Elementary Math to Close the Digital Skills Gap

Pia Niemel

and Antti Valmari

Tampere University of Technology, PO Box 527, FI-33101, Tampere, Finland

University of Jyv

askyl

a, Jyv

askyl

a, Finland

Keywords:

K-12 Computer Science Education, Computing in Math Syllabus, Digital Skills Gap, Professional Develop-

ment of Software Professionals, Effectiveness of Education, Continuous vs. Discrete Math.

Abstract:

All-encompassing digitalization and the digital skills gap pressure the current school system to change. Accor-

dingly, to ’digi-jump’, the Finnish National Curriculum 2014 (FNC-2014) adds programming to K-12 math.

However, we claim that the anticipated addition remains too vague and subtle. Instead, we should take into

account education recommendations set by computer science organizations, such as ACM, and deﬁne clear

learning targets for programming. Correspondingly, the whole math syllabus should be critically viewed in the

light of these changes and the feedback collected from SW professionals and educators. These ﬁndings reveal

an imbalance between supply and demand, i.e., what is over-taught versus under-taught, from the point of view

of professional requirements. Critics claim an unnecessary surplus of calculus and differential equations, i.e.,

continuous mathematics. In contrast, the emphasis should shift more towards algorithms and data structures,

ﬂexibility in handling multiple data representations, logic; in summary – discrete mathematics.

1 INTRODUCTION

21st century society is digitizing rapidly and job des-

criptions of current professions are changing accor-

dingly. Digitalization triggers pressure to change the

current education system. Both domestic and multi-

national governing bodies have recognized the skills

gap of computer science and the growing need for a

digitally ﬂuent workforce. Consequently, the EU has

outlined a strategy for improving e-skills for the 21st

century to foster competitiveness, growth, and jobs.

Just-published technical reports provide guidance for

educators and politicians at the European level (Re-

decker and Punie, 2017; Bocconi et al., 2016), high-

lighting the pervasive and ubiquitous nature of digi-

talization. Digital literacy, responsible use of techno-

logy, and civic participation are thus relevant to ever-

ybody. In consolidation, digitally skillful workers are

more likely to keep their positions and, if displaced,

are reemployed more quickly than employees without

digital skills (Peng, 2017).

The skills gap concerns not only the number of

SW professionals but also the quality of their skills.

The STEM shortage paradox highlights the peculia-

rity of having hard-to-ﬁll open positions and at the

same time an excess of graduates who cannot ﬁnd a

job (Harris, 2014; Smith and White, 2017). One ex-

planation is the skills mismatch, and in compliance

with this, employers point out the candidates’ inca-

pability of breaking down problems into managea-

ble chunks and solving them, and the gaps in techni-

cal, data modeling, and analytical skills. Accordingly,

data base, data management, data analysis and statis-

tics skills outnumber other requested digital skills of

job advertisements in the US (Beblav

y et al., 2016).

The discussion of the role of computer science

(CS) in education is global. A number of countries

all over the world have introduced CS into their K-12

curricula. In line with others, the FNC-2014 compri-

ses algorithmic thinking and programming as parts of

the mathematics syllabus (Finnish National Board of

Education, 2014). In pursuit of consistent CS sup-

port, the entire math syllabus should be reviewed al-

ong with these newly introduced additions. This study

then asks:

• RQ1: What elementary math syllabus areas

should be strengthened for the anticipated CS

emphasis?

• RQ2: Are there math syllabus areas that are cur-

rently overemphasized from this viewpoint?

First, this study reviews the discourse of CS as a

scientiﬁc discipline and the learning targets of mat-

hematics in anticipation of supporting CS. In the Re-

lated Work section, we list already-existing directi-

154

Niemelä, P. and Valmari, A.

Elementary Math to Close the Digital Skills Gap.

DOI: 10.5220/0006800201540165

In Proceedings of the 10th International Conference on Computer Supported Education (CSEDU 2018), pages 154-165

ISBN: 978-989-758-291-2

 2019 by SCITEPRESS – Science and Technology Publications, Lda. All r ights reserved

ves and recommendations of institutions that aim at

building a ﬂexible future work force, such as ACM.

There, we focus on suggested math courses in parti-

cular. For comparison, we check the elementary-level

math and computing syllabi of current strong perfor-

mers in CS, i.e., the UK and US. The Results and Dis-

cussion section cross-exposes these recommendations

with feedback from in-service software engineers by

focusing on the evaluated proﬁtability of the curri-

culum topics. To conclude, we propose hypothetical

math learning trajectories for a CS support.

2 CS&SWE VS. ICT

Most natural sciences and engineering disciplines rely

on calculus, differential equations, and linear algebra

as a mathematical foundation appropriate for continu-

ous phenomena. Systems relying on such phenomena

can be adequately tested. For instance, a bridge does

not need tests for all possible loads between zero and

a maximum value. Testing the maximum load under

typical and extreme weather conditions sufﬁces.

In contrast, Parnas highlights the different nature

of software (Parnas, 1985). Unlike bridge load tests,

testing a piece of software with typical and extreme

values does not guarantee expected behavior with un-

tested values. Furthermore, software is rarely concise

enough to be tested inside out, and unlike mathema-

tical theorems, it is not comprehensively checked by

other experts in the ﬁeld. Thus, frequent errors and

failures are common (Charette, 2005).

