madmaths2190.weebly.commadmaths2190.weebly.com/.../final_assignment.docx · Web viewLesson 6. Length: 1 hour ... A second challenge of mathematical English is understanding words

c3110764 Prue Tinsey

Topic

Statistics and Probability

Data Collection and Representation

Time Allocation: 3 weeks (9hours)

9 x 1hr lessons (see attached teaching plan)

Syllabus OutcomesA student:• communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols• recognises and explains mathematical relationships using reasoning• collects, represents and interprets single sets of data, using appropriate statistical displays

Key Words: axis, axes, horizontal, vertical, interpretation, represent, conversion, sector, scale, key column, picture, statistics, distribution, range, score, mean, mode, median, tendency, tabulate, histogram, data, polygon, tally, analyse, event, sample, survey, census, sigma, random, data, chance, tally, discrete, frequency, location, interval, outlier, cluster.

Knowledge and Skills

Students:

Investigate techniques for collecting data, including census, sampling and observation (ACMSP284)• define ‘variable’ in the context of statistics as something measurable or observable that is expectedto change either over time or between individual observations [L]• recognise variables as numerical (either discrete or continuous) or categorical [L]

- identify examples of categorical variables (eg colour, gender), discrete numerical variables(eg number of students, shoe size) and continuous numerical variables (eg height, weight)(Communicating) [N, CCT]

- recognise that data collected on a rating scale (Likert-type scale) is categorical, eg 1 = dislike, 2 = neutral, 3 = like (Communicating)

• investigate and determine the differences between collecting data by observation, census andsampling [L, N, CCT]

- identify examples of variables for which data could be collected by observation, eg directiontravelled by vehicles arriving at an intersection, native animals in a local area

- identify examples of variables for which data could be collected by a census and a sample,eg a census to collect data about the income of Australians; a sample for TV ratings(Communicating) [N, CCT]

- discuss the practicalities of collecting data through a census compared to a sample, includinglimitations due to population size, eg in countries such as China and India, a census is conducted only once per decade (Communicating, Reasoning) [CCT, A]

Explore the practicalities and implications of obtaining data through sampling using a variety ofinvestigative processes (ACMSP206)• collect data using a random process, eg numbers from a page in a phone book, or from a randomnumber generator [N]• identify issues for which it may be difficult to obtain representative data either from primary orsecondary sources [CCT, EU]

1Kathryn Holmes EDUC2190


- discuss constraints that may limit the collection of data or result in unreliable data, eg proximity to the location where data could be collected, access to ICT or cultural sensitivities that may influence the results (Communicating, Reasoning) [CCT, EU, IU]

• investigate and question the selection of data used to support a particular viewpoint, eg theselective use of data in product advertising [CCT, EU]

Identify and investigate issues involving numerical data collected from primary and secondary sources (ACMSP169)• identify the difference between data collected from primary and secondary sources, eg datacollected in the classroom compared with data drawn from a media source [L]• explore issues involved in constructing and conducting surveys such as sample size, bias, type ofdata required and ethics [L, N, EU]

- discuss the effect of different sample sizes (Communicating, Reasoning) [N, CCT]- describe, in practical terms, how a random sample may be selected to collect data about anissue of interest (Communicating, Problem Solving) [CCT]- detect and discuss bias, if any, in the selection of a sample (Communicating, Reasoning) [N,CCT]

• construct appropriate survey questions and a related recording sheet to collect both numerical andcategorical data about an issue of interest [L, CCT, PSC]

- construct a recording sheet that allows efficient collection of the different types of dataexpected (Communicating, Problem Solving) [L, CCT, PSC]

- refine questions in a survey after a trial (Communicating) [L, CCT, PSC]- decide whether a census or sample is more appropriate to collect the data required to

investigate the issue (Problem Solving) [L, CCT, PSC]• collect and interpret information presented as tables and/or graphs about an issue of interest fromsecondary sources, eg the relationship between wealth or education and the health of populationsfrom different countries; sporting data [L, N, IU, EU]

- interpret and use scales on graphs including those where abbreviated measurements are used,eg ‘50’ on a vertical axis measured in thousands is interpreted as ‘50 000’ (Reasoning) [N]- analyse a variety of data displays used in the print or digital media and in other school subject

areas, eg share movement graphs, sustainable food production (Problem Solving) [N, ICT,CCT, SE]

