Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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PREPARATORY MATHEMATICS

1. Introduction

Maths anxiety can be a key inhibitor for some adult learners considering higher education. In 2004 Dr Paul

Robinson and I designed a Primer Mathematics module for mature students intending taking the FLASHE

(FLexible AccesS to Higher Education) Higher Certificate in Electronic Engineering at IT Tallaght. This 12 week

module was lab-based using the CALMAT software package, Glasgow Caledonian University, to provide the

students with a scaffolding to aid their reflection on their learning in the form of a weekly reflective diary. That

approach, and the positive impact for students were documented and can be accessed at [1] and [2]. Around this

time, through my involvement with Adults Learning Mathematics an International Research Forum, I investigated

more rigorously mathematics anxiety experienced by adult learners, and the strategies for dealing with it.. In

particular, I became aware of approaches outlined in Maths Study Skills (Bass, 2008) which I recognised as

potentially adaptable for the needs of adult learners in Ireland.

Therefore, when asked to write a 12-hour module as part of the Preparatory Certificate in Third Level at IT

Tallaght (a 10 credit, NFQ Level 6 Special Purpose Award for potential students returning to various courses), I

designed a mathematics intervention which would build on the key elements of the earlier success of the Primer

Mathematics initiative, and my increased knowledge of approaches to addressing mathematics anxiety.

The resulting intervention emphasises the building of student confidence and mathematics self-belief over twelve

contact hours rather than solely addressing any perceived student deficits in mathematical content.

Explicitly, the overall aims of the intervention are to provide students with:

– insight into their own mathematics self–image and an opportunity to build confidence in their

mathematical ability

– basic foundation for the basic type of mathematics that is needed for courses in higher

education in IT Tallaght.

Figure 1 gives an overview of the elements that have combined to ensure that the intervention has had a

significantly positive impact. Key to success of the module has been the emphasis on dealing explicitly with maths

anxiety in the first session, together with student reflection and weekly feedback loop, and making use of a short

statistics project, the context of which is specific to the individual student.

Figure 1: Overview of Preparatory Mathematics intervention

Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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ITT Dublin has offered the preparatory Course since 2009 with very positive impact as evidenced by the students’

evaluative feedback. More critically however, progression rates for adult learners are consistently higher for those

who took the module than those who did not, and compares very favourably with the overall student cohort.

2. Key aspects of Preparatory Mathematics module.

Several key aspects of the intervention are now described in more detail to enable other educators reproduce this

approach

2.1 Reflection and team teaching

A key element of the intervention is the creation of the dynamic feedback environment that contributes

significantly to the impact on individuals. This is achieved in the first instance by employing a team-teaching

strategy in the sessions. During the sessions, one of the 2 lecturers leads the learning whilst the other lecturer

monitors the students learning continuously and intervenes with individual learners as needed. The second

lecturer also plays a key role in acting on feedback immediately as described below.

Students are expected to engage in a process of reflection which is part of a weekly feedback loop:

(i) Students are asked to complete a reflection sheet ( Appendix 1 and Link 1 ) after each session which uses two prompts:

My reflections……… e.g., key points learned, high point of the class, low point of the class, my mood, my learning, ……….)

What have I learned from these reflections? For example one thought to carry forward, aspects of learning of topic which are still of concern, a question to ask for the next class and so on.

(ii) At the start of the next session, these reflection sheets are collected by the lecturers. Then one of the 2

lecturers reads and summarises the reflections immediately so that any issues that are raised are dealt

with at some stage during that session. In this way students experience an immediate constructive

reaction to their feedback and so are part of a mathematics learning environment that is different to a

‘chalk/talk/get left behind’ environment that they may have experienced in their past.

2.2 Building mathematical confidence session

The first session of the module which deals with mathematics anxiety and confidence building has been reported

by students as a key element of the module. For the benefit of those who wish to replicate this approach:

(i) the presentation used for the session has been included in Appendix 2 and at Link 2,

(ii) a short video expanding on how the session is conducted is available at Link 3,

(iii) a guide on how to structure of the session is presented in Table 1.

Table 1. Preparatory Course: Mathematics Session structure

Section heading

Slide number

Description Time (mins)

Introductions 1 Introduction of the team of 2 lecturers to students. Distribution of calculators, course notes and reflection sheets.