As we will discuss later, computer scientists have

suggested topics such as logic, formal grammar, and

set theory as an appropriate mathematical basis for

mastering software and improving its quality. In addi-

tion, the importance of algorithmic thinking has been

discussed extensively. In traditional engineering de-

gree programs, classic mathematics and physics are

included early on. The rationale is to develop a suit-

able mindset, that is, a way of thinking that facilita-

tes a more profound learning of engineering topics.

The basis is laid already in elementary school phy-

sics and mathematics. Similarly, professional compu-

ter science and software development need a suitable

mindset that should be developed before studying the

bulk of the software topics. However, because soft-

ware cannot be appropriately mastered with tools sui-

ted for continuous phenomena, this mindset is not the

same as that of, say, an electrical engineer.

The discussion of the educational needs in Fin-

land suffers from a poor distinction between Informa-

tion and Communication Technology (ICT), Compu-

ter Science (CS), and Software Engineering (SWE).

For more than a decade, the Finnish mobile phone

company Nokia was very successful and its educa-

tional needs had a remarkable impact on the Finnish

educational discourse. In addition to SW engineers,

Nokia needed expertise in the ﬁelds of hardware, ra-

dio technology, and signal processing. Therefore,

ICT and SWE were emphasized instead of CS, with

SWE largely perceived via analogy to traditional en-

gineering, less through its relation to CS. As a conse-

quence, Finnish scholars and educators have only par-

tially conceived the special character of CS and SWE

as disciplines distinct from ICT, thus requiring a dif-

ferent educational foundation, which implies changes

in the math syllabus as well.

To clarify the conceptual difference, we deﬁne the

relation of CS to SWE more closely. Parnas equates

it to the relationship between physics and electrical

engineering (Parnas, 1999, p. 21): physics belongs

to the natural sciences, which target an understanding

of a wide variety of phenomena; electrical engineer-

ing is an engineering discipline striving to create use-

ful artefacts. Although electrical engineering is ba-

sed on physics, it is neither a subﬁeld nor an exten-

sion of it. Analogously, CS is a science, and SWE

is an engineering discipline based on CS. Therefore,

CS degrees must focus on the underlying computatio-

nal phenomena and the acquisition of new knowledge

of these, while SWE degrees concentrate on imple-

menting trustworthy, human-friendly software cost-

effectively.

In regard to math, the latest speciﬁcations of

ACM&IEEE explicate the similarity of required skills

both in CS and SWE (ACM&IEEE, 2013; Ardis et al.,

2014). Even if CS is more scientiﬁc as a discipline

and more deeply grounded in math, SW engineers be-

neﬁt from more theoretically-oriented CS education

and discrete math to be able to implement quality soft-

ware. Hence, the conceptual difference does not di-

verge the required math and computing fundamentals.

Consequently, Meziane and Vadera concluded, ’There

is very little difference between the SE and CS pro-

grams currently offered in English Universities’ (Me-

ziane and Vadera, 2004).

3 RELATED WORK

3.1 ACM Recommendations

The standards developed by the Association for Com-

puting Machinery (ACM) are used as a premise in

curriculum planning in a number of Finnish univer-

sities. The CS concepts introduced in the ﬁrst courses

are important either for their own sake or for further

Elementary Math to Close the Digital Skills Gap

155

topics. Obviously, the ﬁrst fundamental concepts are

also the most evident candidates when considering to

advance some basics at the elementary school level.

3.1.1 CS Knowledge Areas of ACM

ACM promotes CS as a discipline and in compliance

prepares normative recommendations for teaching CS

at the tertiary level. ACM (ACM&IEEE, 2013) intro-

duces Curriculum Guidelines for Undergraduate De-

gree Programs in Computer Science (ACM-CS2013).

The material is divided into Knowledge Areas (KA)

and further to Knowledge Units (KU) that match with

no particular course. Instead, courses may incorpo-

rate topics from multiple KAs. Topics are divided into

Core and Elective, and the Core is further subdivided

into Tier-1 (to be fully completed) and Tier-2 (at mi-

nimum 80% coverage). The KAs with the most Tier1

hours are:

1. Software Development Fundamentals (43 h)

2. Discrete Systems (37 h)

3. Algorithms and Complexity (19 h)

4. Systems Fundamentals (18 h)

The natural ﬂow of concepts is to introduce soft-

ware development fundamentals (SDF) and simulta-

neously strengthen the mathematical foundation with

Discrete Systems (DS). In descending order of allo-

cated hours, algorithms and complexity (AL) come

next, where mastering common algorithms is consi-

dered general CS knowledge. Complexity conside-

rations consist of evaluating the algorithm efﬁciency

based on execution time and consumed resources. Sy-

stems Fundamentals (SF) give an insight into system

infrastructure and low-level computing by acquain-

ting students with computer architecture, main HW

resources and memory, and, e.g., sequential and pa-

rallel execution.

From the list above, items 2 and 3 link closely with

math. According to ACM, DS comprises the follo-

wing areas in descending order of emphasis (Tier-1

+ Tier-2 hours): Proof Techniques (11), Basic Logic

(9), Discrete Probability (8), Basics of Counting (5),

Sets, Relations, and Functions (4), and Graphs and

Trees (4). AL in turn consists of basic and advanced

KUs of Analysis, Strategies, Fundamental Data Struc-

tures, Automata, Computability, and Complexity.

