Lajos Pósa (pronounced Posha) is one of the most prominent mathematics educators in Hungary.The authors introduce readers to the Pósa method for teaching mathematics, an approach consistent withexploratory methods promoted in the United States. The authors present two scenarios (with extensions)that were posed during the workshop, both with different strategies and goals

Scanga, Bridget C.

Walker, Janet M.

OSU Libraries Digital Journal Publishing (Ohio State University)

Problem Solving theHungarian WayBridget C. Scanga & Janet M. Walker, Indiana University of PennsylvaniaAbstract: Lajos Po´sa (pronounced Posha) is one of the most prominent mathematics educators in Hungary.The authors introduce readers to the Po´sa method for teaching mathematics, an approach consistent withexploratory methods promoted in the United States. The authors present two scenarios (with extensions)that were posed during the workshop, both with different strategies and goals.Keywords: Problem solving, inquiry, discourse, teaching methods1 IntroductionIn the summer of 2018, I had the most incredible opportunity to study abroad as a student in theBudapest Semesters in Mathematics Education (BSME) program. I am a secondary mathematicseducation major at Indiana University of Pennsylvania and was encouraged to apply for thisprogram during my junior year. Because I hope to create future classroom experiences which allowstudents to learn and discover mathematics in interesting and enjoyable ways, I was excited aboutthis opportunity.1.1 What is Budapest Semesters in Mathematics Education?Budapest Semesters in Mathematics Education (BSME) is a study abroad program for undergraduatestudents and recent graduates who desire to learn more about teaching mathematics (for moreinformation, see page 63). Sixteen applicants were chosen from all over the United States andCanada to attend this 6-week seminar. We spent the first five weeks attending classes in the centerof Budapest, which included a crash course in Hungarian, technology in secondary math education,games and manipulatives, and problem-solving using the Po´sa method. In the final week of theprogram, we moved from a dormitory in the city to a home in the mountains of Ma´trafu¨red to joinapproximately 100 Hungarian children at a summer camp for gifted students from grades 5–9. Atthe camp, we observed and participated in lessons and activities designed to engage gifted middleand high school mathematicians as we observed teaching methods inspired by Lajos Po´sa.1.2 Who is Lajos Po´sa?Lajos Po´sa (pronounced Posha) is one of the most prominent mathematics educators in Hungary.His methods were originally developed for gifted students to discover concepts through individualexploration, but were soon used successfully with many different levels of students. One of themain aspects of his method is to have students experience how professional mathematicians engagewith mathematics. Figure 1 shows the 2018 BSME scholars with Lajos Po´sa (center). I am the thirdscholar from the right in the back row.Ohio Journal of School Mathematics 81 Page 57Fig. 1: Lajos Po´sa with the 2018 BSME scholars.The purpose of this paper is to introduce readers to the Po´sa method for teaching mathematics.Po´sa’s vision for how and what mathematics should be taught is valuable for teachers in theUnited States since it encourages the exploratory methods that support deep student learningconsistent with recommendations set forth by the Common Core (2010). In this paper, we presenttwo scenarios posed during the workshop that use a strategy that requires students to follow asequence of questions to find a solution.2 Scenario #1: Regions with pie cuttingThe first scenario is posed to students as:With one straight cut you can slice a pie into two pieces. How many pieces of pie can becreated from three cuts? (a cut must be a chord of the circle)The structure of Po´sa’s strategy is to engage students in a progression of questions that build uponstudents’ previous knowledge to enable them to develop a general solution.The most common question from students at the start of this problem, and a logical one aftersome thought, is: “Do all the pieces have to be the same shape?” If the student does not posethis question, then the teacher can step in to offer it. The goal of the teacher’s line of questioningis to create an internal monologue in the student that mimics the teacher. Teachers encouragethis gradual trade-off to student questioning by asking, “What do you think I will ask you next?”This anticipation of future questions becomes embedded in the discovery process, with studentsultimately developing their own sequence of questions without the aid of the teacher. Many times,students start out with a solution similar to that shown in Figure 2, with all lines intersecting at thecenter of the circle.Page 58 Ohio Journal of School Mathematics 81Fig. 2: A common student solution.Although the pie-cutting scenario has one specific answer, it is meant to be the springboard for asequence of follow-up questions. For example, once students have offered an initial answer, theteacher may ask, “Is this the greatest number of pieces that can be created from three cuts? How doyou know?” Students then must develop a logical defense for their claim. The advantage of posingsuch a problem to a