C&Pa Design for Learning: Pāngarau/Mathematics 3
‘Design for learning,’ one of our six teaching standards
- curriculum knowledge,
- pedagogical knowledge
- assessment information
- an understanding of each learner’s strengths, interests, needs, identities, languages and cultures.
We pay particular focus to the following elaborations within each of the learning areas:
- Select teaching approaches, resources, and learning and assessment activities based on a thorough knowledge of curriculum content, pedagogy, progressions in learning and the learners.
- Design and plan culturally responsive (culturally sustaining), evidence-based approaches that reflect the local community and Te Tiriti o Waitangi partnership in New Zealand.
- Harness the rich capital that learners bring by providing culturally responsive (sustaining) and engaging contexts for learners.
- Design learning that is informed by national policies and priorities
By now you will be realising that to ‘design for learning’ effectively requires a number of important considerations;
- Knowledge of your ākonga and their identities, their culture, their community, and their tūrangawaewae as your starting point.
- Knowledge of the curriculum learning areas and the progressions within these.
- Knowledge of the pedagogical approaches which incorporate a focus on developing the key competencies and understanding and knowledge of the content specific to each discipline.
Pāngarau: Mathematics
Whole Number: Addition/Subtraction
Learning outcomes:
- Describe the operations of addition and subtraction;
- Explain how to effectively introduce and develop an understanding of the operations including the role of problem- solving, mental strategies, algorithms, estimation, and modelling;
- Explain how children might solve problems that use addition or subtraction using a range of strategies; and
- Consider and describe the role of traditional written algorithms.
Read: Van de Walle, Karp, & Bay-Williams (2020) pages 184-197 of the course text which outlines the main types of addition and subtraction word problems.
Mental and Invented Strategies for Addition and Subtraction:
Ākonga will solve addition and subtraction problems in different ways. It is important that they are encouraged to explain how they solved the problems. This helps us to understand what stage they are at and how to help them in their number development.
Read: Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction (pp. 15-31). Portsmouth, NH: Heinemann.
Mental methods and strategies:
Strategies are given names such as bridging to ten, compensating, sequence methods, and split tens. Different texts and readings sometimes use slightly different terms to explain the same strategy.
Algorithms are traditional written strategies for computation that have been developed over time. An emphasis has moved away from teaching these as learned procedures to developing mental strategies and number sense.
Read: Van de Walle, Karp, & Bay- Williams (2015) Chapter eleven (pages 271-300) of the course text which presents invented strategies for addition and subtraction
For more examples of problem types for addition and subtraction problems refer to Numeracy Project Book 5: Teaching Addition, Subtraction, and Place value (available electronically from nzmaths.co.nz: Numeracy Projects: Books).
Progressions in counting and subtraction:
Follow these links to the curriculum elaborations and the learning progression frameworks for addition and subtraction, trace through the progressions and take note of how these develop.
- Record the mathematical language associated with the operations of addition and subtraction. What are some of the difficulties that children might face with some of these words?
- What is meant by counting all, counting on, the commutative property, and a number fact?
- Why is zero (kore) a challenging number?
- How are addition and subtraction related?
Mental Strategies and Written Algorithms
- Solve each problem below using two different mental strategies. Record your methods
- Solve each problem using a standard written algorithm
- Which method should be the most efficient? Why?
a) 199 + 99 = b) 481 – 19 = c) 1005 – 99 =
- Compare the efficiency of your strategy with that of using an algorithm.
Whole Number: Multiplication/Division
Learning outcomes:
- Explain the operations of multiplication and division;
- Describe how to effectively introduce and develop an understanding of the operations including the role of problem-solving, mental strategies, algorithms, estimation, and modelling;
- Solve problems that use multiplication and division
Read: Young-Loveridge, J. & Mills, J., (2010). Multiplicative thinking: Representing multidigit multiplication problems using arrays. (pp. 41-50) in Averill & Harvey (2010)
This article outlines the reasons why many young ākonga encounter problems when they engage with multiplication. Consider why it is so important that conceptual understanding precedes the teaching of formal methods. As you read this chapter think about how ākonga can be scaffolded to unpack a rich conceptual understanding of multiplication.
