Week 3: Project Euler

Task Discussion

• 
  Lisa Ritt   Feb. 3, 2013, 9:24 p.m.

  Project Euler Task:
  
  PART 1:  This really has me battling something that I have to realize about myself. I have little patience! However, if I KNOW that I have the ability to accomplish a task & I feel its a worth while thing to do..or solve..or its going to help someone, I"m IN. I've thought this through. I have to admit, if I know that most genious folks (folks that I consider to be experts in their field) consider a problem UNSOLVABLE, I tend to think I would steer away from it. UNLESS the problem  really negatively effected someone OR a community of people. Thats me! 

  PRO:

  -"It's the journey, not the destination" There is much to be said in allowing your brain to practice things.
  -There were many occasions in human existence when people thought a problem was unsolvable, but then, the problem was solved...maybe not by the same person,,,but by the next generation or the next...but...the foundations were maybe laid originally by a scholar beginning to think outside the box....working on an "unsolvable equation"
  -Its important for young people to feel a sense of community & responsiblity towards future generations of people. Working on problems that may seem unsolvable in our world, maybe will NOT be unsolvable in the future...in many cases because of research and technology available currently...This creates the building blocks for future solutions!
  -in Math, we always say "show your work." ..& usually students get credit if some of the work is correct but maybe 1 step was wrong...which made the answer wrong... still was worth while & fantastically important to continue to work on!

  CONS:

  -students may feel frustrated by NOT ever having an answer...but I think the right teaching method can create an expectation of the journey (not the answer) being really important!

  Its funny...when I first read this task, I really thought I'd have more CONS...but the more I thought about it, the more I found something that I think would instill good habits in the kids by doing this!

  Also, for me personally, I don't have enough patience. Its a constant stuggle! In our world, knowledge & answers to questions is very much at our fingertips. Most of us our trained these days to find the quickest way to an answer or problems' solutions. BUT, understanding MATH on just about every level, has to do with how each step in figuring out the problems relate to each other...why do you do this or that & realizing there are lots of wasy to figure something out. 

  Computers take the fastest route to an answer...like a GPS. But, the journey to the answers in Math is so much more important. But, how much time the journey takes can be overwhelming to me...and many others I think....which is why technology is wonderful....Using technology allows you to have more time to work on more problems. BUT- if you are using technology and not really understanding the MATH behind it, whats the point? In my opinion, using technology is best in reiterating the step by step HUMAN way...but if you never understood the HUMAN STEPS befor eyou let the technology take over, you are truly missing out!

   

  PART  2: The popularity thing is so interesting to me. People like feeling that they can perhaps solve a problem thats considered UNSOLVABLE by others! Its like a competitive thing. ...like breaking a record of some kind. I'll tell you though, this really has me battling something that I have to realize about myself. I have little patience & I'm not always super competitive! 

  I think some healthy competition is always a good motivator...even things as simple as last year, class A got an average of 85% on this test...do you think you guys can beat that? ...OR having teams compete against each other to come up with different ways to solve the same problem. When a student feels they are unique, especially intellectually, thats HUGE!

  Also, exercising your brain muscles is really important :)

• 
  Maria Droujkova   Feb. 4, 2013, 7:21 p.m.
  In Reply To:   Lisa Ritt   Feb. 3, 2013, 9:24 p.m.

  Lisa, I found your comments very interesting, because you discussed problems altogether unsolvable. Yes, you would think there would be more cons... Problems on Project Euler's site are solvable (some of them even by hand, though it would take a really, really, really long time), but mostly solvable by writing computer programs. However, given the direction of your thoughts, maybe you will  be interesting in the Math Pickle project. The author, Gordon Hamilton, invites young kids to tackle "million-dollar problems" - those no mathematician has yet solve. And also problems that have been proven unsolvable! Check it out: http://www.mathpickle.com/K-12/$1,000,000_Problems.html

  The point you brought up about technology reiterating what humans do - for learning purposes - is rather deep. It was one of the big threads at the first Computer-Based Mathematics Summit organized by Wolfram Foundation. Here is Conrad Wolfram's talk, where he touches on it a bit: http://www.youtube.com/watch?v=60OVlfAUPJg

  http://www.youtube.com/watch?v=60OVlfAUPJg

  Explaining math to the computer (aka programming) may not follow human thinking. I bet you have experienced some of that interfacing with Scratch and GeoGebra. There are pros and cons to learning "computer-native" ways of thinking about math... Of course, when you just use software other people made, you may not learn any math whatsoever, human or computer.

