Connect with Maths~ Teaching maths through problem solving
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This document provides guidance for teachers on facilitating student problem solving and reasoning in mathematics through key teacher actions. It discusses posing challenging tasks, being clear about learning goals, and orchestrating mathematical discussions. Some important points are: teachers should select tasks that cannot be solved easily to encourage higher-level thinking; setting clear learning goals helps guide instruction; and facilitating productive class discussions involves techniques like revoicing students' reasoning and using wait time. The overall message is that teachers should aim to create an environment where students can grapple with open-ended problems and learn from each other through collaboration and explanation of their mathematical thinking.
![From Impactful practices (from NCTM)
• Imagine classrooms where [children] are
working collaboratively, as well as
independently, using a range of resources.
• Imagine classrooms where the interactions
among students and with their teacher, are
focused on making sense of the mathematics…
• These are thinking and reasoning classrooms.](https://image.slidesharecdn.com/cwm-teachingmathsthroughproblemsolving-160527060240/85/Connect-with-Maths-Teaching-maths-through-problem-solving-4-320.jpg)
Connect with Maths~ Teaching maths through problem solving
- 1. Teaching maths through problem solving: Facilitating student reasoning Louise Hodgson 2016
- 2. Learning intention • For participants to become aware of key teacher actions that promote student problem solving and reasoning.
- 3. From the Australian National Numeracy Review Report May 2008 From the earliest years, greater emphasis be given to providing students with frequent exposure to higher-level mathematical problems…in contexts of relevance to them, with increased opportunities for students to discuss …and explain their thinking.
- 4. From Impactful practices (from NCTM) • Imagine classrooms where [children] are working collaboratively, as well as independently, using a range of resources. • Imagine classrooms where the interactions among students and with their teacher, are focused on making sense of the mathematics… • These are thinking and reasoning classrooms.
- 5. Guided by key teacher actions 1. Being clear about the purpose of the lesson. 2. Posing challenging mathematical tasks 3. Orchestrating mathematical discussions
- 6. Key teacher action 1 Be clear about your purpose ‘Without explicit learning goals, it is difficult to know what counts as evidence of students learning, how students learning can be linked to particular instructional activities and how to revise instruction to facilitate students’ learning more effectively’. ‘Forming clear explicit learning goals sets the stage for everything else’. (Smith & Stein, 2011, p. 14)
- 7. In the Australian curriculum V8.0 The proficiency strands: reasoning, problem-solving, understanding and fluency describe how the content is explored.
- 8. From the Australian Curriculum: Mathematics Students are reasoning mathematically when they: • explain their thinking, • justify strategies used and conclusions reached, • adapt the known to the unknown • prove that something is true or false • compare and contrast related ideas and explain their choices.
- 9. What might teachers do to facilitate student reasoning? Turn and Talk with a neighbour
- 10. What might teachers do to facilitate student reasoning? • Encourage students to solve problems in more than one way • Allow students to develop their own approaches • Encourage collaboration between students • Use students’ explanations as the prompt to explaining the mathematical intent of the task and lesson. Fraivillig (2004)
- 11. Children are more likely to reason if they have developed a strategy for themselves than if they are performing a procedure they have been taught. Such thinking for themselves takes time. It comes only when children are working on tasks that they do not know how to solve. (Sullivan & Davidson, 2014)
- 12. What is problem solving? For a question to be a problem, it must present a challenge that cannot be resolved easily. Problem solving is a process of accepting a challenge and striving to resolve it.’ (Polya, 1962)
- 13. Problem solving • ‘Solving problems is a practical art like swimming, or skiing, or playing the piano:... if you wish to learn swimming you have to go into the water, and if you wish to become a problem solver you have to solve problems.’ (Polya 1962)
- 14. Problem Solving In mathematics education, problem solving has been emphasised since Polya’s work in the 1940s. 1. Understand the problem 2. Devise a plan 3. Carry out the plan, and 4. Look back.
- 15. Quotes from PISA in Focus 37 • Teachers’ use of cognitive-activation strategies, such as giving children problems that require them to think for an extended time, presenting problems for which there is no immediately obvious way of arriving at a solution, and helping students to learn from their mistakes, is associated with students’ drive.
- 16. What implications does this have for your teaching? What questions do you have? Turn and Talk with a neighbour
- 17. Key teacher action 2 Select Challenging tasks • Tasks that are cognitively challenging provide the “grist” for worthwhile discussions. (Smith & Stein, 2011) • Children need opportunities to grapple with a task for which the solution method is not known in advance. NCTM 2000 • Productive discussions that highlight key mathematical ideas are unlikely to occur if the task on which students are working requires limited thinking and reasoning. (Smith & Stein, 2011)
- 18. WHY CHALLENGING TASKS ARE IMPORTANT?
- 19. Comparing Two Mathematical Tasks Patterns in the hundreds square Patterns on the hundreds chart Grade 1: Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a number line (ACMNA013)
- 20. Patterns in the hundreds square
- 21. Patterns on the hundreds chart
- 22. Comparing two mathematical tasks 1. Think privately about the purpose of each task. 2. Talk with your elbow partner about how they are the same and how they are different.
- 23. Comparing two mathematical tasks How are patterns in the hundreds square and the patterns on the hundreds chart the same and how are they different?
- 24. Comparing Two Mathematical Tasks Put 9 counters on a ten frame How many ways can you make 9? AC:M Foundation: Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond (ACMNA002)
- 25. Comparing two mathematical tasks 1. Think privately about the purpose of each task. 2. Talk with your elbow partner about how they are the same and how they are different.
