using challenging and consolidating tasks to … · using challenging and consolidating tasks to...
TRANSCRIPT
Using challenging and consolidating tasks to
improve mathematical fluency
James Russo and Sarah Hopkins, Monash University
Overview1. The role of prompts in augmenting challenging tasks2. An example: How many fingers?3. Structuring lessons involving challenging tasks4. The role of consolidating tasks5. More examples: Lessons with challenging and
consolidating tasks (WORKSHOPPED)6. Summary and concluding points (WORKSHOPPED)
1. The role of prompts in augmenting challenging tasks• Challenging tasks are complex and absorbing mathematical problems
which contain enabling and extending prompts, in an aim to include all students in the lesson (see Sullivan & Mornane, 2013).
• Challenging tasks (Sullivan et al., 2011):• Must contain at least one enabling and extending prompt developed prior to
the delivery of the lesson.• Must be solvable through multiple means
• Multiple methods/ approaches (e.g., trial and error, working backwards).• Multiple representations (e.g., concrete, pictorial, abstract)
• May have multiple solutions (i.e., they may be open tasks)• Should take considerable time to complete (i.e., at least 10 minutes engaged
with the task, most likely longer)
1. The role of prompts in augmenting challenging tasks• Prompts:
• Fundamental aspect of Challenging Tasks• Enabling prompts: Enabling prompts are designed to reduce the cognitive
demand of the task by 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)
• Extending prompts: Extending prompts expose students to an additional task that requires them to use similar mathematical reasoning, conceptualisationsand representations as the main task, with a view of increasing the level of cognitive demand (Sullivan et al., 2006)
1. The role of prompts in augmenting challenging tasks• The purpose of prompts is to augment student engagement with the
task, without undermining the Primary Learning Objective (PLO). In fact, prompts should generally reinforce the PLO (see Russo, 2016).
• The idea is that all students can work towards this PLO, albeit at different levels of depth.
• Secondary Learning Objectives (SLOs) are traded-off or added to the task when students access enabling and extending prompts. Ideally, there is an obvious connection/ synergy between the PLO and the SLOs.
1. The role of prompts in augmenting challenging tasksThe rolls of prompts in augmenting learning objectives is best understood by considering an example of a Challenging Task.
2. An Example: How many fingers?
How many fingers are in the room right now? How did you work it out?Show me.How did you count them?Can you find another way of counting them?(Russo, 2015)
2. An example: How many fingers?• Primary Learning Objective: To develop methods for representing worded
problems:• Concrete (e.g., Act it out with materials; physically model problem)• Quasi-abstract (e.g., draw the problem situation)• Abstract (e.g., conceptualise the problem situation as an unsolved number
sentence, such as 26 x 10 = ?).
• Secondary Learning Objectives: Skip-counting sequences• Skip-counting by 5s is an efficient means of counting the number of objects in a
collection• Skip-counting by 10s is an efficient means of counting the number of objects in a
collection
AbstractPictorialConcrete
2. An example: How many fingers?
• Enabling Prompt – Alternative A (focus on problem representation):• How many people are in the room right now? Show me.• Removes Secondary Learning Objective whilst maintaining the Primary
Learning Objective. • By removing the focus on skip-counting patterns, students are able to focus
on the PLO (i.e., representing worded problems). Alternatively, if skip-counting was the primary learning objective, a more appropriate enabling prompt may be a picture of some fingers (see next slide).
2. An example: How many fingers?
Enabling Prompt Alternative B (focus on skip-counting):How many fingers are there in the photo below?To accurately ascertain the number of fingers, students will have to skip-count, or use multiplication. Counting by 1s will not yield the correct answer (as some fingers are obscured).
2. An example: How many fingers?
• Extending Prompt 1: • I forgot to tell you, thumbs don’t count as fingers.• Adds Secondary Learning Objective whilst maintaining focus on the Primary
Learning Objective.• Additional SLO: Students have to flexibly modify their existing problem
representation to cope with this new information. Also, introducing students to more challenging counting sequences (i.e., counting by 4s and 8s can be efficient), and potentially the distributive property of multiplication (10 times-tables minus 2 times-tables = 8 times times-tables)
2. An example: How many fingers?
• Extending Prompt 2: • How many fingers are in the school right now? (remember, thumbs don’t
count as fingers). How did you count them? Can you find another way of counting them?
• Adds Secondary Learning Objective whilst maintaining focus on the Primary Learning Objective.
• Additional SLO: Students have to use a more abstract representation to engage with this more challenging problem. It also exposes students to larger numbers.
3. Structuring lessons involving challenging tasks• Most common is the Task-First Approach (initially three, now five
stages):1. Launch (5 mins; teacher introduces task, context, potentially cues relevant
mathematics)2. Explore (20 mins; students explore task, either individually or collaboratively)3. Discuss (15 mins; student-centred discussion of task, unpacking student
approaches and linking to relevant mathematical content/proficiencies)4. Consolidate (15 mins; students engage in supplementary tasks, usually less
challenging, aiming to consolidate understanding and/or generaliseunderstanding to another context)
5. Summarise (5 mins; teacher-led summary of main mathematical ideas, linked to examples of student work)
4. The role of consolidating tasks
• What are consolidating tasks and how do they support lessons involving challenging tasks?:
• Supplementary tasks to consolidate student understanding following the discussion phase of the challenging task, and prior to the teacher summary.