• 
  Green Machine   Feb. 3, 2013, 6:29 p.m.

  Part 1: I believe that it is a fantastic idea to give scholars math exercises that "cannot" be solved without computer tools. I placed cannot in quotations because I believe that the scholars should be required to solve certain math exercises without the use of a calculator. I would allow the use of a calculator to solve such math exercises once they can eloquently explain and defend the reasoning behind their answer. I believe that the use of such mathematical exercises increases critical thinking, oratory, and explanatory skills. Calculators are magnificient tools and I am an advocate of technology infusion into the classroom, but I also believe greater importance lies with teaching a scholar how to operate without these tools. We must remember that technology can malfuction due to a multitude of variables and that someone who mastered math, programmed technology to carry out the abilities of someone with a pencil, paper, and a brain.

  Computer tool free math exercises may be used a exercises of the week, constructed response activities, and extension or challenge exercises. In my opinion, computer tools should be a second option for scholars; depending on their intellect to effectively complete exercises. The computer tools should be used to check answers, reinforce concepts, or for exercises that take entirely too long to computer by hand. The only negative aspect of giving scholars such exercises to solve is that some scholars may become frustrated if they may not be able to solve it. That could also motivate some scholars to want to increase their mathematical knowledge or abilities, so they can solve such exercises. We must remember that history has proven that monumentus feats have occured through time without the use of computer tools. Some of these feats still cannot be explained, even with the use of our advanced computer tools. That makes me think about the capabilites of hue-man beings and what we can do if properly motivated.

  Part 2: I believe that people like this project so much because of the mental challenge. The exercises on the website allow people to take the basic skills or the foundations of mathematics and computer science; while applying them to a riddle or task. I believe it is also well liked due to the nature of many the exercises. They provide you with hints to help solve them and many of the answers can most likely be discovered once you realize the numerical pattern or sequence that occurs. I could not imagine having to manually find the sum of all the even Fibonacci numbers less than four million. Just imagine the all the writing and then addition you would have to complete. I also believe that people enjoy these exercises because they stress the importance of steps and algorithms when solving complex exercises. They also allow for dialogue amongst peers in the classroom and improves team working skills, depending on how you have scholars work on such exercises.
  
  I think an idea that we could borrow is a website similar to Eulers but it has leveled exercises. Say for instance levels 1 - 5, where one is the easiet and five the hardest. This would allow for the exercises to be properly categorized and scholars can find challenges appropiate to their intellect. I would also use some of the exercises in the classroom as a challenge, where the scholar who solves it properly with a detailed explanation would win some sort of incentive. Depending on the level of difficulty, I would use one for each week or each month. Looking at the exercises on the website, I would most likely use them for a monthly prize and use easier exercises for a weekly challenge. Many of the exercises on the site could be used for concept extensions or infusing them one a lesson with computer programming and mathematical logic. I will definitely be using some of these exercises when I return to the classroom.

• 
  Maria Droujkova   Feb. 4, 2013, 7:28 p.m.
  In Reply To:   Green Machine   Feb. 3, 2013, 6:29 p.m.

  Finding the right amoung of challenge is so hard! Too much, and you frustrate people. Too little, and people get bored. As you say, systems that do LEVELS well are so valuable - maybe because they get it just right... One idea that I find relevant here is FLOW, the optimal "channel" between boredom and frustration. Teaching people to build their own flow is one of harder teacher tasks.

  http://en.wikipedia.org/wiki/Flow_(psychology)

  

• 
  Katherine Hanisco   Feb. 2, 2013, 9:57 p.m.

