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On Mathematical Reasoning
- being told or finding out
Mathias Norqvist
Department of Mathematics and Mathematical Statistics
Umeå 2016
This work is protected by the Swedish Copyright Legislation (Act 1960:729)
ISBN: 978-91-7601-525-4
ISSN: 1102-8300
Electronic version available at http://umu.diva-portal.org/
Printed by: Print & Media, Umeå University
Umeå, Sverige 2016
I love deadlines.
I like the whooshing sound
they make as they fly by.
- Douglas Adams
i
Table of Contents
Table of Contents i!
Abstract iii!
Sammanfattning v!
Acknowledgements vii!
List of papers ix!
1!Introduction 1!
1.1!Aim 2!
2!Mathematics in school 3!
2.1!Syllabi 3!
2.2!Rote learning 3!
2.2.1!Textbooks 4!
2.2.2!Teaching 5!
3!Theoretical frameworks 7!
3.1!Theory of didactical situations in mathematics 7!
3.2!Productive struggle 8!
3.3!Problem solving 10!
3.4!A framework for mathematical reasoning 12!
3.4.1!Reasoning sequences 12!
3.4.2!Creative Mathematically Founded Reasoning 13!
3.4.3!Algorithmic Reasoning 14!
3.5!Cognitive demand 16!
4!Memory and cognition 17!
4.1!Individual variation in cognition 17!
4.1.1!Working memory 17!
4.1.2!Fluid intelligence 18!
4.1.3!Cognitive tests 18!
4.2!Observing brain activity 19!
4.2.1!The technique behind an fMRI image 20!
4.2.2!A downside to fMRI 21!
4.2.3!fMRI and Mathematics 21!
4.2.4!Brain functions connected to mathematics 22!
4.2.4.1!Memory retrieval 22!
4.2.4.2!Novel and complex tasks 23!
4.2.4.3!Calculation and number sense 23!
4.3!Summary 24!
5!Method 25!
5.1!Task design 25!
5.1.1!Design by researchers 25!
5.1.2!Authentic, design by teachers 29!
5.2!Data collection 29!
5.2.1!Behavioral-experiments 29!
ii
5.2.2!fMRI-experiment 30!
5.2.3!Observations 31!
5.3!Methods of analyses 31!
5.3.1!Statistical 32!
5.3.2!Qualitative analysis 32!
6!Summary of the articles/Result 34!
6.1!Study 1 – Learning mathematics through algorithmic and creative
reasoning 34!
6.2!Study 2 – The affect of explanations on mathematical reasoning tasks 35!
6.3!Study 3 – Learning mathematics without a suggested method: Durable
effects on performance and brain activity 36!
6.4!Study 4 – Unraveling students’ reasoning: analyzing small-group
discussions during task solving 37!
6.5!Additional result 39!
7!Discussion 41!
7.1!How will the task design influence students’ solutions process,
mathematical reasoning, and brain activity? 41!
7.2!How will students’ cognitive variation affect their solution rate and does
task design matter? 43!
7.3!How can these results influence teaching practice? 43!
7.4!Limitations and implications for further studies 45!
7.5!Conclusion 47!
8!References 49!
iii
Abstract
Background
Research has for many years pointed out the inefficiency of rote-learning and
the importance of regarding concepts and mathematical properties while
struggling with mathematics tasks (e.g., Hiebert, 2003; Schoenfeld, 1985;
Stein, Grover, & Henningsen, 1996). From a theoretical viewpoint, Brousseau
(1997) suggested that students have to consider such important aspects while
constructing solutions by themselves and that teachers have to develop
situations where this is possible for the students. The added effort needed
from the students could however be cognitively demanding to the point that it
will be overwhelming, in particular for cognitively less proficient students.
Therefore, students’ cognitive abilities are important to consider when
constructing tasks or didactical situations. The aim of this thesis is therefore
to examine how task design and students’ cognitive abilities will influence
students’ mathematical reasoning, student outcome and students’ brain
activity.
Methods
Three of the four included studies are done with a between-groups design
where data is analyzed statistically to search for significant differences in test
results between the different practice conditions. In these studies, practice
tasks were designed by researchers to promote special types of reasoning
(algorithmic reasoning (AR) and creative mathematically founded reasoning
(CMR)) and in one study an explanation on why the solution method is
working was also provided. The practice data from these three studies are also
analyzed as an additional result, not part of the included studies. The last
study was based on observations of students work on tasks designed by
teachers to unravel how student reasoning evolves during the solution
process. Here we transcribed audio recordings from four student-groups’
when they solved tasks constructed by their teacher. We then coded the
solution process by utilizing the framework on mathematical reasoning
suggested by Lithner (2008).
Results
The overall results suggest that creative tasks are more effective than
algorithmic tasks when it comes to memory retrieval and reconstructing
practiced solution methods. There are also clear indications that AR is more
taxing on cognitive abilities during the test than creative tasks (where practice
performance seems to be more important). During practice the dependence of
cognitive abilities is however higher when working with creative tasks.
Furthermore, task design is important for which type of reasoning that the
student will use. However, student-group characteristics (i.e., motivation and
iv
persistence) are also important both when choosing reasoning type and for
task-progression.
Conclusion
Since mathematics students spend a lot of time doing tasks it is important to
study these tasks from a learning perspective. The results in this thesis points
to a few important issues regarding task design and the result of different
types of reasoning. First, since creative reasoning seems to be more effective
than algorithmic reasoning, it would be good for students to encounter more
of this type of task in textbooks as well as in teacher presentations. Second,
cognitive abilities are important for mathematics but there is a difference
where the student’s cognitive abilities are taxed (i.e., algorithmic reasoning
will put higher strain on cognition during the test while creative reasoning will
be highly demanding during practice). This difference in cognitive strain
seems to be related to a deeper encoding during creative practice than during
algorithmic practice. CMR also seems to be more beneficial than AR for
cognitively less proficient students. While the teacher can reduce students’
cognitive load by for instance directing focus to the important properties
during practice, this may not be done during tests (at least not to the same
extent). Third, even though algorithmic tasks do not prohibit the use of
creative reasoning, it is much less likely to occur than algorithmic reasoning.
To ensure that creative reasoning will take place, the task must be designed
for this purpose.
Since creative tasks can put focus on one or more important mathematical
properties and provide deeper understanding than algorithmic tasks,
implementation in school practice can be essential if we want students to
become mathematically literate.
v
Sammanfattning
Bakgrund
Forskning har under många år pekat på ineffektiviteten av utantill-lärande
och vikten av att dels reflektera över koncept och matematiska egenskaper och
dels bli tvungen att kämpa med matematikuppgifter (se t.ex. Hiebert, 2003;
Schoenfeld, 1985; Stein, Grover, & Henningsen, 1996). Brousseau (1997)
föreslog från ett teoretiskt perspektiv att elever måste överväga sådana viktiga
aspekter medan de konstruerar egna lösningar och att lärare bör utforma
situationer där detta är möjligt för eleverna. Den extra ansträngning som
eleverna måste lägga ner kan dock vara kognitivt belastande till den grad att
det blir överväldigande, speciellt för elever med lägre kognitiv kapacitet.
