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The Significance of Problem Solving in Mathematics Education

Problem solving in mathematics is a crucial skill that involves understanding, planning, executing, and reflecting on solutions. It is not just about algorithms but also about creativity and real-life applications. Integrating problem solving in online math teaching can enhance understanding, fairness in exams, and creative thinking, but it also poses challenges in level setting and assessment. Examples from statistics and calculus courses illustrate the benefits and complexities of incorporating problem solving in math education.

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The Significance of Problem Solving in Mathematics Education

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  1. The importance of problem solving in maths (online) teaching Ksenia Shalonova International Foundation Programme, University of Bristol

  2. The importance of problem solving in maths education In 1945 George Pólya (Hungarian mathematician) published the book How To Solve presenting four basic principles of problem solving in mathematics: 1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back In the 21stcentury can computers substitute students in problem solving?

  3. Students and computers - who is better in which category? There are four levels of learning maths (Knill et al, 2004): 1. 2. 3. 4. know the definition master the algorithm understanding the concepts be innovative Problem solving usually requires mastering all four levels. The first two levels (definitions and algorithms) can be completed by computers.

  4. What is problem solving in maths? Is it applied mathematics? Is it any open-ended task? Is it a (formal) proof? Is it always related to a real-life problem? • • • • Students are engaged in Problem Solving in mathematics when they are trying to achieve the result without knowing a straightforward way to do so.

  5. Including problem solving into (online) maths teaching Advantages: • Achieving more fairness in an open book exam in case students use symbolic computation. • Helping to gain deep understanding of maths concepts. • Promoting creative thinking within mathematics. Disadvantages: • Posing a problem at the right level is not straight forward. • It can be difficult to evaluate problem solving skills in formative assessments.

  6. Example 1 (exam question in Applied Statistics, UG unit) Several worried students after the exam asking questions, “Which method did we need to use?”, “Was my method correct?” etc. Some concerns expressed by the external examiner as adding an open-ended question into the exam is not a common practice. • • It was one of the best answered questions in the exam as about 85% of students got 7-10 marks for this question.

  7. Example 2 (exam question in Introductory Mathematics, IFP level 3) Several worried students after the exam asking questions, “Was I supposed to use a definite or indefinite integral?”, ”Do they give the same answer?”. External examiner’s comments were positive, “I really liked those questions for solving real life problems; they are creative and cover a good range of topics.” • • The split between using definite and indefinite integral was approximately 30%-70%. Level 3 students discovered the Second Fundamental Theorem of Calculus. The definite integral gives an explicit solution to the initial-value problem, whereas the indefinite integral gives a procedure for finding a solution when an explicit antiderivative can be found.

  8. Example 3 (coursework in Applied Statistics, IFP level 3) Numerous studies in pedagogy confirmed that students find statistics hard to understand. Students very often view statistics as a set of definitions with recipe-based solutions (as an algorithm). How to add creativity into stats units in order to incorporate problem solving? • • • Students had to compare two regression models (a student’s Conclusion section is below). Students were never taught how to compare two models in stats and had to discover this themselves.

  9. Example 4 (coursework in Applied Statistics, IFP level 3) In statistical analysis problem solving can be proving opposite arguments with the same data or convincing someone of their perspective.

  10. Example 5 (question in Introductory Mathematics, IFP level 3) This is a classical task in applied mathematics, but the challenge is that students are not supposed to use differential equations in the Introductory Mathematics unit.

  11. Conclusions • It is a matter of how students perceive the problem, rather than a lecturer’s perspective. Discovery and challenge are often associated by students with problem solving. • Students seem to enjoy units that incorporate problem solving (Applied Stats UG unit got 4.9 out of 5, 100% of students recommend Applied Stats unit level 3). • A comment from a student taking Introductory Mathematics unit, “Initially I though that this is a very easy course for me as I studied all this maths in China, but then I realised that it is actually not as simple as I thought”.

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