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2 Ice breaker video such as “ Abbot and Costello” maths joke http://www.schooltube.com/video/130f49947f551d61e5a8/. (The point of this is that the students first interaction with maths on the module results in laughter, hopefully relaxing the students and reducing anxiety). Brief discussion of the subtlety of place number system

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Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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Section heading

Slide number

Description Time (mins)

3 Buzz session 1: Students are asked to introduce themselves to the people next to each other in the room.

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3, 4,

5, 6

Buzz session 1, activity. This activity is based on the puzzle from Transum.org, http://www.transum.org/Software/SW/Starter_of_the_day/starter_March29.ASP In which you are shown a number and asked to select from a set of 4 actions: Stand up if it odd; fold your arms if it is a multiple of 4; hum if it is a square; put your hands up if it is prime. This is used in the session as follows as follows: Students are asked to think about what they see on the next slide for one minute on their own. They are then to consult with the students next to them for one minute. The whole group are then (after a countdown by the lecturer ) asked to do what they think the instructions tell them to do. (The point of this is to again reduce anxiety, to build a group dynamic and also to make the point that in higher education a prime resource for learning is the knowledge of other students). A brief discussion to clarify terminology such as prime etc. takes place with the emphasis being on encouraging students to start a dictionary of mathematical terms which they will be assembling as they learn the language of mathematics. The puzzle game is then repeated in the same way as before but with 30 seconds for own reflection and group discussion ahead of giving an answer as a whole class group.

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Maths Study Skills

7 The next section uses ideas based on Math Study Skills by Alan Bass[3] to help students investigate the concept of maths anxiety and to identify strategies they can employ to reduce this and to build confidence in their mathematical ability, slides 8, 9 & 10.

8,9,10 Buzz session 2 activity: Students are formed into clusters of three or four s and are asked to appoint one person to record the result of their deliberations and a second person to report these back to the whole group. The students are told they are going to be posed a question and they have 2 minutes to consider their own response in silence and then will have a further 5 minutes to formulate a list of three to five responses from the group.

The class is then given the query: Why is maths so tough? After the 8 minutes allotted, each group calls out their responses and the lecturers compile a cumulative list on the whiteboard. (There is normally about 30 separate items on the list. Among typical concerns are having to remember formulae, getting left behind, being consumed by ‘maths with letters’, maths not being useful in the real world). The lecturers then remove the concerns by referencing how the module is going to run and by emphasising the importance that will be placed on responding to students feedback dynamically, that essentially we will be starting from the very beginning and that via the statistics project the connection between mathematics and their own lives and interests will be shown.(Normally the list can be reduced to one comment which is usually about learning requiring effort (which it does!)).

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11-16 Discussion on how to address maths anxiety successfully and how to maximise the impact of the time and effort given to studying mathematics.

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17-18 Highlight free learning resources available on the internet (maths support centre, Khan Academy) and on the college V.L.E. (course specific screen casts).

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Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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Section heading

Slide number

Description Time (mins)

Statistics 19 Introduction to measures of central tendency and deviation using a human bar chart and median activity as follows: Students asked to remember their shoe size. Students are then asked to arrange themselves in order of shoe size around the edge of the lecture theatre. Median shoe size is found by getting students to count up the arranged list , remember their position on this ‘count up’ list then count back down the list remembering their position on this ‘count down’ list. Where the crossover in positions on these two lists occurs is where the median is located. Students then form a human bar chart by standing in lecture theatre rows in their shoe sizes and hence pick out the mode as the shoe size that occurs most often. Finally the concept of deviation (range initially), is introduced by way of the difference between the lowest and highest shoe size. (This exercise is always enjoyed by students) Students are then told that they are to start considering an aspect of the lives or interest using numbers from which they will be asked to do a project to find such measures of central tendency and deviation as part of the module.

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Calculators The students are introduced to using their calculators (provided as part of the course), with emphasis on following the order of precedence for performing calculations.

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Reflection 20 Students are given their first reflection sheet, told how to use it. The importance of filling in the reflection sheet before the next session is stressed.

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2.3 Preparatory Mathematics: module content and structure

Aims:

A. The first aim of this enabling mathematics module is to bring prospective students back into contact with mathematics in a way which will boost their confidence in their mathematical ability. In particular the module aims to provide the students with insight into their own mathematical self –image and an opportunity to build confidence in their mathematical ability.

B. The secondary aim of the module is to provide a basic foundation for the type and level of mathematics that is needed for courses in higher education in ITT Dublin.

Learning outcomes:

At the end of the module the student will be able to:

1. gather, organise, present data, and calculate summary statistics using appropriate software 2. apply number sense appropriately for numerical problems of the type seen in applied courses

in higher education making use of a calculator. 3. calculate numerical expressions accurately involving precedence, taking account of scientific

notation 4. parse and manipulate algebraic expressions 5. solve linear equations in context, and 6. identify and build their mathematical strengths to develop the mathematical confidence

necessary to undertake the mathematics modules in higher education in ITT Dublin.

Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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Contents:

The material in the course has been carefully chosen in 4 basic areas: number, data manipulation, algebra

and linear laws. This is to ensure that students start with concrete concepts in number manipulation as

well as data presentation and interpretation. In addition, students are introduced to a simple level of

symbol-manipulation in a brief introduction to algebra that will help them to see its easy application to

their later studies regarding linear law manipulation.

Assessment:

Summative, one- 1 hour open book test 50%

Assignment on data collection and manipulation 50%.

Indicative Time allocation: The 12 hours for the module are allocated over 6 weeks, see Table 2.

Table 2. Session planning

Week 1 2 3 4 5 6

First hour Building confidence

Statistics Statistics Arithmetic for college

Arithmetic for college

Algebra introduction

Second hour Statistics Excel lab on Statistics

Excel lab on Statistics

Excel lab on Statistics

Arithmetic for college

Linear laws introduction

It is important to note that learning outcomes 1, 2, 3, and 6 are prioritised such that, depending on how the

student group is succeeding with the content, the allocation of hours to content may be adjusted accordingly.

There being little benefit in rushing to complete the content, emphasis is placed instead on students achieving

success and building confidence as a consequence.

Module content materials: The workbooks and notes used by the students are available at the following links: Link 4, Link 5, Link 6 , Link 7

Assessment details:

Both aspects of the assessment technique used in the module (project and formal tests) are typical of IT Tallaght

mathematics modules, so it is important that the Preparatory maths students are assessed in the same fashion.

The design of the statistics project is of the ‘shell’ type suggested by my colleague James Reilly in IT Tallaght, Appendix 3 In this approach, the students are given a data gathering task of selecting a sample of 50 measurements from any population of their own choice, e.g. sports data, technical data, financial data, sales data, population/census data. The students are told that the data can be obtained by conducting their own survey or from the web, a book, magazine etc. This approach has been particularly successful for the students on the Preparatory Mathematics module as they make connections between the own lives/ interests and mathematics, and see the potential relevance to learning and applying more mathematical ideas in their future.

REFERENCES.

[1] P. Robinson, C O’Sullivan, P. Coman, 2006 “Teaching Semester 1 Mathematics in a Flexible-Delivery-Mode

Higher Certificate in Electronic Engineering”, Proceedings of the IMA Conference, Mathematical Education of

Engineers, 11-12 April 2006, Loughborough University, Editors S Hibberd & L Mustoe, December 2006, ISBN 978 0

905091 18 3 Institute of Mathematics and its Applications.

[2] P. Robinson, C O’Sullivan, P Coman., 2007, “Engaging adult learners with mathematics in a flexible-delivery-

mode higher certificate in electronic engineering.” Proceedings of 14th

International Conference of Adult Learning

Mathematics (ALM14):The Changing face of Adults Mathematics Education: Learning from the Past, Planning for

the Future.

[3] Math Study Skills, Alan Bass, Addison-Wesley (E) ISBN 13: 9780321513076 ISBN10: 0-321-51307-X

Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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Appendix 1:

Preparatory Mathematics 2015: Three minute reflection

Student Name .

Week Number: .

Date: .

Topic/issue being explored

My reflections……….

(key points learned, high point of the class, low point of the class, my mood, my learning,

……….)

What have I learned from these reflections? (One thought to carry forward, aspects of

learning of topic which are still of concern, a question to ask for the next class …….)

Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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Appendix 2: Introductory Session slides:

Buzz session 1

• Introduce yourself to the people around

you.

• Now be ready for the next task………

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Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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Maths Study Skills

Math Study Skills

Author: Alan Bass

Pyblkisher: Addison-Wesley (E)

ISBN13: 9780321513076

ISBN10: 0-321-51307-X

Contents:

What Makes Math Different

Learning Styles

Math Anxiety

Managing Your Time

Your Class Notebook ,

Your Textbook and Homework

Class Time and Note Taking

Retention and General Study Strategies

Test Taking

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Buzz session 2

• Form into teams of 3 or 4 people.

• Now be ready for the next task………

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Chapter 2:

What Makes Math Different• Why is maths so tough?

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The list of issues

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Maths Study Skills

Math Study Skills

Author: Alan Bass

Publisher: Addison-Wesley (E)

ISBN13: 9780321513076

ISBN10: 0-321-51307-X

Contents:

What Makes Math Different

Learning Styles

Math Anxiety

Managing Your Time

Your Class Notebook ,

Your Textbook and Homework

Class Time and Note Taking

Retention and General Study Strategies

Test Taking

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Discussion based on Ideas from “Maths Study Skills” by Alan Bass

• Attitudes towards Maths.