Därför är elevers kognitiva kapacitet viktigt att beakta när man konstruerar
uppgifter eller didaktiska situationer. Syftet med denna avhandling är därför
att undersöka hur uppgiftsdesign och kognitiva färdigheter kan inverka på
elevers matematiska resonemang, testresultat och hjärnaktivitet.
Metoder
Tre av de fyra inkluderade studierna är genomförda med en mellangrupps-
design där datamaterialet analyserats statistiskt för att hitta signifikanta
skillnader i testresultat mellan olika träningsgrupper. I dessa studier hade
forskarna designat uppgifter som skulle främja olika typer av matematiska
resonemang (algoritmiska resonemang (AR) och kreativa matematiskt
grundade resonemang (CMR)). I en studie fanns även en förklaring tillgänglig
som beskrev varför lösningsmetoden fungerade. Träningsdata från dessa tre
studier har också analyserats som ett ytterligare resultat, utanför de
inkluderade studierna. Den sista studien baseras på elevers arbete med
uppgifter i klassrummet för att reda ut hur elevers resonemangssekvens
utvecklas under lösningsprocessen. Vi transkriberade ljudinspelningar från
fyra elevgrupper när de löste uppgifter som konstruerats av deras lärare.
Sedan kodade vi lösningsprocessen genom att tillämpa Lithner’s (2008)
ramverk om matematiska resonemang.
Resultat
Det övergripande resultatet visar att kreativa uppgifter är effektivare än
algoritmiska vad gäller minne och rekonstruktion av tränade lösnings-
metoder. Det finns också klara indikationer på att AR ställer högre krav på
kognitiva färdigheter under testen än vad CMR gör (där träningsresultatet är
viktigare). Under träningen är däremot kravet på kognitiva färdigheter större
när man arbetar med kreativa uppgifter. Vidare är uppgiftsdesignen viktig för
vilken typ av resonemang studenterna kommer att använda. Studentgruppens
egenskaper (d.v.s. motivation och envishet) är också viktigt, både för vilken
resonemangstyp eleverna väljer samt för progressionen i uppgiften.
vi
Slutsats
Eftersom matematikelever använder mycket tid till att lösa uppgifter är det
viktigt att studera dessa uppgifter från ett inlärningsperspektiv. Resultatet i
denna avhandling pekar ut några viktiga saker vad gäller uppgiftsdesign och
resultat av olika typer av resonemang: 1) Eftersom CMR verkar vara
effektivare än AR så vore det bra om eleverna mötte mer kreativa uppgifter i
såväl läroböcker som i lärares presentationer. 2) Kognitiva färdigheter är
viktiga för matematik, men det är skillnad när elevernas kognitiva färdigheter
belastas, d.v.s. AR är mer belastande under testen medan CMR ger högre
belastning under träningen. Denna skillnad i kognitiv belastning verkar bero
på en djupare inkodning under den kreativa träningen än under algoritmisk
träning. Dessutom verkar de lågpresterande eleverna dra mer nytta av CMR
(jämfört med AR) än de högpresterande eleverna. Även om läraren kan
reducera den kognitiva belastningen genom att exempelvis rikta fokus mot de
viktiga egenskaperna under träning så kan läraren inte göra detta under ett
test (åtminstone inte i samma utsträckning). 3) Även om algoritmiska
uppgifter inte förhindrar CMR är det mindre sannolikt att detta skulle
förekomma än AR. För att säkerställa att CMR ska ske måste uppgiften vara
designad för det.
Eftersom kreativa uppgifter kan sätta fokus på en eller flera viktiga
matematiska egenskaper, samt ge en djupare förståelse än algoritmiska
uppgifter så är det nödvändigt att omsätta dem i skolpraktiken om vi vill att
våra elever ska bli förtrogna med matematik.
vii
Acknowledgements
Året var 1971. En ung kvinna klev in i en taxi för att åka till förlossningen, ca
130 km bort, för att få sitt första barn. Barnet i fråga beslutade att en så lång
resa inte alls kom på fråga, så taxichauffören blev tvungen att svänga in vid
den lilla sjukstugan i det lilla samhället i Västerbottens inland, där en liten
pojke sedan föddes. Ungefär 39 år senare påbörjade den pojken, nu själv far
till en son och med en till på väg, en intellektuell resa som han varken hade
planerat eller packat för. Precis nu, cirka sex och ett halvt år senare, börjar den
första delsträckan av denna akademiska resa närma sig sitt slut. När jag tänker
tillbaka på mina år som doktorand finns det förstås en massa människor som
hjälpt och stöttat mig på många sätt och jag ska, så gott jag förmår, försöka
tacka er alla här.
Redan innan jag blev antagen som doktorand fick jag vara med i uppstarten
av projektet Att lära matematik med imitativa och kreativa resonemang och
fick umgås med två av mina blivande handledare. Det slog mig då att båda två
tog vara på mina idéer även om jag var en nykomling. När jag sedan blev
antagen blev handledarna tre och de har kompletterat varandra på ett utmärkt
sätt. Jag har lärt mig massor av er och uppskattar verkligen att ni tagit er tid
att följa, ledsaga och stötta denna inbitne slöfock så att avhandling och
forskningsprojekt blivit av och blivit bättre. Johan Lithner, Bert Jonsson och
Nina Rudälv, jag tackar er från djupet av mitt hjärta.
När forskarskolan startade mötte jag ett antal människor i samma läge som
jag. Tillsammans har vi läst, funderat, analyserat och rest världen runt. Jag
vill tacka er alla för ett trevligt resesällskap i såväl flyg och vulkanbuss som
avhandlingsresan. Jag vill här även passa på att tacka den dåvarande
ledningen i forskarskolan, P-O Erixon och Carina Rönnqvist, som stöttade alla
oss vimsiga höns, nej doktorander, genom de första åren.
Jag blev även en del av Umeå Forskningscenter för Matematikdidaktik
(UFM) och fick lära känna alla de trevliga och roliga människorna där. Hela
UFM har varit oerhört värdefulla men jag vill rikta ett speciellt tack till Tomas
Bergqvist som snabbt informerade mig om att jag absolut inte ska jämföra mig
med Johan vad gäller effektivitet. Alla ni övriga i UFM ska veta att ni också
gjort ett intryck på mitt liv och min forskning. Tack ska ni ha!
