is it accessible to high school students, but it is a fun and interesting problem to investigate. I follow with only a small insight in order to leave those curious enough with plentyto discover, and investigate. . . Have fun!</span><br />
<br />
<br />
Here we will investigate one possible approach.<br />
<br />
<br />
Taking the original equation, and completing the square we get,<br />
<br />
$$<br />
\begin{eqnarray*}<br />
a^{2}+b^{2}+c^{2}+d^{2} & = & abcd\\<br />
a^{2}-2abcd+(bcd)^{2}+b^{2}+c^{2}+d^{2} & = & abcd-2abcd+(bcd)^{2}\\<br />
(a-bcd)^{2}+b^{2}+c^{2}+d^{2} & = & -abcd+(bcd)^{2}\\<br />
& = & (bcd-a)bcd\\<br />
\Rightarrow(bcd-a)^{2}+b^{2}+c^{2}+d^{2} & = & (bcd-a)bcd<br />
\end{eqnarray*}<br />
$$<br />
<br />
and<br />
$$<br />
a'=bcd-a<br />
$$<br />
<br />
or equivalently,<br />
$$<br />
a_{n}=bcd-a_{n-1}<br />
$$<br />
<br />
We chose to complete the square using 'a', but what about b, c, or<br />
d, after all addition and multiplication are commutative. Well it<br />
similarly follows for those. Rewriting the original equation we get,<br />
$$<br />
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}x_{2}x_{3}x_{4}<br />
$$<br />
<br />
thus,<br />
$$<br />
x_{i}^{'}=\prod_{_{\begin{array}{c}<br />
j=1\\<br />
j\neq i<br />
\end{array}}}^{4}x_{j}-x_{i}<br />
$$<br />
<br />
We see that,<br />
$$<br />
\begin{eqnarray*}<br />
a_{1} & = & bcd-a_{0}\\<br />
a_{2} & = & bcd-bcd+a_{0}\\<br />
& = & a_{0}\\<br />
a_{3} & = & bcd-a_{0}<br />
\end{eqnarray*}<br />
$$<br />
<br />
which has a period of two. Thus, we can't just endlessly continue<br />
iterating the same respective term, but from 'part II' we can see<br />
we can change it up. One way could be,<br />
$$<br />
(a,b,c,d)\rightarrow(a',b,c,d)\rightarrow(a',b',c,d)\rightarrow(a',b',c',d)\rightarrow(a',b',c',d')\rightarrow(a'',b',c',d')<br />
$$<br />
<br />
and so on, but this blows up rather quickly.<br />
<br />
Another way could be to evaluate,<br />
$$<br />
x_{n}x_{n-1}x_{n-2}-x_{n-3}<br />
$$<br />
and assigning it to,<br />
$$<br />
x_{n+1}<br />
$$<br />
consequently generating the sequence,<br />
$$<br />
\{x_{1},x_{2},x_{3},x_{4},...,x_{n}=x_{n}x_{n-1}x_{n-2}-x_{n-3},...\}<br />
$$<br />
<br />
of which any four consecutive terms in the sequence satisfies the<br />
original equation.<br />
<br />
For example, using the trivial solution $\{2,2,2,2\}$ yields,<br />
<br />
$$<br />
2<br />
$$<br />
<br />
$$<br />
2<br />
$$<br />
<br />
$$<br />
2<br />
$$<br />
<br />
$$<br />
2<br />
$$<br />
<br />
$$<br />
6<br />
$$<br />
<br />
$$<br />
22<br />
$$<br />
<br />
$$<br />
262<br />
$$<br />
<br />
$$<br />
34582<br />
$$<br />
<br />
$$<br />
199330642<br />
$$<br />
<br />
$$<br />
1806032092550706<br />
$$<br />
<br />
$$<br />
12449434806576800059248920402<br />
$$<br />
<br />
$$<br />
4481765860945171681908664776799089162954814190172722<br />
$$<br />
<br />
$$<br />
\vdots<br />
$$<br />
<br />
and we can go on indefinitely.
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</script>Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-88706715961774477602016-01-01T00:07:00.003-08:002016-01-01T00:07:40.372-08:00Happy 10/9!*8!*7!-6!*5!/4!+3!*2!+2!*(1!+0!)!)Happy 10/9<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="18856ac1-125a-4851-aae9-e5d06f45271c">!</gs>*8<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="74d91a74-1d36-46ef-a1a6-ecad329144c4">!</gs>*7<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="cdfecaa3-2388-4b6b-ac08-509aa80160ad">!</gs>-6<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="7a8c8dfd-029e-46f4-9baa-ebec09eafd5e">!</gs>*5<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="bc944e85-3470-44ce-b422-e27b2cb8a529">!</gs>/4!+3<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="09041b44-6e1e-48ce-bb51-fad27e1a908e">!</gs>*2<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="5da4f1cf-8119-46bf-bdfb-6fe9ee7da18b">!</gs>+2<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="3fd1f73b-6675-4140-8b83-3aade8e1d3ba">!</gs>*<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5fa47696-7f36-47ee-9847-bdf772d806ff" id="ec54e70e-474c-4adb-b6fd-a65a0f7b7195">(</gs>1!+0!)!)<br />
<br />
Wishing you all the best this upcoming year. May it be your best year yet!<br />
<br />
Cheers,<br />
--Angel AgüeroAnonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-833259323557480032015-12-01T22:41:00.000-08:002017-03-28T01:44:10.406-07:00Testimonials<div style="background-color: #eee8dd;">
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<span style="font-family: "times new roman";">November 22, 2016</span></div>
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<span style="font-family: times new roman;">Working with Mr. Angel Aguero was truly a highly learning experience. Not only was I able to learn college level Statistics in a very short amount of time, but I learned it in a very friendly and genuine environment. Mr. Aguero is a very caring, loving, and highly intellectual individual. He is well-rounded in many different areas of academics and has a drive to teach in the most practical of ways. I will recommend working with Mr. Aguero to anyone who is interested in being challenged and provoked to think outside of the box in an unconventional way. When I found myself struggling with a math problem or found myself having a difficult time understanding a particular concept, Mr. Aguero was able to break down the problem or concept and explain it in a way that was easily understandable and practical to apply to everyday life. Mr. Aguero, thank you for your amazing ability to teach and tutor. I will be coming back for more! </span><br />
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<span style="font-family: times new roman;"> Best Regards,</span><br />
<span style="font-family: times new roman;"><br /></span>
<span style="font-family: times new roman;"> Roberto Acosta, RN-BSN, DNP Student </span></div>
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<span style="font-family: inherit;"><span style="color: black;">To Whom It May Concern: </span><br style="color: black;" /><br style="color: black;" /><span style="color: black;">Angel Aguero served as a geometry tutor for our fifteen (15) year old son. Our son had undergone major medical issues that resulted in him following behind at school. He was faced with what appeared to be the </span><span style="color: black;"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="59f5e960-d3ed-49ad-83ea-52d2b1319be9" id="4dc399e5-662a-43fd-adfc-d29560921f05"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="3fa29ed1-310f-41d9-8531-6722cdab9bdf" id="39be231d-5c37-4308-a150-cf0f0b5e50ab"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="90a04fd6-5809-4180-99f0-79d419bb93da" id="ecfeba2d-ad2f-450b-b443-1553cf10905c">unsurmountable</gs></gs></gs></span><span style="color: black;"> task of learning a semester’s work of geometry within a two (2) week time period. If he failed to pass the exams that covered the semester course material, he would have to repeat the course. The strong possibility of him failing the course served as yet another difficult obstacle to a </span><span style="color: black;"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="e9ebed66-eb78-4296-bb62-21089841cb0e" id="dc87edc7-11f2-46c5-b052-df1d779eb53a"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="6cef4f7a-0760-4217-88e5-3018663ba9d1" id="938a4039-c085-46cf-a299-44f3a2df0748"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="7322c1e7-270c-4fde-96bd-eff303c5a459" id="72fd4f8b-c798-400b-80cb-4aacb105cf1e">months</gs></gs></gs></span><span style="color: black;"> marked with health struggles. In an attempt to assist him, Mr. Aguero was highly recommended to us as an outstanding tutor. We contacted him and he immediately agreed to assist our son. He coordinated time to meet him during weekends and evenings. Moreover, he went above and beyond by meeting with him on a holiday weekend. His commitment to assisting our son was beyond measure. Mr. </span><span style="color: