C&Pa Pāngarau/Mathematics 4
…consider ways that we as Kaiako can keep the culture and language of our tamariki alive in the mathematics classroom, and for that classroom to reflect back their culture and lived experiences through culturally sustaining pedagogy.
The cultural knowledge of Māori, Pāsifika and other diverse ākonga is often excluded from the classroom. Unfortunately research shows us that the cultural capital that shapes schooling practices and curricula in mathematics remains predominantly Eurocentric and is a subject area which is often viewed as “culture-free”. Deficit theorising leads to ineffective pedagogical actions for Māori and Pāsifika ākonga and a loss of identity as someone capable of achieving at school (Hunter & Hunter, 2018; Spiller, 2012; Turner, Rubie-Davis, Webber, 2015).
3.1. Culturally Sustaining Pedagogy in Mathematics
English as an Additional Learner (EAL)
‘culturally sustaining mathematics teaching’ (commonly described as culturally responsive)
The aim of the culturally sustaining teaching approach is to ensure that as many ākonga as possible are kept actively engaged and motivated to learn mathematics, make mathematical progress, and enjoy learning and doing mathematics.
Read: Averill, R., Taiwhati, M. & Te Maro, P. (2010). Knowing and understanding each others’ cultures. In R. Averill & R. Harvey, (Eds.), Teaching primary school mathematics and statistics: Evidence-based practice. (pp. 167-180), Wellington: NZCER Press.
Culturally sustaining pedagogy
…extend beyond ethnicity to include a diverse group of ākonga who may not be able to gain equitable access to the learning for other reasons.
Rather than changing what our ākonga bring from their homes, we can value what they bring. We can build on their experiences and who they are in an explicitly inclusive manner which ensures that they feel a part of a community who can do, understand, and enjoy mathematics.
Read: Abdulrahim, N. A., & Orosco, M. J. (2020). Culturally responsive mathematics teaching: A research synthesis. The urban review, 52 (1), 1-25. Click here
Caring and the English as an additional language learner
Read: Brown, C. L., Cady, J. A., & Taylor, P. M. (2009). Problem solving and the English language learner. Mathematics Teaching in the Middle School, 14(9), 532-539. Click here
As you read this reading reflect on the following questions:
What aspects in the mathematics class does Min struggle with?
Why is mathematics not culturally neutral?
What strategies can a kaiako use to help EAL learners in mathematics?
In ‘discussion intensive inquiry mathematics classrooms’ careful attention needs to be given to making sure that all members of the classroom are able to access the discussions. Often central to the discussions are the mathematical problems ākonga are asked to solve. The next reading will challenge you to think about differentiating the problems you use in culturally responsive ways. As you read this article think about how you could draw on the home background and culture of the ākonga to write and use appropriate problems and activities.
Read: Wilburne, J. M., Marinak, B. A., & Strickland, M. J. (2009). Addressing cultural bias. Mathematics Teaching in the Middle School, 16(8), 461-465. Click here
Commonly the exploring of funds of knowledge related to mathematics of non-dominant communities involves interviews with parents to establish mathematics activities present in household routines. Parents frequently have initial difficulties identifying household activities involving mathematical learning outside of cooking and construction.
Using Cultural Funds of Knowledge to design group worth mathematics tasks
When we consider using tasks flexibly, we are not talking about individualized instruction; we are not suggesting three different lessons within one mathematics lesson with different tasks for different ākonga. We are talking about worthwhile, group-worthy tasks that can be used interactively with a diverse group of learners. These tasks are high ceiling, low floor tasks with multiple entry points for a range of learners to enter at. (Boaler, 2016)
3.2. An Introduction to Fractions
Learning Outcomes:
- Select appropriate models to help students develop an understanding of fractions;
- Introduce fraction concepts to students in a sequenced and meaningful way;
- Use problems that involve suitable contexts appropriate for the teaching of fraction concepts and skills
- Teach fraction computations in a meaningful way.
- Consider the role of “big ideas” in mathematics teaching/learning.
Number Triad:
Once ākonga can competently match words to models and vice versa, the symbols can be introduced. Instructional activities can be used to support the process of linking symbols with concrete materials (including meaningful actions on these materials) and verbal interactions.
The words numerator (the top line of the fraction) and denominator (the bottom line of the fraction) do not need to be specifically taught to young tamariki.
The words have no common reference for tamariki. Whether the words are used or not, it is clear that the words themselves will not assist young tamariki in understanding the meaning of common fractions. However, it is important that tamariki understand what the top and bottom numbers represent.
Read Van de Walle, Karp, & Bay-Williams (2015). Chapter 14: Developing Fraction Concepts (pp. 377-414) in the set text.