As I started working at the math-department I shared workroom with a
woman that also had an impact on my view on research. Therefore, I would
like to thank Gulden Karakok for her encouragements during my first year as
a PhD-student and for her patience with my, at the time, quite rusty English.
När Gulden lämnade Umeå så flyttade ett blivande extrasyskon in i
arbetsrummet. Vi har under många år delat både rum och målsättning och nu,
så här på upploppet, så har vi båda slitit vårt hår över kappor och artiklar. I
vårt rum har vi avhandlat allt från barn, månggifte, resfeber, träning,
matlagning och undervisning, till avhandlingar och artiklar. Vi har både läst
viii
och undervisat kurser tillsammans och jag hoppas verkligen att vi kan
fortsätta med det i många år framöver. Anneli Dyrvold, du har varit en klippa
och jag är djupt tacksam för att jag fick göra denna resa med dig.
Jag tackar även ’fikagänget’ och alla andra på institutionen som kommit
med glada tillrop under min forskarutbildning. Det är ni som är skälet till att
jag trivts så bra med mitt jobb de senaste 6 ½ åren. Jag vill dock rikta ett
speciellt tack till Lisa Hed och Berit Bengtson som varit ett extra stort stöd
under slutet av mitt avhandlingsarbete.
Hela projektgruppen förtjänar också ett stort tack. Förutom Johan och Bert
vill jag speciellt tacka Carina Granberg, Janne Olsson och Tony Qwillbard för
många nyttiga diskussioner och ett bra samarbete som jag också hoppas ska
fortsätta. Jag vill även tacka Moa Eirell för hennes värdefulla hjälp med
datainsamlingen till min egen studie och Hendrik van Stenbrugge som jag
stångats med under den långa process som mynnade ut i avhandlingens sista
artikel. Nu skulle man kunna tro att jag har glömt Yvonne Liljekvist, min sol i
Karlstad som har varit en mycket god vän och hjälp under min doktorandtid,
men så är inte fallet. Yvonne, du har varit en värdefull stöttepelare under dessa
år och vi kommer säkert att fortsätta vårt samarbete när jag kliver på nästa
delsträcka (vad det än må vara).
Den unga kvinnan som jag nämnde i början och den unge man som då var
delvis ansvarig för situationen förtjänar givetvis ett enormt tack. Britt-Marie
och Börje, mamma och pappa, utan er hade inte denna bok blivit skriven. Jag
passar även på att skicka en liten hälsning till min syster Anna och hennes
familj. Ni tillför glädje i mitt liv. Ett tack riktas också till grabbarna i EyeWood
som varit mitt mentala andningshål under alla dessa år.
Slutligen vill jag tacka de som är viktigast i mitt liv, min älskade Maria och
våra söner, Andreas och Viktor, som knappt träffat mig under denna intensiva
sommar. Maria, jag kan inte på något sätt tacka dig nog för att du 1) drog med
mig till Umeå och in på en lärarutbildning, 2) tyckte att det var ok att jag gick
ner i lön samt förlorade min ferietjänst när jag blev doktorand och 3) tagit
ansvar för och roddat allt under senvåren och sommaren medan jag lätt
panikartat jobbat för att få denna bok i tryckfärdigt skick. Jag lovar att
gottgöra dig på något sätt…om inte annat om ca. fem år när det är din tur. !
Du är underbar!"
Om du som läsare tycker att jag har glömt bort dig så är det inte med
mening. Jag skyller i så fall på tillfällig afasi och hoppas att även du tar del av
min stora tacksamhet. Tack allihop!
Sävar, augusti 2016
I may not have gone where I intended to go,
but I think I have ended up where I needed to be.
Douglas Adams
ix
List of papers
I. Jonsson, B., Norqvist, M., Liljekvist, Y., & Lithner, J. (2014). Learning
mathematics through algorithmic and creative reasoning. The Journal
of Mathematical Behavior, 36, 20-32.
doi:10.1016/j.jmathb.2014.08.003
II.
Norqvist, M. (2016). Do explanations increase the efficiency of
procedural tasks?. Manuscript in preparation.
III. Wirebring, L. K., Lithner, J., Jonsson, B., Liljekvist, Y., Norqvist, M., &
Nyberg, L. (2015). Learning mathematics without a suggested solution
method: Durable effects on performance and brain activity. Trends in
Neuroscience and Education, 4(1-2), 6-14.
doi:10.1010/j.tine.2015.03.002
IV. Van Steenbrugge, H., & Norqvist, M. (2016). Unraveling students’
reasoning: analyzing small-group discussions during task solving.
Manuscript in preparation.
x
1
1 Introduction
Mathematics is one of the few school subjects that to its content does not differ
that much across the world. The sum of two and three is always five1 and if you
differentiate the function ! " # "$%&'()*"+ you will end up with the
derivative !," # -" % &'( " . "$%/0&)*"+ regardless of which country you
are educated in. There is however much difference regarding the way this
uniform subject is taught. Teachers are more or less directed by syllabuses,
textbooks are used to differing extent, and more or less responsibility is left to
the student. What is common for all mathematics teaching is however the use
of mathematics tasks to practice and hopefully learn the mathematics that will
be used further on in both higher mathematics and everyday life. In this thesis
the starting point will be the tasks and relating to them both design, learning
outcome, and influence on student work and brain activity will be studied. As
mathematics tasks are used to such a large extent, the impact and efficiency
of these tasks are important to study and understand.
In this thesis I will begin with a short description of school-mathematics by
discussing the content of the syllabi and comparing this to teachers’
presentations and textbook content. This is done to set the stage for a
discussion about task design and its influence on the mathematical reasoning
that students choose or are directed to. Students use a lot of time practicing
by doing tasks and their reasoning during this task solving process will affect
their learning. Learning by rote is quite common in schools all over the world,
and might, if dominating, be one problem when trying to learn mathematical
theories or heuristics as it implies that things are learned without reflection.
When discussing mathematics teaching and the classroom work, Brousseau’s
(1997) Theory of Didactical Situations in Mathematics offers insight in which
roles students and teachers could have in the classroom to increase problem
solving activities and encourage students’ own knowledge construction. How
this should be acted out in the classroom can of course be discussed
extensively, but I will give an overview on some previous findings that will lead
to the framework of mathematical reasoning by Lithner (2008), which is at
the center of all the papers in the thesis.
The framework defines two major types of reasoning, imitative and
creative. Imitative reasoning is closely related to rote learning while creative
reasoning is based on students’ own construction of solutions and therefore
more connected to understanding relations and justifying choices on a more
mathematically fundamental level. Both types will be discussed more
extensively later on in this thesis.