black;"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="00aa7d5a-1328-490b-a3ab-f4db0aa82a99" id="d11c92c1-0cdf-40e9-92c6-daaf1e21b5a5"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="ac42d08b-a6e6-4fd1-9d93-ef15ffab823f" id="ece633dc-f04a-41b4-b855-2e6f946fb73d"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="e0ae1d8c-caaf-4f5b-b14e-0fb092d83e40" id="ae705881-4278-40ac-a5d8-d53b565a86e9">Aguero‘s</gs></gs></gs></span><span style="color: black;"> teaching ability resonates when considering that our son was in jeopardy of failing the course. Through his diligence, our son secured an 86 in Geometry. A goal that seemed </span><span style="color: black;"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="344d9f8d-1ac4-401c-ba05-33448f99bd6c" id="d2868223-1df0-4416-b2c0-e307ba3a3c2f"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="2c040daf-c9a8-4730-b910-9a9cfc7ddfbe" id="2a7eff9e-d19d-429e-acc3-0594849ca03b"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="02aed277-f795-45ab-97a8-ca341dace828" id="8c83e709-9323-4bcc-a07b-a5babb41d0eb">unsurmountable</gs></gs></gs></span><span style="color: black;"> was made possible through the hard work and commitment of Angel Aguero. Our son credits his insightful instruction and patience with allowing him to complete the course and achieve an excellent grade. </span><br style="color: black;" /><br style="color: black;" /><span style="color: black;">We are humbled and thankful that we can recommend Angel Aguero. He is an amazing individual and instructor with a commitment to assisting students achieve their goals. </span><br style="color: black;" /><br style="color: black;" /><span style="color: black;">Should you have any questions or need additional information, please do not hesitate to contact us at the above telephone numbers."</span><br style="color: black;" /><br style="color: black;" /><span style="color: black;">Sincerely,</span><br style="color: black;" /><br style="color: black;" /><span style="color: black;">M & L Gregory"</span></span></div>
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<span style="font-family: inherit;">"I first made the acquaintance of Angel Aguero when we <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="f20e2113-6965-48b8-b6aa-d5c611ea2303" id="5f565b68-df73-4817-8ba7-184d8e10de49"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="316a21a0-1da8-43de-82fc-2457281ef36a" id="cd643fe7-9579-423f-b942-e3800593b2ae">where</gs></gs> in a number theory class early on during our undergraduate studies. From that point on I and many others, including the professors, knew that he had a special talent for mathematics which covered the understanding of new concepts, refinement of older ones, solutions to problem sets, demystifying the underlying logic behind the proofs and illuminating the great ideas that came behind their results. In the classes Angel never rested upon his perch to look down upon us, rather Angel always made every effort to bring us up to his level of understanding. He was always ready to help patch up a proof, motivate solutions to problem sets and help to guide the discussions along as best he could when the topics became difficult. I and many others have had the privilege to take senior and graduate level courses with Angel where his contributions to <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="81996584-b81e-493f-b549-e8f669a3afa2" id="1a40f392-76b7-438e-b875-92ac37606e64"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="cf03b29c-998d-478f-86dd-595b4c7cc855" id="9847342b-b6ad-400d-9d88-09803877c8d1">classroom</gs></gs> where an integral part of the learning that took place. I feel very fortunate to have made his acquaintance as a fellow math student and as a friend. His loyalty extends far beyond mathematics and I highly endorse his friendship and employment"</span></div>
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<span style="font-family: inherit;">Jaime P. <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="f658c4a5-48c5-45c5-b15a-a173f0ff948c" id="70f3ede9-e3a5-4279-a1be-735cf87f50bd"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="b959bdb7-c37d-4b3d-a045-123607509bd9" id="074b68ec-5f88-45d0-8229-d9d6111202ad">Lepe</gs></gs>, B.S., M.S.</span></div>
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<span style="font-family: inherit;">Mathematics Lecturer, University of Texas at El Paso</span></div>
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<span style="font-family: inherit;">"I am a college student working on my Masters. Despite that, <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="5797734b-5d36-450d-925b-6e5e64070ad4" id="c811ebe4-c2be-43e5-8593-b7e50e10c19a"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="a39531f4-989d-4e03-aaa6-30f7e2df21dd" id="22e5437a-1dd0-4f62-a507-d6cd1038279a">math</gs></gs> still intimidates me. One of my courses, to my surprise, had <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="902ea763-e76b-4beb-8ac4-ef1de0129808" id="89e824bc-1080-49d7-9352-0064699489f6"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="7c83f19c-ad6b-44fe-b7ce-79646b4cec11" id="f1483886-ff27-47f2-851b-8c3d49a87746">math</gs></gs> in it. I was in a panic. I turned to the person I know I can trust when it comes to math: Angel Aguero. He came and guided me through the intricacies of statistics. He made what the professor made complex into simplicity itself. Thanks to him, I passed with an A, for which I am so, so thankful. I am so grateful to him for everything he's done, not just in terms of mathematics, but in terms of his character. He is a good man, excellent father, loyal friend. I can't say enough about Angel without running out of superlatives of 'wonderful'. Thanks to him, I understand so much more, and know that I can always count on him."</span></div>
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<i><span style="font-family: inherit;">--Rick Aragon, Librarian, Film Critic. Graduate student in Library Sciences</span></i></div>
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<i><a href="http://rickscafetexan.blogspot.com/" style="color: #663399; cursor: text;"><span style="font-family: inherit;">http://rickscafetexan.blogspot.com/</span></a></i></div>
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<span style="font-family: inherit;">"I met Angel in the latter part of 2014. He was a tutor at a West El Paso learning center. </span></div>
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<span style="font-family: inherit;">Working toward an alternative certification to teach high school English, I had to pass the math portion of the Texas Higher Education Assessment (THEA). </span></div>
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<span style="font-family: inherit;">With a clear goal in mind, I swallowed my pride and sought help at the local tutoring center. It was hard to find a seat in the mists of kiddos. Most were more than half my age. It was also hard to see the facilitators eye to eye after they realized I couldn't get through the pre-assessment test. </span></div>
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<span style="font-family: inherit;">I urged them not to be discouraged and to have faith that, what I didn't know in content, I would make up with hard work. They gave me a shot. </span></div>
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<span style="font-family: inherit;">Enter Angel. </span></div>
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<span style="font-family: inherit;">Angel broke down mathematical complexities to very simple arithmetic. There was no terror. Regardless of whether you were left brain or right brain, his lessons were fun and engaging. </span></div>
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<span style="font-family: inherit;">I don’t remember what side of the brain I am. I must have forgotten to carry the 7. </span></div>
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<span style="font-family: inherit;">In all honesty, Angel is a brilliant mind who will go out of his way to make sure you succeed. He doesn't do it as a self-serving endeavor. He does it because he loves to see other people appreciate numbers and formulas just as much as he does. I will never forget how Angel met me after tutoring to make sure the lessons were sticking. I passed the <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="f626966a-24c2-4758-b406-a5c9f8469e38" id="e0bc525d-eba2-4373-aa68-a21930e8b14c"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="a666411b-4310-481e-ba19-3a2b3c68e0d5" id="f425b368-4722-4562-b3a6-1693b5f2ff33">test but</gs></gs> more importantly, I found a new love for numbers." </span></div>