In sum, algorithms and data structures are at the

center of gravity together with the programming ba-

sics of SDF.

3.1.2 The Most Relevant Math to Support CS

ACM-CS2013 highlights the tight and mutual inter-

dependence between math and CS. However, instead

of being prepared for every kind of career option,

ACM-CS2013 focuses on the common denominator.

Thus, only directly relevant requirements are speci-

ﬁed, such as elements of set theory, logic, and dis-

crete probability comprising the KA of DS. On the

other hand, ACM-CS2013 states that “while we do not

specify such requirements, we note that undergradu-

ate CS students need enough mathematical maturity

to have the basis on which to then build CS-speciﬁc

mathematics”. It also mentions that “some programs

use calculus . . . as a method for helping develop such

mathematical maturity” (ACM&IEEE, 2013).

The recommendations make a distinction between

such mathematics that is an important requirement for

all students in the faculty and mathematics that is re-

levant only to speciﬁc areas within CS, exemplifying

this with linear algebra that “plays a critical role in

some areas of computing such as graphics and the

analysis of graph algorithms. However, linear alge-

bra would not necessarily be a requirement for all

areas of computing” (ACM&IEEE, 2013).

If it were decided to emphasize discrete math in-

cluding logic in the elementary school math curri-

culum, then an age-appropriate and tested subset of

ACM Basic Logic could be found in the National Cur-

riculum and GCSE Mathematics of the UK. The UK

has already emphasized discrete math for a longer pe-

riod, see section 3.3. Logic is deployed frequently

in programming, not only when implementing con-

ditions in selection and iteration statements. Subse-

quently, university-level logic targets more sophisti-

cated and far-reaching knowledge than this. In conse-

quence, Basic Logic of DS introduces normal forms,

validity, inference rules, and quantiﬁcation.

Although probability is linked more weakly to the

programming fundamentals than logic, it gives readi-

ness for various prominent topics, such as the analy-

sis of average-case running times, randomized algo-

rithms, cryptography, information theory, as well as

games. Its basics should cover conditional probabi-

lity, independent and dependent events, and multipli-

cation and addition rules.

3.2 SWEBOK Recommendations

The Guide to the Software Engineering Body of

Knowledge (SWEBOK) of the IEEE breaks down

the mathematical foundations into smaller knowledge

areas (Bourque et al., 2014). In the review, we focus

on both Chapters 13 and 14 of the guide, i.e., Compu-

ting and Mathematical Foundations.

Computing Foundation in Chapter 13 is inclu-

ded because it comprises algorithms and data struc-

tures. Data structures have various classiﬁcations,

CSEDU 2018 - 10th International Conference on Computer Supported Education

156

e.g., linear–nonlinear, homogeneous–heterogeneous,

stateful–stateless. For instance, linear structures or-

ganize items in one dimension (lists, stacks), compa-

red to the two or more hierarchies (trees, heaps) of

non-linear structures. Well-designed data structures

accelerate data storage and retrieval. The efﬁciency

of algorithms depends signiﬁcantly on the selection

of a suitable data structure. Appropriate data struc-

tures can foster algorithm development. When the

effects are combined, performance and memory con-

sumption may range from poor to extremely efﬁcient.

Chapter 14 highlights CS as an applied maths to-

pic. The foundational KAs concentrate on logic and

reasoning as the essences that a SW engineer in par-

ticular must internalize. The chapter describes mat-

hematics as a tool of studying formal systems, wi-

dely interpreted as abstractions on diverse applica-

tion domains. These abstractions are not restricted to

numbers only, but include, e.g., symbols, images, and

videos.

The following subtopics constitute the foundatio-

nal KAs of math. Our assumption is that the order

implicates their importance. We divide these topics

into continuous (c) and discrete (d):

1. Sets, Relations, and Functions (c/d)

2. Basic Logic (d)

3. Proof Techniques (d)

4. Basics of Counting (d)

5. Graphs and Trees (d)

6. Discrete Probability (d)

7. Finite State Machines (d)

8. Grammars (d)

9. Numerical Precision, Accuracy, and Errors (c)

10. Number Theory (d)

11. Algebraic Structures (d)

One obvious observation is a notably smaller por-

tion of continuous math compared to traditional engi-

neering education. In particular, calculus, differential

equations, and linear algebra are missing. Instead, se-

veral topics target a better position of underlying lo-

gic (2,3); and primers for data types, data structures

and algorithms (1,4,5,9,11). In addition, subtopics of

Basics of Counting (4), and Discrete Probability (6)

and Number Theory (10) scaffold a deeper understan-

ding of probability and cryptography. Numerical Pre-

cision, Accuracy, and Errors (9) section reveals un-

derlying HW and memory speciﬁcs that have an effect

on, e.g., the resolution of measurements and impossi-

bility of expressing most real numbers precisely.

3.3 K-12 Math and Computing Syllabi

of the UK and US

For comparison, we went through the National Cur-

riculum (UKNC) and General Certiﬁcate of Secon-

dary Education (UKGCSE) of the UK (Department of

Education, 2014; GCSE, 2015), and the Core Curri-

culum of US (USCC) (Core Standards Organization,

2015). The logic basics are present in the syllabi of

both, with a comprehensive subset. Yet Boolean logic

is currently included in the computing curriculum of

UKNC, not in math. However, Boolean logic would

ﬁt well in the math syllabus as a consistent continuum

of inequalities.