When introducing tasks that are more cognitively demanding, individual
variation in cognitive abilities can be crucial. There are many cognitive
1 Provided that the calculations are made in a base ≥ 6 and modulo 5 or higher.
2
abilities that could influence mathematics learning but working memory and
non-verbal problem solving ability are two that has been proven to be closely
related to mathematical achievement (e.g., Primi, Ferrão, & Almeida, 2010;
Swanson & Alloway, 2012). These two cognitive constructs are used to match
participants into equally proficient groups before our experiments, and in the
following analyses the cognitive measures will contribute to the results. The
cognitive constructs will be discussed later on in the thesis and at this time
functional magnetic resonance imaging (fMRI) will also be addressed. fMRI
has been used to help make connections between active brain regions and
cognitive processes and this is also what it is used for in one of the papers in
this thesis. The connection between different cognitive processes and
mathematical reasoning is important for the results of our experiments and
the conclusions that can be drawn from them.
1.1 Aim
The overall aim for the thesis is to tie all these perspectives together and
build on previous research to extend the knowledge on how task design,
mathematical reasoning and cognitive abilities can affect the learning of
mathematics. This will be done by comparing and combining the results from
the four included papers to discuss the following questions.
1) How will the task design influence students’ solution process,
mathematical reasoning, and brain activity?
2) How will students’ cognitive variation affect their solution rate and does
the task design matter?
3) How could these results influence teaching practice?
3
2 Mathematics in school
Mathematics is a global school subject that has become one of the measures
for school achievement, with tests like PISA and TIMSS. Newspapers have
reported on decline or rise in results of these tests, both over time and between
countries, and this has put some focus on mathematics education worldwide.
In Sweden, decline in PISA results intensified the political debate on how
education should be governed and executed. This debate included the syllabus
for mathematics as well as for education in general both in primary and
secondary school.
2.1 Syllabi
In Swedish syllabi, up until the early 90’s, the aim of school mathematics was
to prepare students for the every-day life with a focus on ability to perform
necessary calculations (e.g., Skolöverstyrelsen, 1970, 1980). The last three
syllabi, from 1994, 2000 and 2011, have changed this focus from pure
calculation to include other abilities as well (e.g., Skolverket, 2000;
Skolverket, 2011a, 2011b; Utbildningsdepartementet, 1994). Since 2000,
seven competencies (i.e., procedures, reasoning, problem solving, modelling,
communication, conceptual understanding, and relating mathematics to the
surrounding world) have been explicitly defined in the syllabus. This change
towards mathematical competence has been seen in other countries as well.
The international movement towards a more competence focused curricula
was in large initiated by the National Council of Teachers of Mathematics
(NCTM, 1989, 2011). The Danish KOM-project defined eight mathematical
competencies that education in mathematics should enhance (Niss & Jensen,
2002). The shift of focus from pure calculating skills to a broader
mathematical competence could be important for mathematical proficiency in
every-day life, as many occupations demand other competencies as well (e.g.,
problem-solving skills and modelling). However, these competencies are not
mutually excluding. Calculating skills and rote learning of certain facts or rules
are also important for an efficient problem solving process, since the focus can
be put on the problem at large instead of each small item that need to be
processed or calculated.
2.2 Rote learning
A lot of things in mathematics are memorized for quick and effortless retrieval
and application when needed. For example, the multiplication table and the
order of operations can be two important things to have quick access to. But
this memorization can also become a hindrance if rote learning is dominating
and if it occurs without understanding. Rote learning can be defined as a
mechanical and habitual repetition of the learning object. If the student lacks
4
understanding of the memorized rules and methods (i.e., why and how the
rule works and is valid) it can be difficult when trying to get a more conceptual
view on mathematics. Also, when rules become too numerous to keep track of
there can also be difficulties if no connection to mathematical properties is
made (e.g., learning integration only by rules is hard as sometimes many basic
techniques have to be used simultaneously). When rote learned rules or facts
become the main knowledge, students will not be able to solve tasks with the
slightest variation from the ordinary (Hiebert, 2003). Hiebert (2003) went so
far as to compare students that had learned mathematics mainly by rote to
robots with poor memory, expressing that one could predict their errors just
by erroneously recall rules or algorithms. This view is shared with Schoenfeld
(1985) that states that the earlier focus on mechanical skills produced dismal
results when students were challenged by more complex problems.
Schoenfeld (1985) found that students tend to use only a small proportion of
their total solution time on analyzing the problem. Instead they rush into a
solution process without good strategies and towards a certain failure. Experts
are more flexible in their problem solving. They use more time for analyzing
the problem and are more prone to revising their choice when they get stuck
(Schoenfeld, 1985). Boaler (1998) demonstrated in a longitudinal study that
students that were taught mathematics with more emphasis on rote learning
did not view mathematics as important in their daily life. These students also
expressed that mathematics was boring, complicated and useless. Boaler’s
other group in this study, which were taught mathematics in a more project
based way (i.e., with more emphasis on mathematical concepts and
construction), expressed a more positive view on mathematics as a useful and
important subject.
Although syllabi have shifted towards a more competence oriented aim,
mathematics textbooks, as we will see in the next section, often stress
algorithmic skills.
2.2.1 Textbooks
Textbooks are used throughout the world to provide students of all ages with
tasks to solve and often also instruction on how this should be done. The high
proportion of classroom time spent reading or solving tasks from textbooks
might vary to some degree, but in most mathematics classrooms, textbooks
are used as a source of information and practice tasks (Mullis, Martin, Foy, &
Arora, 2012; Wakefield, 2006). However, textbooks often tend to send an
implicit message that the focus of mathematics is to swiftly perform
mechanical computations of correct answers rather than to encourage a
conceptual learning of mathematical structures (Lithner, 2004; Newton &
Newton, 2007; Shield & Dole, 2013). Many textbooks also include more tasks
then is reasonable for a student to complete within the frame of the course
5
(Jäder, Lithner, & Sidenvall, 2014). It is also apparent that many of the more
demanding tasks are located at the end of each chapter (Jäder et al., 2014).
This implies that a selection must be done and if students are supposed to
make this selection of tasks they tend to choose the first task sets (Sidenvall,
Lithner, & Jäder, 2015), which increases the proportion of routine tasks
further. In a cross-national study of textbooks from twelve countries Jäder et
al. (2014) concluded that 79% of the tasks could be solved completely by
imitating or following given instructions while only 9% of the tasks required
more extensive conceptual knowledge and justification. If the textbooks
mainly promote algorithms and rote learning then the mathematical
foundation and conceptual knowledge will, most probably, not be developed.
Here the teacher has an important mission to fill the gap between mechanical
calculations often presented in the textbooks and the conceptual
understanding and competencies that the syllabi often calls for.