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<i><span style="font-family: inherit;">--Concerned citizen, Luis Carlos Lopez</span></i><br />
<i><span style="font-family: inherit;">Journalist at <a href="http://www.elpasotimes.com/">El Paso Times</a></span></i></div>
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<span style="font-family: inherit;">"As a high school student, I was struggling to understand conic sections. My Algebra teacher presented it in such a way that involved memorizing several convoluted formulas. When I went to Angel, he explained conic sections much more concisely. Angel’s style of teaching is more tailored to cement learning, as opposed to the confusing test-taking curriculum at school. He has the ability to decompose a complicated concept, and present it in such a way that makes it easier to understand and follow."</span></div>
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<i><span style="font-family: inherit;">--Rene Garcia, Franklin High School,<span style="background-color: transparent; line-height: 1.6;"> El Paso<gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="1c7f48b1-9d9b-4cd3-905e-c9a99bdf7801" id="77f189b6-86da-4922-ba4e-6cab9f5f4082"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="9a9600e2-fdc7-47c9-8d76-2179171d3754" id="cd0c0f86-82f6-4950-b846-e5755ce2f292">,</gs></gs>TX</span></span></i><br />
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<i><span style="background-color: transparent; line-height: 1.6;"><span style="font-family: inherit;">Massachusetts Institute of Technology, Cambridge, MA</span></span></i></div>
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<span style="background-color: transparent; font-family: inherit; line-height: 1.6;">"Angel the genius."</span></div>
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<span style="font-family: inherit;"><span style="background-color: transparent; line-height: 1.6;"><i>--Joaquin Ordonez, Engineer </i></span><span style="background-color: transparent; line-height: 1.6;"><i>and Project Manager at </i></span><span style="color: black;"><i><span style="background-color: initial;">Austin Water Utility</span></i></span></span></div>
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<span style="font-family: inherit;">"I have been working with Angel for over a year and he has been the reason why my math grade went from a 85 when I was a sophomore in algebra 2 to a 97 in pre-calculus as a junior. He has gone through the trouble of helping me on various projects and Angel not only explains <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="c34470e9-c890-42cd-844f-c956be487b82" id="c5efad21-3fa2-498b-8a5c-16b00e4614b2"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="4e0d9de5-ee48-40f6-864b-60526e2841e9" id="30ecce41-4757-4301-8d3b-f8f46cae4765">problems but</gs></gs> breaks them down by deriving all the formulas that are needed to solve it. He takes the time that teachers at <gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="9dfd0b0b-1f4a-4c77-9b22-c27938e97371" id="8d0c1e13-65f7-49ca-a5c9-1ba914345768"><gs class="GINGER_SOFTWARE_mark" ginger_software_uiphraseguid="f835c427-1e22-488e-9459-80cf5b6a5f2f" id="b4239130-7b96-46f4-9058-e39c6722825c">school</gs></gs> cannot provide to students. He has changed my view on math as it is one of my favorite subjects now."</span></div>
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<i><span style="font-family: inherit;">--Eduardo Cabrera, International Baccalaureate Magnet Program at E.P.I.S.D.</span></i></div>
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Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-58023925315051683852015-05-10T01:54:00.000-07:002015-05-10T02:00:27.770-07:00Mathematics Education in the Age of TechnologyIt is no mystery that students are having increasing difficulty learning algebra in schools. One proposition referenced in<br />
<br />
<a href="http://www.edutopia.org/blog/math-as-social-justice-lever-jose-vilson">http://www.edutopia.org/blog/math-as-social-justice-lever-jose-vilson</a>,<br />
<br />
has been to eliminate some requirements, or make the content easier. Clearly they don't understand the value of learning algebra, or higher mathematics even when it isn't their area of study, but that is content of a future post.<br />
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By surrendering to this mind set, we are as educators, failing to give our students the credit they deserve. Where then, lies the problem?<br />
<br />
In my experience it seems that implementation and delivery of the tools available to educators is a source of some of the difficulties our students have with mathematics. I believe our teachers would benefit from more training that would bring them up to date with the new and sophisticated tools we have now. Many of them are unprepared to deliver an effective mathematics education in this day and age where students rely on 'googling' and tools like mathematica to to hand them out answers to problems without even having to attempt them. I recently even had a new student ask me, "Why should I have to bother learning algebra, when Google gives me all the answers." Then there are also the 'accountability' requirements that forces educators to 'teach to the test', of which I talk more about more here:<br />
<br />
Part I: <a href="http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of.html">http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of.html</a><br />
<br />
Part II: <a href="http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of_29.html">http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of_29.html</a><br />
<br />
Before I mention what I believe is a possible solution, lets take a look at what type of questions our students today are having trouble with.<br />
<br />
On the 2014 Texas end of course algebra I standardized exam question 20 was one of the top 2 most missed questions with only 32% of students answering correctly, was not multiple choice and read,<br />
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"20. There are 156 laptops and desktop computers in a lab. There are 8 more laptops than desktop computers. What is the total number of laptops in the lab?"<br />
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These is a surprisingly simple question, yet a large number of students missed it. What then can we do about this?<br />
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In Notices of the AMS Volume 61, Number 6, Dr. Igor Rivin discusses the principle that "A computer program IS a proof." . He goes on to write, "The justly acclaimed book 'Structure and Interpretation of Computer Programs' by H. Abelson and G. J. Sussman introduces the fundamentals of computer programming and, together with a companion book on 'Structure and Interpretation of Mathematics', this would constitute the core of a modern introduction to mathematics." I agree. Technology is all around us, our children use it everyday, and yet they don't really know how to use it outside of Facebook, YouTube, and 'googling'. Teaching mathematics is the perfect opportunity to conceptualize mathematical ideas, while at the same time teaching our students the true beauty of the technology that surrounds us.<br />
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I have successfully applied these methods while volunteering my time at local public housing projects helping troubled youth, living in poverty, with their schooling. One young man who had previously failed the algebra I eoc exam was my first student at the projects. I constructed a project from a problem I had seen, I taught him to write basic python code to help us conceptualize the problem. However, before he could write effective code, he had to understand the problem. Not long after he completed it with great success, I developed some groundwork and put together some notes for future use, and he went on to score 'Advanced' on the eoc exam. He now loves math and computer science and wants to study programming in college. I went on to have further success with this method getting much more mileage from this project, not only with troubled kids, but students of all ages, levels, and economic status. In case you are curious here are those notes(with my code at the end)--a bit crude but these are for my personal use. . .<br />