This section of the chapter describes the different fraction models and how to help ākonga make sense of common fractions. It is important that you understand the different types of models that can be used for fractions. How would you explain what the top and bottom numbers mean? Consider how you use benchmarks such as one half when thinking about fractions.
Read Zhang, Clements & Ellerton, (2015). Engaging Students with Multiple Models of Fractions
Early Fraction Understanding:
It is important to understand the early understandings of fractions that form the foundation of later, more complex understandings. The activities below will help you to consider what misunderstandings ākonga may bring to their early fractions thinking.
Exploring further
Explore the “second tier” material in nzmaths to understand the learning trajectory for ākonga learning about fractions concepts. Use this link as a starting point, click on any of the arrows under “number and algebra” to see an elaboration of the achievement objectives at this level.
Fraction misconceptions
What do these ākonga understand or not understand about fractions?
• 1 is bigger than 4/3 because it’s a whole number
• ⅕ is smaller than 1/10
• ¾ – ⅓ = 2/1
Consider this problem….
Isha’s parents made naan for Isha’s baby brothers first birthday. They cut the naan into pieces for everyone to share. First they made 3 naan and cut them into halves. Then they made 2 more naan and cut them into quarters. How many pieces of naan did they have altogether?
This problem describes naan which have been cut into halves and quarters. Would you rather have a quarter of a naan or a half? EXPLAIN WHY!
What big idea and associated understandings are being developed through this problem for young learners?
Watch this series of videos of a kaiako teaching this problem. Consider how the kaiako supports the learners to make connections to the big idea.
NZ curriculum exemplars
A mathematics exemplar is an annotated sample of ākonga work produced in response to a set task. Each exemplar illustrates ākonga work based on a particular topic
Fractions exemplars:
Go to Mathematics Exemplars – Fractions. Read through the examples at Levels 1, 2, and 3. Examine ‘What the Work Shows’, ‘The Learning Context’ and ‘Where to Next?‘ sections for each exemplar.
Early Fraction Understanding:
Fractions that are equivalent are different names for the same value. For example, 1/2 and 2/4 are equivalent since they both represent the same number. Models, such as paper strips, are important for developing this conceptual knowledge and understanding.
Read Van de Walle, Karp & Bay-Williams, Pages 399-400 of chapter 14, this takes you through the process of how to help ākonga develop an understanding of equivalence. This understanding is developed through modelling and looking for patterns, and leads to finding equivalent fractions by multiplying the numerator and denominator by the same number.
The reading Fractions across the curriculum outlines fractions learning pathways – as you consider this reading, compare these pathways to those outlined in New Zealand’s numeracy framework. This reading explores the importance of children developing fraction sense – in other words how can we teach fractions in ways that develop student’s conceptual understanding of fractions rather than just teaching rules and procedures?
Read Ontario Ministry of Education. (2018). Fractions across the curriculum (Capacity Building Series No. Special Edition #47).
Fraction computation
A sound understanding of fractions and of the number operations is needed before ākonga begin adding, subtracting, multiplying, or dividing with fractions. Estimation plays a very important part. Although we may use a calculator for some computations, we should use estimation skills to make sense of the answer.
Addition and Subtraction:
Think about a common misconception. When tamariki apply whole number thinking to fraction addition they believe that: 1/2 + 1/3 = 1/5
How might you help a tamaiti overcome this misconception?
Multiplication and Division
Ākonga can make sense of fraction problems that involve the operations of multiplication and division if they relate understandings to whole number problems.
Consider the following: 2/3 x 24 = and 2 x 3/4 = .
Write an appropriate word problem for each of the equations above.
It is important that you work through these pages in the text and complete the examples. Take note of the variety of contexts that are used for the word problems and how the computation written in symbols matches the word problem. You should make sense of the problems and use drawings to support your sense making. At this stage, do not revert to using the algorithms that you were taught at secondary school.
If your own fraction computation skills need strengthening, websites like Skills Wise, or Quiz-Tree can be helpful. There are many other fraction quizzes available on the internet should you need more practice.
Read: Van de Walle, Karp, & Bay-Williams (2015)Chapter 15: Developing Fraction Operations(pp. 415-445)
Fraction Tasks:
Adding and Subtracting fractions:
Think about a group of tamariki you have worked with recently, perhaps on practicum. Write a word problem using a suitable context for these tamariki for each of the following equations:
a. ¼ + ¼ = ?
b. ¾ + 1/8 = ?
c. 9/10 – 1/5 = ?
d. 5/9 + 4/7
Show or explain how a ākonga might solve each of the above problems. Use a diagram to support the problem solving process.