- identify features on graphical displays that may lead to misleading interpretation, eg displacedzeros, the absence of labelling on one or more axes or misleading units of measurement(Communicating, Reasoning) [L, N, CCT]

• use spreadsheets or statistical packages to tabulate and graph data [ICT]- discuss ethical issues that may arise from collecting and representing data (Reasoning) [CCT,

EU]

Construct and compare a range of data displays including stem-and-leaf plots and dot plots(ACMSP170)• use a tally to organise data into a frequency distribution table [N]• draw and interpret frequency histograms and polygons [N]

- select and use appropriate labels and scales on the horizontal and vertical axes (Communicating, Problem Solving, Reasoning) [N, CCT]

• construct and interpret dot plots and stem-and-leaf plots, including stem-and-leaf plots with two digit stems [N]

- explain the importance of aligning data points/values when constructing dot plots and stem and-leaf plots (Communicating, Reasoning) [CCT]

• interpret a variety of graphs, including divided bar graphs, sector graphs and line graphs [L, N]



- calculate the percentage of the whole represented by different categories in a divided bar graph or sector graph (Problem Solving) [N, CCT]

- compare the strengths and weaknesses of different forms of data display (Reasoning)- identify and explain which graph types are suitable for the type of data being considered,

eg sector graphs and divided bar graphs are suitable for categorical data, but not for numericaldata (Communicating, Reasoning) [N, CCT]

- draw conclusions from data displayed in a graph, eg ‘The graph shows that the heights of allchildren in the class are between 140 cm and 175 cm and that most are in the group 151−155 cm.’ (Communicating, Reasoning) [N, CCT]

• construct divided bar graphs, sector graphs and line graphs with and without ICT [L, N]- calculate the angle at centre/length of bar required for each sector/section of sector

graphs/divided bar graphs (Problem Solving) [N, CCT]

Background Information

Students studying in Stage 4 can be expected to have some prior knowledge of both dot plots and linegraphs as these types of graphs are first introduced in Stage 3. Students construct, describe andinterpret column graphs in Stages 2 and 3; however, Stage 4 is the first Stage in which histograms,divided bar graphs and sector (pie) graphs are encountered.

Statistical data is part of everyday life. Data may be displayed in tables and/or graphs, and may appearin all types of media. Graphs provide a visual overview of the topic under investigation. Studentsshould be aware that while many graphs are accurate and informative, some can be misleading.

Students need to experience interpreting a wide variety of graphical representations, including columngraphs, line graphs, dot plots, stem-and-leaf plots, divided bar graphs and sector

Language

In everyday language the term ‘pie chart’ is often used in reference to sector graphs.



Teaching PlanLesson 1

Length: 1 hour

Outcomes:

Investigate techniques for collecting data, including census, sampling and observation (ACMSP284)

• define ‘variable’ in the context of statistics as something measurable or observable that is expected

to change either over time or between individual observations [L]

• recognise variables as numerical (either discrete or continuous) or categorical [L]

- identify examples of categorical variables (eg colour, gender), discrete numerical variables

(eg number of students, shoe size) and continuous numerical variables (eg height, weight)

(Communicating) [N, CCT]

- recognise that data collected on a rating scale (Likert-type scale) is categorical, eg 1 = dislike, 2 =

neutral, 3 = like (Communicating)

Language Issues/ Pedagogical Strategies:

In this lesson, students will be introduced to a lot of new terminology, such as, census,

sampling, variable, measurable, observable, discrete, continuous, categorical and numerical.

If teachers do not define this mathematical jargon during this lesson then students are going

to struggle throughout the rest of this topic, because these words reoccur often and form the

basis for this topic. It is important that teachers define these terms early on in the lesson.

They should then get students to write up a definitions page in their work books and

encourage students to write the definition in their own words, so as to make sure that they do

fully understand. Teachers should also make an effort to embed this terminology throughout

their mathematics lessons, so that students can recognise and witness it used in context.