• Reinforce the primary learning objective (or the ‘Big Idea’ that is the focus of the lesson).
• The consolidating task(s) offers students the opportunity for problem-based practice working on tasks which are similar in structure to the main task, however involve a lower level of cognitive demand.
• Challenging Tasks engage students with a particular mathematical concept (or ‘Big Idea’), whereas consolidating tasks aim to develop mathematical fluency in relation to this concept/ Big Idea.
4. The role of consolidating tasks
• Consolidating tasks can vary in structure and emphasis. Different consolidating tasks are likely to be more or less appropriate for a particular challenging task.
• Consider these two challenging tasks, and their associated consolidating tasks. We will work through the problems together.
5. More examples: Tower Task (F-3)
• Primary Learning Objective: Towers of a given height can be represented in a number of different ways.
• Big Idea: Equivalence. “Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value.” (Charles & Carmel, p. 14).
Tower Task (adapted from Sullivan, 2017)
1) Create the towers like the ones in the picture.
2) Can you create two towers of equal height using all of these smaller towers? Record your solution.
Enabling prompt1) Create the towers like the ones in the picture.
2) Can you create two towers of equal height using all of these smaller towers? Record your solution.
Extending promptCan you do it another way? How many solutions can you find?
Consolidating Task1) Create the towers like the ones in the picture.
2) Can you create a tower that is exactly 10 blocks tall from these smaller towers? Record your solution.
3) There are exactly nine solutions. Can you find them all?
5. More examples: Twos, threes, fours and fives; which numbers will survive? (1-4)• Primary Learning Objective: Overlaying multiple skip counting
sequences will result in some numbers being covered more than once (i.e., numbers with many factors) and some numbers not being covered at all (i.e., potential prime numbers).
• Big Idea: Patterns. “Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways” (Charles & Carmel, p. 17).
Twos, threes, fours and fives; which numbers will survive?Starting at 0, I skipped counted by 2’s to 40, crossing off the numbers as I went. Then I did the same thing, but instead skip counted by 3’s. Next, I did it by 4’s. Finally, I skip counted again, but counted by 5’s. Some numbers were crossed off more than once, but some numbers survived – they weren’t crossed off at all. Can you guess which 10 numbers survived? Now check if you are right.
Enabling promptHint about the skip-counting patterns
Note:
It is less important that students independently generate the skip-counting patterns – the primary focus is on examining the interrelationships between the patterns.
This is the rationale behind including this image as an enabling prompt. Generating skip-counting sequences is a secondary learning objective which we are willing to sacrifice to focus the learning on the primary learning objective.
Extending promptWhat if I also skip counted by 6’s, 7’s, 8’s, 9’s and 10’s? Would all 10 numbers still survive? How many more numbers would get crossed off?
What if I kept skip-counting to 100 by 2s, 3s, 4s, 5s, 6s, 7s, 8s, and 9s? What numbers do you think might survive? Now check if you’re right.
Do you notice anything interesting when looking at the list of numbers that survived?
Consolidating tasks• Starting at 0, I skip counted by 2’s to 20, crossing off the numbers as I went.
Next, starting at 0, I skip counted by 3’s to 20, crossing off the numbers as I went. Some numbers were crossed off more than once, but some numbers survived – they weren’t crossed off at all. Can you guess which 7 numbers survived? Now check if you are right.
• Starting at 0, I skip counted by 2’s to 20, placing a counter on all the numbers I landed on. Next, I skip counted by 5’s to 20, again placing a counter on all the numbers I landed on. Finally, I skip counted by 10’s to 20, again placing a counter on all the numbers I landed on? What are the numbers with three counters on them – the numbers I landed on three times?
• Starting at 0, I skip counted by 3’s to 20, placing a counter on all the numbers I landed on. Next, I skip counted by 5’s to 20, again placing a counter on all the numbers I landed on. What is the only number with two counters on it?
6. Summary and concluding points
• Well-designed Challenging Tasks ensure that all students are in a position to focus on (and have success with) the Primary Learning Objective. Challenging Tasks, therefore, support collective learning.
• However, student learning experiences will vary depending on which Secondary Learning Objectives they engage with, which will also impact the depth to which they engage with the Primary Learning Objective. Challenging tasks, there, also support individualised learning.
• Consolidating task(s) offers students the opportunity for problem-based practice working on tasks which are similar in structure to the main task, however involve a lower level of cognitive demand.
• Challenging Tasks engage students with a particular mathematical concept (or ‘Big Idea’), whereas consolidating tasks aim to develop mathematical fluency in relation to this concept/ Big Idea.
