I recently ran into the following interesting problem: Can you find distinct integers such that
$$
\begin{eqnarray*}
a^{2}+b^{2}+c^{2}+d^{2} & = & abcd?
\end{eqnarray*}
$$
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!
Here we will investigate one possible approach.
Taking the original equation, and completing the square we get,
$$
\begin{eqnarray*}
a^{2}+b^{2}+c^{2}+d^{2} & = & abcd\\
a^{2}-2abcd+(bcd)^{2}+b^{2}+c^{2}+d^{2} & = & abcd-2abcd+(bcd)^{2}\\
(a-bcd)^{2}+b^{2}+c^{2}+d^{2} & = & -abcd+(bcd)^{2}\\
& = & (bcd-a)bcd\\
\Rightarrow(bcd-a)^{2}+b^{2}+c^{2}+d^{2} & = & (bcd-a)bcd
\end{eqnarray*}
$$
and
$$
a'=bcd-a
$$
or equivalently,
$$
a_{n}=bcd-a_{n-1}
$$
We chose to complete the square using 'a', but what about b, c, or
d, after all addition and multiplication are commutative. Well it
similarly follows for those. Rewriting the original equation we get,
$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}x_{2}x_{3}x_{4}
$$
thus,
$$
x_{i}^{'}=\prod_{_{\begin{array}{c}
j=1\\
j\neq i
\end{array}}}^{4}x_{j}-x_{i}
$$
We see that,
$$
\begin{eqnarray*}
a_{1} & = & bcd-a_{0}\\
a_{2} & = & bcd-bcd+a_{0}\\
& = & a_{0}\\
a_{3} & = & bcd-a_{0}
\end{eqnarray*}
$$
which has a period of two. Thus, we can't just endlessly continue
iterating the same respective term, but from 'part II' we can see
we can change it up. One way could be,
$$
(a,b,c,d)\rightarrow(a',b,c,d)\rightarrow(a',b',c,d)\rightarrow(a',b',c',d)\rightarrow(a',b',c',d')\rightarrow(a'',b',c',d')
$$
and so on, but this blows up rather quickly.
Another way could be to evaluate,
$$
x_{n}x_{n-1}x_{n-2}-x_{n-3}
$$
and assigning it to,
$$
x_{n+1}
$$
consequently generating the sequence,
$$
\{x_{1},x_{2},x_{3},x_{4},...,x_{n}=x_{n}x_{n-1}x_{n-2}-x_{n-3},...\}
$$
of which any four consecutive terms in the sequence satisfies the
original equation.
For example, using the trivial solution $\{2,2,2,2\}$ yields,
$$
2
$$
$$
2
$$
$$
2
$$
$$
2
$$
$$
6
$$
$$
22
$$
$$
262
$$
$$
34582
$$
$$
199330642
$$
$$
1806032092550706
$$
$$
12449434806576800059248920402
$$
$$
4481765860945171681908664776799089162954814190172722
$$
$$
\vdots
$$
and we can go on indefinitely.
Sunday, April 16, 2017
Friday, January 1, 2016
Happy 10/9!*8!*7!-6!*5!/4!+3!*2!+2!*(1!+0!)!)
Happy 10/9! *8! *7! -6! *5! /4!+3! *2! +2! *( 1!+0!)!)
Wishing you all the best this upcoming year. May it be your best year yet!
Cheers,
--Angel Agüero
Wishing you all the best this upcoming year. May it be your best year yet!
Cheers,
--Angel Agüero
Tuesday, December 1, 2015
Testimonials
November 22, 2016
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!
Best Regards,
Roberto Acosta, RN-BSN, DNP Student
To Whom It May Concern:
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 theunsurmountable months Aguero‘s unsurmountable
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.
Should you have any questions or need additional information, please do not hesitate to contact us at the above telephone numbers."
Sincerely,
M & L Gregory"
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
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.
Should you have any questions or need additional information, please do not hesitate to contact us at the above telephone numbers."
Sincerely,
M & L Gregory"
"I first made the acquaintance of Angel Aguero when we where classroom
Jaime P. Lepe
Mathematics Lecturer, University of Texas at El Paso
"I am a college student working on my Masters. Despite that, math math
--Rick Aragon, Librarian, Film Critic. Graduate student in Library Sciences
"I met Angel in the latter part of 2014. He was a tutor at a West El Paso learning center.
Working toward an alternative certification to teach high school English, I had to pass the math portion of the Texas Higher Education Assessment (THEA).
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.
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.
Enter Angel.
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.
I don’t remember what side of the brain I am. I must have forgotten to carry the 7.
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 test but
"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."
--Rene Garcia, Franklin High School, El Paso,
Massachusetts Institute of Technology, Cambridge, MA
"Angel the genius."
