Book: Teaching Primary School Mathematics and Statistics: Evidence-based Practice

Harvey, R., & Averill, R. (2010). Teaching primary school mathematics and statistics : evidence-based practice. NZCER Press.

[C&Pa topic Learning to count]

Chapter 1: Learning to Count

Pg 17

Activity: Count in another language
If appropriate, ask a child who is a confident speaker of another language to teach you, and possible some other children, how to count in their language. This is one way you can acknowledge the expertise of children who speak other languages.
What difficulties arose for you in learning how to count in another language?

Pg 18

Activity: Puppet
To help children who are learning to count, use a puppet. Have the puppet make mistakes while counting in a way that shows some of the difficulties the children are having. Ask the children to help the puppet learn to count.

Pg 21

Counting to find out how many

Gelman and Gallistel (1978) counting principles developed as children become more competent at object counting:

• The one-one principle: as the child processes each object in the set they will need to say the net number in the counting sequence.
• Stable order principle: the words in the counting sequence must be used in a consistent manner.
• Cardinal principle: the number you end on tells you how many objects are in the set.
• The order irrelevance principle: the order in which the objects in a group are counted is not important It is important that each object is counted exactly once. In practice this means that if a collection is counted in more than one way, the cardinality will be the same for each count. This principle also requires understanding that in a count the sequence word does not get attached to a particular object.
• The abstraction principle: over time children realise they can count all sorts of sets of objects, and the skills develop from counting physical objects to more abstract skills, such as counting ideas.

Pg 22

Activity: Preparing to count on
This exercise is aimed at helping children to using the ‘ counting on’ strategy.
Example: Let’s count up to 12. Stop. Now let’s count four more numbers: 13, 14, 15, 16.
It is useful to illustrate the count 1, 2, 3, 4 by raising fingers as the numbers 13, 14, 15, 16 are said.

Learning to use symbols

The concept of eight remains the same regardless of language; however the spoken word, written words and symbol used depend on what language is being used.

Pg 25

As adults we are able to co-ordinate our use of counting skills in an automatic way, but for a young child perfecting these skills it is a demanding task that develops over several years.

Pg 27

Chapter 2: Handling numbers

… we are all powerful mathematical thinkers, that mathematics in the mind is a human construct built on embodied experiences in a physical word (schemas), and that we conceptualise numbers in unique and personal ways using physical metaphors such as colletions of things, positions or movement in space, or pieces that can be put together.

…teachers [need to] listen to children’s mathematical explanations rather than listen for particular responses.

Pg 29

Schema theory, metaphor theory and the embodied mind

Worthington and Carruthers (2003) identify 10 mathematical schemas: envelopment, trajectory, enclosure, transporting, connecting, rotation, going through a boundary, oblique trajectory, containment and transformation. Schema theory explains how children make connections between one situation and another using these schemas. A child may repeatedly place items in bags and boxes, for example, fascinated with containment.If a teacher models addition with pegs in bags, this child might readily word with this representation of numbers.

Lack of ease in handling numbers may indicated schematic dissonance (teachers’ use of inappropriate schemas) or schematic impoverishment (children’s lack of opportunities to explore numbers in a range of mathematical schemas in sufficient depth).

Pg 30

The four grounding metaphors for handling numbers (the 4Gs)

Grounding metaphor	Representation of numbers	More is…	Strategic thinking	Equipment
1. Object Collection	Seeing numbers as collections	Larger	Subitsing, counting all from one, counting on by adding more, counting back by taking away (collections growing or shrinking in size)	Counters, dots, dice, fingers, sets of individual objects, pictures of things, sets (marbles, stickers, etc), stringed beads
2. Object construction	Seeing numbers as parts of wholes	A greater proportion	Bridging through 10, adding or subtracting in chunks, “front end” strategies, place value (100s, 10s and 1s), multiplication arrays	Place value blocks, tens frames, 100s charts, towers of 10, mone, fraction walls and fraction towers.
3. Measuring stick	Seeing numbers as equal-sized segments or units of a ruler-like continuum	Longers	Skip counting as equal-sized segments, adding or subtracting in segments	Cuisenaire rods, Metric lines, rulers, unifix or centicubes, decade-sectioned bead strings, thermometers
4. Motion along a path	Seeing numbers as positions along a connecting tract	Further along	Counting on by moving in one direction, counting back by moving in the opposite direction, skip counting as hops along the track	Number lines, tracks, and trails (e.g., snakes and ladders board), clock face, spinner, hopscotch court.

Pg 37

…children’s responses illustrated the metaphorical differences in their reasoning [exploring 38 investigation]….