6. Summary and concluding points
Can you think of a consolidating task for this problem?
6. Summary and concluding points
• Remember, the primary learning objective…• Primary Learning Objective: To develop methods for representing
worded problems:• Concrete (e.g., Act it out with materials; physically model problem)• Quasi-abstract (e.g., draw the problem situation)• Abstract (e.g., conceptualise the problem situation as an unsolved number
sentence, such as 26 x 10 = ?).
How many fingers?
How many fingers are in the room right now? How did you work it out?Show me.How did you count them?Can you find another way of counting them?(Russo, 2015)
References• Charles, R. I., & Carmel, C. (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. Journal of
Mathematics Education, 7(3), 9-24. • Russo, J. (2016). Teaching mathematics in primary schools with challenging tasks: The Big (not so) Friendly Giant. Australian Primary Mathematics
Classroom, 21(3), 8-15.• Russo, J., & Hopkins, S. (2017a). Class challenging tasks: Using cognitive load theory to inform the design of challenging mathematical tasks.
Australian Primary Mathematics Classroom, 22(1), 21-27. • Russo, J., & Hopkins, S. (2017b). Examining the Impact of Lesson Structure when Teaching with Cognitively Demanding Tasks in the Early Primary
Years. In S. L. A. Downton, & J. Hall. (Ed.), Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia(pp. 450-457). Melbourne, Australia: MERGA.
• Russo, J., & Hopkins, S. (2017c). Student reflections on learning with challenging tasks:‘I think the worksheets were just for practice, and the challenges were for maths’. Mathematics Education Research Journal, 29(3), 283-311.
• 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.
• Sullivan, P. (2017). A comparing sequence. Encouraging Persistence, Maintaining Challenge Project. EPMC.• Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane, A., Roche, A., & Walker, N. (2014). Supporting teachers in structuring mathematics
lessons involving challenging tasks. Journal of Mathematics Teacher Education, 1-18. • Sullivan, P., Cheeseman, J., Michels, D., Mornane, A., Clarke, D.M., Roche, A., & Middleton, J. (2011). Challenging mathematics tasks: What they
are and how to use them. Paper presented at the maths is multidimensional (Mathematical Association of Victoria), Melbourne, Australia: MAV.• Sullivan, P., & Mornane, A. (2013). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. Mathematics Education
Research Journal, 25, 1-21. • Sullivan, P., Mousley, J., & Zevenbergen, R. (2006). Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms.
International Journal of Science and Mathematics Education, 4(1), 117-143.
Consolidating with an additional challenging task: Peter Sullivan’s (2017) counter problem• Primary Learning Objective: Exploring the part-whole idea; that is,
that any given whole can be represented by a number of different parts.
• Big Idea: Equivalence. “Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value.” (Charles & Carmel, p. 14).
Source: Sullivan, 2017
Five students have 21 counters between them. All have different numbers of counters. How many counters might the students each have?
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Enabling Prompt: Removing a step in the problem
Five students have 21 counters between them. All have different numbers of counters. Josh has 7, Jill has 6. How many counters might Tyson, Nellie and Veer have?
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Josh has 7
Jill has 6Tyson has ? Nellie has ?
Veer has ?
Extending prompt: Being systematic
Five students have 21 counters between them. All have different numbers of counters. How many counters might the students each have?
Convince me that you have found all the possibilities?
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• Three people go fishing. They catch 11 fish between them. No one catches the same number of fish. How many fish might each person catch?
CEM 2016
Consolidating the learning
Five students have 21 counters between them. Two pairs of students have the same numbers of counters. How many counters might the students each have?
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Consolidating the learning
Christmas Shopping (1-3)
• Primary Learning Objective: When adding a sequence of numbers, partitioning and regrouping into lots of 10 can make addition easier.
• Big Idea: Basic Facts and Algorithms. “Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones.” (Charles & Carmel, p. 16).
Christmas ShoppingJeffrey did some Christmas shopping for his two sisters. He decided to get them both tickets to see Katie Perry in concert. The tickets cost $99 each. He also got his dog, Sook, a plastic bone for $2. How much money did he spend on his Christmas shopping?
Enabling PromptSimpler problem: 19 + 19 + 2 =
If we partition the 2 into 1 and 1, we can rewrite the number sentence as
19 + 19 + 1 + 1= ?
Can you see any rainbow facts?
Extending promptJeffrey forgot that his 4 cousins and their dog Fletcher were also going to be at his house on Christmas day. He decided to buy his cousins entrance to Luna Park, which cost $49 each. For Fletcher, he bought a new collar for $5. How much more money did poor Johnny have to spend?
Christmas Shopping: Consolidating tasks
• 9 + 4 = ?• 9 + 7 = ?• 3 + 9 = ?• 19 + 6 = ?• 29 + 29 + 2 = ?• 19 + 9 + 19 + 3 = ?• 49 + 39 + 2 = ?• 4 + 18 + 19 + 9 = ?• 5 + 19 + 18 + 19 = ?