Solving open problems with students as a first research experience

Solving open problems with students as a first research experience Abstract In this article, we share some experiences about working with students on research projects. Instead of giving them problems from our own projects to work, we use open problems from journals. Using these resources we have worked with several undergraduate and some graduate students giving them a taste of their first research experience. Here we describe how these projects were conducted and the various criteria we thought about when we found the right problems for the students to work on. We also give a list of problems that our students worked on and the books and articles the students may read in order to get ready for the projects. I. Undergraduate research experience Undergraduate research is an important part of the undergraduate curriculum and many colleges and universities encourage this initiative. In small colleges that primarily focus on teaching, providing the research experience in the undergraduate level is a challenge for some faculties. During the years that we have been supervising research projects of undergraduates, the students have presented their research at different meetings and on some occasions we have been asked questions like, how to start working with students who have never had experience in research, where to find open problems that are solvable by undergraduate students and so on. In addition to supervising undergraduate projects, we also have had some graduate students who have undertaken research projects while working on their master’s degree in mathematics education. We have found the graduate students also to be well motivated. Teachers dedicate a lot of time designing and implementing research projects for students. Therefore, it is desirable to have highly motivating projects, especially when the students are doing research for the first time. We have found that instead of assigning them—for the first time—problems from our own projects to work, it is more motivating for them when we assign open problems from journals. We have also found two interesting consequences of those types of projects: the first is that that the students are motivated by the fact that the solution to the open problem they solve will be published in an academic journal. The second consequence is that when the students finish the project with us, they know where they can find new problems for their own projects and work independently. There are several journals that publish open problems at various levels and areas of mathematics. Our main focus is recursive sequences and therefore our main resource is the Fibonacci Quarterly (The Fibonacci Association). Other journals are, for example, Mathematics Magazine (Mathematical Association of America), The American Mathematical Monthly (Mathematical Association of America), The College Mathematics Journal (Mathematical Association of America), MathProblems Journal (University of Prishtina) and La Gaceta (in Spanish) (Real Sociedad Matemática Española). We have had positive experiences when the students work on open problems using recursive sequences. The problems on these sequences are understandable (even if some of them are very hard to solve). We have noticed the problems on recursive sequences, like Fibonacci and Lucas, are very interesting and are therefore very helpful in stimulating student curiosity in research. From our own experience we noticed that, once the students understand the problem, they are willing to try to find a solution. We have noted that, even when the problem is extremely hard, they want to keep trying—motivated by their curiosity. We would like to note that some of these students (undergraduate or graduate) who started their research career by solving problems, later worked on joint projects either with us, with other researchers or even by themselves. We should note here that many problems from Fibonacci Quarterly are a good fit for students but it is up to the advisor to choose the right problem for a student to work on. Several problems, even in the elementary section, are very hard for the undergraduate and graduate students. Many of those problems require expertise in areas like combinatorics, number theory, generating functions, functional analysis, etc. 2. Background and suggested problems to work with students If the students are interested in doing research involving solving open problems from the Fibonacci Quarterly, we recommend that they study the book Fibonacci and Lucas Numbers with Applications by Koshy (2001). We especially recommend Chapter five of the mentioned book. Other important sources that the students can use are Benjamin (2003), Knott, and Vajda (1989). We now recall that the Fibonacci numbers Fn and Lucas numbers Ln satisfy the recursive relations: Fn=Fn−1+Fn−2withF0=0andF1=1 Ln=Ln−1+Ln−2withL0=2andL1=1. The Binet formulas for Fibonacci numbers and Lucas numbers