class of students is that there will likely be one or more students who discoverthat a pie can be cut into seven pieces. If students do not reach this conclusion on their own, offergentle hints like “Try to see if you can cut it so that there is one more piece.” This same question canbe offered even after they have shown a pie can be cut into seven pieces. Students must then provethat no more than seven pieces can be created, which leads into the next question of how manypieces can be shown with four cuts.Explanation: With three straight cuts through the center of the circle, a pie can be cut into sixequal slices. Since the slices do not necessarily need to be equal, the pie can actually be cut intomore. One cut divides the pie into two regions. The second cut makes at most two additionalregions, splitting the pie into at most four regions. The third cut divides the pie into at most threemore regions. Therefore, the greatest number of regions made by three cuts is 2 + 2 + 3 = 7 regions.A line can divide a plane into at most two half-planes. To create the maximum number of regionswith each line in the circle, each line must divide as many regions as possible into half-planes. Thisis achieved by drawing each new line to intersect all previously drawn lines. The result is a seriesof cuts similar to those shown in Figure 3.Fig. 3: One possible diagram of a pie cut into seven pieces.Once students understand the first scenario, they are then asked, “How many regions can be createdfrom four cuts, five cuts, six cuts?” These scenarios are significantly more difficult to draw andencourage different problem-solving strategies including the use of non-pictorial representations. IfOhio Journal of School Mathematics 81 Page 59students are not already doing so, the teacher may suggest the use of a table consisting of numberof cuts and corresponding numbers of regions. Once the table is complete, it is relatively easy forstudents to recognize that the pieces of pie are increasing in a pattern as suggested in Figure 4.Fig. 4:Maximum number of regions created by n cuts.Ask students to continue this table a few more rows and ask them about what patterns they see.Teachers may wish to encourage students to create an equation which describes the maximumnumber of regions created as a function of the number of possible cuts. If students have not hadexperience before with recursive sequences, this would be an optimal time to introduce them toterminology and notation. One form of this recursive sequence is:pn = pn−1 + n, with pn the maximum number of regions made with n cuts (with p0 = 1).Ask students to relate this equation to the context of the original problem. One possible response is,“To find the number of regions that can be made from n cuts, add together the number of regionsfrom the previous cut (pn−1) and the number of cuts to be made, n.” Therefore six cuts can make:p6 = p6−1 + 6 = p5 + 6 = 16 + 5 = 22 regions.Fig. 5: One possible diagram of a pie with six cuts (22 regions).In order to differentiate for more advanced or more curious students, the teacher can ask “Howmany regions can be created from thirty cuts?” Students quickly realize that a recursive sequenceisn’t an efficient tool to answer such a question. The question prompts them to search for an explicitequation for the relationship (pn =(12) (n2 + n+ 2)which yields 466 pieces for 30 cuts). The proofPage 60 Ohio Journal of School Mathematics 81of this equation lies in identifying and using series, which are not taught until higher-level calculuscourses, however a teacher might want to challenge certain students to try to prove it. The proofcan be found at: http://mathworld.wolfram.com/CircleDivisionbyLines.htmlA key component of the Po´sa method is not for students to prove concepts, but for studentsto have an experience before formality; time to develop personal connections with the mathematicsbefore defining them in a more mathematical context. The value of this activity is that it encouragesstudents to use their own problem-solving abilities to pose questions and answer them. Modelingproblem solving through a sequence of questions, students are led gradually from concrete toabstract reasoning.3 Scenario #2: Regions with overlapping circlesA second scenario and a challenging extension question for students is: “Can this formula, or anadaptation of it, be used to describe the number of regions created from overlapping circles ratherthan cuts?” An identical questioning process to that presented with linear cuts is presented. Forexample, how many regions are formed from three intersecting circles?Fig. 6: A diagram of regions formed from three overlapping circles.Explanation: Consider the table in Figure 7 which displays the relationship between the numberof circles and the maximum number of regions created including the space outside of the circles.Similar to our last scenario, the maximum number of regions is created by forcing each new circleto intersect all previous circles at two points (instead of one, which is the maximum for a lineintersecting another line).Ask students to compare this table to that of the first scenario. How is this table the same asthe table in Figure 4? How is this table different from the table in the original question? Studentsmight notice that both tables start out the same on the first