Multiplying and dividing fractions:
Think about a group of tamariki you have worked with recently, perhaps on practicum or micro teaching. Write a word problem using a suitable context for these tamariki for each of the following equations:
a. 3 x 2/5 = ?
b. ¾ x 12 = ?
c. ¾ x 4/5 = ?
d. 6 divided by 2/3 = ?
e. 2/3 divided by 1/6 = ?Show or explain how a ākonga might solve each of the above problems. Use a diagram to support the problem solving process.
3.3. Decimal Fractions
Learning Outcomes:
- Describe how the base ten system extends to include decimal fractions
- Select appropriate models and contexts to help ākonga develop an understanding of the concept of decimal fractions
- Describe suitable learning experiences to help ākonga to be able to compare and order decimal fractions
- Describe suitable learning experiences for developing ākonga understandings of operations on decimals
- Recognise and explain common misconceptions with decimal understandings
This is why we refer to 4.326 as a decimal fraction. The role of the decimal point (ira ngahura) needs to be understood; it is to separate the whole number part of the numeral from the fractional part.
Read recommended text: Van de Walle, Karp, & Bay-Williams (2020) Chapter 16: Developing Decimal and Percent Concepts (pp.448-465) in the recommended text.
It is important that the ākonga learn to read and say the decimals correctly, for example: 0.25 reads as “zero point two five”. Difficulties can occur when comparing decimal fractions if ākonga are saying them incorrectly. For example: Which decimal fraction is larger 0.4 or 0.25?An ākonga who is saying these decimals incorrectly may say that 0.25 is larger than 0.4. Why?
Adding and subtracting decimal fractions
Addition and subtraction of decimal fractions must be based on sound understandings of place value, common fractions, and decimal fractions. Ākonga should not be merely ‘lining up’ the decimals based on the decimal point. They should be encouraged to apply the same mental strategies that they use for whole numbers to decimal fractions. The Empty Number Line (ENL) can also be used to support their mental strategies and explanations. Estimation is a most important skill to help ākonga with addition and subtraction of decimals.
Read: Van de Walle, Karp, & Bay-Williams (2020) Chapter 16: Computation with Decimals (pp. 464-479) in the recommended text. As you read, complete the tasks in the text. This will help you develop your own understanding of decimal fractions and give you insights into how tamariki understandings may develop.
Activity: Adding and Subtracting Word Problems
Write word problems to match each of the following equations:
2.7 + 3.09 = ? 4.18 – 2.9 = ? 0.67 – 0.49 = ?
4.6 + 1.95 = ? 7.36 – 6.8 = ?
2. Explain how a ākonga might solve each of the above equations using a mental strategy.
3. Find the matching achievement objective(s) from the Mathematics and Statistics curriculum statement.
4. Anticipate some possible misconceptions that might emerge when tamariki are solving these and describe how you would respond specifically.Choose ONE of your word problems and answer the questions above in the Stream forum for this week. Try and choose one that hasn’t already been shared by another group member.
Multiplying and dividing decimal fractionsĀkonga should be encouraged to make sense of word problems that involve the operations of multiplication and division. Once again, estimation and mental computation should be emphasised. It is suggested that ākonga begin with a problem that has a whole number as one of the factors. It is important that you complete the problems in this section of the text (pp. 464-477) so that you assess your number sense when working with decimals.
Your focus should also be on the concept and those ‘Big Ideas’ described at the beginning of the chapter.
Activity: Multiplying and dividing word problems
- Write a word problem that matches the following equation: 4 x 3.6 = ?
- Give an estimate for the problem.
- Explain how a solution could be modelled on the number track or with decimal pipes (Check the glossary if you are unfamiliar with this equipment or check out Pipe Decimals Animations.)
- Explain how ākonga could make sense of the following equations:
0.5 x 6 = ? 0.5 x 0.2 = ?
6 ÷ 0.3 = ? 0.8 ÷ 0.04 = ?Activity: Making sense of problems
1. Write an equation to match the word problem below.
2. Explain how a ākonga might model a solution to the problem.
The Problem
I have 5 metres of ribbon that is to be cut into equal lengths that measure 0.2 of a metre. How many lengths of ribbon can be cut?Misconceptions about decimal fractions:
Evidence from mathematics education research has shown that both tamariki and adults experience difficulties with decimal fractions and their operations. The main difficulties are:
- fully understanding the meaning of decimal numbers;
- choosing the appropriate operation for a given situation; or
- trying to make sense of decimals by building on what is already known about whole numbers and place value.
Read: Martinie, S. L. (2014). Decimal Fractions: An Important Point, Mathematics Teaching in the Middle School MTMS, 19(7), 420-429. Retrieved Jun 2, 2021, from https://pubs.nctm.org/view/journals/mtms/19/7/article-p420.xml
This journal article provides insights into some of the misconceptions that ākonga have about decimal fractions. It presents ākonga responses to decimal questions that indicate some common misconceptions. The article also provides some practical activities which may help tamariki understand decimal fractions.