are given by: Fn=αn−βnα−βandLn=αn+βnwhereα=1+52andβ=1−52. As a new advisor for undergraduate student research, it is natural to ask, how to help students without helping too much or, what happens if the students are asked to work on research without enough help from the advisors? So we help students by asking them similar questions as Polya asked in Polya (2014). We now provide a list of some problems from the Fibonacci Quarterly that may help the students in their preparation before starting their research project. After each problem we explain how it can be solved and their difficulty level. The complete solutions of all the problems are cited in the bibliography section. We recall that to solve the following problems one might need the resources in Benjamin (2003), Koshy (2001), Knott, and Vajda (1989). Other problems that may help prepare students for research are those that Higuita solved when he was an undergraduate student. Problem B-1081 (Michaels-Arfor, 2012). Prove that Fn+12Fn−12−Fn4=(−1)n[2L2n+(−1)n]/5. The students only need to know Chapter five of Koshy (2001). Problem B-1133 (Watson, 2014). Determine the value of the following infinite series S given by 12·1+11·3−12·3+13·5+13·8−15·8+18·13+18·21−113·21+…. This problem can be solved by students with calculus background. In particular it requires that the students know about limits, series (partial sums) and telescopic series. Problem B-1190 (Harbol & Tiscareno, in press). Let n≥1 be a positive integer. Compute Fn+2FnFn+1(Fnn+Fn+1n−Fn+2nFnn−1+Fn+1n−1+Fn+2n−1)+Fn+3Fn+1Fn+2(Fn+1n+Fn+2n−FnnFnn−1+Fn+1n−1+Fn+2n−1)+2Fn+Fn+1Fn+2Fn(Fn+2n+Fnn−Fn+1nFnn−1+Fn+1n−1+Fn+2n−1). This problem is a good fit for a student with calculus and discrete mathematics background. Even if the problem does not need calculus, it is recommended that the students have some experience in writing proofs of theorems along with some idea about computer programming. Problem B-1154 (Lindberg, 2015). Find a closed form for the expression ∑i=0nLi2Li+12. This problem is a good fit for students with calculus and discrete mathematics background. It is recommended that the students have some experience in writing proofs of theorems before undertaking the project to solve this problem. Problem B-1165 (Adams, 2016). For an integer n≠0, find the value of LF3nFF3n−1FF3n−2+LF3n−1FF3n−2FF3n+LF3n−2FF3nFF3n−1. This problem is also a good fit for students with calculus and discrete mathematics background. The problem does not require calculus but it is recommended that the students have some experience in writing proofs of theorems. Problem B-1192 (Ching, in press). Let Mn be an n × n matrix given for all n≥1 by: (F110…000F2F11…000F3F2F1…000…⋯…⋱………Fn−1Fn−2Fn−3…F2F11FnFn−1Fn−2…F3F2F1). Prove that det(Mn)={0,ifniseven,1,ifnisodd. This problem is a good fit for students with discrete mathematics and linear algebra background. The solution requires knowledge of principle of mathematical induction and determinants. Problem H-766 (McAnally, 2016). Let n=m+2. For m≥1, prove that ∑h=1m∑i=1h∑j=1i∑k=1jFk4=4Fn4+n4−5n2100. This problem is considered to be at an advanced level by the journal. We recommend that this problem only be assigned to undergraduate students in their senior year or some math education students who are pursuing a master’s degree. Problem B-1196 (Wang, in press). Prove that Ln+25−Ln+15−Ln−15−Ln−25Ln+23−Ln+13−Ln−13−Ln−23=5Fn+25−Fn+15−Fn−15−Fn−25Fn+23−Fn+13−Fn−13−Fn−23. This problem is a good fit for a student with calculus and discrete mathematics background. It is recommended that the students have some experience in writing proofs of theorems. Problem B-1130 (Graves, 2014). Prove that for all positive real numbers m ∑k=1nFk2m+2k3m≥4mFnm+1Fn+1m+1n2m(n+1)2m. This problem can be worked by advanced students with some knowledge of real analysis or functional analysis. 3. Beyond solving open problems Once the students have solved the open problems, we may encourage them to think about ways to further their research by asking the right questions. For example, if they replace Fibonacci numbers with Lucas numbers or if they change Fibonacci numbers to Fibonacci polynomials, they may obtain results that are deeper and more interesting (see the ‘Conjecture’ below and Flórez et al. (in press)). The students who solved (Harbol and Tiscareno, in press) went on to discover that a similar result holds for Lucas numbers. They also used a computer program to test various other identities obtained by changing problem (Harbol and Tiscareno, in press) slightly. Their experiments yielded the results in Theorem 3.1 that they have since proved. We note that these results also hold when Fibonacci numbers are replaced with Lucas numbers. Theorem 3.1 If α is the Golden Ratio then, (1) limn→∞1Fn+1(Fn+1n+Fn+2n−FnnFnn−1+Fn+1n−1−Fn+2n−1)=α (2) limn→∞[1−Fn+2FnFn+1(Fnn+Fn+1n−Fn+2nFnn−1+Fn+1n−1+Fn+2n−1)−2Fn+Fn+1Fn+2Fn(Fn+2n+Fnn−Fn+1nFnn−1+Fn+1n−1+Fn+2n−1)]=α A natural question that follows from Ching (in press) is: what is the