   

  Part 1: I like the idea of giving students math problems that can’t be solved without a computer. I talked about this a little bit in my response to the Scratch task – whether or not students ever learn a specific computer language, the concept of looking at a problem and knowing how to break it down into a set of algorithms is a very important skill. I used to be an actuary at an insurance company, and even though programming was not part of my job description, I used algorithms for complex computations on an almost daily basis. So I can say from experience that knowing how to solve math problems with a computer is extremely useful!

  On a more abstract level, I think that these problems provide the opportunity to talk about math in a historical context. Number theory developed without computers and there are many ideas to explore that don’t require any computation at all, but computers have allowed mathematicians to look at problems that were impossible before computers. For example, the exploration of prime numbers was very different before and after computers. I know that there usually isn’t time to work that sort of thing into a busy curriculum, but I love the idea of including relevant math history because I think it adds a layer of richness to the material.

  As for cons, I think that including these kinds of problems too often could take the focus off the math. Obviously students would have to understand the math in order to solve the problems, but if they are a major part of instruction I think that other important methods and concepts could get pushed aside.

  Part 2: As Megan mentioned, I think one of the big draws to this project is that some people just really enjoy solving puzzles. They’re problems that make you think, but as far as the math goes, many of them are accessible to anyone who has a good grasp of high school math. I also think there is an appeal for people who enjoy competing with others (or themselves) to solve problems to move up levels. Including math games or activities where students can earn points and move up levels could make students motivated about the material. 

• 
  Maria Droujkova   Feb. 3, 2013, 11:30 a.m.
  In Reply To:   Katherine Hanisco   Feb. 2, 2013, 9:57 p.m.

  It might be interesting to compare and contrast the motivation that comes from solving puzzles t the motivation that comes from progress through levels, earning points, gaining badges and other "gamification" ideas.

  The history of math and technology is fascinating. Abacus and its versions (invented independently on all continents) led to big breakthroughs both for math and for its applications... Prime numbers is an excellent example, with all the cool patterns the computers reveal, like the Ulam Spiral.

  

• 
  SueSullivan   Feb. 1, 2013, 8:24 p.m.

   

  PART 1

  I looked at Project Euler's first 5 problems.  When I took CS201 (Introduction to Programming in Java) last spring, we wrote code to solve similar types of problems (I also remember doing the same thing in 9th grade when a year of Basic was mandatory).  In these cases, programming isn't essential to solve the problem in a literal sense; programming skills are essential if you want to solve the problem in a reasonable amount of time (obviously these don't fall into the 'most problems' category that the introduction mentions).  I looked at problem 50 (Sum of Consecutive Primes below 1 million); this also could be done without programming, although it would take most people a very, very long time.

  I think that it's appropriate to give students problems that require the use of computer tools, as long as the task is within the student's Zone of Proximal Development/frustration tolerance limits.  If the student is capable, a teacher should certainly encourage the student to explore.  I don't feel that using computers to solve problems makes students 'lazy' or anything like that, for students need to know how to solve a problem themselves before they can instruct a computer how to solve it.

  I consider programming (by itself) to be similar to learning to speak another language, and in this way, it has very little to do with math and more to do with giving instructions in a way that will be understood.

  However, it seems that due to standardized testing/NCLB, many public schools have to 'teach to the test' and have little or no room in their curriculum for anything else (such as teaching programming that's not required by law).  I'm not trying to start a political argument - this statement is based solely upon my personal observations (in classrooms, with other parents, bloggers, etc.). 

  Another drawback to having students rely on programming to solve problems in a timely fashion is the fact that technology can evolve so rapidly.  A school may commit to (and invest in) teaching students a particular programming language (or other technology) in elementary school, only to have it rendered obsolete by the time the students are in high school.  Some students will adapt; others will not.  Math by itself isn't subject to the external factors that influence technology.