Strategies for multiplication:
The strategies for multiplication are more complex than addition and subtraction and rely on ākonga to have sound understandings of place value.
Ākonga need to be able to estimate the product when working with large numbers and should use representations to support their thinking.
As you read the chapters from the Van de Walle et al. text, pay particular attention to pages 311-321 which outline some strategies for multiplication and introduces the area model for two-digit by two-digit multiplication. Make sure that you make sense of how to do this. Pages 317-321 examine and make sense of the standard algorithm for multiplication. This is taught LAST when the ākonga can easily use a range of invented strategies.
Read: Van de Walle, Karp, & Bay-Williams (2020)
· Chapter 9, pages 197-215 and
· Chapter 12
These are key chapters that you will return to often to inform your teaching of multiplication and division. As you read through these sections, work through the two problems on page 314 and activities 12.1 and 12.2 on pages 318-319.
Types of Division Problems:
Division problems are usually introduced through a social context of sharing.
Partitive Division
This type of division problem can clearly be modelled by sharing objects equally in to the required number of sets (one for you, one for you, one for you, etc).
Ākonga who solve problems by direct modelling will often need to physically represent or draw a circle to represent each set
Measurement Division
In measurement division problems the ākonga is finding out how many piles of objects can be formed.
Division Problems:
1. Write two division word problems. Write one as a partitive division problem and the other as a measurement division problem.
2. Solve each of the following problems and state whether it is a partitive division problem or a measurement division problem.
There were 27 videotapes to be placed in boxes. Each box holds exactly 9 videotapes. How many full boxes of videotapes will there be?
Twenty-eight apples are to be shared equally amongst 7 people. How many apples will each person receive?
3. For each problem, explain how ākonga might model a solution to the problem.
Read: Jong, C., & Magruder, R. (2014). Beyond Cookies: Understanding Various Division Models: Reflect and Discuss
This article explores partitive and measurement division problems and strategies for writing division story problems. Think about how you can transfer new learning here to an Aotearoa NZ context.
Remainders
It is a good idea to have problems that involve division with remainders (whakawehenga whai toenga). Ākonga often deal with remainders in creative and reasonable ways. For example ask a young ākonga to solve the following:
Problem: Mrs Smith has 21 jellybeans to share equally among her four child. How many jellybeans will each child get?
For some problems, the remainder is the answer to the problem.
Problem: There are 27 children going on a class picnic. For now, the teacher has only 6 cars available, each car is able to take 4 children. How many children do not have a ride to the picnic?
Read: Lamberg, T., & Wiest, L. R. (2012). Conceptualizing division with remainders.
Orchestrating a Problem Solving Lesson
This proposed lesson plan format is intended for ‘group worthy rich tasks’ as discussed in module two, open tasks with multiple entry points, in the words of Jo Boaler, ‘low floor, high ceiling’ tasks. It is expected that the task will be challenging and require a group effort to persevere through productive struggle to find a way to solve the task. Children will collectively draw on their prior knowledge and individual strengths to contribute to the groups efforts.
When our ākonga participate in collaborative problem solving consider the ways in which māori and pasifika values might be fore fronted to support participation and inclusion in productive mathematics learning. These values include for Māori, whakaako, whakawhanaungatanga, kotahitanga, manaakitanga and tautoko and for Pasifika, reciprocity, service, respect, inclusion, family, relationships and belonging.
When we use talk moves such as think, pair, share or revoice what Pasifika value are we upholding? When we develop socio group participatory norms, such as leaving no one behind and developing a shared understanding, what Māori value are we upholding?
Task: Form a collaborative planning group with fellow kaiako pitomata and have a go at designing a lesson for your mathematics micro teaching number three, use the lesson plan format provided, the elements of the lesson sequence outlined below and the following problem. Post your plan in the forum for this week.
Danielle’s whānau had arranged buses to transport their 374 whanau to a family reunion down the line. They organised 16 buses. Danielle’s papa was in charge of snack boxes for the buses and needed to know how many people were on each bus? Can you help Papa work it out?
Use the lesson outline here and see below for anticipate, big idea, curriculum decisions, solve and share, notice and connect, and reflect to design your lesson. Then, plan the launch in detail. The lesson design and planning process is a crucial process to help you make sense of the elements of successfully orchestrating a lesson.
