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How I changed my approach to testing to promote learning Geraldine O’Flynn. Scoil Chríost Rí, Portlaoise

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What the research says When assessment is integrated with instruction, it informs teachers about what activities and assignments will be most useful, what level of teaching is most appropriate and how summative assessment provides diagnostic information. (McMillan 2000) An assessment activity can help learning if it provides information to be used as feedback, by teachers, and by their [students] in assessing themselves and each other, to modify the teaching and learning activities in which they are engaged. Such assessment becomes ‘formative assessment’ when the evidence is actually used to adapt the teaching work to meet learning needs. (Black et al. 2002) Feedback is formative only if the information fed back to the learner is used by the learner in improving performance. If the information fed back to the learner is intended to be helpful but cannot be used by the learner in improving her own performance it is not formative. It is rather like telling an unsuccessful comedian to ‘be funnier’. (William, D., 1999)

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Trying something new In the past I gave my students a test at the end of the topic, marked it, handed it back and moved on to the next topic. Now it’s different … This time I presented them with an open activity and let them work in groups. Normally I would have a lot of routine questions on the test.

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Some thoughts… We had finished the work on patterns from the Teaching and Learning Plans; my students were able to draw a graph, and generalise the pattern. I thought they would do well with this. At first I thought they had performed well, but on closer inspection I saw lots of mistakes.

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The Task The First Year students in your school are planning a sponsored charity cycle to raise money for Portlaoise hospital. They came up with the following ideas: Case 1: How about we ask for a €5 donation and 50c a mile Case 1: How about we ask for a €5 donation and 50c a mile Case 2: Ask people to sponsor €1 a mile Case 2: Ask people to sponsor €1 a mile Case 3: We won’t make enough; ask them to sponsor €2 a mile Case 3: We won’t make enough; ask them to sponsor €2 a mile · Make a table and graph of each of the sponsor plans and write an equation that can be used to calculate the money a sponsor owes, when given the total distance the student cycles. · Compare the plans and write an argument for the sponsor plan you would recommend. Remember that your argument should be supported with maths!

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Examples of student work

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Examples of student graphs

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My observations on their work They were having difficulty with the following: Start values on the graphs Scaling the axes Generalising The prescriptive use of T1, T2, T3 was confusing Examining the relationship between the miles and the amount raised.

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Examples of students’ conclusions The sponsor plan I would recommend is the Case 3 – €2 a mile. I chose this because I thought it would raise the class more money and it’s a reasonable price. So if the cycle was 2o miles, you’d get €40. In the second option for 20 miles you’d only get €20. For the first option it’s €0.50 a mile so that’s €10 and add your €5, so that’s €15. So I would recommend the third one. (Student, Scoil Chríost Rí). I think Case 1 because each time they are adding on €0.50 and they already have a €5 donation, so the more miles they do the more money they get. (Student, Scoil Chríost Rí)

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My observations on their comments The students are comparing the sponsor plans, but only for a scenario where they would cycle 20 miles. The first comment reveals that the student can generalise in words but not in symbols (it’s €0.50 a mile so that’s €10 and add your €5). They seem to have no idea why we are using tables, graphs, equations and how these representations help us solve problems. None of them used the same scale and axes to compare the graphs.

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What I should have asked them: Will Case 3 always be better? Is there a condition under which Case 1 or 2 will raise more money? What if the sponsored cycle was for 100 miles, which sponsor plan would raise the most money? Is it easy to compare the plans? What would make it easier?

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Revisiting the assessment task I realised there were huge gaps in the students’ understanding, so I came at the assessment task from a different angle and asked them to match the story with the table, graph and equation. As they worked I went around and asked them questions, to see if they could make connections between the figures in the table and the graph and the equation and the story. [Note: You can see what happened in the lesson in the accompanying video.]

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Matching exercise MilesAmount 00 12 24 36 48 510 612 714 816 918 1020 y = 0.5x + 5 Sponsor €2 for every mile MilesAmount 00 11 22 33 44 55 66 77 88 99 10 A €5 donation and €0.50 a mile y = x Miles Amount (€) 05 15.50 26.00 36.50 47.00 57.50 68.00 78.50 89.00 99.50 1010.00 Sponsor €1 for every mile y = 2x

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What to do next? In the follow-on tasks which I gave to the students I looked out for the following: - Can they represent mathematical ideas using tables, graphs, equations? - Are they scaling axes properly? - Can they see the significance of the slope and the point of intersection?

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Additional reflection A next step would be to discuss discrete and continuous data as they all drew a line to connect the points on the graphs without thinking of the context of the task. Also after students have engaged with a few of these tasks, I will ask them- What do all the tasks have in common? So that they are able to recognise the properties of linear relationships.

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Follow-on activities (1) Jade and Emma were comparing network offers. Star -Mobile offers 30 free texts at the beginning of the month and an additional 3 free texts each night. Asteroid offers 2 free texts at the beginning of the month and an additional 5 free texts each night. Which network would you advise the girls to go for? Use maths to support your answer.

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Follow-on activities (2) Sarah, Jack and Sophie were offered savings plans by their Dad for their birthday. They could choose which plan they wanted. Plan 1: Their Dad would save €4 every day for them. Plan 2: Their Dad would put in €10 upfront and then put €3 in every day. Plan 3: Their Dad would put in €20 upfront and then put €2 in every day. Which plan would you advise them to choose? Under what conditions would Plan 1 be the better choice? Plan 2? Plan 3? Is there a condition that makes no difference which plan you choose?

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How I’ll manage Activity 2 I’ll get students to work in pairs. Before they start working out the problem I’ll ask them which plan they would pick for themselves and to explain their choice to their partner. If students are having difficulty getting started I’ll ask them how we approached the sponsor plans, what tools did we have to help us solve the problem? When students have come to a conclusion having used their maths, I’ll ask them to compare that to their initial choice. I’ll ask students to nominate someone to report back their conclusions and justify these to the class. Then we’ll look at the approaches the various groups took and see which strategies were helpful and why.

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Extending the learning- (Activity 2) I’ll ask students questions like: How much will Plan 1, Plan 2, Plan 3 amount to after a year? How can we use the graph, table, general expression to make predictions? After how many days will there be €76 in Plan 3 – how did you work this out, is there any other way? If there was a fourth plan where the Dad invests €30 upfront and €4 a day, what would this look like on a graph? What would the general expression in words be? I might show them the graph of another savings plan and ask them to explain the plan in words.

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Making connections I might write up two linear equations on the board and ask students to describe these saving plans in words: y = 3 + 8xy = 7x + 7 If they are struggling I’ll ask – is the Dad paying something upfront? How do you know? How much is being paid in every day? How do you know? Which plan is better, why? Is there a time when they have the same amount of money? How did you work this out? What shape would the graphs be, how do you know? Without a table, sketch the graph. For homework I’ll get them to write equations for their own savings plans and see if their partner can describe them in the next lesson.

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4.4 – Matching Equations and Graphs Math 9. 3 students participate in a 5-km walk for charity. Each student has a different plan to raise money from his/her.

4.4 – Matching Equations and Graphs Math 9. 3 students participate in a 5-km walk for charity. Each student has a different plan to raise money from his/her.

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