  PART 2

  I think that Project Euler is popular because it encourages the use of programming so that problems can be solved as as quickly and efficiently as possible.  Using programming/technology saves time.  Being able to solve a problem relatively quickly might mean that someone can use their 1-hour lunch break (or child's 1-hour nap time) to get satisfaction from solving a challenging (yet mentally stimulating) math problem.  Others may use their 'time savings' for other learning tasks (such as more math problems).

  In the classroom, using technology to quickly perform tasks (that students already know how to do 'manually') allows more time to be spent on other things, whether that's exploring math or other subjects, playing a community-building game, or simply taking a couple of minutes to get out of one's seat to stretch and refresh.  

• 
  Maria Droujkova   Feb. 3, 2013, 11:37 a.m.
  In Reply To:   SueSullivan   Feb. 1, 2013, 8:24 p.m.

  Programming as a literacy or language is an interesting metaphor for it. As a way to give instructions - yes, not much math to learn there. As a way of thinking - ?

  Teaching is very political in nature, and technology of math ed is no exception. I expect some political arguments to surface as we discuss issues. 

  You discuss time as a critical variable. I would like to contrast two situations.

  1 - Using a ready-made algorithm to do something you could do by hand. For example, punching x²+2x+1=0 into Wolfram|Alpha and reading the answers.

  2 - Creating an algorithm to do a task. For example, programming something (Basic, Scratch, Geogebra...) so you give a, b, c and the program solves ax²+bx+c=0

  The first situation "just" saves time and allows us to get on with more interesting math problems. The second can do more for math learning.

• 
  SueSullivan   Feb. 3, 2013, 5:19 p.m.
  In Reply To:   Maria Droujkova   Feb. 3, 2013, 11:37 a.m.

  Thanks for the examples - they provide a very clear and concise example of the difference between the two situations!

• 
  Gina Mulranen   Jan. 31, 2013, 11:06 p.m.

  Part 1
  I think that using computer tools is a great way to challenge students in a new way with concepts they have already mastered. I think that by introducing computer tools before the student has enough time to practice the concept will cause the student to lose some understanding of how and why the concepts works. For example, I have students graph linear functions by hand for weeks before I show them how to graph them on the graphing calculator. Then we use the graphing calculator to graph multiple lines and explore the transformations, since they are very familiar with slope and the y-intercept at that point. Using this example also explains my other view of using computer tools. Before teaching transformations on linear functions, I have the students graph linear functions with different slopes and y-intercepts on the graphing calculator and take notice of what was happening to the parent function. After this activity using technology, the students were able to discover the different transformations. This is when using computer tools can be a part of the learning process.

  Giving math problems that can’t be solved without computer tools would be an interesting challenge for those students who have the depth of understanding necessary to understand how the computer tool works. For example, the construction that we did in GeoGebra would be a great way for students to problem solve using the properties of a square that they learned. The con that I see coming from an assignment like this for middle school is the frustration level of 5^th and 6^th graders (and their parents!) as they spend hours troubleshooting with the program. I would have to incorporate in-class lessons to teach the students how to use the program, which is very hard to squeeze into my curriculum. I only see my students 2 or 3 days a week since I teach in an enrichment program, so I do not have a lot of class time to work with. I have been assigning projects involving spreadsheets, but have been approached by parents that the tutorials I provide are not like classroom instruction and I need to teach the technology in order to expect the students to use it. So I been recently frustrated with the technology that I have been trying to incorporate.

  Part 2
  This was an easy question for me to answer. I am one of the teachers at my school that runs the Math Counts club and team. The Math Counts club consists of students that love the opportunity to just do challenging math problems without the pressure of grades or a formal class structure. It’s for students that really love math and the challenges it presents when problem solving. A lot of the problems we do the students work on their own to solve and then share different ways to solve the same problem. It is incredible to hear how students can find these interesting ways to solve a problem, and so quickly! I learn just as much as they do! I have seen how much these students love the challenge when presented with a problem that is unique from a question they would see in their math class. It is like a puzzle for them, working with the parts of the problem until it fits. It is also competitive, which I also saw in the statistics of the Project Euler website. Seeing how many people have solved the problems gives students, like myself, the motivation to solve it. The students practice hard all year for the Math Counts competition, where they compete as a team and individually. The most exciting to watch is the two buzzer rounds that are on a stage in front of an audience and the first person to get the answer right moves on. You should see how fast the students can get an answer! From my experience with Math Counts, I think that the Project Euler sites attracts those people who enjoy a challenge, competition, and the thrill of solving the puzzle, especially when you are competing with people all over the world!

