Engaging students in meaningful mathematics problem-solving is the intention of many education stakeholders around the world. Research suggests that the implementation of high-stakes problems in mathematics teaching is one way to strengthen students’ conceptual understanding. Many carefully crafted open-ended problems constitute high-stakes problems, and proper use of such problems in teaching and learning not only encourages learners’ flexible thinking but also helps detect their misconceptions. However, what is less practiced and understood is: how exactly one should aim to implement such problems in a classroom setting. Teaching pre-service middle school teachers for a few years using high-stakes (mostly open-ended problems) has given me insights that may be useful to teachers around the world. In this paper, we share my experience of teaching with high-stakes problems. We will demonstrate how user-friendly interactive graphing tools can be used in the creative process of problem-solving

Basyal, Deepak

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CCU Digital Commons

Coastal Carolina University CCU Digital Commons Mathematics and Statistics College of Science Summer 6-2023 Engaging Students with High-stakes Problems Deepak Basyal dbasyal@coastal.edu Follow this and additional works at: https://digitalcommons.coastal.edu/mathematics-statistics  Part of the Educational Technology Commons, Elementary Education Commons, Junior High, Intermediate, Middle School Education and Teaching Commons, Other Mathematics Commons, and the Other Teacher Education and Professional Development Commons Recommended Citation Basyal, Deepak, "Engaging Students with High-stakes Problems" (2023). Mathematics and Statistics. 4. https://digitalcommons.coastal.edu/mathematics-statistics/4 This Article is brought to you for free and open access by the College of Science at CCU Digital Commons. It has been accepted for inclusion in Mathematics and Statistics by an authorized administrator of CCU Digital Commons. For more information, please contact commons@coastal.edu. Mathematics Education Forum Issue 411MATHEMATICSEDUCATIONFORUMYEAR 23 JUNE 2023 SPECIAL ISSUE ISSUE 41COUNCIL FOR MATHMATICS EDUCATIONKATHMANDU, NEPALMathematics Education Forum Issue 4150Engaging students in meaningful mathematics problem-solving is the intention of many education stakeholders around the world. Research suggests that the implementation of high-stakes problems in mathematics teaching is one way to strengthen students’ concep-tual understanding. Many carefully crafted open-ended problems constitute high-stakes problems, and proper use of such problems in teaching and learning not only encourages learners’ flexible thinking but also helps detect their misconceptions. However, what is less practiced and understood is: how exactly one should aim to implement such problems in a classroom setting. Teaching pre-service mid-dle school teachers for a few years using high-stakes (mostly open-ended problems) has given me insights that may be useful to teachers around the world. In this paper, I share my experience of teaching with high-stakes problems. We will demonstrate how user-friendly interactive graphing tools can be used in the creative process of problem-solving. IntroductionA mathematical problem refers to any circumstance that requires the use of mathematical tools and concepts to find a solution. To comprehend mathematical concepts thoroughly and utilize them in practical ways, students must undergo the process of problem-solv-ing. Students learn to  solve problem in mathematics primarily through ‘doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning’ (Copley 2000, p. 29). Problems that allow for various solution methods and the meaningful application of alge-braic notations, along with the use of visual aids, can contribute significantly in mathematical problem-solving (Woodward et al., 2012). Studies show that open-ended problems which have multiple correct answers possible have the potential to develop fluency, flexibility, and originality in students’ problem-solving skills (Kwon et al, 2006). In this regard, Johnson and Kaplinsky (n.d) created mathematics problems known as “open middle problem” to enhance students’ problem-solving abilities and critical thinking skills, and typically allows for multiple approaches to reach a solution. These types of problems may seem uncomplicated and procedural at first but can be more difficult and intricate than they appear. They offer several ways of solving them, with the objective of obtaining the best or optimal answer. Klerlein and Hervey (2019) propose that effective problems are those