2.2.2 Teaching
In a study on 200 teachers’ implementation of reform-based syllabi, Boesen
et al. (2013) found that there was an emphasis on procedural activities during
mathematics lessons both in teacher presentations and during students
individual (or small group) work. Focusing on the teacher presentations,
Bergqvist and Lithner (2012) concluded that routine tasks and simplifying
explanations were commonly used. Teachers often used quite thorough
explanations when presenting new topics but often without verifications or
connections to intrinsic mathematics (Bergqvist & Lithner, 2012). It seems
logical, that if we want students to become skilled at solving novel tasks and
at justifying their solution methods, teachers must demonstrate this during
presentations. Studies have shown that students often motivate their choice
of solution methods by looking at superficial properties and poorly memorized
algorithms (Bergqvist, Lithner, & Sumpter, 2008; Hiebert, 2003; Lithner,
2000, 2003). Stein et al. (1996) also saw this behavior among the students
that they studied. Many students preferred to use known procedures even if
the procedures did not fit the task at hand. Stein et al. (1996) also concluded
that many tasks lost their challenging quality due to poor help from the
teacher or a shifted focus towards the correctness of the answer.
An important factor for the the task to maintain its complexity and its
conceptual challenge is the type of help the teacher provides the students with.
Henningsen and Stein (1997) conclude that teachers that select appropriate
and worthwhile tasks, press for justifications, and “support students’ cognitive
activity without reducing the complexity” will help students to reach further.
The teacher also has an important role in showing the class what high-level
performance should look like and in giving appropriate time constraints to the
tasks (Henningsen & Stein, 1997; Stein et al., 1996).
6
If teachers want students to become problem solvers and students expect
to learn an algorithm or simple rules there is a problem. Add to this that
algorithms, although effective and secure, according to Brousseau (1997) are
designed to avoid meaning and there won’t be that much room left for the
mathematical concepts and properties that teachers probably wish to
communicate.
7
3 Theoretical frameworks
This thesis has the aim to analyze how task design and mathematical
reasoning can affect the learning of mathematics. To do this there is a need to
frame it with theories and frameworks that are relevant for the coming
analysis. I will do so by starting off with Brousseau’s (1997) theory of didactical
situations, which lurks at the center of most of the included studies in this
thesis. Brousseau’s thought that students need to take responsibility of the
task solving process to learn the intended knowledge has also been discussed
by others (e.g., Bjork & Bjork, 2011; Hiebert & Grouws, 2007; Jonsson,
Kulaksiz, & Lithner, 2016) as the importance to struggle with central
mathematical concepts. This struggle can of course be accomplished in
different ways and engaging in problem solving is one of the ways that have
been studied and discussed extensively ever since Pólya (1945) wrote his
famous book ‘How to solve it’. However, OECD (2015) concluded that teachers
in average have seven hours a week to spend on lesson preparation and as
Blum and Niss (1991) infer, mathematics teachers are afraid that problem
solving will take too much time. They also mention that problem solving can
be viewed as a challenging and slightly overwhelming project to embark on for
many teachers, since additional non-mathematical knowledge is necessary. A
slightly more reasonable effort could instead be put into constructing or
adjusting ordinary tasks so that they require more justification and conceptual
understanding rather than procedural skills. One way of doing this is to apply
Lithner’s (2008) framework on mathematical reasoning to task design. This
framework will be described and the types of reasoning that Lithner suggests
will be defined, together with a new type that is more connected to task design
than to student reasoning. Later on Lithner’s framework will also be
connected to student cognition, as tasks’ cognitive demand may differ as Stein
et al. (1996) suggests. This will then lead us into chapter 7 that describes
cognition and its influence on mathematics education.
3.1 Theory of didactical situations in mathematics
Learning mathematical concepts and strategies, to be able to construct or
reconstruct solution methods if the old ones are insufficient or forgotten, is a
major goal in mathematics learning, but as algorithms often are provided the
aim instead becomes to memorize and make use of them (Hiebert, 2003)
In Brousseau’s (1997) Theory of Didactical Situations (TDS) one of the
central ideas is that if students are to learn mathematics they have to construct
the key concepts by themselves. Brousseau (1997) argued for a task or problem
design where the students have to construct at least some part of the solution
by themselves as this is central for a learning that goes beyond mere
memorization of a method.
8
However, Brousseau argues that much of the work in classrooms is done
interacting with teaching materials, peers, and the teacher. The interaction
with this milieu can, if done poorly, destroy the learning opportunity, for
example by having the solution method given in the textbook or by having a
peer telling the answer. Thus, Brousseau took into account that the milieu was
important to relate to when discussing the learning of mathematics.
For the student to be able to go beyond given algorithms the teacher must
arrange the devolution of a good problem (Brousseau, 1997). This requires
that the student gradually takes responsibility for the solution process which
ends with the construction of a justified solution. If this devolution occurs the
student will enter an a-didactical situation devoid of the teacher’s didactical
intentions and where the teacher is separated from both the student’s progress
and learning. Here the responsibility for solving the task falls completely on
the student and the teacher has to release control of the solution and the
learning to the student. For this devolution to take place a mutual relationship
that states what responsibility the student will have during this process and
what the teacher’s duty is will implicitly be agreed upon (e.g., what effort the
student should give the problem before calling for help or what kind of support
the teacher will give when called for). This informal and often non-spoken
agreement is called a didactical contract (Brousseau, 1997). Brousseau
underlines the shared responsibility of teacher and student if the contract is
broken. The teacher has the responsibility for the results by designing solvable
problems that give rise to a natural a-didactical situation. Simultaneously, the
student has to accept a problem solving situation where the solution method
has not yet been taught. This can be hard for some students to accept at first
as many of them are used to apply given algorithms to solve tasks (Hiebert,
2003), not having to struggle with mathematical properties or concepts.
3.2 Productive struggle
Imagine for a moment that you visit a friend in a city where you have not been
before. She picks you up at the station and you walk together, first to her
apartment and then to a restaurant to eat dinner. All the time, as you walk
through the city, you talk and have a wonderful time. Suddenly you realize that
you have no idea where you are, no idea of how you got there or how you would
find your way back if your friend suddenly would leave. If you took this walk
with your friend every day for a week you would probably learn the way even
though you don’t know anything else about the city. Now, imagine that you
come back a year later, the restaurant has moved and so has your friend. How
will you find the way? Well with a map of course. This could be a little tricky
and you’d have to notice more things (e.g., architectural markers and street
names). It could be a bit of a struggle but eventually you’ll get to the right
9
place. After finding your way this time you would be more confident when
visiting this particular city again.