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<div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;">
<a href="https://www.scribd.com/doc/264713631/Projectile-Motion-Project" style="text-decoration: underline;" title="View Projectile Motion Project on Scribd">Projectile Motion Project</a> by <a href="https://www.scribd.com/AngelAguero" style="text-decoration: underline;" title="View AngelAguero's profile on Scribd">AngelAguero</a></div>
<iframe class="scribd_iframe_embed" data-aspect-ratio="0.7729220222793488" data-auto-height="false" frameborder="0" height="600" id="doc_49285" scrolling="no" src="https://www.scribd.com/embeds/264713631/content?start_page=1&view_mode=scroll&access_key=key-jdTGKKR8TJpdv5hKaD65&show_recommendations=true" width="100%"></iframe>
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Finally, I agree this is a, "difficult problem with nuanced roots and a hard uphill battle," but we have the tools to make a huge difference. Once again, I believe the problem is mostly in implementation and delivery, and not in the students' abilities. However, this takes hard work, dedication, and patience from educators who roll this out as they may likely not see immediate rewards. The biggest challenge however is, this takes a lot of work to make it work!Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com2tag:blogger.com,1999:blog-5751956832291925224.post-87516435374927604132015-05-02T04:56:00.004-07:002015-05-02T12:44:55.359-07:00Everyday AlgebraRecently I have ran more and more often into the following statement.<br />
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“Well, another day has passed. I didn't use algebra once.”<br />
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I even saw it printed on a t-shirt! It seems an increasing number of individuals need justification to not care about, or bother learning algebra. I could similarly remark, “Another day has passed, and I didn't perform the Heimlich maneuver or a single chest compression!” Yet, universally most would agree they are useful skills to know. While most of us did not perform open heart surgery today, algebra is one of those fields that are so fundamental to the things we do everyday, that is is virtually impossible to not use algebra at least once in the course of a day. Like language, mathematics allows us to communicate with our peers and perform basic everyday tasks we likely take for granted.<br />
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What is Algebra? A quick 'Google' search, or any standard dictionary tells us, “In its most general form algebra is the study of symbols and the rules for manipulating symbols.” Wow! That doesn't sound like anything I did today, or want to do ever! However, Algebra originating from Arabic's Al-jebr, and meaning, “reunion of broken parts”, is about bringing ideas together. Yes, that is right, bringing ideas together, and making sense of them.<br />
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Let us briefly take a look at the simplest type of algebra problem, namely, solving the following equation for $x$,<br />
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$$ax+b=c$$<br />
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Note: Don't let my use of symbols, or mathematical jargon scare you, you don't need to understand any of that, I am simply going to attempt to illustrate where we use these ideas in our everyday lives.<br />
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Now, to solve this equation for $x$<br />
we subtract b from both sides, and then divide both sides by a. Mathematically we say, “We use the additive inverse of 'b' followed by the multiplicative inverse of '$a$' together with the algebraic field properties over the reals to find an element $x$ such that $f(x)=c$, whenever $f(x)=ax+b$.” ... Whatever that means. What is important is that in essence we are taking inverses, i.e., doing and undoing things, and applying rules to functions.<br />
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Let us now look at a few examples of functions in everyday life. One common example is a soda machine. The buttons on the machine are its inputs(the domain), and the respective drinks are its output (the range). Mathematically,<br />
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$$f([pressin\_button\_picturing\_drink])=[Dispensing\_that\_drink]$$<br />
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Another example involves driving. It turns out that the speed at which the vehicle travels depends on how far we press the gas pedal with our foot. Once again this is a function with its input(domain) being the amount the pedal is displaced from its normal position, and its output(range) being the speed at which it travels. Mathematically,<br />
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$$f([amount\_pedal\_is\_displaced])=[speed\_of\_the\_vehicle]$$<br />
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Hence, anyone who drives uses algebra, regardless of whether they are aware of that fact or not! Even the act of stapling a stack of papers is an exercise in algebraic problem solving. Its domain being the size of the stack of papers, the rule being the act of stapling the papers, and its range the stapled stack of a particular size. Before you say I am pulling your leg, let us examine this situation. Can we evaluate an inverse operation, and apply the inverse operation to the original rule thereby undoing the operation and restoring the original domain? Allow me to direct your attention to the following Image retrieved from: http://i.imgur.com/396M1.jpg:<br />
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<a href="http://3.bp.blogspot.com/-EIuCtaCoABE/VUS64nbHcII/AAAAAAAAAG4/Hdwnr6NTbWw/s1600/InverseStapler.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-EIuCtaCoABE/VUS64nbHcII/AAAAAAAAAG4/Hdwnr6NTbWw/s1600/InverseStapler.jpg" height="240" width="320" /></a></div>
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Mathematically,<br />
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$$f^{-1}(f(x))=x$$<br />
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i.e., the stapler applies a staple to the stack of papers, and staple remover restores the stack to its original unstapled condition--except for the holes of course. Then there are also finances, whether it is about buying our favorite snack to paying our bills every time. When we buy something for say, \$1.78, we know that \$1.78+[Ourchange]=[Billwepaidwith], assuming we didn't swipe our card. The budgeting of our expenses, and calculating how much money we can spare towards entertainment the following weekend is also an algebra problem. You may say, “Wait that is an arithmetic problem,” and you would be correct if you only did that problem once, or it was the exact same problem every time, but the values vary, introducing variables into your calculations, thus, making it an algebra problem which most people know how to deal with in spite of the changing values.<br />
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Fundamentally, this is all there is to algebra! The ideas of algebra are really this simple. They are not a scary monster we must avoid at all cost. They are the stuff of thought and the rules that govern our everyday lives--so much that it is impossible for anyone to say, “Well, another day has passed. I didn't use algebra once.” and be right--chances are they did even if they didn't realize it! The better we get at manipulating the essence of our ideas the better we become at bringing our ideas together and creating new ones. Our native language is our means of communicating our thoughts with others, while Mathematics is a language that allows us to manipulate and work with our ideas. Like any other language mastering its rules, syntax, and grammar allows us to put together an elegant flow of ideas and create beautiful prose in the language of mathematics.<br />