behaviour of the sequence generated by the determinants of the matrices in Problem B-1192 when their non-zero entries are replaced by their corresponding Lucas numbers? This question can be formally stated as follows. Conjecture. If Nn is the matrix obtained by replacing Fi in Mn in problem B-1192 by Li−1 with 1≤i≤n, then det(Nn)=an−1, where an=an−1+2an−2,n≥2,a0=2, and a1=3, i.e, an is equal to [5(2n)+(−1)n]/3 for n≥0. The student who solved (McAnally, 2016) went on to work on more advanced topics later on. He started working on his first research project during his second year of college and due to his expertise in this area and academic maturity, he later worked on a paper involving generalized Fibonacci polynomials. Working on this project we discovered that over 100 identities that are true for Fibonacci numbers, also happen to be true for the generalized Fibonacci polynomials (see Flórez et al., in press). 4. Feedback from some former students Student 1. As a graduate student, I had the opportunity to conduct research under the guidance of my teachers, Drs. Flórez and Mukherjee. I remember being in a classroom with all of my work written across several white boards and finally figuring out that the solution I had was correct. What a great feeling! After the first problem that I solved was published, I presented my work at several conferences. This led me to meet several other math lovers that I worked on my second research project therewith. There were days where I was ready to give up my research project after being frustrated of not figuring it out. Looking back, I am so thankful that I stuck to it and discovered the joys of doing mathematics research. Student 2. While I was working on my M.A.E. in Mathematics, I was challenged by Dr. Flórez to solve an open problem posed in the Fibonacci Quarterly. I worked on several related problems provided in order to learn what identities and tools would be necessary to solve my problem. Once I was ready, I began trying to solve the equation. After over a month of preparing and practicing, I discovered that I knew everything I needed in order to solve the problem. Throughout this process, I was reminded that I love solving problems like that and I have continued trying to solve other open problems like the one I submitted to Fibonacci Quarterly. The icing on the cake is that my solution was published. Seeing my name printed in an international mathematics journal is incredible still to this day and has fuelled my desire to continue to push myself to research new and budding branches of mathematics in order to be the best mathematician and teacher I possibly can. Student 3. My final opinion of doing research is that it is extremely satisfying. At first, it seems as if there’s no point to what you are attempting to learn. Then you arrive at some massive question that is completely overwhelming and seems to make no sense, but thanks to the initial preparation, you feel ready to attack it. You learn everything possible related to that one problem, then you begin slaving away at this one big question. You work long and hard, get frustrated, want to give up, but continue working. Then all of a sudden things start to line up and make sense and the problem seems to fall together right in front of you. You finally reach some solution. All the work paid off. It is one of the most satisfying and rewarding feelings to be done and be able to travel and share your work with others. But then it is time to move on, and the cycle starts all over again. But you know there is a reward at the end, so you keep pushing the knowledge of mathematics and begin to truly find new results. Student 4. Conducting research as undergraduates helps set us apart from our peers across all academic disciplines. In doing research, we have had the opportunity to present our work to a variety of audiences in different states. It is especially rewarding when professors in graduate programmes approach us about our work. Working on research problems is addictive; we oftentimes have to force ourselves to not look into research questions until our other work is completed. It was initially intimidating looking at a complex problem with the expectation that we would solve it; however, the satisfaction of completing these challenges and formulating new challenges for others is unmatched in any other academic realm. We encourage other undergraduates to take on the task on conducting their own research as it is very rewarding. Student 5. Usually, in college you learn some theorems and the homework consists of some applications of those theorems. However, when you try to solve an open problem from a journal you do not know what theorems to use. You never know if the way that you choose to solve the problem is the correct one. Solving problems from journals opens up a new world, opens up your creativity and you are free to ask your own questions. You ask, for example, is there a similar solution for this problem in other areas of mathematics that I can use here? Is it possible to generalize the solutions that were found? Solving problems from journal helps you find the real power of inequalities, derivatives, approximations, identities, etc. 5. Conclusion We believe it is important to motivate the undergraduate students to start thinking about solving open problems early on in their college years. This can then make them more curious about what else they can do in research and ask questions like: ‘how can this problem be extended?’ . It also gives them time to work on more advanced topics. In fact they can participate in a research project which eventually leads to the publication of a full-length journal article before they graduate from college. This last part is good for the students as it prepares them for graduate school. Another important observation we have made is that on some occasions it is better to have two students work on an open problem. Each have their own strengths that help them learn from each other. Rigoberto Flórez is an Associate Professor at The Citadel, The Military College of South Carolina. He got his Ph.D. in Mathematics from Binghamton University (SUNY). His research area is Combinatorics with a focus on Matroids and Graph Theory. He has published several papers and earned several academic awards. He likes working on mathematical projects with undergraduate and graduate students. His students have presented their research in local and national conferences and won awards doing the same. He is bilingual (English and Spanish). Antara Mukherjee is an Associate professor at The Citadel, The Military College of South Carolina. She received her Ph.D. from The University of Oklahoma. Her main research interests include Topology, Number Theory and Combinatorics. She is interested in working with undergraduate students on different mathematical problems. Her students have presented their research in local and national conferences and won awards doing the same. Acknowledgements We would like to thank the referee and the editor for their helpful comments. We would also like to thank all our students who worked tirelessly on the problems mentioned in the paper and completed them successfully. References Adams J . ( 2016 ) Solution to problem B-1165 . Fibonacci Quart ., 54 , 85 – 86 . Benjamin A. ( 2003 ) Proofs that Really Count: The Dolciani Mathematical Expositions . Washington, DC : Mathematical Association of America , p. 27 . Ching H . Solution to problem B-1192. Fibonacci Quart. (in press). The Fibonacci Association , The Fibonacci Quarterly, http://www.fq.math.ca/list-of-issues.htm Flórez R , McAnally N , Mukherjee A. Identities for generalized Fibonacci polynomials. https://arxiv.org/abs/1702.01855 (in press). Graves R . ( 2014 ) Solution to problem B-1130 . Fibonacci Quart ., 52 , 183 – 184 . Harbol M , Tiscareno L . Solution to problem B-1190. Fibonacci Quart. (in press). Higuita R . Solutions to problems B-1084, B-1086, B-1087, B-1088, B-1089, B-1090, B-1091, B-1101, B-1102, B-1103, B-1104, B-1106, B-1107, B-1108, B-1112, B-1116, B-1117, B-1119 and B-1120. Fibonacci Quart. Knott R . Fibonacci and golden ratio formulae. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormulae. Koshy T . ( 2001 ) Fibonacci and Lucas Numbers with Applications . New York : John Wiley . Lindberg K . ( 2015 ) Solution to problem B-1154 . Fibonacci Quart ., 52 , 277. Mathematical Association of America, Mathematics Magazine, http://www.maa.org/press/periodicals/mathematics-magazine Mathematical Association of America, The American Mathematical Monthly, http://www.maa.org/press/periodicals/american-mathematical-monthly Mathematical Association of America, The College Mathematics Journal, http://www.maa.org/press/periodicals/college-mathematics-journal/the-college-mathematics-journal McAnally N . ( 2016 ) Solution to problem H-766 . Fibonacci Quart ., 54 . Michaels-Arfor S . ( 2012 ) Problem B-1081 . Fibonacci Quart ., 50 , 84 . Polya G . ( 2014 ) How to Solve It . Princeton, NJ : Princeton University Press . Real Sociedad Matemática Española, La Gaceta. http://gaceta.rsme.es/index.php. University of Prishtina, MathProblems Journal, http://www.mathproblems-ks.org/ Vajda S . ( 1989 ) Fibonacci and Lucas Numbers, and the Golden Section. Theory and Applications , Ellis Horwood Series: Mathematics and its Applications, New York : Ellis Horwood Ltd., Chichester; Ellis Horwood: Halsted Press (John Wiley and Sons, Inc.) . Wang W . Solution to problem B-1196. Fibonacci Quart. (in press). Watson J . ( 2014 ) Solution to problem B-1133 . Fibonacci Quart ., 52 , 277 – 278 . Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications 2017. This work is written by US Government employees and is in the public domain in the US. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Teaching Mathematics and Its Applications: International Journal of the IMA Oxford University Press

Solving open problems with students as a first research experience