This every-day example illustrates the importance of the need to struggle
with central objects or concepts. There is however an important difference
between a struggle and a productive struggle. The former could be achieved
by having students doing mathematics in a dark room or by giving the
students extremely difficult tasks. This struggle would be considered un-
productive or undesirable. Tasks that impose a surmountable productive
struggle with intrinsic mathematical ideas may give a more lasting impression
or knowledge. This is exactly what happens when students are working with
well-designed CMR-tasks (Jonsson et al., 2016). The student will have to
struggle since there is no apparent way of solving the task and if the task
design is good this struggle will not be overwhelming and negative but rather
manageable and positive for learning.
The importance of struggle for learning has been noted by researchers in
both educational science (e.g., Hiebert & Grouws, 2007; Jonsson, Norqvist,
Liljekvist, & Lithner, 2014; Niss, 2007) and cognitive psychology (e.g., Bjork
& Bjork, 2011; Pyc & Rawson, 2009; Wiklund-Hornqvist, Jonsson, & Nyberg,
2014). Within the realm of educational science, the importance of productive
struggle for learning of mathematical concepts are discussed. Hiebert and
Grouws (2007) give an overview of the significance of having to put some
effort into learning something. They argue that the effort that is directed to
the task at hand will be beneficial for learning. In a study where students
practiced solution methods by either given algorithms or by constructing
algorithms Jonsson et al. (2014) saw that the more effortful construction
process was more beneficial for learning than using a given method. They
discussed if this had to do with the struggle itself or if the design of the practice
and test tasks could influence this (i.e., that the constructive practice tasks
were similar to the test tasks while the imitative practice tasks were not).
Jonsson et al. (2016) took this discussion further and examined if an effortful
struggle was more influential for learning than practicing in the same way as
being tested (i.e., transfer appropriate processing). The conclusion was that
an effortful struggle that focused construction of the solution method was
more beneficial. This resounds well with findings in cognitive psychology
where there is much evidence for the benefits of struggling with important
concepts and structures. Pyc and Rawson (2009) concluded that effortful
retrieval from memory will be more beneficial for learning than easy retrieval.
While practicing, this effortful retrieval can be achieved by repeated testing.
Wiklund-Hornqvist et al. (2014) showed that repeated testing was more
efficient for later retrieval than re-reading information. The repeated testing
invokes more afterthought than just re-studying a concept and this is also
10
considered as an effortful and productive struggle2. Bjork and Bjork (2011)
summarizes the importance of what they chose to call desirable difficulties in
the following statement: “Conditions of learning that make performance
improve rapidly often fail to support long-term retention and transfer,
whereas conditions that create challenges and slow the rate of apparent
learning often optimize long-term retention and transfer”. To create some
challenge or productive struggle, novel tasks (e.g., problem solving tasks) can
be utilized.
3.3 Problem solving
One of the competencies that are stressed as important in many mathematics
syllabi (NCTM, 2011; Skolverket, 2011b) is to become a proficient problem
solver. In the Swedish syllabus this competence is formulated as the ability to
“formulate, analyze, and solve mathematical problems and also evaluate
strategies, methods, and results”.
A student always brings prior knowledge into every task-solving situation.
It could for example be that the student has seen similar tasks before and
therefore know how to embark on solving the task. Schoenfeld (1985)
describes four different categories of knowledge that contribute to problem
solving. The first is resources, the content knowledge that the student has
acquired during previous schooling and that could be of importance for the
particular task at hand. This could for example be knowledge about how to
subtract 9 from 4 or how to differentiate the function f(x) = 1/x. If a student
experience gaps in her resources, let’s say that she does not know how to add
fractions, the learning of algebra could be impaired.
Schoenfeld’s second category is heuristics, the strategies and techniques
needed to solve the problem. Here we talk about methods of solving tasks and
in what order procedures should be done. The idea of heuristics in problem
solving was first formulated by George Pólya in 1945. Pólya (1945) formulated
four heuristic principles that could be applied to all problem solving:
understand the problem, devise a plan, carry out the plan, and revise your
work. More specific heuristics has of course also been formulated (e.g., ‘draw
a figure’ or ‘try to solve a simpler task’).
Schoenfeld’s third category is control (or as he later renamed it meta-
cognition), in this case control over which strategies and resources to select
and use. This includes reflecting on your own thoughts and on your available
knowledge to choose wisely. All teachers have seen examples of lacking
control. It might be that students use the addition strategy of common
denominators when multiplying fractions or that students solve non-existing
equations while simplifying algebraic expressions. A student with good
2 For a review on the testing effect, see Dunlosky, Rawson, Marsh, Nathan, and Willingham (2013).
11
control will make the most of her resources so that she will be able to solve
novel tasks in a more efficient way.
Lastly the student’s belief systems, his or her personal view on
mathematics, is important for how the student will tackle novel tasks
(Schoenfeld, 1985). This would include the thoughts you have about
mathematics as a subject and your mathematical abilities. A student whose
mathematical beliefs are poor will probably be more prone to give up on novel
tasks and will have a harder time to control his or her resources and heuristics.
Additionally, Jackson, Garrison, Wilson, Gibbons, and Shahan (2013)
argued that contextual aspects also can be important to consider during task
setup, to bridge the eventual gap of information if there are contextual
features that are unfamiliar to the student. They argue that key contextual
features should be explicitly addressed to make task solving more effective.
Schoenfeld elaborated on Pólya’s four steps by describing the problem
solving procedure as containing six possible stages: reading the task,
analyzing the task, exploring methods, making a plan, implementing the plan,
and verifying the result. Not all of these stages have to be present during the
solving of a problem. For example, the analysis of the task could be enough to
generate a plan on how to proceed and then the exploration phase would be
un-necessary (Schoenfeld, 1985). Schoenfeld also discovered that there was a
significant difference between novice problem solvers and experts in how
much time they put into analyzing a problem. Novice problem solvers typically
decided quickly on an approach and pursued it even if there was clear evidence
that the strategy was not bringing them closer to a solution. Experts put more
time in analyzing the task, formulate and implement a plan, and verifying it to
be able to go back and re-think the strategy if needed (Schoenfeld, 1992).
However, Blum and Niss (1991) indicates that teachers seems to think that
even though problem solving is important it will i) need additional knowledge
about other subjects and ii) take much time to implement. Time is also
expressed as an issue by Boris (2003) when comparing teachers’
mathematical beliefs and their practice. Another, and maybe more time
efficient, way to focus the important mathematics during task solving could be
to engage in tasks that promote creative mathematically founded reasoning.
Creative tasks do not have to be as challenging as problem solving and can
include elementary reasoning as well as more elaborated reasoning. This will
however require that the task-design put emphasis on a particular
mathematical hurdle that the students need to learn, and that the task does
not reveal the solution method for the student. This is basically what creative
mathematically founded reasoning tasks does as we will see in the following
section.
