FOR PRO. S . F

Contents

1. Introduction............................................................................3

2. Justification for selection of topic...........................................4

3. Problem definition and survey of relevant literature...............5

4. Draft title, research aims and objectives................................8

5. The proposed research design..............................................9

6. Methodology.........................................................................11

7. Conclusion............................................................................15

8. Timetable of activities...........................................................16

9. References...........................................................................18

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Introduction This report addresses a proposed research study into the benefits of craft (knitting and

crochet) as a tool for the effective teaching of mathematics to undergraduate computing

students. In keeping with the module learning outcomes the paper will demonstrate insight

into the relevance of research within the service and technological economies. The paper

will apply this understanding to a critical and reflective appraisal of the research process.

The proposed study will set out to test the hypothesis that crafts such as knitting are an

effective tool for teaching a range of complex mathematical concepts, as claimed by

Belcastro (2009), Belcastro and Yackel (2008), Korbey (2013) and Taimina (2009), and

that there is a degree of urgency in addressing a knowledge gap among students within

many British university computing departments. This gap in mathematical training has

been specifically articulated by academics such as Hurley (2009) and, in regard to the

wider UK workforce, by business leaders, for example the CBI audit of 2008. Hurley

describes one business leader announcing “I only hire Mathematicians and Engineers -

Computer Science graduates do not know how to solve problems” (2009).

In proposing and investigating this study the paper will demonstrate that the research is

justified and that a feasible design has been formulated, one that includes ethical

considerations, sampling and reliability evaluation. A range of philosophical approaches

have also been evaluated and the proposed methodological approach outlined and

rationally justified in the report. The following section will now describe the relevance of the

proposed research topic and the reasons for its selection.

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Justification for selection of topic

The computer scientist and mathematician Hurley (2009) states “Software engineers need

a broad Mathematical Education. Computer Science graduates have no Mathematical

equipment in which to analyse what they are trying to do”. The problem of mathematics

education has been widely acknowledged by business leaders and academics in the

United Kingdom, for example Warren Viant, head of computer science at the University of

Hull states “there are very few people on any games courses who are good enough to get

a job in most traditional game development” (Hall, 2013). Viant identifies this problem as

connected to a lack of mathematical ability. The National Numeracy organisation, a UK

based charity, also states “The UK needs a numerate population in order to build a strong

economy and to compete globally. We are currently failing to achieve this” (National

Numeracy, 2014).

Table 1. UK numeracy skills by age group (National Numeracy, 2014).

The widespread shortage of mathematically skilled workers has far reaching implications

for an economy that must generate future programmers and technological innovators if it is

to survive amidst mathematically accomplished Chinese and Asian work forces (Jerrim,

2011). As both a lecturer in Computing and an academic with a keen interest in

mathematics and logic the topic of innovative mathematical teaching is doubly significant.

The implication of knitting and crochet for teaching mathematics is an emergent academic

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field, investigated by academics such as Korbey (2013), Yackel and Belcastro (2008), it

addresses a pertinent question, If crafts-people who define themselves as non

mathematical can solve advanced mathematical problems, why are so many computing

students unable do the same? The wider implications of this question will be explored in

the following survey of relevant literature.

Problem definition and survey of relevant Literature

The subject of teaching children, which is formally known as 'pedagogy' has always been

emotive and ideologically non-neutral (Meighan & Harber, 2007: 212), indeed, in the 1990s

the so-called American school “maths wars' raged over some of the tensions between

ideologically opposed approaches. 'Andragogy', or the teaching of adults, is arguably no

less contentious, and it would be naïve to approach such a topic without acknowledging

the tensions inherent in any discussion of 'best' practice for teaching adults mathematics.

Pitici (2009) writes that “Mathematics education is a rapidly expanding area of research

that is riddled by controversial topics'. And yet the need for improved mathematical skills is

acute. According to the Confederation of British Industry's (CBI) 2008 audit, over half of

the 735 UK business leaders consulted said they were 'concerned that they would not be

able to find enough skilled people with the right qualifications in future”.

Table 2. Proportion of the English population with skills at GCSE of grade “C” or above (National

Numeracy.org, 2014).

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Texts that clarify some of the reasons behind this problem include Hazzan et al (2011) who

state that, “Since programming is a problem-solving process, problem-solving skills must

be a core idea of any introductory computer science course”, however, Hazzan et al go on

to state that “whereas the teaching of programming languages is usually well-structured

within a curriculum, the development of learners’ problem-solving skills is largely implicit

and less structured “(63). This lack of overt instruction is arguably a legacy of the

pedagogic philosophy known as Constructivism.