Lesson 2



Length: 1 hour

Outcomes:

• investigate and determine the differences between collecting data by observation, census and

sampling [L, N, CCT]

- identify examples of variables for which data could be collected by observation, eg direction

travelled by vehicles arriving at an intersection, native animals in a local area

- identify examples of variables for which data could be collected by a census and a sample,

eg a census to collect data about the income of Australians; a sample for TV ratings

(Communicating) [N, CCT]

- discuss the practicalities of collecting data through a census compared to a sample, including

limitations due to population size, eg in countries such as China and India, a census is conducted

only once per decade (Communicating, Reasoning) [CCT, A]


This lesson will consist of similar mathematical language as the previous lesson. Therefore it

is important that teachers continue to embed this jargon into their lessons, so that the students

become used to hearing it in context. The teacher should also encourage discussion amongst

the students, so that they have a chance to use the terminology, which will help them to

understand if they are having any difficulties with the language.



Lesson 3

Length: 1 hour

Outcomes:

Explore the practicalities and implications of obtaining data through sampling using a variety of

investigative processes (ACMSP206)

• collect data using a random process, eg numbers from a page in a phone book, or from a random

number generator [N]

• identify issues for which it may be difficult to obtain representative data either from primary or

secondary sources [CCT, EU]

- discuss constraints that may limit the collection of data or result in unreliable data, eg proximity to

the location where data could be collected, access to ICT or cultural sensitivities that may

influence the results (Communicating, Reasoning) [CCT, EU, IU]

• investigate and question the selection of data used to support a particular viewpoint, eg the

selective use of data in product advertising [CCT, EU]


This lesson will also consist of similar mathematical language as the previous lesson.

Therefore it is important that teachers continue to embed this jargon into their lessons, so that

the students become used to hearing it in context. The teacher should also encourage

discussion amongst the students, so that they have a chance to use the terminology, which

will help them to understand if they are having any difficulties with the language.



Lesson 4

Length: 1 hour

Outcomes:

Identify and investigate issues involving numerical data collected from primary and secondary sources

(ACMSP169)

• identify the difference between data collected from primary and secondary sources, eg data

collected in the classroom compared with data drawn from a media source [L]

• explore issues involved in constructing and conducting surveys such as sample size, bias, type of

data required and ethics [L, N, EU]

- discuss the effect of different sample sizes (Communicating, Reasoning) [N, CCT]

- describe, in practical terms, how a random sample may be selected to collect data about an

issue of interest (Communicating, Problem Solving) [CCT]

- detect and discuss bias, if any, in the selection of a sample (Communicating, Reasoning) [N,

CCT]


Teachers in this lesson should initially highlight the difference between a primary and

secondary source, and give many examples of each so that students are able to confidently

recognise the difference. These terms should also be added to the students list of definitions

for the topic, so that they can look up the meaning at any time. Students should then be given

time to investigate and identify the difference between data collected from a primary and

secondary source, to better help with their understanding of these words and their concepts.

Teachers should then focus on terms such as sample size, bias, random sample. A strategy

that teachers could use to teach this language is to ask the students what they think these

words mean, and create a mind-map from these ideas. These words are such that students are

likely to have heard them before in everyday language so may have an idea of their meaning,

and this strategy encourages them to use higher-order thinking to come up with the terms

mathematical meanings.



Lesson 5

Length: 1 hour

Outcomes:

• construct appropriate survey questions and a related recording sheet to collect both numerical and

categorical data about an issue of interest [L, CCT, PSC]

- construct a recording sheet that allows efficient collection of the different types of data

expected (Communicating, Problem Solving) [L, CCT, PSC]

- refine questions in a survey after a trial (Communicating) [L, CCT, PSC]

- decide whether a census or sample is more appropriate to collect the data required to

investigate the issue (Problem Solving) [L, CCT, PSC]


This lesson will consist of similar mathematical language as in previous lessons. Therefore it


continue to hear it being used in context. The teacher should also encourage discussion

amongst the students, so that they have a chance to use the terminology, which will help them

to understand if they are having any difficulties with the language.