Object collection

One hundred is too many numbers. Thirty-eight is somewhere in the middle. You count, you say 1,2,3… 99, 100 and you know where 38 is. You can draw 100 dots all over the paper and count them and find 38 and put a ring around it.

Object construction

you put ten tens together to make one hundred. It’s like 50 cents and 50 cents – that’s 100 cents and that’s a dollar.

Motion along a path

I can show you on the number line, you start from 1 and go all the way along to 100. I can jump like this 10, 20 30, 40, that’s too far, I have to go back.

Working with a measuring stick

38 is 3 orange long ones [tens] and 8 of these [ones]. I line them up. Line them up… there, you can see.

pg 39

Children can make sense of complex mathematical structures by using their preferred or primary metaphors.

Children’s metaphors for thinking about numbers may not extend usefully beyond 100 even at years 7 and 8. … they need to understand and believe that their thinking can be extended through meaningful schemas and metaphors to encompass new genres and magnitudes of numbers.

Pg 40

Chapter 3: Multiplicative thinking: Representing multidigit multiplication problems using arrays

pg 41

Many students have difficulty understanding the meaning of multiplication. They often regard the answer to a multiplication equation as a basic fact to be recalled. They don’t relate this to what the equation itself represents in terms of structure and pattern. Basic facts such as 2 x 3 = 6 are frequently learnt by rote, with the focus on getting the equation correct to the detriment of understanding what the equation is all about.

pg 42

… Pesek and Kirshner (2000) have shown, once students are taught to use standard written algorithms, it can be extremely difficult to then try to help them develop conceptual understanding.

Structures and pattern are at the heart of learning multiplication (Mulligan & Mitchelmore, 2019). Multiplication is a process involving groups of groups and arrays provide a powerful showing the structure and pattern of multiple groups. Evidence has shown that students appreciation of structure and pattern is very important in relation to their understanding of mathematics (Mulligan & Mitchelmore, 2019).

Low achievers don’t seem to notice structure and regularity in mathematics the way high achievers do. It is therefore important to the teachers draw students attention to structure pattern, because it can bring about substantial improvements and the mathematical learning and understanding.

Activity 3.1: Modelling multiplication

1. Give the children a multiplication equation (e.g., 3x 5). Write the problem on the whiteboard or in the modelling book.
2. Ask the children to write a word problem/story showing the meaning of the equation. The children will need to discuss in their groups what they think 3 x 5 means in order to be able to agree on a suitable story.
3. Ask the children to draw a picture, or model with materials, the word problem/story. The picture, or model, will need to show three groups with five objects in each group.
4. Ask the children to share the word problem/story and picture/model with others in the group/class. At this point, encourage discussion based around the relationship between the written equation (3x 5) and the picture/model (as three groups of five). Keep pictures, or take photographs of models, for future reference.

Materials may include: unifix cubes, blocks, counters, plastic animals.

Pg 44

Its generally accepted in New Zealand schools that “3× 5″‘ means “three groups of five”. As the Numeracy Project Book 6 points out, “this is a convention that is not shared across all cultures” (Ministry of Education, 2007, p. 12), and it may need to be clarified with the children. Māori and some other cultures may read it as three replicated five times. It is important for teachers to model consistency in the use of mathematical language during their lessons in order to avoid confusion later.

Pg 45

As children become aware of the meaning of multiplication, they will need to have a knowledge of basic facts to draw on for solving problems. This is an important part of moving towards multiplicative thinking. The children will then be able to utilise these known facts to derive others that are unknown, using multiplicative properties.

Pp 46

Linking multiplication to measurement and geometry shows why understanding number is so crucial to developing an understanding of other strand, mathematics. Teachers might like to develop this idea further at this point. For example, you could measure a desktop or a textbook. Alternatively, you could take the children outside and look for rectangles in the environment (e.g, the tennis court), which you could measure and work out the area for. How does finding the area of rectangular objects relate to understanding arrays in multiplication?

Pg 51

Chapter 4: Bridging the gap: Challenging conceptions of fractions and decimals

52

A common approach to teaching decimals and fractions has been to supply students with reconstructed rules that can help them when they are representing, comparing or operating with fractional numbers. As discussed in the previous chapter in the context of multiplication algorithms, creating a dependence on rules in early learning experiences can inhibit students’ ability to construct meaning, and mask many of the underlying interconnections important for conceptual understanding (Skemp,2006).

Teachers are better able to foster the development of complex networks of understanding if they are aware of the possibilities for mathematical learning offered by a variety of representations, but also if they understand the potential difficulties learners might encounter (Ball, Thames, & Phelps, 2008). One common difficulty is establishing relationships between decimals, whole numbers and the place value system. An undeveloped understanding of relationships between decimals and fractions may hinder a learner’s construction of representations of fractions as decimals.