The most significant historical theorists of mathematical teaching are Polya, (1945) and

Skinner (1958), while, as stated, the current most dominant methodology is constructivism,

as outlined by Piaget (1945), Vygotsky (1978) and Seymour Papert (1971). In short

constructivism is the idea that learning is best experienced as an active discovery process

as opposed to a process of transmission in which a teacher 'pours' knowledge into a

passive student (Piaget, 1945). The assumptions embedded in constructivism may explain

the lack of overt mathematical problem solving instruction for both school and University

level students.

Zevenbergen (1996) argues that core aspects of the constructivist model may be at odds

with real-life experiences of adult learners. “At this point in time” Zevenbergen states, “the

constructivist epistemology has assumed a dominance with mathematics education. This

is evidenced in the amount of journal articles which adopt a constructivist framework.”

Constructivist theory, according to Zevenbergen, believes students will inherently grasp

the transition from concrete constructs to abstract concepts, arguably relying upon an

idealised notion of cognitive development. Students who do not make this conceptual

transition can be pathologised as atypical in their cognitive development, instead of

representing a significant social and cultural group. But, as Meighan & Harber are at a

pains to point out, it is 'the system that is the problem not the user' (2007: 435).

Zevenbergen (1996) as well as Usher and Edwards (1994) indicate the need for a move

away from positivist models of the student as an innately logical rationaliser, to models that

are subjectively and experientially grounded. This review has researched evidence that

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craft is an increasingly valuable resource for such a theoretical shift, offering the teaching

of mathematics a useful alternative to the hardcore constructivism that has dominated

mathematical pedagogy. Belcastro and Yackel write “The mathematics that arises in fiber

arts such as knitting, crocheting, cross-stitch, and quilting is wide-ranging and includes

topology, graph theory, number theory geometry, and algebra” (Belcastro and Yacke, 2008:

15).

There are a number of other useful references to support this hypothesis, Iseri (Petici,

2009) writes about using paper models to explore the curvature of various surfaces,

locating the problem within an historical context, suggesting student-friendly ways for

teaching the notion of curvature of both Euclidean and non-Euclidean surfaces. In the

same text, Leron and Hazzan discern four different ways of distinguishing between

intuitive and analytical thinking. They do it by placing mathematical thinking in a broader

cognitive context that considers general theories of human learning.

Figure 1. Hyperbolic Coral Reef, an example of hyperbolic geometry constructed in crochet

(Tylicki, M., 2014)

Korbey (2013), Belcastro and Yackel (2008) have gone some way to outlining a systematic

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approach to the use of knitting and allied crafts for mathematics teaching, it is their

synthesised theoretical perspective that offers the proposed research project a

philosophical and methodological framework. At the same time, the specific mathematical

domain that the proposed research addresses requires further and wider research, as

indicated in some of the texts cited here. Belcastro and Yackel (2008) do not address the

specific mathematical context of programmers, which is predominantly algorithmic and

heuristic (Hazzan, 2011), while games programmers and industrial designers may also be

concerned with hyperbolic geometry and topology. Taimina (2009) provides another

valuable perspective on hyperbolic geometry and knitting, but it is not a specifically

pedagogic or computational text. The proposed research therefore has scope for

significant innovation.

Draft title, research aims and objectives

Having established in the previous section that there are many ideological tensions in the

subject of andragogy (the teaching of adults), it is particularly important to emphasise as

objective an approach as realistically possible. While Meighan and Harber (2007) and

many others have argued that all research is embedded with ideological values, the

project will try to use the most open and neutral methods it possibly can, this is reflected in

the draft title which seeks to maintain a scientific frame of reference. The draft title of the

project is:

“An empirical investigation into the role of craft as an effective tool for the teaching

of mathematics to computing undergraduates.”

Though somewhat long-winded as a sentence, the title will serve provisionally to direct the

project and maintain its conceptual boundaries, meaning, it will not spill into investigating

other categories of learners or attempt to investigate, for example, knitting for improving

motor skills. The question may be refined and articulated more elegantly over the six

months of the proposed research duration, but the core area of research will remain

exactly as defined in the draft title.

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The proposed study aims to test the hypothesis that crafts, in particular knitting and

crochet, are effective tools for teaching mathematics to computing students. The study will

focus on algebra, algorithms, hyperbolic geometry and calculus because of their particular

relevance to computing and their potential interest to undergraduates.

The research will expand on the work done by Taimina (2009), Belcastro and Yackel

(2008) but, by specifically focusing on programming and other IT subjects, such as

database design (which uses relational calculus) the proposed study will be significantly

original, contributing a new perspective to the themes explored by the authors cited here.

The research objectives that will support the testing of the project hypothesis are outlined

below.