--Joaquin Ordonez, Engineer and Project Manager at Austin Water Utility
"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 problems but school
--Eduardo Cabrera, International Baccalaureate Magnet Program at E.P.I.S.D.
Sunday, May 10, 2015
Mathematics Education in the Age of Technology
It is no mystery that students are having increasing difficulty learning algebra in schools. One proposition referenced in
http://www.edutopia.org/blog/math-as-social-justice-lever-jose-vilson,
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.
By surrendering to this mind set, we are as educators, failing to give our students the credit they deserve. Where then, lies the problem?
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:
Part I: http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of.html
Part II: http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of_29.html
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.
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,
"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?"
These is a surprisingly simple question, yet a large number of students missed it. What then can we do about this?
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.
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. . .
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!
http://www.edutopia.org/blog/math-as-social-justice-lever-jose-vilson,
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.
By surrendering to this mind set, we are as educators, failing to give our students the credit they deserve. Where then, lies the problem?
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:
Part I: http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of.html
Part II: http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of_29.html
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.
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,
"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?"
These is a surprisingly simple question, yet a large number of students missed it. What then can we do about this?
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.
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. . .
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!
Saturday, May 2, 2015
Everyday Algebra
Recently I have ran more and more often into the following statement.
“Well, another day has passed. I didn't use algebra once.”
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.
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.
Let us briefly take a look at the simplest type of algebra problem, namely, solving the following equation for $x$,
$$ax+b=c$$
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.
Now, to solve this equation for $x$
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.
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,
$$f([pressin\_button\_picturing\_drink])=[Dispensing\_that\_drink]$$
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,
$$f([amount\_pedal\_is\_displaced])=[speed\_of\_the\_vehicle]$$
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:
Mathematically,
$$f^{-1}(f(x))=x$$
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.
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.
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:
$$F\propto\frac{\triangle P\cdot r^{4}}{\eta\cdot L}$$
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!
The pdf version:
“Well, another day has passed. I didn't use algebra once.”
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.
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.
Let us briefly take a look at the simplest type of algebra problem, namely, solving the following equation for $x$,
$$ax+b=c$$
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.
Now, to solve this equation for $x$
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.
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,
$$f([pressin\_button\_picturing\_drink])=[Dispensing\_that\_drink]$$
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,
$$f([amount\_pedal\_is\_displaced])=[speed\_of\_the\_vehicle]$$
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:
Mathematically,
$$f^{-1}(f(x))=x$$
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.
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.
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:
$$F\propto\frac{\triangle P\cdot r^{4}}{\eta\cdot L}$$
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!
The pdf version:
Wednesday, April 29, 2015
Mathematics Problems in an Age of Accountability and High Stakes Testing II
This is the second part of my previous post, 'Mathematics Problems in an Age of Accountability and High Stakes Testing'. 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.
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:
Given the sequence 2, 8, 18, 32, 50, . . . What is the next term in the sequence?
- 60
- 72
- 120
- 192
Lets think about this for a second, and try to work out an answer. With a little effort we can come up with say,
$$f(n)=(-\frac{1}{10})n^{5}+\frac{3}{2}n^{4}-\frac{17}{2}n^{3}+\frac{49}{2}n^{2}-(\frac{137}{5})n+12$$
which results in
$$ {2,8,18,32,50,60, . . . } $$
But,
$$f(n)=2n^2$$
gives
$$ {2,8,18,32,50,72, . . . } $$
However,
$$f(n)=n^5-15n^4+85n^3-223n^2+274n-120$$
gives,
$$ {2,8,18,32,50,192, . . . } $$
But then there is,
$$f(n)=(\frac{2}{5})n^{5}-6n^{4}+34n^{3}-88n^{2}+(\frac{548}{5})n-48$$
that gives,
$$ {2,8,18,32,50,120, . . . } $$
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.
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!
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.
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.
I close with another ill posed problem, though this one probably intentionally. This one reminded me of the types of problems I've seen on standardized test, and has recently, in one variation or another, been making waves over the internet. The problem is the following simple sequence asking for the value of the '?',
8 = 56
7 = 42
6 = 30
5 = 20
3 = ?
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 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.
Friday, April 24, 2015
Mathematics Problems in an Age of Accountability and High Stakes Testing.
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?
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.
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.
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.
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.
Take for example the following sequence:
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.
Now, how about this sequence:
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.
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,
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.
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.
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.
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.
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.
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.
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.
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.
Take for example the following sequence:
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
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.
Now, how about this sequence:
2, 4, 6, 8, 10, 12, 14, 16, 18, ...
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.
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,
1, 2, 3, 4, 5, 6, 7, 8, 9, ... ,
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.
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.
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.
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.
"Good, he did not have enough imagination to become a mathematician."
—David Hilbert's response upon hearing that one of his students had dropped mathematics to study poetry.
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