… students explore creative and varied models in order to promote rich mathematical discussion. They should encounter both discrete (using units or sets; e.g., counters) and continuous (using regions or lines;
e.g., rotating regions) representations.

Pg 55

Part-to-whole thinking

Many students’ experiences with fractions involve partitioning a whole into parts (i.e., finding a fraction of a set or a shape). Providing opportunities that require students to create a representation of a whole from a fraction allows them to gain a better understanding of how a fraction is defined by its relationship to the whole.

To communicate their ideas, students could use materials, paper representations, drawings, words and symbols to describe or demonstrate the whole they have constructed from the given portion. Sharing their results allows for multiple opportunities not only to consolidate relationships to the referent whole, but also to explore the interrelationships between representations and develop proportional thinking.

Pg 56

The “nestedness” of place value

Part of the difficulty students have when placing fractional numbers on a number line is a lack of perception of continuity of the line. …decimal fractions are found inside other decimal fractions – a little like nested boxes or Russian dolls. For example ,multiple layers of meaning for the number 10 can be expressed as one ten, ten ones, 100 tenths, 1000 hundredths, as well as a tenth of 100 and so on.

… by stretching out or zooming in on a number line, students can glimpse the way in which an infinite range of numbers exists between and within numbers

… challenge conceptions students might bring to this area, such as believing that decimals belong to the left of zero on the number line, or trying to make symmetry around the decimal point and invent a “oneths” column before the tenths.

Pg 65

Chapter 5: Developing early algebraic reasoning

Authors: Glenda Anthony and Jodie Hunter

pg 66

What is early algebra?

Algebraic reasoning can be differentiated from arithmetical thinking through the shift in focus from a procedural perspective of operations and relations to a structural perspective. As expressed by Mason (2008), “algebraic thinking is required in order to make sense of arithmetic, rather than just performing arithmetic instrumentally”(pg. 58).

Pg 67

Exploring number properties

… for example, when you add zero to a number it doesn’t change. However, children often get confused about when these properties work and when they don’t. For example, children who know that a + b = b are not always sure if a – b = b is a true or a false statement (Anthony & Walshaw, 2002).

When children “genuinely understand arithmetic at a level at which they can explain and justify the properties they are using as they carry out calculations they have learned some critical foundations of algebra”(Carpenter, Frankie, & Levi, 2003, p. 2).

Testing out conjectures can lead students to make new observations and helps them make sense of how numbers work.

Pg 70

In order to provide arguments that don’t depend on specific numbers or cases, students need opportunities to learn to justify by using different forms of reasoning. Arguments that include verbal, numerical and graphical strategies can help students to develop their understanding of justification.

Table: Recording the combinations of monkeys in the trees

Monkeys in the big tree	0	1	2	3	4	5
Monkeys in the little tree	5	4	3	2	1	0

The number sentence problem above could be represented by w + y = 5. If we introduce a new context not restricted to whole number solutions, what are some of the other possible solutions? Can w and y have the same value? How could you represent all the solutions on a graph?

Communicating reasoning involves learning what counts as legitimate justification (Carpenter et al., 2003).

Pg 75

Chapter 6: Algebra – More than just patterns

Authors: Chris Linsell and Lynn Tozer

Algebra cannot be taught in isolation, because prerequisite skills and knowledge of number must be considered fi we are going to teach it with understanding.

What big kids do

At [New Zealand curriculum maths] Level 4, if we are expecting students to form and solve equations, what strategies do they actually use? … Basically, students solve equations by the use of known basic facts, counting techniques, guess and check, cover-up, working backwards or formal operations.

… inverse operations are involved in all the successful strategies (except guess and check) for equations involving two or more steps.

It is not surprising, then, that when students attempt to solve a two-step equation (e.g. 3n – 5 = 7), many are reduced to using guess and check or a learnt procedure that applies only to a specific structure. The majority of students who make sue of inverse operations to solve equations of this type successfully do so by working backwards. This strategy is illustrated by students perceiving the previous equations as stating that some unknown number, n, is multiplied by 3 and then has 5 subtracted to give the answer 7. The value of n is found by reversing the process with a series of arithmetic computations, 7 + 5 =12 and then 12 / 3 = 4.

Pg 78

Higher algebra requires students to perceive equations as objects and act formally on them, and so failure to do this limits students’progress in mathematics.

Not surprisingly, instant recall of basic facts, including multiplication and division, is crucial for using the more sophisticated strategies.