Research Objectives:

1. Research and outline the predominate current teaching models.

2. Generate empirical evidence that there is a problem through the testing of a

significant sample population and comparison with historical records and other

countries.

3. Establish a formalised craft approach for the teaching of Algebra and Calculus

4. Pilot this approach in workshops (first testing participants on their core maths skills)

5. Conduct experimental workshops to test the effectiveness of a formalised craft

based approach.

6. Analyse and establish the statistical significance of the results.

The proposed research design

According to Saunders et al (2012) it is the researcher's understandings and associated

decisions in relation to the outer layers of the 'research onion' that provide the context and

'boundaries within which data collection techniques and analysis procedures will be

selected'. In order to develop an effective research design the implications of these

elements, from the outside layers to the innermost ones have been evaluated, including

data collection techniques and analysis procedures. Explicit consideration of the inner

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elements of Saunders' onion have been particularly valuable in developing an appropriate

and logically coherent research design.

The outermost layer of Saunders' research onion describes the overarching philosophical

orientation of the research process. The project in question will be underpinned by a

scientific and empirical design, as described by Denzin and Lincoln (1994: 99). While the

author acknowledges the many critiques of such an approach, for example Kuhn (1962)

who questions the very motion of objectivity, it is also important, given the ideologically

loaded history of mathematical pedagogy, to try and remove any overt ideological

orientation from the research process. The best design strategy for minimising subjective

input is therefore quantitative, one that will involve numeric data and statistical hypothesis

testing.

Table 3. The layers of Saunders' 'Research Onion'.

The inner layers described by Saunders et al are concerned with appropriate methods.

Researchers can choose to use a single data collection technique and a corresponding

analysis procedure, in what is called a 'mono method' or a 'multi method' approach which,

in the context of quantitative research, would entail using more than one quantitative data

collection technique, such as a questionnaire and structured observation, with the relevant

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associated statistical analysis procedures. Such a multi method quantitative approach is

proposed for this project. The aim being to obtain testable results as well as structured

observation of the workshop process. These observations can be categorised and coded

and therefore processed into statistically analysable data sets.

The core of the research onion is concerned with data collection methods. In keeping with

its scientific methodology the project will use stratified random probability sampling, in

which the entire target population (Computing BSc students) of 40 UK universities will be

invited to participate, but will be divided into mathematically representative strata (these

will be discussed in more detail in the methodology section). An online survey interface will

be used for the test process and for instigating invitations to participate in workshops. The

interface will be embedded with a mechanism for random stratified probability sampling.

This ensures maximum generalisability and wider proportionate representation of, for

example, age groups, abilities and genders. The following section will go into more detail

about the specific philosophical framework and the associated methods and

considerations the proposed research design will deploy.

Methodology

An empirical research methodology involves moving from general explanations to specific

data (De Groot, 1969). Steenhuis et al (2006) also state that the “positivist and

postpostivist approach can also be viewed as nomothetic, i.e. it emphasizes quantitative

analysis of a few aspects across large samples in order to test hypotheses and make

statistical generalizations” (Steenhuis et al, 2006). This project will emulate such a

methodological approach, moving from the hypothesis that crafts such as knitting are

useful tools for teaching mathematics to the specific analysis of data relating to the

outcomes of knitting and maths workshops that will be delivered to a proposed sample of

100 computing students.

Sampling method

In order to generate the sample a list of 40 colleges and universities will be drawn from the

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target population of UK Higher Education institutions using three types of information.

Firstly the classification will involve the university’s Research Assessment Rating (a formal

measure of 'quality profiles' for each institution’s submission of research activity), secondly,

by the grades required for entry to a Bachelor of Science computing degree, and thirdly by

geographic location. The sample method will attempt to ensure institutions are drawn from

representative regions and categorical strata across the U.K.

The sample of 40 institutions will represent 7 different locations, the North of England, the

Midlands, Wales, the South East, the South West of England, Scotland and London, split

according to the student population. Institutions will be selected randomly, yet

proportionately from each of the regions. The sample therefore will be drawn to maximize

representation of different types of institutions and students from different regions of the

United Kingdom.

The testing process

The first part of the online survey will involve a specialised maths test, this will aim to

establish the degree to which participants can solve complex algorithmic and algebraic

problems. The outcomes of the tests will define which computing students will be put into

the pool for random stratified sampling. The automated system will select students who

achieve below average results. From that pool stratified students will be selected by

chance and invited to take part in the knitting workshop. At the end of each workshop

participants will be tested again, the test will cover the same mathematical ground but with

different numeric values randomly generated for each question.

Accessing and analysing data

The first phase of data will be collected via an interactive interface written in HTML and

PHP by the author, the data will be stored in a secure relational database. The data can

then be retrieved and queried. A range of descriptive statistics can initially be obtained to

facilitate the second selection process, stratifying participants according to university, the

result of their maths test and educational achievement (for example, Maths A levels etc).