12
3.4 A framework for mathematical reasoning
As much of this research is based on the research framework for mathematical
reasoning that Lithner (2008) suggested, it seems appropriate to give a short
summary of the different aspects of it. The framework provides a basis to
analyze student’s reasoning, primarily with respect to the distinction between
using available (memorized or given) solution methods and constructing the
solution. It can also be used to classify mathematical tasks with respect to the
mathematical reasoning they promote and/or assess. The reasoning promoted
by the task is depending on the individual’s prior knowledge and the text,
guidance, or examples that are available at the time of task solving. The
reasoning sequence starts with the given task and continues to an answer and
the reasoning that is carried out is the product of the task, the individual’s
thoughts, and the milieu.
3.4.1 Reasoning sequences
When solving a mathematical (or maybe any other) task you have to decide
where to start. Schoenfeld (1985) observed that novice problem solvers often
put less time into preparation and choosing than experts. While the experts
put a lot of thought into preparation the novices were quicker to dive into an
unprepared and often unsuitable problem solving process. The solving
process could, as Lithner (2008) suggests, be seen as a directed graph where
implementation of a solution strategy (edges) are connected by instances
(vertices) which indicate both a momentary state of knowledge and of the
subtask (see Figure 1). These subtasks comprise both explicitly written
subtasks and implicit subtasks that the reasoner formulate during the solution
process. The edges consist of solution processes that are more or less
outspoken, where the reasoner is implementing the strategy of choice for the
specific subtask. This implies that the task at hand can be solved or answered
along different paths through the graph.
Figure 1: Reasoning sequence as retrieved from Lithner (2008).
Let me give you a simple example of this with the following task: Jane has a
salary of €1800 per month. How much will her salary be if she will get a 15%
raise? Depending on the prior knowledge of percent this task could be solved
by at least three paths.
13
Path 1:
Find what 1% of €1800 is.
a. 1%: €1800/100 = €18
Find what 15% of €1800 is.
b. 15%: 15 · €18 = €270
Add €270 to €1800.
c. €1800 + €270 = €2070
Answer: Jane’s salary will be €2070.
Path 2:
Find what 15% of €1800 is.
d. 0,15 · €1800 = €270
Add €270 to €1800.
c. €1800 + €270 = €2070
Answer: Jane’s salary will be €2070.
Path 3:
Calculate the new salary by finding 115% of the old salary.
e. 1,15 · €1800 = €2070
Answer: Jane’s salary will be €2070.
If we should try to draw this simple example (provided that the task solver
does not make any other assumptions or calculations than what is given here)
the graph would look like Figure 2.
Figure 2: Reasoning graph of the task-solving example above.
3.4.2 Creative Mathematically Founded Reasoning
Lithner (2008) identifies two major reasoning types, imitative and creative
reasoning, and then proceeds to divide these into sub-categories. When there
is not enough information at hand to solve the task with a known solution
method (i.e., by an algorithm or by recalling memorized answers) it can still
be solved but another type of reasoning must be used. At least some parts of
the reasoning sequence must then be constructed by the task solver and
argued for by connecting it to the intrinsic mathematical properties
important for the task. Reasoning that involves both novelty and
mathematically founded arguments is called Creative Mathematically
14
founded Reasoning (CMR). CMR is defined by Lithner (2008) as follows:
Creative mathematically founded reasoning (CMR) fulfils all of the
following criteria.
1. Novelty. A new (to the reasoner) reasoning sequence is created, or a
forgotten one is re-created.
2. Plausibility. There are arguments supporting the strategy choice and/or
strategy implementation motivating why the conclusions are true or
plausible.
3. Mathematical foundation. The arguments are anchored in intrinsic
mathematical properties of the components involved in the reasoning.
The creativity here should not necessarily be seen as something
extraordinary or ingenious but rather as the construction of a, for the task
solver, new reasoning sequence (Lithner, 2008). As an example, a task that
asks for the area of a triangle with a given height and base could be considered
a creative task (denoted CMR-task) if there is no provided formula or if the
students have not done this previously. The students would have to base their
reasoning on what they already know (e.g., the area of a parallelogram) and
then consider the triangle to be half a parallelogram. After this we could also
ask the students to formulate the rule or formula by themselves. Since
mathematics is an ingredient in other school subjects, CMR could also be
applied in them. For example, Johansson (2015) showed that CMR can be an
important element when learning physics.
3.4.3 Algorithmic Reasoning
As a contrast to CMR, reasoning that is connected to performing a recalled
procedure without connecting it to mathematical properties is called
Algorithmic Reasoning (AR). Lithner (2008) defines AR as follows:
Algorithmic reasoning (AR) fulfils the following two conditions.
1. The strategy choice is to recall a solution algorithm. The predictive
argumentation may be of different kinds (see below for examples), but
there is no need to create a new solution.
2. The remaining reasoning parts of the strategy implementation are trivial
for the reasoner, only a careless mistake can prevent an answer from
being reached.
A task would be categorized as promoting algorithmic reasoning (denoted
AR-task) if it is reasonable to think that the solution method could be retrieved
from memory by the solver, or if the solution method is available in the
instructions or a worked example (Lithner, 2008). AR-tasks seems to be quite
15
common in textbooks across the world and at all levels of mathematics
education, from compulsory school to university (e.g., Jäder et al., 2014;
Lithner, 2004; Newton & Newton, 2007). An example of an AR-task could be
when students are asked to calculate the area of a triangle where the height
and base are provided and the formula, 1 # *2 % 3+4-, is written at the top of
the page. The focus will be to apply the formula correctly and will likely not
include considerations of mathematical properties like that the triangle is half
of a parallelogram (hence, the division by 2).
Lithner (2008) differentiates between different types of algorithmic
reasoning, depending on what type of AR-information the task solver make
use of. Commonly, the supplied AR-information will put focus on how to solve
the task and not why the task can be solved in the given way. In study 2, I
discuss and test another type of AR, eXplained Algorithmic Reasoning (XAR),
which concerns the reasoning that occurs when a student has access to both a
solution method and an explanation on why the solution method is valid. It is
important to distinguish between a description that tells how a solution
method should be applied and an explanation on why the solution method is
valid. The former would be categorized as AR-information since it gives
explicit instruction on how to perform the calculations, without explicit
connection to the mathematical properties. The latter would be more than AR
information since it not only describes but also justifies the solution method.
The justification included in XAR could be similar to the justification that is
constructed during CMR with the difference that in XAR the justification is
available from the start and in CMR it is constructed as a part of the reasoning
sequence.