Students who can use formal operations to solve equations correctly understand the equals sign as meaning “is equivalent to”. Students unable to use formal operations usually understand the equals sign as meaning “calculate the answer now.”

pg 79

Building fundamental understanding at Levels 1 and 2

At Level 1… it is important to appreciate that an equation is “a formula affirming equivalence of two expressions connected by the sign = ” (Skyes, 1982, pg 326) … develops an understanding of the meaning of “equation” as both sides of the equals sign being in balance or having the same value.

The semantics of word choice associated with the equals sign deserves careful consideration. 2 + 3 = 5 is typically read as “two plus three equals five”…suggests a process or computation when compared with “two plus tree is equal to five”, which reinforces balance or equivalence.

The importance of fine semantic details, such as the transition from the use of “and” to “plus”, cannot be overemphasised.

Pg 81

The exploration of result unknown ( 2 + 3 – _ ), change unknown (2 + _ = 5), and start unknown ( _ + 3 = 5) is also important.

Pg 91

Chapter 7: Exploring Shapes

Author: Andy Begg

If two circles intersect, how many points are on both?

A typical response … is “two”, because the circumference is assumed to beh the circle; however, there are other possibilities, including

• “1” if the circles just touch
• “many, or an infinite number” (points in the overlapping areas)
• all” (if the two circles are the same size and overlap completely)
• all the points in the small circle (if the larger completely covers the smaller)

All five answers are reasonable – they result from creative thinking (finding alternatives) and logical thinking (justifying).

An important aspect of geometry is learning to visusalise, to imaging shapes in nonstandard positions (e.g. a triangle, square or cube “standing” on a corner) and to draw diagrams.

Pg 99

mathematical shapes and solids are fun to explore, and most people enjoy playing with different words to describe them. Sometimes in teaching mathematics we formalise too early instead of having fun.

Pg 101

Chapter 8: Teaching practical measurement

Author: Michael Drake

Learning to measure is not a simple process, and is something that children need to focus on throughout their years of primary schooling. One reason for this is the wide range of attributes we can measure, such as length, temperature, time, capacity, mass, colour, area, volume, speed – the list seems endless. A second reason is that learning to measure does not involve mastering a single procedure, but a collection of complex processes.

Pg 103

Early experiences

Early measurement experiences for children don’t require the use of number because they tend to be language-based.

However, even at this stage there needs to be an emphasis on developing an understanding of ways to compare.

• the same length
• high (teitei), deep (hōhonu), wide (whānui)
• hot (wera), warm (mahana), cool (whakamātao), cold (makariri), freezing (pātiotio)
• long (roa), longer (roroa), longest (roa rawa)

Measuring by counting

…while children are developing their confidence with the counting process we also challenge their understanding by teaching them to use counting for measuring things, a mathematical situation in which a number of their hard-learnt understandings do not in fact apply.

Pg 104

Counting confusions

… in measuring, the critical concept to develop is that of the unit, and that all units need to be identical if the count is to be meaningful.

Pg 106

Developing a concept of scale

The third process children need to develop an understanding of involves the concept of a scale, and its use in measurement. However, using classroom rulers too early can lead to children misunderstanding what they show.

Common misunderstandings among the younger and less able children related to the role of zero, whether to align the start of the measurements at the end of the rule (or at zero or one), and whether to count the marks or the spaces.

pg 107

Much research (e.g. Bragg & Outhred, 200a, 200b) identifies that children need to be carefully scaffolded into the use of measurement scales. … when introducing a scale it is important to introduce measurement conventions and move away from comparison or counting conventions. Children need to understand that using a measurement scale involves a different process with different rules.

Pg 113

Chapter 9: A place to stand: Investigating space through mapping the environment

Author: Ruth Pritchard and Chanda Pinsent

There is a world of difference between the large-scale physically experienced reality of a three-dimensional (3-D) space and a reduced two-dimensional (2-D) representation of that space produced in a drawing or a map. … they require not only the decoding of implicit social and cultural information, but also the interpretation of visual conventions and an understanding of aspects of proportion, measurement and scale.

Pg 119

Activity 9.3: Practice

Provide opportunities for students to practise using, generating and finding co-ordinate pairs in different contexts:

• walking on a large map with established gridlines
• using a large 3-D space with imagined gridlines
• using 2-D representations of Cartesian planes such as can be found in textbook exercises
• locating landmarks on maps from their local area playing games involving co-ordinates (e.g., Battleships).

Chapter 10: Statistical literacy: Learning to read between the lines

…Although collecting and presenting data is an important aspect of statistics, developing statistical literacy is a more enduring lifelong skill.

pg 128

Statistical literacy as a life skill becomes increasingly important as more and more data needing critical examination become available. The ability to make sense of data that affect them enables people to draw sensible conclusions upon which to make informed decisions (Books, Bond, Sparrow, & Swan, 2004; Walman, 1993).