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As one example of communicating a more advanced idea with the language of mathematics consider the following. It turns out there are three primary factors determining the resistance to blood flow within a blood vessel, namely, vessel radius and length, and blood viscosity. We could write tirelessly page after page about how a blood vessel that is twice the length of another, but having equal radii will have twice the resistance to blood flow, or that if the viscosity of the blood doubles, then resistance to flow will also double, or that with increased radius comes reduced resistance, etc. Or we could mathematically say all that and more, like a master poet, with a single formula:<br />
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$$F\propto\frac{\triangle P\cdot r^{4}}{\eta\cdot L}$$<br />
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Mathematics allows us to become better at working with our thoughts, hence improving our rhetorical skills. It makes us better thinkers and better writers as a result of being able to take apart an argument, analyze it, and put it back together through both deductive and inductive principles which are at the foundation of mathematical thought. For example, we don't exercise because we may find the need to lift 400 pounds at work, but rather because a healthy body allows us to do every day things more efficiently. Similarly, exercising our brains by entertaining a mathematical thought or puzzle makes us better thinkers, and as a result better at making decisions. Those who communicate clearly, and can express their ideas better are likely to get further ahead in their careers, as well as closer to their dreams. This is why so many strive to be better writers, and better speakers--mathematics only facilitates this by making us better at logical thinking, and since it is already the stuff of thought, even unknowingly, our days can not pass without engaging in it at least once in our day--every day!<br />
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<i>The pdf version:</i><br />
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<div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;">
<a href="https://www.scribd.com/doc/262679045/Everyday-Algebra" style="text-decoration: underline;" title="View Everyday Algebra on Scribd">Everyday Algebra</a> by <a href="https://www.scribd.com/AngelAguero" style="text-decoration: underline;" title="View AngelAguero's profile on Scribd">AngelAguero</a></div>
<iframe class="scribd_iframe_embed" data-aspect-ratio="0.7729220222793488" data-auto-height="false" frameborder="0" height="600" id="doc_82168" scrolling="no" src="https://www.scribd.com/embeds/262679045/content?start_page=1&view_mode=scroll&access_key=key-G8w4yBNO4r8fCefORGZN&show_recommendations=true" width="100%"></iframe>
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</script>Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-81581475395113840652015-04-29T05:02:00.001-07:002015-04-30T04:16:44.442-07:00Mathematics Problems in an Age of Accountability and High Stakes Testing II<div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;">
This is the second part of my previous post, '<a href="http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of.html"><i>Mathematics Problems in an Age of Accountability and High Stakes Testing</i></a>'. Here I will address the issue of what constitutes an ill posed math question and what we should be doing about it. Also, although I may include some, not necessarily straightforward mathematical content, you may readily skip over it without loss of content or continuity--The more mathematically inclined are encouraged to verify my statements and calculations. </div>
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Let's examine our first example of an ill posed problem. As you may recall from my previous post, we have to be careful with problems involving sequences. Not too long ago a student brought me one of her STAAR test practice worksheets. In particular, she needed help with one problem that read:</div>
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<i>Given the sequence 2, 8, 18, 32, 50, . . . What is the next term in the sequence?</i></div>
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<ol type="a">
<li style="text-align: left;"><i>60</i></li>
<li style="text-align: left;"><i>72</i></li>
<li style="text-align: left;"><i>120</i></li>
<li style="text-align: left;"><i>192 </i></li>
</ol>
The student had selected 'c' for her answer, but her teacher marked it wrong, and circled the correct answer as 'b'.<br />
<br />
Lets think about this for a second, and try to work out an answer. With a little effort we can come up with say,<br />
$$f(n)=(-\frac{1}{10})n^{5}+\frac{3}{2}n^{4}-\frac{17}{2}n^{3}+\frac{49}{2}n^{2}-(\frac{137}{5})n+12$$<br />
which results in<br />
$$ {2,8,18,32,50,60, . . . } $$<br />
<br />
But,<br />
$$f(n)=2n^2$$<br />
gives<br />
$$ {2,8,18,32,50,72, . . . } $$<br />
<br />
However,<br />
$$f(n)=n^5-15n^4+85n^3-223n^2+274n-120$$<br />
gives,<br />
$$ {2,8,18,32,50,192, . . . } $$<br />
<br />
But then there is,<br />
$$f(n)=(\frac{2}{5})n^{5}-6n^{4}+34n^{3}-88n^{2}+(\frac{548}{5})n-48$$<br />
that gives,<br />
$$ {2,8,18,32,50,120, . . . } $$<br />
<br />
It turns out ANY of those choice of answers are a possible continuation of the given sequence! They all work(check for yourself). Thus, they are all correct.<br />
<br />
How is this possible? It turns out that given any finite set of numbers, the set does not necessarily define any single sequence. How then can we in good conscience count our students' answer against them when we have this situation, or how can we in good conscience mislead a student for the sake of cranking out statistical data which, as a result of 'teaching to the test', no longer represents what it originally intended. We can't!<br />
<br />
We have to rethink how we implement these accountability tools so that our children's learning is not crippled in the process. The data collected with these tools was once meant to represent how well our students were prepared academically, now it has simply been reduced to how well a student can fill in a bubble sheet.<br />
<br />
Let's contemplate the example above once again. A problem like this provides a great opportunity for deep thought. This seemingly boring and straight forward problem is of great investigative value, because with a little contemplation, investigation, and ingenuity it becomes a fascinating and exciting project. Most people would be satisfied after finding one solution, but as educators we can lead our students into using their imagination and creativity to ask, "Are there any others?" As you recall I posted three more. Then we ask, "Can we find more?" And the answer to this is a big YES! We can take it further in this day and age of technology, and use a computer to investigate its behavior, generate neat diagrams, and find other interesting solutions. In the process our students learn to really use the technology they have grown accustomed to, while at the same time, gaining an understanding, and appreciation for mathematics. After gaining the deep understanding that comes from hard work and discovery, questions from the STAAR test will become a piece of cake. After all, investigating a problem like this one, as deeply as I describe touches upon most of the content knowledge required to master about 80% of what is on the test. Once again I stress that mastery of the material will result in mastery of the test and not the other way around.<br />
<br />
I close with another ill posed problem, though this one probably intentionally. <span style="line-height: 24px;">This one reminded me of the types of problems I've seen on standardized test, and has recently, in one variation or another, </span><span style="line-height: 24px;">been making waves over the internet.</span><span style="line-height: 24px;"> The problem is the following simple sequence asking for the value of the '?',</span><br />
<blockquote class="tr_bq" style="border: none; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