Sets can illustrate nested number sets of natural

numbers (N), integers (Z), and reals (R) that match

with variable types (unsigned, int, ﬂoat) in program-

ming. However, due to differences in how, e.g., reals

appear in both, we note that this juxtaposition is prone

to misconceptions. For instance, in:

i n t x = 1; f l o a t y=x / 2 ;

division may produce a value of zero depending on

the used language. All the same, not every int is a

ﬂoat, in contradiction of the math subset relation of

Z R. In addition to primitive types, sets are the ba-

sic mathematical abstraction of containment, and are

thus relevant for programming as a cognitive tool. A

group of numbers may be introduced as a set, a vec-

tor or a matrix, and the same group operations apply.

Therefore, set theory would be useful in any mathe-

matics curriculum designed to support programming.

Currently, sets are a part of UKNC, but absent from

USCC and FNC-2014.

Linear algebra basics are included in the USCC

as matrices and basic operations; and as vectors and

transformations in UKNC, whereas they are missing

from the FNC-2014. For example, linear algebra ba-

sics could be a beneﬁcial addition even if supported

by ACM-CS2013 only as an elective math topic, be-

cause matrices are extensively exploited in the ﬁelds

of statistics, data analysis, games, and graphics, for

instance. The need for matrices is increasing, because

topicality of their application areas and because

many libraries in, e.g., Python exploit them extensi-

vely. As a topic, matrices and vectors belong together,

and various transformations (such as scaling, transla-

tion, reﬂection and rotation) are main operations on

image manipulation and animations.

Matrices are extensively exploited, e.g., in ma-

chine learning, data analysis, pattern recognition, and

game engines for 2D/3D-transformations. All sugge-

sted math syllabus areas remain at the preliminary le-

Elementary Math to Close the Digital Skills Gap

157

Table 1: Math Syllabi (KS=key stage, G=grade, HS=high

school. Each key stage covers several grades ranging from

two to four. The GCSE exams follow KS4.

UKNC USCC

Logic (in CS)

KS2: logical reaso-

ning to explain how

simple algorithms

work

KS3: Boolean logic

(AND/OR/NOT)

and its applica-

tion in circuits and

programming

Sets

Prob

KS3: enumerate sets,

unions/intersections,

tables, grids and Venn

diagrams

KS4: data sets from

empirical distribu-

tions, identifying

clusters, peaks, gaps

and symmetry, ex-

pected frequencies

with two-way ta-

bles, tree and Venn

diagrams

G6: data sets,

identifying clus-

ters, peaks, gaps,

symmetry

G7: random sam-

pling to generate

data sets

HS: interpreting

differences in

shape, center

and spread of a

distribution

Vectors

Mat-

rices

KS4: (in Geome-

try) translations as 2D

vectors, addition and

subtraction of vectors,

multiplication with a

scalar, diagrammatic

and column represen-

tations

GCSE: transformati-

ons & vectors

HS: addition,

subtraction, mul-

tiplication of

matrices, multi-

plication with a

scalar, identity

matrix, transfor-

mations as 2x2

matrices

vel in UKNC and USCC and we propose the same: in

logic truth tables and Boolean logic in order to sim-

plify several simultaneous conditions; in sets, Venn

diagrams and basic operations of union, intersection

and cut with at most three sets; and in matrices, trans-

formations of translation, reﬂection, rotation and en-

largement and ﬁnding an inverse matrix. This new

math knowledge should be carefully bridged with the

prior knowledge with lots of visual exercises and by

starting early enough. Table 1 illustrates in which or-

der these topics are handled in the UKNC and USCC.

Thus, in lieu of the ACM DS Logic subset, a

readily ﬁeld-tested elementary syllabus is found in

GCSE CS (GCSE, 2015). It contains the following

topics:

• binary and hexadecimal notations

• binary addition and shift

• Boolean values (true, false)

• Boolean operators (AND, OR, NOT); truth tables

Sets prompt types in programming and they can be

utilized in abstracting both primitives and collections.

UKNC speciﬁes the syllabus of sets followingly:

• sets visualized by Venn diagrams

• set operations: subset, proper subset, intersect,

and union, combinations of these

• sets represented as lists, and

• set and its complement

In addition, in the CS syllabus of the GCSE clear le-

arning targets for algorithms are set: at a minimum,

binary search and merge sort (GCSE, 2015).

4 METHOD

This study complies with the scope of curriculum the-

ory (Pinar, 2012), and its key question of what kno-

wledge is most valuable and how this knowledge is

constructed as consistently as possible. Here, we are

concerned with the educational and sociological as-

pects due to the aim of improved employability and

ﬁlling the digital skills gap. This study is restricted to

elementary math and compares the FNC-2014 to the

UKNC and USCC (Department of Education, 2014;

English Department for Education, 2013; Core Stan-

dards Organization, 2015) and to the recommendati-

ons given by the ACM and IEEE (ACM&IEEE, 2013;

Bourque et al., 2014). The comparison exploits con-

tent analysis in searching for the math syllabus antici-

pated to be the most useful for CS students.