- 26. How many different ways can you make 9? Record your findings so that you can describe them to others
- 28. Enabling prompt(s) (for students experiencing difficulty): • Can be posed to students who have not been able to make progress on the learning task. Enabling prompts can involve slightly varying an aspect of the task demand, such as: – the form of representation, – the size of the numbers, or – the number of steps (Sullivan, Zevenbergen, & Mousley, 2006)
- 29. Extending prompt (for those who finish quickly): • Some students might finish the learning task quickly. The intention is such students be posed “extending prompts” that extend their thinking on an aspect of the learning task. (Sullivan, Zevenbergen, & Mousley, 2006)
- 30. Mathematical tasks: A critical Starting point for instruction Not all tasks are created equal and different tasks will provoke different levels and kinds of thinking. (Stein, Smith, Henningsen, & Silver, 2000)
- 31. Mathematical tasks • If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.
- 32. In this approach • Children are encouraged to find solutions to problems by thinking for themselves before instruction from the teacher. • This is “fundamental to opportunities for problem solving and reasoning” (Sullivan et al. 2015, p. 108)
- 33. Key teacher action 3 Orchestrate mathematical discussions 1. Anticipate 2. Monitor 3. Select 4. Sequence 5. Connect Smith & Stein, 2011
- 34. Specific teacher moves to guide discussion • Using wait time • Revoicing • Asking students to state someone else’s reasoning • Asking students to apply their own reasoning to someone else’s reasoning • Prompting students for further participation
- 35. Teach children how to participate in a discussion Idea: Sentence starters cue students to know what to say Explain to me what you meant by.... How is your way different from… This also helps students to learn what to listen for so they can contribute to the discussion. Reinforcing norms support student persistence and participation (Kazemi and Hintz, 2014)
- 36. Norms… • show how you get your answers • keep trying even if it is difficult (it is meant to be) • explain your thinking • listen to other students
- 37. FINDING THE DIFFERENCE BETWEEN TWO NUMBERS A Sample grade 2 lesson
- 38. From the Australian Curriculum This lesson addresses the following descriptor from AC for year 2 • Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting (ACMNA028) • Explore the connection between addition and subtraction (ACMNA029) • Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030) There is also potential for students to come to Understand number relationships, to be more Fluent with the mental calculations, to find an efficient solution by Problem Solving, and to develop Reasoning by explaining their strategy.
- 39. A sample grade 2 lesson • Rationale for the lesson: • Mental computation often involves strategies to make the process easier. The focus in this lesson is on different strategies for partitioning (breaking a number into convenient parts) and bridging (making up to convenient numbers like 10, 100, etc) to solve number problems mentally.
- 40. Explain goals for lesson We are learning that here are many ways to find the difference between two numbers
- 41. In this lesson, I need you to • show how you get your answers • keep trying even if it is difficult (it is meant to be) • explain your thinking • listen to other students
- 42. The Zone of Confusion • A space where students can be encouraged to enter for powerful problem solving processes to occur
- 43. My friend Sophie is 18 years old. Her sister Georgia is 25. How much older is Georgia? Introductory task/s
- 44. How Many more? At the speedway the Ford car did 89 kilometres per hour (kph). The Holden car did 111 kilometres per hour (kph). How much faster was the Holden travelling. Explain two different ways you could work this out in your head.
- 45. If you are stuck • The Ford car did 25 kph. The Holden car did 18 kph. How much faster was the ford car? • The Ford car did 11 kph. The Holden car did 8 kph. How much faster was the ford car?
- 46. If you are stuck • The Ford car did 11 kph. The Holden car did 8 kph. How much faster was the ford car?
- 47. If you have finished • The Ford travelled 1035 kilometres. The Holden travelled 989 kilometers. How many more kilometers does the Holden need to travel to catch up to the Ford? • Explain how you can work it out in your head.
- 48. The zone of confusion
- 49. Some final words… • Posing challenging tasks requires a different lesson structure • Lessons should foster the sense of a classroom community to which all children contribute with the intention that children learn from each other • The experience of engaging with the task happens before instruction • Few rather than many tasks • All children are given time to engage sufficiently to participate in the lesson summary. (Sullivan and Davidson), 2014
- 50. What implications does this have for your teaching? What questions do you have? Turn and Talk with a neighbour
- 51. References • Elham K., & Hintz, A. (2014). How to structure and lead productive Mathematical Discussions. USA. Stenhouse. • Fraivillig, J. (2004). Strategies for advancing children’s mathematical thinking. Teaching Children Mathematics, 7, 454-459. • NCTM, (2000). The Teaching Principle. Retrieved from http://www.nctm.org/standards/content.aspx?id=26808 • Pólya, G. (1962). The teaching of mathematics and the biogenetic law. The Scientist Speculates, 352-356. • Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313-340. • Stein, M. K., Smith, M. S., Henningsen, M. A., & Edward, A. Silver. 2000. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development. • Smith, M., & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. Reston V.A. NCTM • Sullivan, P., Zevenbergen, R., & Mousley, J. (2006). Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms. International Journal for Science and Mathematics Teaching, 4, 117-143. • Sullivan, P., & Davidson, A. (2014). The role of challenging mathematical tasks in creating opportunities for student reasoning. In J. Anderson, M. Cavanagh & A. Prescott (Eds.). Curriculum in focus: Research guided practice (Proceedings of the 37th annual conference of the Mathematics Education Research Group of Australasia).(pp. 605 – 612). Sydney. • Sullivan, P., Walker, N., Borcek, C., Rennie, M. (2015). Exploring a structure for mathematics lessons that foster problem solving and reasoning. In M. Marshman, V. Geiger & A. Bennison (Eds.), Mathematics Education in the Margins: Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia. (pp. 41 -57). Sunshine Coast.