Building units of work with PPS lessons and mastering proof

In the second half of 2022, two teachers, who are expertly experienced in the Powerful Problem-Solving (PPS) approach, completed the final two master series units. In a PPS lesson, teachers present the class with a single challenging problem, allow them to work on the problem in groups at easel whiteboards, and bring the class together for a well-planned and orchestrated discussion about the mathematics, using the students’ work as the centre of discussion. You can check out the Alcoa Maths Enrichment Program blog to get a detailed explanation of the process and many articles where teachers discuss their lessons implementing this approach.

In the last two Master Series units, the teachers learnt how to connect this approach with other explicit forms of teaching, and how to take students further in terms of developing the ability to prove solution. In unit 3 they followed a 5-phase plan for the unit of work:

1. PPS Lesson 1 – Diagnostic: Least sophisticated task – what do they know about a core concept?
2. Concept attainment: Activities to build students’ understanding of core concept.
3. PPS Lesson 2 – Consolidation: More sophisticated task drawing on same core concept.
4. Consolidation and expansion: Activities to embed concept and connect it with other related concepts.
5. PPS Lesson 3 – Application: Most sophisticated task, extending the core concept and pushing other PPS skills (e.g., reasoning).

Unit 3 Integrating PPS lessons into units of work – Karl Schoeppner, Oakwood Primary

Karl was running an academic extension class with top students from Yr 5 & 6, once a week. He built a unit on these core concepts:

• Fractions, percentages, and decimals are all representations of numbers that can be found on a number line,
• They can be converted between and compared.

In his PPS Lesson 1, students were tasked to work in small groups sort the following numbers on a number line (numbers were printed and cut out, and students hung on a clothesline so they could be adjusted as they reconsidered during discussions):

Students discussed solutions and mistakes in a safe manner. They were then given individual follow up tasks, arranging other numbers on a line, including some percentages and decimals.

In the concept attainment phase, Karl focussed on the visualisation of equivalent fractions, as this was the point that some had missed in their first tasks. The following two tasks were used (among others):

Here, Karl has cleverly used sets where the total number of units is the common denominator between the fractions. It is easy for students to think of and find 1/2 and 1/3 of the collection in the first example, but because the total collection is 12 (a common multiple) it automatically directs students’ attention toward the understanding of how equivalent fractions work.

They also played a 4 corners game placing various fractions in the categories:

• 0 to 1/2
• Greater than ½ to 1
• Greater than 1 to 1½
• Greater than 1½ to 2

PPS Problem 2, brought in percentages and decimals. Again, they had to order the numbers on a line.

Subsequent lessons (in the consolidation / extension phase) focussed on converting between the three forms and culminated with the following application problem:

This is a more challenging problem that requires them to take their knowledge of conversions and relations between fractions, decimals and percentages and apply it to a novel situation.
Karl had the following final reflections on the work:

Learning was smoother. Students picked up on new learning quickly. For example, students quickly grasped the concept of multiplying or dividing both numerator or denominator to convert fractions. By linking in the fact that percent can be interpreted as “per 100” students could work out for themselves and explain to me how they might go about converting a fraction into a percentage. By linking tenths and hundredths to place value, they were then able to explain to me how they would go about converting fractions and percentages to decimals.

Unit 4 Mastering Proof – Lisa

Lisa worked with a year 3 class, she was acting as Deputy, so it was not her class and she just had the opportunity to work with the class a few times. Her aim was to introduce the idea of proof to the students. In her first lesson with the students, she gave them the following task, without discussion of proof or convincing arguments:

The class quickly got to work and were on task for the lesson. Some students still needed reminding about symmetry. Others were ready for the extending prompt: how do you know you found all the different necklaces? A couple of groups did find all solutions; however, they did not explain how they knew or what system or strategy they used to solve the problem. To end the lesson, she introduced students to Matt Sexton’s VLS Framework. As a class, they went through the two solutions on the NRich website and discussed which had convinced us, relating both examples to the framework.

The Visual, Language and Symbol framework
(Sexton, 2019, adapted from Fuson et al. 1997, Lesh et al. 1987, & Mason et al. 2010)

The student solutions that were discussed on the NRICH task webpage.

Lisa discussed with the students about what forms were used in each solution and how this helped them to be more convincing.

In her second lesson with the students, they began by revisiting the VLS framework and students reflected on their symmetrical necklace solution, identifying which circle in the framework it best fit and how they could make their work more convincing. Most students were in the Visual circle, using concrete materials or colouring squares to represent beads on the necklace. As a class they discussed how using a combination of 2 or 3 elements – visual, language and symbols – made it easier to explain their strategy and for other people to understand their work, therefore ‘proving’ their solution was correct. Lisa then put students into random groups again and gave them a new problem (below) from Origo Stepping Stones Year 3 Module 9. As an extending prompt, students were asked if they could find more than one solution to the problem.

Lisa has provided several work samples from the students to show the difference in the level of their justification between task 1 and task 2, due to the use of the VLS framework. She has assessed the level of communicating reasoning based on the framework for Communicating reasoning – from Novice to Expert developed by the NRICH Primary team summarised here:

If you are interested in learning in training in the Powerful Problem-Solving approach, Scitech provides free intensive training over the course of a year to Perth metropolitan schools (currently investigating remote options for regional schools). It includes resources, personalised coaching and relief funding so that teachers can work in pairs observing and discussing their lessons, with the coach. Find out more on the Lighthouse Maths Application page.

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Keep thinking and enjoy the struggle!
Shyam Drury (Coordinator of Lighthouse Maths)

Authors are participants of Lighthouse Maths, proudly supported by Chevron.