In addition to the comparison, the effectiveness of

the university-level SWE studies reﬂects back to the

curriculum design. We do not collect any new data but

reuse the data of existing studies (Lethbridge, 2000;

Puhakka and Ala-Mutka, 2009; Surakka, 2007; Kit-

chenham et al., 2005). The results of the previous

studies are cross-correlated to conﬁrm their validity

in order to draw conclusions about the most proﬁta-

ble math topics.

5 RESULTS AND DISCUSSION

In this section, we ﬁrst review the feedback from the

ﬁeld: SW professionals evaluate the curriculum to-

pics according to their proﬁtability in working life.

Having being informed of both the previous section’s

CSEDU 2018 - 10th International Conference on Computer Supported Education

158

recommendations and criticisms of the current rea-

lization, we summarize the necessary math syllabus

content and bridge the learning trajectories from ele-

mentary to higher-education math.

5.1 Feedback From SW Engineers

To evaluate the effectiveness of their education, SW

engineers have scored the proﬁtability of a plenty of

curriculum topics (Lethbridge, 2000). An imbalance

between supply and demand was discovered and as a

remedy, the author recommends putting less emphasis

on the topics of minor importance – or teaching them

in a way that makes them more relevant to SWE stu-

dents. The study was run in year 1997 and repeated

in 1998. The differences between outcome remained

modest. In 1998, the sample size was N 181, and

the survey consisted of 75 topics of CS, SWE, etc.

A few years later, in 2004, Kitchenham & al. con-

ducted a research focusing on the curricula and gra-

duates of four UK universities (Kitchenham et al.,

2005). The methodology was somewhat different and

so was the obtained list of the most under-taught to-

pics. The ﬁndings regarding mathematics were, ho-

wever, the same. Then in 2009, a decade after Let-

hbridge’s original research setup, Puhakka et al. pu-

blished an analogous study conducted in Tampere

University of Technology (Puhakka and Ala-Mutka,

2009, N 212). Out of the original 75 subtopics,

three were removed because of their not being com-

mon in Finnish curricula. Both sub-ﬁgures of Fig. 1

illustrate the differences between math-related per-

ceptions among SW professionals in the examined co-

horts of US and Finland. First, we observe that the re-

sults correlate surprisingly well, taking into account a

timespan and continent switch. The scientiﬁcally sig-

niﬁcant values of R

are 0.88 in the upper, and 0.91 in

the lower ﬁgure.

The green circles in sub-ﬁgures designate the

areas considered either useful (the upper) or in need

of more emphasis (the lower) to build work-life com-

petences of SW professionals. The lower sub-ﬁgure,

however, demonstrates the rarity of topics in need

of more emphasis. Negative values indicate a post-

graduate knowledge loss, whereas positive values a

knowledge gain, in other words, inadequate learning

of such topics in higher education.

The latter sub-ﬁgure is visually telling.

Only algorithms and data structures are in need

of more emphasis. In addition to these, the Let-

hbridge top-ten consists of no other mathematical but

instead such items as negotiation, human-computer

interaction, and leadership.

In comparison with both previous surveys, Su-

Figure 1: The comparison of usefulness and adequacy of

math education evaluated by SW professionals (Lethbridge,

2000; Puhakka and Ala-Mutka, 2009, N 181; N 212).

rakka separates the sample into the cohorts of SW

engineers, academics (professors, lecturers) and stu-

dents, see Fig. 2.

The winner is again clear: algorithms and data

structures, also the prominence of discrete math com-

pared with continuous math is unchallenged, yet the

bias has an academic ﬂavour. Discrete math scores

highest among professors and lecturers (3.1).

5.2 CS-supportive Math for Elementary

In constructing a strong basis for CS, both ACM and

SWEBOK emphasize discrete math, conﬁrmed by the

feedback from the ﬁeld. After programming basics,

ACM values discrete systems as the second most,

and algorithms, data structures, and complexity as the

third most prominent KAs, whereas the in-service SW

engineers value this area the highest. In SWEBOK,

nine out of eleven math KAs comprise discrete math.

UK, spearheading in CS, invests in discrete math alre-

Elementary Math to Close the Digital Skills Gap

159

Figure 2: The math areas perceptions [1(not important),

4(very important)] of Surakka’s engineers, academics, and

students contrasted with Lethbridge and Puhakka et al.;

N 11, 19, 24, 181, 212; respectively.

ady at the elementary level and in addition, provides

CS as a separate subject with more in-depth topics.

Algorithmic thinking

The referenced studies categorize algorithms and data

structures as part of the CS Core. In programming-

oriented math, data structures can be seen as an appli-

cation of set theory, e.g., sets conceptualize collecti-

ons. In programming, collections are of various ty-

pes: a set is an unordered collection of values, a list

an ordered collection, and a map a collection of va-

lues identiﬁed by keys, which may also be interpreted

as a representation of a mathematical function.

Denning equates algorithmic and computational

thinking (Denning, 2009), which he in turn associ-

ates with general problem solving (Denning, 2017).