Lesson 6



Length: 1 hour

Outcomes:

• collect and interpret information presented as tables and/or graphs about an issue of interest from

secondary sources, eg the relationship between wealth or education and the health of populations

from different countries; sporting data [L, N, IU, EU]

- interpret and use scales on graphs including those where abbreviated measurements are used,

eg ‘50’ on a vertical axis measured in thousands is interpreted as ‘50 000’ (Reasoning) [N]

- analyse a variety of data displays used in the print or digital media and in other school subject

areas, eg share movement graphs, sustainable food production (Problem Solving) [N, ICT,

CCT, SE]

- identify features on graphical displays that may lead to misleading interpretation, eg displaced

zeros, the absence of labelling on one or more axes or misleading units of measurement

(Communicating, Reasoning) [L, N, CCT]

• use spreadsheets or statistical packages to tabulate and graph data [ICT]

- discuss ethical issues that may arise from collecting and representing data (Reasoning) [CCT,

EU]


This lesson will consist of similar mathematical language as in previous lessons. Therefore it


continue to hear it being used in context. The teacher should also encourage discussion

amongst the students, so that they have a chance to use the terminology, which will help them

to understand if they are having any difficulties with the language.

It may be important to redefine words such as ‘relationship’ and highlight the

meaning of this word when it comes to comparing and analysing data, as students are

unlikely to have seen this word in this context.

Lesson 7



Length: 1 hour

Outcomes:

Construct and compare a range of data displays including stem-and-leaf plots and dot plots (ACMSP170)

• use a tally to organise data into a frequency distribution table [N]

• draw and interpret frequency histograms and polygons [N]

- select and use appropriate labels and scales on the horizontal and vertical axes (Communicating,

Problem Solving, Reasoning) [N, CCT]

• construct and interpret dot plots and stem-and-leaf plots, including stem-and-leaf plots with two digit

stems [N]

- explain the importance of aligning data points/values when constructing dot plots and stem and-

leaf plots (Communicating, Reasoning) [CCT]


This lesson once again introduces a lot of new terminology such as range, stem-and-leaf

plots, dot plots, frequency distribution table, frequency histogram, frequency polygon. The

language in this lesson refers a lot more too mathematical processes such as placing raw data

into a frequency distribution table for example. Therefore, it is crucial that students are aware

of the meaning behind the language, so that they can grasp a strong understanding of this part

of the topic. I think it is important to show them one step at a time, so for example show them

what a stem-and-leaf plot looks like and how to represent raw data in this format, and only

once they are confident doing this, move onto the next graph such as dot plots. Highlight to

students memory trigger strategies such as the fact that stem-and-leaf plots look like a plant

with a main stem and leafs, so that it helps them to visualise and easily recall what a stem and

leaf plot looks like and how it works.

Lesson 8



Length: 1 hour

Outcomes:

• interpret a variety of graphs, including divided bar graphs, sector graphs and line graphs [L, N]

- calculate the percentage of the whole represented by different categories in a divided bar graph or

sector graph (Problem Solving) [N, CCT]

- compare the strengths and weaknesses of different forms of data display (Reasoning)

- identify and explain which graph types are suitable for the type of data being considered,

eg sector graphs and divided bar graphs are suitable for categorical data, but not for numerical

data (Communicating, Reasoning) [N, CCT]

- draw conclusions from data displayed in a graph, eg ‘The graph shows that the heights of all

children in the class are between 140 cm and 175 cm and that most are in the group 151−155 cm.’

(Communicating, Reasoning) [N, CCT]


This lesson once again introduces a lot of new terminology such as bar graphs, sector graphs

and line graphs. These new terms define mathematical graphs and data displays such as

placing raw data into a line graph and analysing the results in this format. Therefore, it is

crucial that students are aware of the meaning behind the language, so that they can grasp a

strong understanding of this part of the topic. An important language issues to address in this

lesson is that a sector graph is merely a synonym for a pie graph. Students are unlikely to

have heard of a sector graph before, however, may have heard it being called a pie graph in

everyday language. It is important that they can make this connection as it can also be used as

a memory trigger strategy. For example, highlighting to students that a sector graph looks like

a pie, and hence why they have heard it be called a pie graph, may help them to learn this

information and hence recall it in a test.

Lesson 9



Length: 1 hour

Outcomes:

• construct divided bar graphs, sector graphs and line graphs with and without ICT [L, N]

- calculate the angle at centre/length of bar required for each sector/section of sector

graphs/divided bar graphs (Problem Solving) [N, CCT]


This lesson will consist of similar mathematical language as in the previous lesson. Therefore

it is important that teachers continue to embed this jargon into their lessons, so that the

students continue to hear it being used in context. The teacher should also encourage

discussion amongst the students, so that they have a chance to use the terminology, which

will help them to understand if they are having any difficulties with the language.