<span style="line-height: 24px;">8 = 56</span> </blockquote>
<blockquote class="tr_bq" style="border: none; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
<span style="line-height: 24px;">7 = 42</span> </blockquote>
<blockquote class="tr_bq" style="border: none; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
<span style="line-height: 24px;">6 = 30</span> </blockquote>
<blockquote class="tr_bq" style="border: none; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
<span style="line-height: 24px;">5 = 20</span> </blockquote>
<blockquote class="tr_bq" style="border: none; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
<span style="line-height: 24px;">3 = ?</span></blockquote>
<div style="border: none; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
<span style="line-height: 24px;">A great many even displayed an irrational rage when the solutions of others did not agree with theirs. It turns out this once again is a poorly posed question open to interpretation. In the link below </span><span style="line-height: 24px;">I discuss this sequence assuming they are using the relation operator for assignment rather than equality, as the former has more interesting consequences. The layman can skip the mathematical details without loss of content or continuity.</span></div>
<div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;">
<br /></div>
<div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;">
<a href="https://www.scribd.com/doc/260182194/Elementary-Sequences" style="text-decoration: underline;" title="View Elementary Sequences on Scribd">Elementary Sequences</a> by <a href="https://www.scribd.com/AngelAguero" style="text-decoration: underline;" title="View AngelAguero's profile on Scribd">AngelAguero</a></div>
<iframe class="scribd_iframe_embed" data-aspect-ratio="0.7729220222793488" data-auto-height="false" frameborder="0" height="600" id="doc_98933" scrolling="no" src="https://www.scribd.com/embeds/260182194/content?start_page=1&view_mode=scroll&access_key=key-CSSxVVDd3RVjJARGulLR&show_recommendations=true" width="100%"></iframe>
<script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js" type="text/javascript">
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HTML: ["input/TeX","output/HTML-CSS"],
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equationNumbers: { autoNumber: "AMS" } },
extensions: ["tex2jax.js"],
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</script>Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com3tag:blogger.com,1999:blog-5751956832291925224.post-36643434103735075642015-04-24T02:29:00.003-07:002015-04-29T12:36:53.597-07:00Mathematics Problems in an Age of Accountability and High Stakes Testing.<div style="border: none; line-height: 0.25in; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
In an effort to improve education, we have entered an era of accountability via high stakes testing. What does this mean to mathematics education, and what impact does this accountability have on learning?<br />
<br />
Both as a private math instructor, and a math learning center instructor, I work with students of all levels and ages. One observation I have made is that the math homework assigned to most of my public school students consists almost entirely of standardized tests worksheets. My students really know how to take a test. They know a great number of 'tricks', even some that don't require them to understand, or even read the question completely! Apparently, they spend a large percentage of class time every day learning test taking 'techniques' and working problems almost strictly from these practice worksheets--time that, I believe, can be better spent actually learning the material.<br />
<br />
When I first begin working with some of these students, and I assign them some straight forward problem to work out, I usually get blank stares. Some even ask me, "What are the choices?" Of course I explain to them I would like them to try and solve the problem by themselves as much as they can. Frequently, thirty seconds later I get a remark saying, " I still don't get it," or "It's still too hard." At this point I ask the student to read the question to me, and surprisingly about 70% of the time they are able to solve the problem after just reading it, or actually writing some work down and trying after reading it. It turns out all they had to do was actually read the problem thoroughly, or actually attempt the problem by writing out and organizing their ideas.<br />
<br />
What happens when the problems aren't so straightforward though? In my experience they seem to give up after about two minutes if they can't arrive at an answer. An observation I don't experience very often with their private school counterparts.<br />
<br />
Conditioning our students to solve math problems this way cheats them from understanding what math really is, and how beautiful it can be. In order to solve more interesting problems in mathematics a student must be willing to invest the time and effort, as well as apply a bit of ingenuity and imagination. Sometimes patterns are straightforward, other times this is far from the case.<br />
<br />
Take for example the following sequence:<br />
<br />
<div style="text-align: center;">
1, 2, 3, 4, 5, 6, 7, 8, 9, ... </div>
<br />
What is the next number in this sequence? If you said, "10" that's great. You saw a pattern that works, in particular, that the nth term is n.<br />
<br />
Now, how about this sequence:<br />
<br />
<div style="text-align: center;">
2, 4, 6, 8, 10, 12, 14, 16, 18, ...</div>
<br />
What is the next number in this sequence? If you said, "20" great again. Once again, you saw a pattern that works, this time however, that the nth term is 2*n.<br />
<br />
While these are acceptable solutions to questions regarding these sequences, a natural question to ask is, "Are these the only ones?" It turns out they are not. A particularly curious and creative student may say that given,<br />
<br />
<div style="text-align: center;">
1, 2, 3, 4, 5, 6, 7, 8, 9, ... ,</div>
<br />
the next number in the sequence is 0 if the sequence represents the product of the digits of natural numbers, or if this instead represents the sequence of palindromes then next number in the sequence is 11. These rules work, hence, they are also correct answers.<br />
<br />
This leads me to my next observation. These standardized testing practice worksheets are filled with questions that aren't even well posed(some samples and explanations in a future post). Yet we continue to give our students these poorly structured questions, and grind into their brains not only what test makers expected them to answer, but that there is no other way. We are, as Sir Ken Robinson describes, "Killing our children's creativity." Instead we should be nurturing our students curiosity and creativity, after all, they are more essential to learning and understanding.<br />
<br />
The point is, there is much more to math than we lead our students to believe, and only by not crippling them, because we care more about how we look statistically on paper, will we be able to hand down the values and skills we already should be handing down--especially since they empowered us and trust us to do so.<br />
<br />
What then is the problem? Should we give up standardized tests altogether. The answer to this is of course no. We need to hold our educators accountable, and to some extent standardized tests can help. The problem, however, is with implementation. First we need to encourage critical thinking, imagination, and creativity by letting teachers teach, and realizing that mastery of the ideas will lead to mastery of the test, not the other way around.<br />
<br />
<blockquote class="tr_bq">
"Good, he did not have enough imagination to become a mathematician." </blockquote>
<blockquote class="tr_bq">
—David Hilbert's response upon hearing that one of his students had dropped mathematics to study poetry.</blockquote>
</div>
<div style="border: none; line-height: 0.25in; margin-bottom: 0.31in; orphans: 1; padding: 0in;">