When solving a problem, it is beneﬁcial to start by

decomposing it to smaller solvables implemented in a

code as sub-routines, for instance. At its simplest, an

algorithm may then be understood as a sub-routine,

a sequence of commands called repeatedly as many

times as desired, e.g., (CSTA, 2016). Computing is

what Wing refers to as automation of abstractions, al-

gorithms being the most prominent class of these ab-

stractions (Wing, 2008).

The gradual division between human-completed

calculation and computer-based computing has been

the watershed between the disciplines of math and

CS. In pondering the difference between the mindsets

of mathematicians and computer scientists, Knuth

points out that computer scientists need to be concer-

ned about algorithms and their computing speciﬁcs,

such as the notion of complexity or economy of ope-

rations. In most programming languages, the com-

puting process comprises a series of sequential state

changes executed assignment-by-assignment, which

is an operation absent in math. Moreover, data struc-

tures in CS are inhomogeneous, which spreads the

spectrum of concerns compared with more convergent

mathematical structures (Knuth, 1985), excluding the

data structures of advanced set theory and logic.

Algorithmic thinking has been brought within re-

ach of school or even pre-school children with mul-

tiple initiatives such as (Liukas, 2015). It may be

well taught even without computers, as demonstrated

by the CS-unplugged movement (Taub et al., 2012),

and algorithmic plays (Futschek and Moschitz, 2010).

Puzzles and games can be thought-provoking, thus

this approach is also exploited by a number of univer-

sities in familiarizing students with algorithms (La-

magna, 2015). Unplugging removes the extra cogni-

tive load of programming details.

Data represented and modeled in multiple ways

Multiple external representations (MERs) elucidate

the data and problem from different perspectives. For

example, a function may be represented as an ex-

pression, a curve, a map from argument set to image

set, a table with two columns, or a function machine.

Flexibility in moving from one representation to anot-

her indicates a deeper understanding of the concept

(McGowen et al., 2000), which facilitates problem

solving. Wilkie and Clark denote representational

ﬂexibility as ﬂuency with the order of operations;

commutative, associative, and distributive laws; and

equivalence of expressions (Wilkie and Clarke, 2015).

In programming, representational ﬂuency is practi-

ced, e.g., with the syntactic diversity of operations,

such as addition: x y, x, y , or x y .

Fig. 3 illustrates the use of the MathCheck lear-

ning tool (Valmari and Kaarakka, 2016) in studying

the relationship between textual and tree representati-

ons. Such exercises aim at training the precedence

and left- and right-associativity rules in particular.

The exercises help students to grasp the distinction

between semantics and syntax by differentiating bet-

ween associativity as a semantic notion and left- and

right-associativity as syntactic notions. Furthermore,

the example in Fig. 3 reveals that the relation opera-

tors ( and , and so on) are neither left- nor right-

associative unlike arithmetic operators ( , , and so

on). Consequently, in x y z, the ﬁrst comparison

result is not passed as an argument to the second, but

instead, a Boolean AND is performed on both. Thus,

drawing as a child of , or vice versa, would be

misleading. Being even, the relation operators must

share the root of a tree as Fig. 3 illustrates. This also

makes it explicit that although y occurs only once,

both comparisons use it as an argument.

In problem solving, the ability to model and ab-

stract the data is crucial. USCC speciﬁes Modeling as

CSEDU 2018 - 10th International Conference on Computer Supported Education

160

Figure 3: A tree representation of a model relation chain,

and a failed student attempt to yield a similar tree.

one syllabus area of HS math (Core Standards Orga-

nization, 2017; Core Standards Organization, 2015).

Modeling links to a broader pedagogical idea of using

open-ended problems of everyday life and it combi-

nes skills from math, statistics and technology, and

’ . . . and an ability to recognize signiﬁcant variables

and relationships among them. Diagrams of various

kinds, spreadsheets and other technology, and alge-

bra are powerful tools for understanding and solving

these problems.’ Although modeling, say, a banking

system for implementation as software is fundamen-

tally different from modeling a physical or statistical

problem, the need to recognize and formalize the es-

sential aspects of the problem is common to all of

them. Modeling requires ’speciﬁcational thinking’,

which is necessary for both SW engineers and their

customers in order to reach a common vision, and

describe use cases and requirements pellucidly. In

FNC-2014, phenomenon-based learning approaches

the anticipated open-endedness in problem setting.

Logic

In CS formalization, Dijkstra described its distincti-

veness with a formula (Dijkstra et al., 1989): CS

math logic. In accordance, he called students to

learn formal math and logic to construct a well-

grounded basis for CS. UKNC points out that already

a novice programmer at the elementary level needs

simple Boolean logic, for example, the operators of

AND, OR and NOT, and combinations, see Fig. 4.

In the same context, logic gates in circuits are intro-

duced. This connection between Boolean logic and

logic gates can be used to create a link with electrical

engineering and physics.

Figure 4: Logic in UKNC.

To skim other logic uses, we reviewed ACM

course descriptions. The chief applications were

proofs, correctness, combinational and sequential lo-

gic of state machines, and in addition to these, logic

of knowledge representation and reasoning that tar-

gets translating natural language (e.g., English) sen-

tences into predicate logic statements. Such a skill

would stand out when applying speciﬁcational thin-

king, check page 7.