three rows, but then the values begin todiffer. They might also notice that the values for the change in the number of regions from the lastcut of the pie increase by one, where that same value with the circles increases by two (after the firstthree rows).Ask students to speculate why this might be. Students may suggest that it is because a linecan only intersect another line once, yet a circle intersects another circle at most two times.Ohio Journal of School Mathematics 81 Page 61Fig. 7:Maximum number of regions created by n overlapping circles.It is also possible to determine recursive and explicit functions for this scenario. The recursiveequation is rn = rn−1 + 2n (with r0 = 1) and the explicit equation is rn = n(n − 1) + 2, where nrepresents the number of circles and rn represents the maximum number of regions formed by thesecircles. The explicit formula can be used to determine the maximum number of regions created byn overlapping circles. Again, a proof of this can be used to challenge gifted or curious students andbe found at: http://www.90thkilmacudscouts.com/maths/circles_lines_soln.htmlHave students compare the original recursive and explicit equations for both scenarios as inFigure 8. How are they different? How are they the same? Speculate as to why you think thesedifferences/similarities happen.Fig. 8: Recursive and explicit equations for each scenario.Both scenarios use a sequence of questions to help students solve problems. Po´sa’s question posingmethod encourages students to explore a problem using a progression of questions posed by theteacher. Students use previous knowledge to determine a general solution.4 ConclusionPo´sa’s strategy of using a sequence of questions is useful for many problem-solving situations andis designed as a typical strategy that mathematicians might use in their careers. Although I onlypresented two scenarios in this paper, this strategy can be used with many other scenarios. Thereare also other strategies that we learned at BSME which I hope to present in the future.Experiencing Po´sa’s methods in the BSME program for engaging students in mathematical learningPage 62 Ohio Journal of School Mathematics 81and helping them to feel the excitement of discovery was an invaluable experience for me. I amvery fortunate to have been able to study these strategies, especially with Po´sa himself. I believethis experience will make me a better teacher, and I am excited to try the strategies soon in my ownclassroom as well as find new, creative methods that present concepts and structure activities whichmake discovery natural and memorable.ReferencesNational Governors Association Center for Best Practices (2010). Common Core State Standards forMathematics, Washington D.C.Budapest Semesters in Mathematics Education. (n. d.). The Prospective Students Information page.Retrieved from https://bsmeducation.com/prospective-students/Bridget C. Scanga graduated from Indiana University of Pennsylvania in May2019 with a Bachelor of Science degree in Secondary Mathematics Education.Ms. Scanga is from Bel Air, Maryland. In college she worked as a tutor for themathematics department, was a member of the math club, and served on theleadership team for IUP’s Fellowship of Christian Athletes.Janet M. Walker, jwalker@iup.edu, is a Professor of Mathematics and theMathematics Education Program Co-coordinator at Indiana University ofPennsylvania where she has taught for the past 23 years. Prior to this shetaught high school mathematics for eight years. Her research interests includethe use of technology in problem solving, gifted education, and standards-based instruction.More information about the BSME ProgramThe Budapest Semesters in Mathematics Education (BSME) program takes place in Budapest,Hungary, and is available to students for the Fall, Spring, and Summer semesters. In thisprogram, students earn college credits through a variety of classes that focus on teachingmathematics “the Hungarian way.” The deadline of the application for the Fall semester isby June 1 of the same calendar year, the Spring semester deadline is by November 1 of theprevious calendar year, and the Summer semester deadline is by April 1 of the same calendaryear.The cost of tuition for the Fall 2019/Spring 2020 is $9,995, and the Summer 2020semester is $4,995. The tuition does not include the costs associated with living in Budapestduring the program or transportation to Hungary. Although BSME does not offer financialaid or scholarships, it is often possible for students to receive some or all of the aidgranted through their home institution. More information can be found on their website,https://bsmeducation.com/.Ohio Journal of School Mathematics 81 Page 63

Problem Solving the Hungarian Way

Lajos Posa (pronounced Posha) is one of the most prominent mathematics educators in Hungary. The authors introduce readers to the Posa's technique of developing numerous methods as a means to foster student mathematical understanding. The authors illustrate the approach through exploration of multiple solution strategies for the 1000 square task

Scanga, Bridget C

Problem Solving the Hungarian Way (Part 2) (pp. 9--15)

https://library.osu.edu/ojs/index.php/OJSM/article/download/6861/5342

Problem Solving the Hungarian Way

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