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Part II here: '</span><span style="font-family: inherit;"><i><a href="http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of_29.html">Mathematics Problems in an Age of Accountability and High Stakes Testing II</a></i></span><span style="font-family: inherit;">' </span></div>
Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-47481743700646891342013-10-17T01:22:00.000-07:002015-04-07T01:27:02.701-07:00R.I.P Mr. T^2On October 15, 2008 the #1 teacher of mathematics in the United States of America, in fact to ever live, took the journey to be at the side of God. <br />
<br />
Mr. Toby H. Tovar Jr. left his mark upon the world through his family and thousands of students.<br />
<br />
Mr. Tovar only days ago came to mind when, while I was teaching to young children, The director of a learning center said to me, “Angel, if I didn't witness you in action, I wouldn't believe a person that can teach the way you do exists!” The only thing that came to mind that moment was, “You obviously never met Mr. Tovar!” If I could ever only be 1/20th as good a teacher as Mr. Tovar was that would my greatest accomplishment and<span style="font-family: inherit;"> I would feel s</span>et for life.<br />
<br />
The last time I saw Mr. T^2 weeks before his passing he gave me a copy of a short autobiographical sketch he wrote which I include below. Those who new him know that the lines that are slant on that paper are not due to his penmanship(he had the best penmanship of anyone I've ever known), but rather my cheap scanner which I couldn't get to scan that page correctly.<br />
<br />
And now in Mr. Toby H. Tovar’s own handwriting his autobiographical sketch. R.I.P. Mr. T we miss you!<br />
<div style="-x-system-font: none; display: block; font-size-adjust: none; font-stretch: normal; font: 14px helvetica,arial,sans-serif; margin: 12px auto 6px;">
<a href="http://www.scribd.com/doc/176931143/Toby-H-Tovar-Jr-autobiography" style="text-decoration: underline;" title="View Toby H. Tovar Jr. autobiography on Scribd">Toby H. Tovar Jr. autobiography</a></div>
<iframe class="scribd_iframe_embed" data-aspect-ratio="undefined" data-auto-height="false" frameborder="0" height="600" id="doc_45882" scrolling="no" src="//www.scribd.com/embeds/176931143/content?start_page=1&view_mode=scroll&show_recommendations=true" width="100%"></iframe> <br />
Mr. Tovar first came into my life when as a freshman at El Paso High school I wanted to enroll in an Algebra I class, but the counselor wouldn’t let me because, “Mexicans can’t do math.” He went with me to the counselors office, told me to wait, shut the door behind me, and after some yelling I could hear through the door he walked out with my new schedule that included his honors probability and statistics course, and his honors algebra course. He believed in anyone and everyone. If the need arose he could teach calculus to a chimpanzee or even a rock. When Mr. Tovar walked into a room you knew it even if you weren’t looking. The room lit up and you could sense something special in the atmosphere—his presence was certainly felt. <br />
When ever I would praise him he replied that he was nothing special; that he couldn’t do what he did alone because frequently he felt he lacked the strength, and every day when he rose at 4:00 a.m. the first thing he would do is say this prayer by St. Ignatius of Loyola:<br />
<blockquote>
Lord, teach me to be generous.<br />
Teach me to serve you as you deserve;<br />
to give and not to count the cost,<br />
to fight and not to heed the wounds,<br />
to toil and not to seek for rest,<br />
to labor and not to ask for reward,<br />
save that of knowing that I do your will.</blockquote>
It was then and only then held by the hand of God that he could finally get up and do his will. <br />
<br />
Shortly before his passing he said to me he had lived a long and wonderful sixty six years full of joy and packed with living. That he had no regrets and that he was ready for what God had in store for him next. It was at that time that I first heard the following quote by Dr. Seuss. <br />
<br />
<blockquote class="tr_bq">
“Don't cry because it's over, smile because it happened.”<br />
― <a href="http://www.goodreads.com/author/show/61105.Dr_Seuss">Dr. Seuss</a></blockquote>
<br />
R.I.P. <strike>Mr.</strike> Dr. Toby H. Tovar Jr.<br />
<br />
<br />Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-65168733511557415912013-08-26T01:02:00.000-07:002015-04-07T01:21:47.593-07:00Back in session . . .<span style="font-family: inherit;">We are once again at the beginning of a standard school year. Today is the first day students return at our local school district. This new school year brings to our school district a new superintendent; taking over, fresh from scandal, the fifth largest school district in Texas. Fortunately for him there is nowhere to go but up—so I hope.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Some changes I would like to see is unity among teachers and administrators. For too long it has seemingly been an us v.s. them attitude among teachers and administrators. However, I don’t believe either is at fault. I feel that Principals have too much pressure from above to enforce ‘teaching to the test’, and other shady policies even if it means no longer exercising practical wisdom.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Teachers, it seems, want to teach and inspire children but are hindered by politics. Administrators seemingly push teachers to ‘teach to the test’. Don’t get me wrong, I believe there is value in repetition and by the numbers, but exclusively drilling students through algorithmic exercises preparing them to only excel at taking standardized multiple choice state exams fails to teach them critical thinking skills. Furthermore, they fail to engage their imagination and creativity.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">The pressure our administrators are put through to achieve a certain score on standardized test, or receive compensation based on attendance is diverting them from what is truly important--Our children's education! I am of course not suggesting that it is ok to not meet basic academic standards or that skipping school is ok. On the contrary, I believe in the importance of a well rounded education and the expectation that children should attain a minimum level of mastery before they move on, as well as the importance of being in school where they may be guided by their teachers towards mastery. Our school's leaders are being led, through pressure to crank out results via statistical data that has strayed from what it originally meant to represent, to put too much emphasis on ‘teaching to the test’. Other predetermined statistical goals have lead to practical wisdom going out the window, e.g., when a child is absent due to serious illness, under pressure to perform at some level on whatever category attendance falls on, administrators encourage parents to home-school their children—even when they are honor roll students.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">These are only some of the issues I am counting on our superintendent to tackle. </span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Though I now slightly digress, let me tell you, I too have experienced the wrath of a principal under pressure to crank out what I believe are now meaningless scores and statistics. On one occasion a principal tried to throw my son out of his school after a brief hospitalization for an appendectomy—that didn't turn out so bad, but on a separate occasion things became so tense with my daughter’s principal that I was moved to write the letter you see below to the entire school board, E.P.I.S.D. administrators and members of the local media! She just wanted my daughter out of her school although she had straight A’s, because she is a severe asthmatic. Even though she didn't know me, she even once told me, “If you listen to me and do exactly what I tell you to do, and how I tell you to do it, your daughter could be the first person in your family to graduate from high school!” Is my lack of education written on my face? Do I give away the fact I have a low IQ by the clothes I wear? Anyhow, if you are curious as to what I wrote in response read the letter below; otherwise just skip it</span>.<br />