Sets, statistics, probability

Sets

A’ A

P(A)+P(A’) = 1

n(A) = 1 (only x, in this case)

B={0,1,2,3}

A={x | x∊B, x>2 } descriptive

C={x | x∊B, x<2},D={}, D=∅

visualized by Venn

listed complemented

A⊂B

proper subset

B⊆B

subset

A∩B

A intersects B

A∪B

A union B

A∩B∩C

A B

Figure 5: Sets in UKNC.

The syllabus areas of sets, statistics and probabi-

lity are inter-related at the elementary level, justifying

a combination of these topics. Sets are missing from

FNC-2014, whereas UKNC deﬁnes a functional sub-

set visualized in Fig. 5. Sets (na

ıve set theory) in

UKNC are a gentle kick-start for the set theory, fa-

miliarizing students with different notations, e.g., the

interchangeable use of either a list or a Venn diagram

(excluding some special cases). A number of basic

concepts are introduced, such as a set and its comple-

Elementary Math to Close the Digital Skills Gap

161

ment, a universe, and a subset. Set operations cover

union and intersection.

Building the knowledge base and gaining expe-

rience of these topics may be initiated, for instance,

by collecting data of concrete phenomena, such as

measuring the heights of students of a class and con-

structing a histogram of the heights of the class. Stu-

dents should be capable of reading and interpreting

these charts. For instance, the shape of the height

histogram should resemble the typical bell-shape of a

normal distribution making it timely to introduce the

concepts of mean, median, and mode in this context.

In addition to histograms, the alternative way of re-

semi-interq:

28/2=14

MED

Statistics

illustrated by e.g.

Pie Bar Line

characterized by

Histogram,

frequency polygon

Cumulative frequency

Box and whiskers

Strong positive

correlation

Negative

correlation

Mean<Mode Mode<Mean

Figure 6: Statistics in UKNC.

presenting this information is to construct a cumula-

tive frequency chart, in the UKNC subset visualized

in Fig. 6, the left bottom corner. Ultimately, informa-

tion could be reduced to a box-and-whiskers chart.

Venn diagrams and relative frequency charts

prompt probability issues. The relative frequency of

an event, e.g., which percentage of students are 140–

150 cm tall, provides an obvious scaffold to inves-

tigate the probability of a randomly-selected student

being 140–150 cm tall. In Venn, the bin of 140–150

cm students can represent the set A, where the com-

plement set of A represents all the students not within

this height category. In the universe of this class (or

any other), a selector will get either a student from

the set A or its complement A with 100% probability,

i.e., P A P A 1. In Finnish elementary math,

probability links closely with statistics in the descri-

bed manner. In contrast, UKNC progresses further by

including the multiplication and addition rules (Fig.

7).

The sun either shines or not, no other options ex-

ist. However, if the sun shines, a bird will sing more

probably. A decision tree assists in constructing the

combined probabilities correctly: the multiplication

rule applies horizontally to each branch at a time, and

Figure 7: Probability in UKNC.

the products are added vertically. In a tree, all the

probability branches of one joint must sum up to one.

In preparation for CS and related math courses of

higher-education including sets, statistics, and proba-

bility, UKNC speciﬁes a valid and deliberately plan-

ned math syllabus for an elementary level that could

be emulated as such in FNC-2014.

5.3 The Learning Trajectories Bridged

from Elementary to Higher-ed Math

Fig. 8 divides into four horizontal layers: Elem.math,

CT, HS math, and Tert.math. Elementary school is

compulsory, the rest elective. Vertical dashed lines re-

present the learning trajectories of discrete math pro-

posed in the previous sections. The learning trajec-

tory of algorithms and data structures is marked with

green to illustrate its prominence. Currently, elemen-

tary math does not specify any other learning targets

of algorithms except the need for algorithmic thin-

king. It starts with problem solving and decomposi-

tion, which in programming implies subroutines. Ul-

timately in algorithms, e.g., the simplest sort and se-

arch algorithms were a natural learning goal. Data

structures are prompted by number sets that corre-

spond with variable types; in programming types can

be, e.g., primitives or collections, such as arrays, lists,

and vectors. Structuring data in various ways, mo-

deling and visualizing it, assists in raising the ab-

straction level and in problem solving. The second

most prominent trajectory is logic. Like algorithmic

thinking, logic is included only as a requirement of

logical thinking, except optionally in programming,

where it is needed for the conditions of selection and

iteration structures. The logic subset illustrated in

Fig. 4 is proposed to partly enhance Y7–9 math, phy-

sics, and native language syllabi. To add further value

to this age range, the UKNC syllabus areas of sets

CSEDU 2018 - 10th International Conference on Computer Supported Education

162

Stat. Fig.6

Prob. Fig.7

S E T S Fig.5

L O G I C S Fig.4

A L G O R I T H M I C T H I N K I N G P.7

Fig.3

number

sense

integers

spatial

imagery

problem

solving

coordinates

shapes

reals

comparison

operators

data

collection

charts

rel.fq

med.

mode

multiple representations

statistics

calc.

rnd

truth

values

logical

operators

condition

selection iteration

variable

problem

decomp.

expression

equation

inequality

function

type, data

structure

recur-

sion

higher-

order

func.

container

array

list

vector

Pythag.

trig.

point,line angle

transfor-

mations

graphs

shapes

Turtle

computer

graphics

animations

Algebra (A)

Arithmetic (N)

Geometry (G)

Abstraction

Automation

Y1-2

Y3-6

Y7-9

Statistics

Probability

testing, debugging, optimizing

Analysis

Logic(L)

Creativity(C)

Elem.math

HS math

Tert.math

course description

MAB5 statistics and probability

(MAB8) statistics and probability II

MAA10 probability and statistics

combinatorics

course description

MAY1 numbers,sequences

MAB4 modeling,patterns

MAB6 commercial math

MAY1 numbers,sequences

MAA8 root,log functions

(MAA11) number theory,proofs Euclidean alg.