If time permits a topic test may be conducted in this lesson or in a tenth lesson for this

topic. Not only should this topic test, examine the skills of this topic but also the language. It

might be beneficial to see where students are having difficulties by asking them to define

common words and give examples. It will be evident whether students have a good grasp of

the language if they do well in a topic test.



What are the language difficulties that are likely to arise when teaching this topic?

What strategies could be employed to overcome these difficulties? How can these

strategies be modified to accommodate diversity among learners?

Language is a pivotal part of mathematics. However, both the mathematical and the ordinary

English language can present significant challenges for students (Otten & Herbel-Eisenmann,

2009). Challenges arise for students when words are used differently inside the mathematics

classroom than they are outside (Otten & Herbel-Eisenmann, 2009). A second challenge of

mathematical English is understanding words which only have mathematical origins

(Rangecroft, 2002). Finally, students also have difficulties understanding mathematics and

interpreting a question if they are unfamiliar with the symbolism that is used to represent

different words (Booker, 2002). As mathematics teachers, we need to create methods in

which we can better teach students so that they become familiar and fluent with in

understanding the complexities of the mathematical language.

The question in appendix 1 is an example of a data and statistics question from a stage

four mathematics text book (Godwin & Powell, 2000). The question highlights a great

example of the importance of knowing and understanding mathematical English. Imagine if

you were a student looking at this question and you had no recall of the meaning of the

words, ‘frequency distribution table’, ‘mean’, ‘mode’, ‘range’ and ‘frequency histogram’,

then let’s face it you would be getting zero out of five for this question. As a teacher you

would mark this student’s work and think that they had a very poor understanding of data and

statistics. However, what if the child did know how to calculate averages, they just didn’t

know that the mean was a synonym for average. What if the student could recognise that 15

was the most occurring score, they just did not know that this was called finding the mode.

Language in mathematics is very important, and many students struggle in tests, simply

because they have a poor understanding of mathematical English.



Statistics and probability is a topic in the stage four syllabus where students are likely

to have difficulty with distinguishing the difference in meaning between words that they use

in their everyday English vocabulary to their altered mathematical meaning (Rangecroft,

2002). An example of this is the word average. In the English language this words has the

meaning typical, common or ordinary (Collins, 2009). However, in statistics we introduce it

as the result obtained by adding several quantities together and then dividing this total by the

number of quantities (Rich, 2004). Just to make matters more confusing for students, we also

give the average a second name, the mean. A second example is the term pie-graph which is

used in more everyday language; however the more mathematical term is a sector graph.

To avoid student confusion teachers should point out the distinctions or the shared

meanings explicitly by saying for example, another word that we use to describe the average

is the mean (Barnett-Clarke & Ramirez, 2004). Teachers should also make sure that students

have written definitions of these terms in their workbooks so that they can look-up the

meaning themselves at any time. Another useful strategy that teacher can encourage in their

classrooms is discussion (Rangecroft, 2002). Even if students think they understand

something in their heads, they often find it challenging to put it into their own words and

explain it to someone else (Lee, 2006). By encouraging students to talk about mathematical

concepts and use the correct mathematical jargon, they are more likely to learn and

understand the meanings of these words whilst also allowing them to grasp an understanding

of where some of their vocabulary difficulties may lie (Rangecroft, 2002) .The key is to try as

many different strategies as possible, to cater for the diversity of all students (Barnett-Clarke

& Ramirez, 2004).

The topic of statistics and probability is an example of a topic where learning and

understanding the mathematical language can be quite difficult. It introduces a lot of new

terminology that students are likely to be unfamiliar with, which can cause confusion and



gaps in students’ knowledge (Rangecroft, 2002). For example, words such as mode, median,

mean and range. There are many strategies to help students understand these words and recall

their meanings. For example, emphasising to students the ‘o’ in mode, can help them to

remember that the mOde is the mOst Occuring, score in the data set. A similar strategy can

be used when introducing the median, by emphasising the ‘d’ that is, the meDian is the

MiDdle score.