• 
  Maria Droujkova   Feb. 3, 2013, 11:46 a.m.
  In Reply To:   Gina Mulranen   Jan. 31, 2013, 11:06 p.m.

  Gina, I expect anyone who tries to use technology can relate to your frustrations! Conventions of, say, a spreadsheet - how you drag a cell down, how you click a cell to enter its code (A3 or whatever), how you write =... to enter formulas... All these conventions are far from obvious, and take time to learn. Scratch, for example, is the successor of some 40 years of trying to make programming accessible to students. It's great in many ways, but is it obvious? Ha!

  Great examples of two tasks where you start by hand vs. start by observing graphing patterns. I like to use large colored screens and graph multiple lines (which different students suggest) on the same coordinate plane, for a similar task.

  Math Counts is a very lovely program, with passionate people behind it. It would be nice if they had something like Project Euler out of their old problems. But the Math Counts community already shares problems a lot. One place that does it well is The Art of Problem Solving, with their Alcumus system: http://www.artofproblemsolving.com/Alcumus/Introduction.php

• 
  Gina Mulranen   Feb. 3, 2013, 12:45 p.m.
  In Reply To:   Maria Droujkova   Feb. 3, 2013, 11:46 a.m.

  We actually just had the 7th and 8th grade Math Counts competition yesterday and even though we practiced and practiced and reviewed different ways to solve all these types of problems, the students still said the problems were very challenging. Even my brightest students said they were not as confident as they felt in class. It is funny that you mentioned the Art of Problem Solving because one of my parents just told me about this website to help with preparing the students for Math Counts competitions. I really like the layout and how it organizes it by concepts because we realized as a team that our weakness was geometry. Using this website, we can work through the different examples and learn different ways to solve these types of problems. Thank you!!!

• 
  MgnLeas   Jan. 29, 2013, 6:19 p.m.

  Project Euler

  Part 1

  I think this answer depends on the students and the class in which you are teaching. For instance, at the elementary level, students are just starting to understand the basic tools of math, addition, subtraction, ect. The problems they are given should not be so complex that they need to turn to computers and algorithms to solve.  However, in middle and high school the math becomes more and more complex. At this point the concepts are what becomes important, not necessarily the work behind it. In these cases using computers may be what the students need. The students could also have the opportunity to write their own algorithm for how to solve a more complex problem without doing all the actual “leg work”.

  Pros	Cons
  1.       Easier to solve more complex problems. 2.       Create algorithms that can be used to solve these problems. 3.       Use of technology in the lesson.	1.       Lose the basic solving skills. 2.       The problem may beyond the grasp of students. 3.       The computer could solve the problem without the student understanding how.

   

  I had the opportunity to take a computer programming class during my undergrad. We all had to complete a final project that was writing a program of our choosing. I had looked for a while for a project and came across The Locker Lottery Problem. For those who have never heard it here is how it goes,

                  There is a school with 1000 lockers and 1000 students. On the first day of school all the lockers are closed. The first student enters school and opens ever locker. The second student comes in and goes to the second locker. He closes this lockers and every second locker. The third student goes to the third locker and closes it. He then goes to every third locker, if it is open, he closes it, and if it is closed he opens it. The fourth student comes in, goes to the fourth locker and again, if it is open she closes it and if it is closed she opens it. This continues until every student has entered the building. At the end, how many lockers are open and which ones and how many are closed and which ones?

  This question stuck with me and I just had to know. It would have taken a long time to work this out by hand. However it took me about two hours to write a program and getting it working to perform the opening and closing of lockers. I had my answer and thanks to the computer had it in a fraction of the time. This is an instance where the use of the computer was, in my opinion, a necessity.