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Abstract In this article, we share some experiences about working with students on research projects. Instead of giving them problems from our own projects to work, we use open problems from journals. Using these resources we have worked with several undergraduate and some graduate students giving them a taste of their first research experience. Here we describe how these projects were conducted and the various criteria we thought about when we found the right problems for the students to work on. We also give a list of problems that our students worked on and the books and articles the students may read in order to get ready for the projects. I. Undergraduate research experience Undergraduate research is an important part of the undergraduate curriculum and many colleges and universities encourage this initiative. In small colleges that primarily focus on teaching, providing the research experience in the undergraduate level is a challenge for some faculties. During the years that we have been supervising research projects of undergraduates, the students have presented their research at different meetings and on some occasions we have been asked questions like, how to start working with students who have never had experience in research, where to find open problems that are solvable by undergraduate students and so on. In addition to supervising undergraduate projects, we also have had some graduate students who have undertaken research projects while working on their master’s degree in mathematics education. We have found the graduate students also to be well motivated. Teachers dedicate a lot of time designing and implementing research projects for students. Therefore, it is desirable to have highly motivating projects, especially when the students are doing research for the first time. We have found that instead of assigning them—for the first time—problems from our own projects to work, it is more motivating for them when we assign open problems from journals. We have also found two interesting consequences of those types of projects: the first is that that the students are motivated by the fact that the solution to the open problem they solve will be published in an academic journal. The second consequence is that when the students finish the project with us, they know where they can find new problems for their own projects and work independently. There are several journals that publish open problems at various levels and areas of mathematics. Our main focus is recursive sequences and therefore our main resource is the Fibonacci Quarterly (The Fibonacci Association). Other journals are, for example, Mathematics Magazine (Mathematical Association of America), The American Mathematical Monthly (Mathematical Association of America), The College Mathematics Journal (Mathematical Association of America), MathProblems Journal (University of Prishtina) and La Gaceta (in Spanish) (Real Sociedad Matemática Española). We have had positive experiences when the students work on open problems using recursive sequences. The problems on these sequences are understandable (even if some of them are very hard to solve). We have noticed the problems on recursive sequences, like Fibonacci and Lucas, are very interesting and are therefore very helpful in stimulating student curiosity in research. From our own experience we noticed that, once the students understand the problem, they are willing to try to find a solution. We have noted that, even when the problem is extremely hard, they want to keep trying—motivated by their curiosity. We would like to note that some of these students (undergraduate or graduate) who started their research career by solving problems, later worked on joint projects either with us, with other researchers or even by themselves. We should note here that many problems from Fibonacci Quarterly are a good fit for students but it is up to the advisor to choose the right problem for a student to work on. Several problems, even in the elementary section, are very hard for the undergraduate and graduate students. Many of those problems require expertise in areas like combinatorics, number theory, generating functions, functional analysis, etc. 2. Background and suggested problems to work with students If the students are interested in doing research involving solving open problems from the Fibonacci Quarterly, we recommend that they study the book Fibonacci and Lucas Numbers with Applications by Koshy (2001). We especially recommend Chapter five of the mentioned book. Other important sources that the students can use are Benjamin (2003), Knott, and Vajda (1989). We now recall that the Fibonacci numbers Fn and Lucas numbers Ln satisfy the recursive relations: Fn=Fn−1+Fn−2withF0=0andF1=1 Ln=Ln−1+Ln−2withL0=2andL1=1. The Binet formulas for Fibonacci numbers and Lucas numbers are given by: Fn=αn−βnα−βandLn=αn+βnwhereα=1+52andβ=1−52. As a new advisor