that are both accessible and extendable, enable individuals to establish connections, foster discussion, and communication, encourage originality and innovation, stimulate questions such as “what if?” and “what if not?”, and incorporate an element of surprise.The purpose of this paper is to discuss the classroom implementation of high-stakes problems that are relevant to middle and high school. Our aim is to present a set of ‘open middle problems’ and provide guidance on incorporating them into the middle school classroom. To solve these problems, students must establish links between mathematical concepts, apply existing knowledge to novel situations, and create and uncover mathematical tools in their unique manner. We use visual aids during this process to help students actively participate in constructing solutions. Problem 1: The line CD passes through the origin is perpendicular to line AB. Find the length of the line CD.Figure 1: Finding the length of the perpendicular line using various techniques.Engaging students with high-stakes problemsDeepak BasyalMathematics Education Forum Issue 4151Inspired by Nerdypoo (n.d), the problem above, although not an open-ended problem, can be solved in many ways. This problem allows students to explore different mathematical concepts that are relevant in middle school and beyond. No pathways are suggested, and students are forced to devise their own plans to get to the solution.  We now present the solution to the problem in four different ways: Using the area of triangle  First, we find the area of the triangle using the CB as a base and AC as a height. So that Area  ΔABC=1/2 CB*AC=1/2 3*4=6. Then, we find hypotenuse AB using the pythagorean theorem, AB^2=AC^2+CB^2. Putting the known values and taking the positive square root, we find AB=√(3^2+4^2 )=5 . Now, using the area of triangle again, but this time, we take AB as a base and CD as a height, we get, Area ΔABC=1/2 AB*CD. As we already know the area and base of the triangle, we now, solve for CD. So,  6=1/2 5*CD. Therefore, CD=12/5. Using similar trianglesΔBAC and ΔBCD are similar triangles because angle A and C are right triangles, and both share the angle B. Therefore, BA/AC=BC/CD. 5/3=4/CD , CD=12/5. Finding the coordinate of D and using distance formulaWe can view the point D as the point of intersection of two lines. First, we can find the equation of lines passing through the point (0,3), and (4,0). By using the slope formula (y_2-y_1)/(x_2-x_1 ), we get the slope to be -¾. Then using the y-intercept (0,3), we find the equation of the line to be y=-3/4 x+3. The perpendicular line will have the slope 4/3, and it passes through the origin. Therefore, the equation of the line become  y=4/3 x. Now, setting them equal, we get -3/4 x+3=4/3 x. Now solving this equation, we get x=36/25. Putting this value in y=4/3 x gives y=48/25. Now using the distance formula between two points (0,0), and (36/25,48/25) gives CD=√((36/25 )^2+(48/25 )^2 )=4/25 √(9^2+12^2 )=4/25 √(81+144)=4/25 √225=4/25 15=60/25=12/5. Using trigonometryIn ΔACB , we have SinB=AC/AB=3/5. Therefore, B=ArcSin(3/5). Now, in ΔCDB, SinB=CD/CB=CD/4. Thus, CD=4SinB=4Sin(arc-Sin(3/5))=4*3/5=12/5.Even though the last solution method may be a little too far to introduce in middle school, we can certainly introduce the topic for high school students. Moreover, the problem can be solved using integrals in the calculus classroom. This method is probably not appropriate to the intended audience of the paper. We ask interested readers to try finding the solution using arc length formula.Problem 2: Using the digits 1 to 9 at most one time each, fill in the boxes to create two points that are equidistant from (4,-1). (Source: Robert Kaplinsky-Open middle mathematics)Figure 2: Open middle problem on equidistant pointsMathematics Education Forum Issue 4152Finding equidistant points would be easier if there are no restrictions. The easier solutions can be found on horizontal and vertical lines. For example (4,0) and (4, -2) would be at equal distances from (4,-1), and there is no need to use the distance formula. Moreover, (3,-1) and (5, -1) are also at equal distances from (4, -1). However, the restriction on digits 1 to 9 will add challenges to solving this problem. In turn, students are forced to find a strategy that will help them find all different possible solutions even with these restrictions. One way to find the solution is to use the distance formula in conjunction with the guess and check method. However, doing the guess and check work with distance formulas demands too much work. We can reduce the hand calculation of the distance between two points