• MATH ANXIETY

– Symptoms of Maths anxiety

– Why Maths anxiety

– THE CURE

Do math every day.

Study smart.

Do NOT skip class

Be organized.

Practice quizzing yourself.

Replace negative self-talk with positive self-talk.

Use your resources.

– THE PHYSIOLOGY OF ATTITUDE

• I am responsible for the grade I make in this course.

• I will be patient and persistent in pursuing this goal.

• I am smart enough to pass math.

• I will pay the price for success.

• Using Time.

“ However difficult your

struggles in maths might

be, I can assure you that

mine are greater.”

Albert Einstein.

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Class time - Before

• Reviewing the last class– Reviewing your notes from the last class.

– Surveying the section that will be covered that day

– Reviewing the material from the last section to prepare

questions.

– Doing one or two problems from a previous section that you

know you can do.

• Punctuality Be on time!

• Get a good seat

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ClassTime:Class time is a workout; no pain, no gain

arrive early

get a good seat,

listen actively

– focus!!!!!!!take good notes,

ask questions.

do not wait get

stuck in!Class

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Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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ClassTime: During Class• Ask Questions!!!!!!!!!!

• "Could you quickly go back to. . . ?"

• "Is that the only way to do that step?"

• "Could that step be done this way?"

• "Could we. . . ?"

• "What's the difference between. . . ?"

• "What if we. . . ?"

• "Why did you have to. . . ?”

• "For this type of problem, is that example about at the level of

difficulty that would be on a test?" (Don't ask this one too

much!)15

ClassTime: After Class

• Do some work in between lectures

– attempt all the skills from the last class

• Reworking notes • Reread the notes.

• Add in extra steps and comments to the notes so

that they are clearer.

• For something you don't understand, put a

question mark by

it and try write a question you could ask about it.

• circle key terms or formula or examples

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Free Help: mathcentre

• www.mathcentre.ac.uk

resources which will help you:

• Quick reference leaflets which provide easily accessible support on key topics.

• Teach-yourself booklets with a more in-depth treatment of important topics and which include theory, worked examples and exercises.

• Practice and revision booklets which contain hundreds of practice exercises with answers.

• On-line exercises, which allow you to self-test or practice basic techniques.

• Mathtutor video tutorials on a wide range of mathematical topics.(Streaming kindly provided by University of Portsmouth Creative Technologies, http://stream.port.ac.uk)

• iPod video segments, useful short clips from our Mathtutor video tutorials that you can download to your video iPod.

• 3GP mobile phone downloads, useful short clips from our Mathtutor video tutorials that you can download to your mobile phone

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Moodle Page:

• Enrolment Key:……………………

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Range

• Human Statistics!!!!!

• Shoe Size

Mean,

Mode,

Median

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Reflection Sheets

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• My reflections……….( key points learned, high point of the course, low point of the course, my mood, my learning, ……….)

• What have I learned from these reflections? (One thought to carry forward, aspects of learning of topic which are still of concern, …….)

Each week and end of module

Exemplar: Preparatory Mathematics Ciaran O’Sullivan, IT Tallaght

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Appendix 3:

Preparatory Mathematics 2015 Statistics Assignment

Data gathering task:

Select a sample of 50 measurements from any population of your choice (e.g. sports data,

technical data, financial data, sales data, population/census data). The data can be obtained

by conducting your own survey or from the web, a book, magazine etc.

Working with the data:

Your assignment is to produce an Excel workbook with:

1. A sheet containing your raw data, the maximum and minimum values of your data, a

frequency table with approximately 6 rows and a histogram of the frequency table. The

source of your data must be clearly indicated so that it can be found by your lecturer. Type

this into a cell near the top of the sheet.

Comments:

Comment on what the shape of the histogram tells you about your sample. Type this into a

cell near the top of the sheet.

2. A sheet containing the data sorted so that the median can be calculated.

Comments:

What is the median of your data set?

Type this into a cell near the top of the sheet.

3. A sheet where the sample mean and standard deviation are calculated.

Comments:

What do the sample mean and standard deviation tell you about your data?

If the sample mean differs markedly from the median, explain why that might be. Type this

into a cell near the top of the sheet.

Your assignment must be uploaded to the Preparatory Mathematics Moodle site in Statistics

Assignment section by the date and time announced on the Prep Maths Moodle page.