for undergraduate student research, it is natural to ask, how to help students without helping too much or, what happens if the students are asked to work on research without enough help from the advisors? So we help students by asking them similar questions as Polya asked in Polya (2014). We now provide a list of some problems from the Fibonacci Quarterly that may help the students in their preparation before starting their research project. After each problem we explain how it can be solved and their difficulty level. The complete solutions of all the problems are cited in the bibliography section. We recall that to solve the following problems one might need the resources in Benjamin (2003), Koshy (2001), Knott, and Vajda (1989). Other problems that may help prepare students for research are those that Higuita solved when he was an undergraduate student. Problem B-1081 (Michaels-Arfor, 2012). Prove that Fn+12Fn−12−Fn4=(−1)n[2L2n+(−1)n]/5. The students only need to know Chapter five of Koshy (2001). Problem B-1133 (Watson, 2014). Determine the value of the following infinite series S given by 12·1+11·3−12·3+13·5+13·8−15·8+18·13+18·21−113·21+…. This problem can be solved by students with calculus background. In particular it requires that the students know about limits, series (partial sums) and telescopic series. Problem B-1190 (Harbol & Tiscareno, in press). Let n≥1 be a positive integer. Compute Fn+2FnFn+1(Fnn+Fn+1n−Fn+2nFnn−1+Fn+1n−1+Fn+2n−1)+Fn+3Fn+1Fn+2(Fn+1n+Fn+2n−FnnFnn−1+Fn+1n−1+Fn+2n−1)+2Fn+Fn+1Fn+2Fn(Fn+2n+Fnn−Fn+1nFnn−1+Fn+1n−1+Fn+2n−1). This problem is a good fit for a student with calculus and discrete mathematics background. Even if the problem does not need calculus, it is recommended that the students have some experience in writing proofs of theorems along with some idea about computer programming. Problem B-1154 (Lindberg, 2015). Find a closed form for the expression ∑i=0nLi2Li+12. This problem is a good fit for students with calculus and discrete mathematics background. It is recommended that the students have some experience in writing proofs of theorems before undertaking the project to solve this problem. Problem B-1165 (Adams, 2016). For an integer n≠0, find the value of LF3nFF3n−1FF3n−2+LF3n−1FF3n−2FF3n+LF3n−2FF3nFF3n−1. This problem is also a good fit for students with calculus and discrete mathematics background. The problem does not require calculus but it is recommended that the students have some experience in writing proofs of theorems. Problem B-1192 (Ching, in press). Let Mn be an n × n matrix given for all n≥1 by: (F110…000F2F11…000F3F2F1…000…⋯…⋱………Fn−1Fn−2Fn−3…F2F11FnFn−1Fn−2…F3F2F1). Prove that det(Mn)={0,ifniseven,1,ifnisodd. This problem is a good fit for students with discrete mathematics and linear algebra background. The solution requires knowledge of principle of mathematical induction and determinants. Problem H-766 (McAnally, 2016). Let n=m+2. For m≥1, prove that ∑h=1m∑i=1h∑j=1i∑k=1jFk4=4Fn4+n4−5n2100. This problem is considered to be at an advanced level by the journal. We recommend that this problem only be assigned to undergraduate students in their senior year or some math education students who are pursuing a master’s degree. Problem B-1196 (Wang, in press). Prove that Ln+25−Ln+15−Ln−15−Ln−25Ln+23−Ln+13−Ln−13−Ln−23=5Fn+25−Fn+15−Fn−15−Fn−25Fn+23−Fn+13−Fn−13−Fn−23. This problem is a good fit for a student with calculus and discrete mathematics background. It is recommended that the students have some experience in writing proofs of theorems. Problem B-1130 (Graves, 2014). Prove that for all positive real numbers m ∑k=1nFk2m+2k3m≥4mFnm+1Fn+1m+1n2m(n+1)2m. This problem can be worked by advanced students with some knowledge of real analysis or functional analysis. 3. Beyond solving open problems Once the students have solved the open problems, we may encourage them to think about ways to further their research by asking the right questions. For example, if they replace Fibonacci numbers with Lucas numbers or if they change Fibonacci numbers to Fibonacci polynomials, they may obtain results that are deeper and more interesting (see the ‘Conjecture’ below and Flórez et al. (in press)). The students who solved (Harbol and Tiscareno, in press) went on to discover that a similar result holds for Lucas numbers. They also used a computer program to test various other identities obtained by changing problem (Harbol and Tiscareno, in press) slightly. Their experiments yielded the results in Theorem 3.1 that they have since proved. We note that these results also hold when Fibonacci numbers are replaced with Lucas numbers. Theorem 3.1 If α is the Golden Ratio then, (1) limn→∞1Fn+1(Fn+1n+Fn+2n−FnnFnn−1+Fn+1n−1−Fn+2n−1)=α (2) limn→∞[1−Fn+2FnFn+1(Fnn+Fn+1n−Fn+2nFnn−1+Fn+1n−1+Fn+2n−1)−2Fn+Fn+1Fn+2Fn(Fn+2n+Fnn−Fn+1nFnn−1+Fn+1n−1+Fn+2n−1)]=α A natural question that follows from Ching (in press) is: what is the behaviour of the sequence generated by the determinants of the matrices