Common fractions and decimal fractions: Making Connections
This topic began with a focus on the importance of ākonga making the connection between common fractions and decimal fractions. In everyday situations there is a need to be able to convert common fractions to decimal fractions and vice versa. The renaming of decimal fractions as common fractions involves reading the decimal fraction using the language of tenths, hundredths, and thousandths.
For example 0.25 is 25 hundredths which can also be written as 25/100.
However, renaming a common fraction as a decimal with understanding is not as straight forward unless the denominator is easily converted to tenths, hundredths, or thousandths. So, for example:
3/5 = 6/10 = 0.6 13/25 = 52/100 = 0.52
Other fractions depend upon ākonga understanding of the quotient concept of fractions.That is, that 3/4 means 3 divided by 4.
Read: Van de Walle, Karp, & Bay-Williams (2020) To reinforce understandings about the connections between common and decimal fractions re-read Connecting Fractions and Decimals (pp. 452-456).
3.4. Flexibility with fractions
Learning Outcomes:
- Explore suitable models and learning experiences to help ākonga develop an understanding of the concept of percentage;
- Develop and describe flexible understandings of proportion and making connections among common fractions, decimal fractions, percentages and ratios;
Introduction: This topic explores the links among common fractions, decimal fractions, percentages and ratios. Flexible understandings of fractions are underpinned by the development of proportional reasoning.
Read: Van de Walle, Karp, & Bay-Williams (2020) Chapter 16: Introducing percents (pp. 471-477) in the set text.
This chapter provides you with a link to fractions, and hundredths in particular, and the importance of applying percent problems in a realistic context.
Models for Percentages:
A model for percentages should display in some form 100 units as a whole. Suitable equipment includes:’
- Slavonic Abacus
- Numberlines and bead strings
- Hundreds Grids
- Rectangular Arrays (e.g. 5×20, 4×25)
Percentage Problems:
Ākonga should learn to solve percentage problems with understanding; this is important given how often we encounter percentages in real-life problem solving situations. They should be encouraged to solve problems mentally and to use estimations before reaching for a calculator. This encourages ākonga to focus on the reasonableness of their results.
Problems involving per cent typically require ākonga to find a percentage of a given number, find what percentage one number is of another number, or to find the total when only a percentage is known.
Ākonga should be provided with problems that not only use real-life contexts but use familiar fractions and relatively easy percentages. The focus is on the understandings of relationships, not complex computational skills. Encourage ākonga to use models or drawings to explain their solutions and mental computations.
Solve the problems on page 474 of the recommended text. Problems like these can be solved mentally – use fraction equivalents where appropriate.Further problems can be found in the Figure-It-Out Number books. One example is Water World in Figure-it-out, Number, Book 3, Levels 3-4 (pp. 18-19). You can access the Teachers’ Notes for answers and additional information for this activity.
Connecting fractions, decimals and percentages:
Ākonga need to see the connections among common fractions, decimal fractions, and percentages. They should not consider them as separate unconnected number systems.
1. Examine the Fraction Exemplar for Level 4.
2. Read through the Kaiako-Ākonga conversation and Where to Next?
3. Read through the Rational Number Unit Plan on stream ‘Whole numbers are not enough’
4. What adaptations might you make to the problems in this unit, think of the tamariki you had in your last practicum?
Ratio and Proportion:
Closely connected with the concepts of fractions and percentages are the concepts of ratio and proportion. A ratio is a multiplicative comparison of two numbers, measures, or quantities. Proportional reasoning is based on multiplicative rather than additive thinking. Instead of looking at an absolute change, one quantity is compared to another. For example:
For every 3 blue counters in a bag there are 7 yellow counters. That is, the ratio of blue counters to yellow counters is 3 : 7. What fraction of the counters in the bag is blue?
If there were 40 counters in the bag how many would be yellow?
If there were 9 blue counters, how many counters would there be in total?
A proportion question:
It takes 10 balls of wool to make 15 beanies. How many balls of wool does it take to make 6 beanies?
The question is effectively: 10 : 15 = ? : 6
What strategies might ākonga use to solve this problem?
Amongst the several ways to answer this, one of the easiest is to recognise that 10 :15 = 2 : 3 = 4 : 6. So the answer is 4 balls of wool.
Rather than teaching ratios as fractions to be manipulated with procedural rules, it is important to develop an understanding or sense of proportion and ratio through a variety of problem types. Ensuring that you allow ākonga to devise strategies that make sense to them.
Developing proportional reasoning:
Fractions problems – including common fractions, decimals, percentages and ratios – should be presented in a wide range of contexts. These situations might include sets, measurements, or rates (e.g. prices). Ākonga should be encouraged to discuss, experiment, and consider the quantities and the relationships in the problem.