The research process will follow the later stages of de Groot's empirical cycle:

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 Deductive consequences of hypothesis as testable predictions.

 Testing the hypothesis with new empirical material.

 Evaluating the outcome of testing

(de Groot, 1969)

In keeping with a hypothesis lead research project, if the theory is not confirmed by the

findings the theory will need to be revised (Saunders et al, 2011).

The earlier stages in de Groot's empirical cycle, of observation and hypothesis formation

(or 'induction') are implicit in the secondary research, in particular that of Taimina (2009),

Belcastro and Yackel (2008). A characteristic feature of the empirical cycle is that the

researcher is a spectator who is not a part of the problem being studied by means of

personal involvement (de Groot, 1969). This implies that there is a separation between the

researcher and the research object, while this may be an idealised notion, it is one that the

project will try to maintain, given the potentially contentious nature of the research and the

historical 'Maths Wars'. The empirical cycle is characterised by de Groot as a stringent

scientific method.

The part of de Groot's cycle that is involved with testing a hypothesis with new empirical

material will be encompassed by knitting workshops. These workshops will be evolved

from the initial secondary research process and a pilot. The workshop pilot will test out the

effectiveness of the new teaching approach developed in the earlier stages of the project.

In keeping with the scientifically rigorous approach the analysis of data will also be

mathematically structured, a range of appropriate tests will be carried out to establish

confidence in the reliability and generalisability of results, for example, the Kruskal-Wallis

test aims to test whether the scores on a variable differ between groups. This will be

appropriately used as there will be more than two groups. The Fisher-eksact-test can be

used to test the independence between variables, meaning the degree to which one

variable might influence another, specifically to test the impact of the knitting workshops on

the mathematical problem solving ability of participants. In summary the proposed

research will unfold as follows, given the hypothesis that knitting is a useful tool for

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teaching advanced maths topics:

1. Research and develop a teaching framework for knitting and maths

2. Test the framework via pilot workshops

3. Establish a random stratified sample

4. Test the chosen participants on their mathematical ability

5. Invite suitable participants to take part in workshops

6. Test their ability after the workshops

7. Statistically analyse the results

8. Re-evaluate the initial hypothesis

These stages clearly bring up a number of ethical considerations, not least of all the

protection of participant data. The following section will outline the formal measures that

will be taken to protect participants from any form of exploitation, loss of privacy or other

forms of abuse.

Ethical implications

The ethical implications of the proposed research will be carefully interpreted

assessed and applied at all stages. Consent forms will be obtained from all

participants as well as from the institutions where they study. Briefing and debriefing

forms will be provided to all students taking part, they will outline what the research

is for and provide contact names and numbers, as well as information about how

the data will be stored and when it will be disposed of.

It will be clear to all participants that they can withdraw at any point from the

research study and that their contribution will be anonymised at all times.

Workshops will be assessed for health and safety, accessibility and privacy. No

children will take part in the study. All data will be held in password protected and

encrypted forms. The data will be destroyed immediately after the study has been

marked by the University of Derby.

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Feasibility considerations

It is always challenging to generate sufficient quantitative data to offer a generalisable set

of results, however, this challenge will need to be met in order to present credible research

outcomes. The project will be embedded with realistic contingency plans, such as a very

large potential sample pool, with 40 UK universities targeted for participation. In the event

of poor uptake, the pool will be widened.

The workshops will be intensive and also present logistic challenges, these will

necessitate contingency plans and very precise time management. The proposed analysis

of data will be relatively swift, as the survey interface is automated and data can be

analysed dynamically and updated by that software. The timetable included in this report

will evidence the carefully researched project timeframe and the use of the time

management tool 'SmartSheet' to optimise the research process.

Conclusion

This report has outlined a proposed research project of relevance to the service and

technological economies. It has evidenced a critical and reflective appraisal of the

research process needed to conduct a credible and valid study into the hypothesised value

of craft for teaching mathematics to computing students.

Throughout the report relevant academic references have been made to support the

arguments and opinions stated within it. The report has provided a rationale for its

methodological approach and a logically coherent appraisal of methods for accessing and

evaluating quantitative data and the proposed hypothesis.

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Timetable of activities

Table 4. Proposed Timetable of Activities (DUMMY VERSION).

The timetable of activities was constructed using the time management tool 'SmartSheet'.

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The timetable takes the form of a Gantt chart showing activities and tasks displayed

against a time frame, the tool was chosen because it facilitates the tracking of tasks

as well as showing their inter-relationships, deadlines and relevant additional

information.

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