  Part 2

  After reading the first 20 problems, and spending more time than I had to spend on attempting a few of them, I can see why this project draws so many people in. These problems are not ones that can be easily solved by many people. With time and patience most people could probably solve them all. Some people like to work on things simply for the joy of solving them. Take Sudoku and Crossword puzzles; there is no other reward to solving them accepts the satisfaction of doing it. Perhaps beating the time it took someone else. (I get competitive with Sudoku myself. A group of friends and I solve them to see who can do it faster.)I used some Sudoku puzzles while student teaching to keep kids occupied while others were still completing work. They got bonus points for being the first 3 to solve it. I could take many things from these problems to liven up some lessons. As I am currently not teaching these are just some ideas, not sure how they will actually work in classes. The first problem Multiples of 3 and 5 this would be an interesting way to get them to learn these. Also problem 7 deals with listing primes, having them have a goal other than just simply listing the primes might make some students more motivated to find them. Some of these problems could be modified so they would not take as much time too. They could also be used for “extra points”. Students could work on them when they finish the tasks for the day. They could even be used as group projects.

• 
  SueSullivan   Feb. 1, 2013, 8:30 p.m.
  In Reply To:   MgnLeas   Jan. 29, 2013, 6:19 p.m.

  It's nice to hear that my sister and I aren't the only people out there who have Sudoku competitions!    She usually beats me though...

• 
  MgnLeas   Feb. 1, 2013, 9:05 p.m.
  In Reply To:   SueSullivan   Feb. 1, 2013, 8:30 p.m.

  I tried to teach my older sister which was a waste of a day. She did not get it and did not talk to me for days! But my younger sister, who is only 16 now, caught on and beats me regularly! It is fun to do math in different ways with outs really thinking about it.

• 
  Maria Droujkova   Feb. 3, 2013, 11:57 a.m.
  In Reply To:   MgnLeas   Jan. 29, 2013, 6:19 p.m.

  Nice layout of pros and cons! You can compare and contrast them easy in a table.

  You bring up  ages (or curriculum levels) as a factor for the use of technology. As you mention, young kids mostly learn shapes, addition and subtraction. The question I find intriguing is - what COULD they learn, given more powerful tools? Could they observe patterns in graphs if you change formulas? I had a six-year-old run up to me with a graphing calculator, all excited about that neat picture he could make (I think he punched sinx and liked the wave). But then he started to change it, like sin2x. A kid who can add can start playing with a spreadsheet... A kid who can work with shapes could explore transformations in GeoGebra, etc.

  Puzzles, oh puzzles! The math (the "whys") of Sudoku is fascinating. Here are a couple of (hard) questions from the Math Circles site:

   

  1. How many Sudoku squares are there?
  2. What is the minimum number of clues that yield a unique solution to a Sudoku puzzle?
• 
  MgnLeas   Feb. 3, 2013, 7:01 p.m.
  In Reply To:   Maria Droujkova   Feb. 3, 2013, 11:57 a.m.

  It is a good point you bring up about age and ability. I see my 2 year old doing gymnastic skills that I see 4 year olds unable to do. He is physically advanced for his age. But as soon as he took his first step we were enrolled in a gymnastic class. I imagine the same would hold true for mental abilities. If you expose children to things they will learn them. The more time they have to play with things the more they learn. And children remember the craziest things! Children now have so many technologies around them. The six year old you spoke of had to be exposed to that calculator for him to be able to learn how to use it.

• 
  Maria Droujkova   Feb. 4, 2013, 6:52 a.m.
  In Reply To:   MgnLeas   Feb. 3, 2013, 7:01 p.m.

  Just so - exposure. Plus interest. I bet your son likes gymnastics-type activities, so he does a lot of them, so he gets better. This is one reason we need to have a wide range of tech for math ed available. So different kids could choose what they like.

  For example, I find that offering several programming languages as a choice (after a brief exposure to each) makes programming courses more appealing. It's not that one language is more interesting than another, but they all have different STYLES.