A suggested lesson outline: for year 3-8, (adjust down for juniors)
Problem Launch:
- 7-10 minutes approx depending on complexity of problem
Establishing Socio Group Norms:
- 2-5 minutes briefly discuss the participatory norms for the independent work, (depending on how established the norms are with your ākonga)
Problem Solve:
- 15-20 minutes approx
- Small groups of 4 mixed ability (Pairs in junior classes)
- Children must work collaboratively, set this up in the norms
- Shared and agreed solution strategy (1 pen, 1 paper)
- Monitor but don’t mini group teach, facilitate where necessary so the group can persevere, productively struggle together to develop their solution strategy.
- Decide how you will sequence the sharing back.
Share Back:
- 15-20 minutes, teacher as facilitator of children’s explanations
- Listen, encourage explanations, use your talk moves, facilitate their talk, don’t teach, sit in the audience.
- Record their strategy on large paper for all to see. Facilitate the use of mathematical practices.
Notice, Connect and Reflect:
- What did you notice as tamariki shared their strategies?
- 5-10 minutes connecting to the big mathematical idea and connections between strategies (an important part of the lesson)
- How are the strategies similar? different? What do you notice about the numbers/operations in this strategy compared to this strategy?
- How can you connect their strategies to the big mathematical idea of the lesson and associated understandings?
Use questioning and talk moves to elicit tamariki thinking around this.
Before you plan the launch…….
Anticipate: Strategies and misconceptions, with your planning team solve the problem as a group, identify as many possible solution pathways that you can that tamariki might use. Identify any misconceptions which might emerge.
Big Idea: Analyse your predicted strategies and misconceptions, what key mathematical ideas emerge? List the key maths ideas and think about what your learning goal/s might be. Now connect these with a big idea from a source, (include the source in your planning). Sometimes you will start with a big idea in response to ākonga needs and/or strengths, of key importance either way is anticipating the solutions and misconceptions.
Curriculum Decisions: How does this connect with the curriculum? What is the intended maths learning from this task? Think about and plan for an achievement objective and associated learning outcomes, these must connect tightly. What Key competency will you develop through the pedagogical approach you are using? What Socio group participatory norms will you focus on? What mathematical practices will you focus on? You will choose one each of these for recording in your plan and reporting on but others will emerge of course.
Solve and Share: When tamariki are working in their groups of four, your job is to rove and listen and monitor the strategies which are emerging, these should usually be strategies you have anticipated. Select and plan how you will sequence the sharing back of strategies in alignment with the learning goals of the lesson. Which group will you start with for sharing back, who will be in the middle and who will be at the end? Use your anticipated strategies to plan this.
When teaching a lesson like this, as the small groups of four share back their strategies with the wider group, represent their explanations for them on paper or have them represent their strategies on paper so everyone can see what they did. As they share, your job is to facilitate their use of maths practices as they explain their strategies using your talk moves and questioning.
Notice and Connect: As they are sharing take note of where connections occur between the strategies and to the big idea. You will need to pre plan and anticipate the connections when you anticipate the strategies, pre plan which strategies connect well to the big idea or demonstrate the big idea.
Read: Smith, M.S., Hughes, E.K., Engle, R.A., & Stein, M.K. (2009). Orchestrating Discussions.
Trajectories of learning, planning a sequence of lessons
…we can’t change to a mixed ability or a strength based grouping approach without key pedagogical tools to support the success of this. A key tool we can use is a ‘low floor, high ceiling tasks’ or ‘complex/rich tasks’ with multiple entry points. Planning series of these lessons, which follow a tight trajectory of learning will help to ensure that tamariki develop sound understandings of big mathematical ideas along with key mathematical concepts over time and all through an inquiry approach.
Building an inquiry community in your mathematics classroom takes work, ākonga need to be taught specifically how to participate in mathematical inquiry effectively. This involves communication, collaboration and participation within a problem solving group, productive struggle and persistence with challenging tasks, socio group participatory norms, and the use of mathematical practices (expressing agreement/disagreement, explaining, justifying, asking questions/seeking clarification, making connections and generalising).