(MAA12) algorithms in math Newtonian alg.

course description

MAB2 expression,equations

(MAB7) math.analysis

MAA2 polynomials

MAA6 derivative

MAA9 integral calculus

(MAA13) diff.equations

course description

MAB3 geometry

MAA3 geometry

MAA4 vectors

MAA5 analytic g.

MAA7 trigonometry

statistics,

prob.

set theory

predicate

logic

algorithms,

data structs

calculus

differential

equations

Discrete Math

Continuous Math

math skills for modern SW engineers

time, resources

Figure 8: Hypothetical learning trajectories bridged from the FNC-2014 elementary to higher education math.

(Fig. 5), statistics and probability (Fig. 6 and 7) were

worth considering in descending order of importance.

However, due to time constraints, adding content to

the math syllabus is problematic. CS, as a separate

subject, would solve the problem. Below elementary

math, the CT layer illustrates the computing enhance-

ment and how the process divides into abstraction, au-

tomation, and analysis phases. In this layer, the math

fundamentals have their computational counterparts.

The math schedule (Y7–9) implies an appropriate in-

troduction order of corresponding CS fundamentals.

The next layer of high school (HS) math is

elective. It divides into A and B math: A is the ma-

jor (ten compulsory MAA* courses, four electives), B

being the minor subject (six compulsory MAB* cour-

ses, two electives) (Finnish National Board of Edu-

cation, 2015). Regrettably, the HS math rigidly tar-

gets the matriculation exam, whose importance has

lately grown as a selection criteria for tertiary educa-

tion, thus extending the CT layer to cover the HS level

is not topical. In HS, algorithms are introduced only

in the elective courses of ’Number theory and proofs’

(MAA11), and ’Algorithms in math’ (MAA12). Clo-

sest to logic is the elective MAA11 with conjunctives

and truth values. In regard to the remaining trajecto-

ries, sets are missing from the FNC-2014, both at the

elementary and HS, whereas the situation of statistics

and probability is brighter. They start already at the

elementary, and HS allocates three courses to the to-

pic: MAB5, MAB8, MAA10. Tertiary math elucida-

tes the required skills for modern SWE by represen-

ting the most prominent topics only.

6 CONCLUSIONS

RQ1: Math syllabus areas to be strengthened? Ac-

cording to the reviewed studies, SW engineers need

stronger algorithms and data structure skills. In ac-

cordance, ﬂuency with multiple representations and

Elementary Math to Close the Digital Skills Gap

163

modeling is considered beneﬁcial in illustrating and

structuring data, thus improving problem solving

skills. To further strengthen the theoretical basis ne-

cessitates the inclusion/teaching of primarily logic,

and secondarily set theory, statistics, and probability.

In increasing discrete math, the UKNC math and

CS provide an exemplar to emulate in elementary

education in Finland. USCC deﬁnes HS Modeling

for structuring data, and the area could be subset age-

appropriately for the elementary level. Modeling as-

sociates also with the use case/requirement speciﬁca-

tions of SWE, which prompts the new term of ’speci-

ﬁcational thinking’.

However, discrete math does not beneﬁt only fu-

ture SW engineers, but all students in becoming gene-

rally educated and acquainted with CS. Even though

continuous and discrete math are posed as opposite, in

practice, they are deeply interconnected and comple-

ment each other. Natural sciences continue to exploit

continuous math as before, so continuous math must

keep a signiﬁcant role in the curriculum. However, to

meet the challenges of digitalization, we believe that

it is appropriate to move some emphasis from conti-

nuous to discrete math.

RQ2: The overemphasized math syllabus areas?

Curriculum planning is a zero-sum game. If the

volume of discrete mathematics were increased, some

areas ought to be decreased correspondingly. The pro-

posal is to move some emphasis from continuous to

discrete math already at the elementary level. For all

the intended content the current time allocation is ex-

iguous, which is why adding CS as a separate subject

is a distinct option.

Further studies

The shifting of more emphasis to discrete math must

be executed in an evidence-based manner, i.e., the le-

arning outcomes must be carefully evaluated in co-

operation with pedagogical experts both in elemen-

tary and higher education. To advance this approach

further, the results should speak for themselves. To

achieve the full potential of discrete math in higher-

education, traditional ’Advanced Engineering Calcu-

lus’ would need its discrete math counterparts, say,

’Programmers’ Introduction to Automata and Formal

Languages’ or ’Set Theory for Software Engineers’,

which indisputably explicate the beneﬁts of the rene-

wed math syllabus for the good of programming.

Is the time ripe for the next Lethbridge iteration to

evaluate the current topics of the SWE curriculum?

ACKNOWLEDGEMENTS

Thanks to the Academy of Finland (grant number

303694; Skills, education and the future of work) for

their ﬁnancial support.

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