Words that are used in the mathematical English vocabulary that have a similar

meaning to their ordinary English origin, can work in the favour of both teachers and

students, for example, the word outlier (Rangecroft, 2002). In ordinary English an outlier is

defined as a person or thing situated away from the main body or system (Collins, 2009). In

statistics an outlier is an observation that lies abnormally distant from other values in the

random sample of a population (Rich, 2004). Teachers can use students’ knowledge of the

ordinary English definition to assist their understanding of the statistical meaning, because

they are very similar.

Not only does the topic of statistics and probability introduce new language it also

introduces diagrams and symbols (Rangecroft, 2002). This can cause students just as much

confusion as new language, when trying to decipher the meaning of symbols and notation.

For example, ‘f’ in mathematics usually means ‘function’, as it does sometimes in statistics

also, however it is more commonly known as the ‘frequency’ (Rangecroft, 2002). Another

example is the symbol x-bar, which is used to represent the sample mean. When introducing

symbols to students it is imperative that they first understand the language surrounding the

symbol (Lee, 2006). For instance, first explain to the students how to calculate the mean of a

sample population, and then introduce to them the terminology, and then only if the students

have understood this then introduce the x-bar notation. We don’t want to turn them off a topic

because of ‘scary’ symbols and words, before they have had a taste of what it is about.



Mathematical English can cause many difficulties for students if they are not taught

strategies to help them overcome the challenges of new terminology. No one strategy is the

correct strategy for all students and so it is important that teachers incorporate a wide variety

of techniques into their lessons to cater for the diversity of their students. Some students may

learn better from writing down a definitions list, some students may learn better verbally and

other may learn best through the creation of word trigger and word association strategies. For

teachers it comes down to knowing how your students best learn and employing strategies to

best cater for their learning needs.

Note: Writing this essay even highlighted to me how confusing the language can be.

For example, I wanted to use the word mean as a synonym for define, but because I am

defining the mathematical meaning of mean, I can’t write the mean means the average or the

average means the mean, because that is just confusing, and when I went and tried to find a

different synonym for mean it gave me words such as angry, nasty etcetera which wasn’t the

context of the word mean that I was after at any rate! Just though I would add this because it

is appropriate as it again highlights how challenging it can be to decipher between the

ordinary English and mathematical English language. So many homonyms!



References

Barnett-Clarke, C., & Ramirez, A. (2004). Language Pitfalls and Pathways to Mathematics.

In R. Rubenstein & G. Bright (Eds.), Perspectives on the Teaching of Mathematics:

sixty-sixth yearbook. Reston, VA: National Council of Teachers of Mathematics.

Booker, G. (2002). Valuing Language in Mathematics: Say what you mean and mean what

you say. Paper presented at the Annual Conference of the Mathematical Association

of Victoria, Melbourne.

Collins. (2009). English Dictionary. Sydney: Harper Collins Publishers Limited.

Godwin, B., & Powell, M. (2000). Spectrum Mathematics Gold Year 8. Oakleigh: Cambridge

University Press.

Lee, C. (2006). Starting to talk in the mathematics classroom Language for Learning

Mathematics: Assessment for Learning in Practice. Berkshire: Open University Press.

Otten, S., & Herbel-Eisenmann, B. (2009). Multiple meanings in mathematics: Beneath the

surface of area. Paper presented at the 31st annual meeting of the North American

Chapter of the International Group for the Psychology of Mathematics Education.,

Atlanta, GA.

Rangecroft, M. (2002). The Language of Statistics. Teaching Statistics,

24(2), 34-37.



Rich, G. (2004). Secondary Maths Dictionary. Sydney: Harper Collins

Publishers Limited.



Appendix

Appendix 1

Question from:

Godwin B, & Powell, M. (2000). Spectrum Mathematics Gold Year 8. Cambridge University

Press

These are the ages of Jade’s friends at her birthday party:

12 15 19 15 18 12 10 15 9 18 19 15 11

15 16 16 15 17 17 19 15 15 10 14

a) Put the data in a frequency distribution table.

b) Find the mean (answer to 1 decimal place).

c) Find the range.

d) Find the mode.

e) Put the data in a frequency histogram.


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madmaths2190.weebly.commadmaths2190.weebly.com/.../final_assignment.docx · Web viewLesson 6. Length: 1 hour ... A second challenge of mathematical English is understanding words