Pg 129

The initial stages

Before children can start making comments about data collected by others, they need to experience the process of collecting an analysing their own data. These experiences need to be based in context that are meaningful and relevant for children.

pg 131

As statistical literacy develops, children become aware that the statements of others can be accepted or refuted by checking data displays.

Pg 137

Supporting the children in their journey

…Topices where tdata are concrete and indisputable are more likely to provide an opportunity for the teacher to channel the children’s focus into using the data rather than relying on personal opinion or experience when making statements.

Possible sources of statistical data relevant to children include:

• school and community newspapers
• current events (including sports results, elections)
• census at school
• the Figure it Out series

The ability to identify individual contributions in the data is important, especially for young children. … Using parallel data (perhaps collected by other classes in the syndicate) maintains familiarity but allows children to be scaffolded into using the evidence presented in the data to justify conclusions…

pg 138

As children develop skills in reading data they need to understand that data can be combined. If two or more sets of parallel data are presented, students need to realise they can combine these to present one picture. This provides an opportunity to introduce or reinforce the use of a numerical scale. These numerical aspects becomes more important as children begin to deal with issues relating to validity, appropriate presentation and sample size.

Activity 10.3 Combining data

Provide parallel data for children to consider. Key questions when combining data/introducing a numerical scale are:

• Using both sets of data, what is the most popular?
• how many …. in example one? How many in example two? How many altogether?
• How many more …. than….?

The issue of fairness in the process of collecting data can be introduced to young children and developed throughout the journey. With appropriate scaffolding issues such as sampling and validity can also be developed.

Activity 10.4: Fairness

Have students recap the data-gathering methods used. Key questions when discussing validity and sampling are:

• How would you know if the data are correct? What information do you need?
• If repeating this survey in a big school, would you need to ask every single person? How could you still make sure you have the views of all age groups?

The ability to critically “read” data presentations is a gradual process that continues throughout children’s schooling. By focusing on analysing the data and questing the validity of the process, children will learn to use not only what is present but also what is missing in drawing conclusions.

Chapter 11: Teaching statistics through investigations

Author: Tim Burgess

…what is statistical thinking?…

Pg 142

A description of statistical thinking has been given by Wild and Pfannkuch (1999), based on what statisticians do when they are problem solving. There are a number of components of statistical thinking, all these all interact and connect when people engage in statistical thinking. The components are as follows:

pg 143

• Recognising the need for data, and knowing that the more data you have the better position you are in, is important
• “Transnumeration” of the data – this means using different representations for the data, so the some of the “stories” in the data might be revealed. These representations include tables, graphs, lists or calculations as a summary of the data. Being prepared to try different representations is important, as some might reveal some interesting patterns in the data that are not likely to be revealed in any other way.
• Recognition of variation in data – understanding that if you get some data from two groups of people there will be both similarities between the groups as well as some differences is a key idea in statistics.
• Being able to reason using statistical models
• Being able to integrate your knowledge of the real-world context related to the data with your statistical knowledge

Pg 144

…students also need to engage in thinking about the Statistical Enquiry Cycle.. to engage in the interrogative cycle (Wild & Pfannkuch, 1999). This means that students need to interact with the data, think about questions that might be answered by the data, interpret what they see and be critical and questioning about what they notice and find out.

For students to develop statistical thinking when problem solving, their teachers need to be able to think statistically and consider how they can encourage their students to do likewise.

Teaching using investigations

Being able to think about data in relation to the Statistical Enquiry Cycle (or Investigative Cycle) … this cycle involves:

• posing a problem
• planning for what data should be collected in order to solve the problem, and how that data should be collected
• collection of the data
• analysing the data by sorting, using a variety of approaches or models, identifying patterns and interpreting the data in order to draw a conclusion or solve the problem
• communicating the findings

pg 145

There are two main approaches to teaching using investigations: the first involves giving the students dat and getting them to pose questions/problems the data might reveal, and continuing with the enquiry cycle from there; the second approach starts right at the beginning of the cycle with the posting of a question or a hypothesis or conjecture, which is to be verified or disproved.

Pg 145

Teacher knowledge needed when using investigations in the classroom

To be an effective teacher of statistics… teacher knowledge required… Burgess (2009)

• specialised knowledge of content – thinking about the questions from a statistical point of view, and consider whether such a a data collection question is feasible.
• knowledge of content and teaching – thinking about what response to give
• knowledge of content and students – to anticipate what they might struggle with from a statistical point of view