<div style="-x-system-font: none; display: block; font-size-adjust: none; font-stretch: normal; font: 14px helvetica,arial,sans-serif; margin: 12px auto 6px;">
<span style="color: #999999;"><a href="http://www.scribd.com/doc/163396705/School-Board-Letter" style="text-decoration: underline;" title="View School Board Letter on Scribd">School Board Letter</a> </span>by<span style="color: #999999;"> <a href="http://www.scribd.com/AngelAguero" style="text-decoration: underline;" title="View AngelAguero's profile on Scribd">AngelAguero</a></span></div>
<span style="color: #999999;"><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772922022279349" data-auto-height="false" frameborder="0" height="600" id="doc_76026" scrolling="no" src="//www.scribd.com/embeds/163396705/content?start_page=1&view_mode=scroll&access_key=key-1tb3k1zjfz4tktmmxytc&show_recommendations=true" width="100%"></iframe> </span><br />
<span style="font-family: inherit;">To the district’s credit, I received a phone call from the campus director, although on personal leave because of a death in her immediate family, at 7:00 a.m. the next morning with an apology and all the other formalities. By 10:00 a.m. the assistant superintendent in charge of special education was knocking on my door. On top of that, my daughter’s principal and I went on to have a wonderful professional relationship and still do to this day. I found out what a wonderful lady she really is and that deep down she really cares for the children in her school.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Once again, it is only the beginning of the school year and teachers and administrators seem optimistic about the new year and the new superintendent. What I gather from teachers and administrators I’ve spoken to is he cares about the children—we’ll see. Most importantly I hope he creates an environment in which teachers and administrators are once again on the same side. And after that tackles the challenge of educating our children by encouraging critical thinking, imagination, and creativity by letting teachers teach, the administrators once again worry about the children and exercise their practical wisdom, and finally realizing that mastery of the ideas will lead to mastery of the test not the other way around.</span><br />
<span style="color: #999999;"><br /></span>
<span style="color: #999999;"><br /></span>
<br />
<blockquote class="tr_bq">
<em><span style="font-family: inherit;">“The most erroneous assumption is to the effect that the aim of public education is to fill the young of the species with knowledge and awaken their intelligence, and so make them fit to discharge the duties of citizenship in an enlightened and independent manner. Nothing could be further from the truth. The aim of public education is not to spread enlightenment at all; it is simply to reduce as many individuals as possible to the same safe level, to breed and train a standardized citizenry, to put down dissent and originality. That is its aim in the United States, whatever the pretensions of politicians, pedagogues and other such mountebanks, and that is its aim everywhere else.”</span></em><br />
<blockquote>
<em><span style="font-family: inherit;">― <span style="color: orange;">H.L. Mencken, 1924</span></span></em></blockquote>
</blockquote>
Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-38279486612014279532013-08-24T02:09:00.000-07:002017-03-28T02:12:30.103-07:00Once more unto the cold night . . .<div style="line-height: 0.19in; margin-bottom: 0in;">
<span style="color: #999999; font-family: "verdana" , sans-serif;">It
is all to easy to create a blog, but to decide what to include in it
is quite the puzzler. Should I talk about birds, cars , games,
or the stars? Should my audience include painters, athletes, or
laborers? Should it specialize, generalize, or be one big incoherent
mess? In any case, here I am scribbling away my fragmented thoughts
with no goal towards achieving clarity or harmony. In the end
these are my thoughts, for me if you will, or those tired of reading
all the great and interesting things out there and now turn to the
worst so that they may experience it all--the good, the great, the
bad, and the ugly. I shouldn't be expected to focus on
just a few things. Robert A. Heinlein said,</span></div>
<blockquote class="tr_bq">
<h1 class="western" style="background-position: initial initial; background-repeat: initial initial; border: none; font-weight: normal; line-height: 0.19in; margin: 0in 0.39in 0.2in; padding: 0in;">
<span style="font-family: "verdana" , sans-serif;"><span style="color: #999999; font-size: small; font-weight: normal;"><i>A
human being should be able to change a diaper, plan an invasion,
butcher a hog, con a ship, design a building, write a sonnet, balance
accounts, build a wall, set a bone, comfort the dying, take orders,
give orders, cooperate, act alone, solve equations, analyze a new
problem, pitch manure, program a computer, cook a tasty meal, fight
efficiently, die gallantly. Specialization is for insects.</i></span></span></h1>
</blockquote>
<span style="color: #999999; font-family: "verdana" , sans-serif; line-height: 0.19in;">I
wholeheartedly agree. Hence, my focus shall be on nothing. I
will let my fragmented thoughts flow free caring not whether there is
a connection from one post to the next, or even if I have made any
sense within a single post. I will take no care not to go off
on tangents and share with you the delightful fact that pi
can be written as,</span><br />
<div style="line-height: 0.19in; margin-bottom: 0in;">
<span style="color: #999999; font-family: "verdana" , sans-serif;">$$\pi
= \int_{0}^{1} \frac{4}{1+x^{2}}$$</span><br />
<span style="color: #999999; font-family: "verdana" , sans-serif;">or perhaps even spontaneously link to what I may feel at the moment is a delightful image... </span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="color: #999999; font-family: "verdana" , sans-serif;"><a href="http://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Van_Gogh_-_Starry_Night_-_Google_Art_Project.jpg/1280px-Van_Gogh_-_Starry_Night_-_Google_Art_Project.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="253" src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Van_Gogh_-_Starry_Night_-_Google_Art_Project.jpg/1280px-Van_Gogh_-_Starry_Night_-_Google_Art_Project.jpg" width="320" /></a></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="color: #999999; font-family: "verdana" , sans-serif;">In the end I may be making this journey alone. My audience consisting of only myself. Comforting my solitude with my own words; my own fragmented thoughts and ideas. That's alright as I may be the only one who can make sense of the disorder within my own head. Not turmoil, but rather a sort of organized chaos nourished with dreams and making sense only to the imagination of that one individual who gave birth to the ideas.</span></div>
<div class="separator" style="clear: both; color: #59756f; text-align: left;">
<span style="font-family: "verdana" , sans-serif;"><br /></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: "verdana" , sans-serif;"><i><span style="line-height: 18px;">“Be who you are and say what y</span></i></span><i style="font-family: Verdana, sans-serif; line-height: 0.19in;"><span style="line-height: 18px;">ou feel, because those who mind don't matter, and those who matter don't mind.”</span></i></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: "verdana" , sans-serif;"><i><span style="color: #999999;"><span style="line-height: 18px;">― </span><a href="http://www.blogger.com/blogger.g?blogID=5732362523302511668" style="line-height: 18px; text-decoration: none;">Bernard M. Baruch</a></span></i></span></div>
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Anonymoushttp://www.blogger.com/profile/14914571534438144849noreply@blogger.com0tag:blogger.com,1999:blog-5751956832291925224.post-67139114106458444122013-04-07T01:47:00.000-07:002015-04-07T02:43:33.468-07:00Test^2Consider,
$$
\begin{equation}\label{eq:gravt}
F=G\frac{mM}{r^2}
\end{equation}
$$
and,
$$
\begin{equation}\label{eq:gravo}
F=mg
\end{equation}
$$
Combining ($ \ref{eq:gravt} $) and ($ \ref{eq:gravo} $) yield,
$$
g=G\frac{M}{r^2}
$$
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