Feb 19, 2016 - knowledge. Below is a list of the months of the year, and how many days in each month. January. 31. February. 28 (29 in leap years). March. 31.
Teaching mathematics in primary schools with challenging tasks: When a quarter is not quite a quarter James Russo discusses challenging tasks and introduces a counter-intuitive conundrum for young students.
T
his article offers a brief overview of teaching mathematics using challenging tasks. It provides an example of a challenging task appropriate for Grade 2, Grade 3 or Grade 4 students. (Ages 5-8)
Challenging Task: When a quarter is not quite a quarter.
What are challenging tasks? Challenging tasks are complex and absorbing mathematical problems with multiple solution pathways, where the whole class works on the same problem (Sullivan & Mornane, 2013). The task is differentiated through the use of enabling and extending prompts. Enabling prompts are designed to reduce the level of challenge through: simplifying the problem, changing how the problem is represented, helping the student connect the problem to prior learning and/or removing a step in the problem (Sullivan, Mousley, & Zevenbergen, 2006). When developing enabling prompts, it is critical that they do not undermine the primary learning objective of the lesson by ‘giving too much away’. Extending prompts are designed for students who finish the main challenge and expose students to an additional task that is more challenging. They require the students to use similar mathematical reasoning, conceptualisations and representations as the main task. When teaching with challenging tasks, the teacher generally begins by launching the challenge and describing what resources students have at their disposal for solving the task (For example the enabling prompts, referred to as the ‘hint sheet’, counting frames, one hundreds charts). Once the challenge has been launched, students then explore the task, either individually or collaboratively and try to find at least one way of solving it. Finally, the teacher facilitates a whole-group discussion where students present their different approaches to solving the task (Sullivan et al., 2014). The teacher closes by offering a brief summary, reiterating the learning objective and presenting a sample of student work which supports this objective (Russo, 2015).
February 2016
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Learning objectives Primary objective:
• For students to appreciate that mathematical
problems can involve counter-intuitive findings.
Secondary objectives:
• For students to use their knowledge of how
the seasons and months are constructed to solve a mathematical problem.
• For students to appropriately represent a challenging mathematical problem.
Challenging Task In England, what is the longest season? Enabling Prompt 1: Easier problem In England, Spring is March, April and May. How many days in Spring? Enabling Prompt 2: Connection to relevant prior knowledge Below is a list of the months of the year, and how many days in each month. January
31
February
28 (29 in leap years)
March
31
April
30
May
31
June
30
July
31
August
31
September
30
October
31
November
30
December
31
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Teaching mathematics in primary schools with challenging tasks: When a quarter is not quite a quarter
Extending Prompt They say Australia is the ‘lucky country’, but I say England is luckier. Throughout their lifetimes, people in England have a whole extra season of Summer, compared to people who live in Australia, who instead get a whole extra season of Winter. Can you prove that this crazy sounding statement is true? How this challenging task supports the learning objectives Success with the challenging task without accessing the enabling prompts will involve students achieving both of the secondary learning objectives. This is because in order to have solved the problem, students would have needed to draw on the relevant knowledge of how the seasons and months are constructed, and would have had to find an appropriate means of representing the problem. However, if the student accesses the enabling prompts (i.e., the the enabling prompts (hint sheet) prior to solving the task, it is not immediately obvious whether students would have achieved the two secondary learning objectives. This is because accessing the enabling prompts partially assists students with both problem representation (for example sequencing the months in a table) and recalling the relevant content knowledge for solving the problem (for example the months in Spring, the days in each month). However, I would argue that this information scaffolds and supports the secondary learning objectives, rather than undermines them. Students who access the enabling prompts are still faced with the task of coordinating all of the information provided, combining this with additional knowledge not provided in the prompts (the months in the other three seasons) and determining an appropriate means of representing the problem (the table begins a possible representation, but does not complete it).
Reasoning through this process allows students to arrive at the counter-intuitive conclusion that, in the northern hemisphere, summer and spring are the longest seasons (92 days), and winter the shortest (90 ¼ days). The extending prompt builds on this counter-intuitive finding through exploring how the slight differences in season lengths (1 ¾ days of summer over winter) aggregate across one’s lifetime (an extra season of summer every 53 years). It also involves students having to consider the relative, rather than absolute, nature of the seasons (they are hemisphere specific), which is also not intuitive for most young students. If you find the above task useful in your classroom, please let the journal editors know, and I will be sure to submit additional challenging tasks in future additions. James Russo, Belgrave South Primary School, Victoria, Australia. References Russo, J. (2015). How challenging tasks optimise cognitive load. In Beswick, K. Muir, T. & Wells J. (Eds.). Proceedings of 39th Psychology of Mathematics Education conference. Hobart, Australia, PME. 4: 105112. Sullivan, P., et al. (2014). Supporting teachers in structuring mathematics lessons involving challenging tasks. In Journal of Mathematics Teacher Education 1-18. Sullivan, P. and A. Mornane (2013). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. In Mathematics Education Research Journal 25: 1-21. Sullivan, P., et al. (2006). Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms. In International Journal of Science and Mathematics Education 4(1) 117-143.
Regarding the primary learning objective, most students who engage meaningfully with this challenging task should develop an appreciation that mathematical problems can have counterintuitive findings. When confronted with this task, the initial ‘gut’ response from many children is to state that the four seasons are of equal length, because each season lasts for three months. The key insight into the problem involves understanding that, because months contain different numbers of days, the seasons can actually be of different lengths.
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February 2016
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