in Problem B-1192 when their non-zero entries are replaced by their corresponding Lucas numbers? This question can be formally stated as follows. Conjecture. If Nn is the matrix obtained by replacing Fi in Mn in problem B-1192 by Li−1 with 1≤i≤n, then det(Nn)=an−1, where an=an−1+2an−2,n≥2,a0=2, and a1=3, i.e, an is equal to [5(2n)+(−1)n]/3 for n≥0. The student who solved (McAnally, 2016) went on to work on more advanced topics later on. He started working on his first research project during his second year of college and due to his expertise in this area and academic maturity, he later worked on a paper involving generalized Fibonacci polynomials. Working on this project we discovered that over 100 identities that are true for Fibonacci numbers, also happen to be true for the generalized Fibonacci polynomials (see Flórez et al., in press). 4. Feedback from some former students Student 1. As a graduate student, I had the opportunity to conduct research under the guidance of my teachers, Drs. Flórez and Mukherjee. I remember being in a classroom with all of my work written across several white boards and finally figuring out that the solution I had was correct. What a great feeling! After the first problem that I solved was published, I presented my work at several conferences. This led me to meet several other math lovers that I worked on my second research project therewith. There were days where I was ready to give up my research project after being frustrated of not figuring it out. Looking back, I am so thankful that I stuck to it and discovered the joys of doing mathematics research. Student 2. While I was working on my M.A.E. in Mathematics, I was challenged by Dr. Flórez to solve an open problem posed in the Fibonacci Quarterly. I worked on several related problems provided in order to learn what identities and tools would be necessary to solve my problem. Once I was ready, I began trying to solve the equation. After over a month of preparing and practicing, I discovered that I knew everything I needed in order to solve the problem. Throughout this process, I was reminded that I love solving problems like that and I have continued trying to solve other open problems like the one I submitted to Fibonacci Quarterly. The icing on the cake is that my solution was published. Seeing my name printed in an international mathematics journal is incredible still to this day and has fuelled my desire to continue to push myself to research new and budding branches of mathematics in order to be the best mathematician and teacher I possibly can. Student 3. My final opinion of doing research is that it is extremely satisfying. At first, it seems as if there’s no point to what you are attempting to learn. Then you arrive at some massive question that is completely overwhelming and seems to make no sense, but thanks to the initial preparation, you feel ready to attack it. You learn everything possible related to that one problem, then you begin slaving away at this one big question. You work long and hard, get frustrated, want to give up, but continue working. Then all of a sudden things start to line up and make sense and the problem seems to fall together right in front of you. You finally reach some solution. All the work paid off. It is one of the most satisfying and rewarding feelings to be done and be able to travel and share your work with others. But then it is time to move on, and the cycle starts all over again. But you know there is a reward at the end, so you keep pushing the knowledge of mathematics and begin to truly find new results. Student 4. Conducting research as undergraduates helps set us apart from our peers across all academic disciplines. In doing research, we have had the opportunity to present our work to a variety of audiences in different states. It is especially rewarding when professors in graduate programmes approach us about our work. Working on research problems is addictive; we oftentimes have to force ourselves to not look into research questions until our other work is completed. It was initially intimidating looking at a complex problem with the expectation that we would solve it; however, the satisfaction of completing these challenges and formulating new challenges for others is unmatched in any other academic realm. We encourage other undergraduates to take on the task on conducting their own research as it is very rewarding. Student 5. Usually, in college you learn some theorems and the homework consists of some applications of those theorems. However, when you try to solve an open problem from a journal you do not know what theorems to use. You never know if the way that you choose to solve the problem is the correct one. Solving problems from journals opens up a new world, opens up your creativity and you are free to ask your own questions. You ask, for example, is there a similar solution for this problem in other areas of mathematics that I can use here? Is it possible to generalize the solutions that were found? Solving problems from journal helps you find the real power of inequalities, derivatives, approximations, identities, etc. 5. Conclusion We believe it is important to motivate the undergraduate students to start thinking about solving open problems early on in their college years. This can then make them more curious about what else they can do in research and ask questions like: ‘how can this problem be extended?’ . It also gives them time to work on more advanced topics. In fact they can participate in a research project which eventually leads to the publication of a full-length journal article before they graduate from college. This last part is good for the students as it prepares them for graduate school. Another important observation we have made is that on some occasions it is better to have two students work on an open problem. Each have their own strengths that help them learn from each other. Rigoberto Flórez is an Associate Professor at The Citadel, The Military College of South Carolina. He got his Ph.D. in Mathematics from Binghamton University (SUNY). His research area is Combinatorics with a focus on Matroids and Graph Theory. He has published several papers and earned several academic awards. He likes working on mathematical projects with undergraduate and graduate students. His students have presented their research in local and national conferences and won awards doing the same. He is bilingual (English and Spanish). Antara Mukherjee is an Associate professor at The Citadel, The Military College of South Carolina. She received her Ph.D. from The University of Oklahoma. Her main research interests include Topology, Number Theory and Combinatorics. She is interested in working with undergraduate students on different mathematical problems. Her students have presented their research in local and national conferences and won awards doing the same. Acknowledgements We would like to thank the referee and the editor for their helpful comments. We would also like to thank all our students who worked tirelessly on the problems mentioned in the paper and completed them successfully. References Adams J . ( 2016 ) Solution to problem B-1165 . Fibonacci Quart ., 54 , 85 – 86 . Benjamin A. ( 2003 ) Proofs that Really Count: The Dolciani Mathematical Expositions . Washington, DC : Mathematical Association of America , p. 27 . Ching H . Solution to problem B-1192. Fibonacci Quart. (in press). The Fibonacci Association , The Fibonacci Quarterly, http://www.fq.math.ca/list-of-issues.htm Flórez R , McAnally N , Mukherjee A. Identities for generalized Fibonacci polynomials. https://arxiv.org/abs/1702.01855 (in press). Graves R . ( 2014 ) Solution to problem B-1130 . Fibonacci Quart ., 52 , 183 – 184 . Harbol M , Tiscareno L . Solution to problem B-1190. Fibonacci Quart. (in press). Higuita R . Solutions to problems B-1084, B-1086, B-1087, B-1088, B-1089, B-1090, B-1091, B-1101, B-1102, B-1103, B-1104, B-1106, B-1107, B-1108, B-1112, B-1116, B-1117, B-1119 and B-1120. Fibonacci Quart. Knott R . Fibonacci and golden ratio formulae. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormulae. Koshy T . ( 2001 ) Fibonacci and Lucas Numbers with Applications . New York : John Wiley . Lindberg K . ( 2015 ) Solution to problem B-1154 . Fibonacci Quart ., 52 , 277. Mathematical Association of America, Mathematics Magazine, http://www.maa.org/press/periodicals/mathematics-magazine Mathematical Association of America, The American Mathematical Monthly, http://www.maa.org/press/periodicals/american-mathematical-monthly Mathematical Association of America, The College Mathematics Journal, http://www.maa.org/press/periodicals/college-mathematics-journal/the-college-mathematics-journal McAnally N . ( 2016 ) Solution to problem H-766 . Fibonacci Quart ., 54 . Michaels-Arfor S . ( 2012 ) Problem B-1081 . Fibonacci Quart ., 50 , 84 . Polya G . ( 2014 ) How to Solve It . Princeton, NJ : Princeton University Press . Real Sociedad Matemática Española, La Gaceta. http://gaceta.rsme.es/index.php. University of Prishtina, MathProblems Journal, http://www.mathproblems-ks.org/ Vajda S . ( 1989 ) Fibonacci and Lucas Numbers, and the Golden Section. Theory and Applications , Ellis Horwood Series: Mathematics and its Applications, New York : Ellis Horwood Ltd., Chichester; Ellis Horwood: Halsted Press (John Wiley and Sons, Inc.) . Wang W . Solution to problem B-1196. Fibonacci Quart. (in press). Watson J . ( 2014 ) Solution to problem B-1133 . Fibonacci Quart ., 52 , 277 – 278 . Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications 2017. This work is written by US Government employees and is in the public domain in the US. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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Teaching Mathematics and Its Applications: International Journal of the IMAOxford University Press

Published: Sep 5, 2018

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