by using the distance formula in the Desmos calculator with the use of variable points (a, b). However, soon we will realize that this will not be an effective method of solving this problem. Therefore, one needs to think strategically to find the solutions to the problem. If we can visualize the distance we are looking for it will make a hypotenuse of the right triangle, there is not even a need for calculating the actual distances. For example, one could try to form a right triangle starting at point (4,-1) with base 3 and height 4. That means the two legs of this right triangle will have lengths of 3 and 4 units so that the newly found point will be located 5 units away from (4,-1). In the picture below, the right triangle on the left side of (4,-1) would have two other vertices at (1,-1), and (1,3), and on the right will be at (7,-1), and (7,3). Both points (1,3) and (7,3) are in equal distance from (4,-1). However, this is still not a valid solution because we are repeating 3. To remedy this situation, we could create the right triangle on the right side, by going 4 units horizontally, and 3 units vertically. So the two other vertices are (8,-1), (8,2). We have found the first set of points (1,3) and (8,2) that are in equal distance from (4,-1). We can find three more sets of points by following similar reasoning. More solutions are available in the Desmos calculator at the following link https://www.desmos.com/calculator/ks41sfqxpmFigure 3: Desmos online calculator solution to problem 2More problems to ponder (Taken from Open Middle Problems)Figure 4: Fractions and DecimalsMathematics Education Forum Issue 4153Figure 5: Minimizing SlopeFigure 6: Comparing functionsFigure 7: Solution of Two Linear EquationsMathematics Education Forum Issue 4154ConclusionsBased on the discussion of the problems above, it is evident that the role of teachers is paramount in creating a classroom in which high-stakes problems are implemented. In a classroom like this, the responsibilities of teachers di!er signi"cantly from those in a con-ventional lecture-based classroom. #e intention of open-ended problems is to encourage productive struggle, build a learning community, provide immediate but balanced feed-back, facilitate idea sharing, and manage classroom discussion. #e challenge starts with se-lection of the problems, setting the classroom environment, and successfully implementing it.  Learning to implement e!ective problem-solving in the classroom is an iterative process, and it should be practiced on a regular basis to master the skill. Teachers should not limit themselves to teaching only computational skills but must strive to o!er classroom expe-rience that would be useful in dealing with real issues in students’ future. #erefore, it is crucial to motivate students to engage in critical thinking and meaningful reasoning, which can have practical applications beyond the classroom and the academic sphere. Educators around the world are invested to create readily available classroom resources. Many resources of similar nature are available on numerous websites. Some of these prob-lems are available in GeoGebra and can be imported to the GeoGebra classroom and Goog-le Classroom. We could even see students’ progress in real-time and help or challenge them as needed. #is allows a greater opportunity to di!erentiate the problems among students of di!erent preparation levels. ReferencesCopley, J. V. (2000). #e young child and mathematics. National Association for the Educa-tion of Young Children, 1509 16th Street, NW, Washington, DC 20036-1426.Kaplinsky, R. (2020). Open middle math: Problems that unlock student thinking, grades 6-12. Stenhouse Publishers. Kaplinsky, R., & Johnson, N. (n.d.). What’s Open Middle? Open Middle. https://www.open-middle.com/whats_open_middle/Klerlein, J., & Hervey, S. (2019). Mathematics as a complex problem-solving activity: Pro-moting students’ thinking through problem-solving. Generation Ready White Paper.Kwon, O. N., Park, J. H., & Park, J. S. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Paci"c Education Review, 7, 51-61.Nerdypoo, K. (n.d.). Finding the Length of a Right Triangle’s Altitude. Open Middle. https://www.openmiddle.com/"nding-the-length-of-a-right-triangles-altitude/Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., ... & Ogbue-hi, P. (2012). Improving Mathematical Problem Solving in Grades 4 through 8. IES Practice Guide. NCEE 2012-4055. What Works Clearinghouse.

Engaging Students with High-stakes Problems

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Engaging Students with High-stakes Problems

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