For example, the area of a triangle could be introduced by giving the
formula (i.e., 1 # *2 % 3+4-+ and showing how to apply it. This would be
considered AR-information since there is no connection to intrinsic
mathematical properties. The introduction could also explain why the formula
is valid by describing and showing that a triangle is half of a parallelogram,
ending with the solution method that the students could apply. This
introduction would be classified as XAR information since it starts off from
the mathematics behind the formula and explains the validity of it from this
point of view. In this way the formula can be logically founded in mathematics
and not only something that you just may have to accept and believe in.
Most textbook information seems to concern the description of solution
methods rather than presenting the reasons for why these solution methods
work (Shield & Dole, 2013; Stacey & Vincent, 2009). However, even though
XAR can be found in textbooks it is always accompanied with solution
methods and/or examples that are highlighted and therefore gives the
impression that they are the most important pieces of information.
Thus, task design can influence which reasoning the student will use. It is
also plausible that a creative task will be more cognitively demanding than an
16
algorithmic task since the student has to construct new solution methods and
not only apply provided algorithms. If cognition is crucial for creative
reasoning to occur, then cognitively less proficient students will have
problems even if the tasks are well designed.
3.5 Cognitive demand
Another way to characterize mathematics tasks is to sort them regarding to
the cognitive demand they impose on the task solver. Stein et al. (1996) made
such a distinction between tasks when trying to find factors that contribute to
the preservation of cognitive demand throughout the solution process. They
defined five categories of tasks based on what was demanded from the
students to be able to solve them (non-mathematical, memorization,
procedure without connection to concepts, procedure with connection to
concepts, and doing mathematics). In their study it became clear that the tasks
with higher cognitive demand (procedure with connection to concepts and
doing mathematics) often lost much of this demand during the teaching and
solution process. Most of this happened when the teacher (or sometimes a
peer) provides help by removing the challenging aspects of the task or when
the focus shifts from the concepts to finding the correct answer. This reduces
a cognitively demanding task to a task where the only aim is to apply the
correct procedure.
There are some connections between the way Stein et al. (1996) categorized
tasks by cognitive demand and the way Lithner (2008) has categorized tasks
by looking at the reasoning they will promote. Tasks that according to Stein et
al. (1996) requires memorization or procedure without connection to concepts
are similar to the tasks that Lithner categorizes as imitative- or algorithmic
reasoning tasks (Lithner, 2008). Here the student can rely on either a
memorized solution method or a method given by the textbook or by a person
close by (i.e., the teacher or a peer). The tasks that require procedures with
connection to concepts or that students engage in doing mathematics are
comparable to Lithner’s creative reasoning tasks. Here the students need to
consider the mathematical properties to solve the task, either by reflecting on
why a known procedure would be appropriate to use or by constructing a new
mathematically founded (and justified) solution method.
Hence, there are implications that cognitive variation is a part of the puzzle,
and if an individual’s cognitive abilities are important for task solving and if
CMR requires a capability to handle higher cognitive demand, cognition could
be decisive in how students learn from solving tasks. If students’ individual
cognitive variation matters it is important to examine this as well as deciding
which measures to use when doing so. This will be addressed in the next
chapter.
17
4 Memory and cognition
4.1 Individual variation in cognition
How a student will handle the requirements of doing mathematics could vary
a lot depending on individual prerequisites. This could for example be how
well a student will be able to concentrate in a noisy classroom or if the student
feels motivated to engage in the sometimes stressful conditions of a test
situation. Doing mathematics sometimes taxes the individual’s cognitive
abilities quite extensively. There are lots of abstract information that need to
be processed and even though most of us use pen and paper to ease the
cognitive strain, high processing power could be important for mathematics
achievement (e.g., Floyd, Evans, & McGrew, 2003; Freund, Holling, & Preckel,
2007). Hence, individual variation of cognitive ability could be an important
factor to consider when studying mathematical tasks and reasoning. In the
project we have therefore chosen to include and control for some measures of
cognitive capacity in our experimental studies.
4.1.1 Working memory
One cognitive construct that is often connected to mathematical thinking is
working memory (WM). This is the ability to simultaneously store and process
information. The multi-component model of WM was first suggested by
Baddeley and Hitch in 1974. Baddeley has made some additions to this model
and it now contains four parts. There is a Central Executive that coordinates
incoming information to the three slave systems, 1) the Phonological Loop
that process auditory information, 2) the Visuo-Spatial Sketchpad that
process visual and spatial information, and 3) the Episodic Buffer that handles
the temporal part of the acquired information so the stories we remember are
episodically coherent (Baddeley, 2000).
The connection between WM and mathematics achievement has been
extensively studied and, in the chapter on working memory in the first volume
of Educational Psychology Handbook, Swanson and Alloway (2012) conclude
that there is much scientific proof of a link between mathematics achievement
and WM. For example, Bull, Espy, and Wiebe (2008) let primary school
children do both tests of mathematics skills and of WM. The tests showed a
high correlation between visuo-spatial WM and math skills in primary school
children. Passolunghi, Vercelloni, and Schadee (2007) also conducted a study
on primary school children with the similar results, that WM is important for
mathematics achievement. However, Swanson and Alloway (2012) also note
that WM is not the only important factor in mathematics learning.
18
4.1.2 Fluid intelligence
WM is also closely connected to a construct that cognitive psychologists refer
to as General fluid intelligence or fluid reasoning (Gf). This is the part of the
human cognition that is devoted to problem solving. It is not unexpected that
these two are interconnected. While solving any problem we need to activate
WM as we need to store and process information at the same time, so the fact
that Gf and WM account for some of the same processes is not strange.
However, WM and Gf are not the same construct. In a study on how time
constraints influence the correlation between WM and Gf, Chuderski (2015)
showed that stricter time constraints on Gf tests produce a higher correlation
between Gf and WM. When time constraints are removed the correlation
between the two decrease. This indicates that there are processes involved in
Gf that are not directly linked to WM.
There are studies that link Gf to mathematics achievement. For example,
Primi et al. (2010) argue that a higher Gf, or at least higher results on Gf-tests
(e.g., Raven’s Progressive Matrices), implicate a steeper mathematical
learning curve (i.e., faster learning). Taub, Keith, Floyd, and McGrew (2008)
conducted a study on children and youths, 5-19 year olds, where they conclude
that Gf is an important factor for mathematics achievement in all age groups.
However, they also mention that other factors play in when it comes to how
well students perform in mathematics. As CMR contains elements of problem
solving, Gf could be influential for their reasoning and in turn affect the
students’ results.
4.1.3 Cognitive tests
Most cognitive functions are studied by using different behavioral tests
designed to test the sought after ability. In the case of WM there are a number
of tests available. Typical for all these is that you are supposed to process some
information whilst remembering other information. For example, reading and
judging the validity of a few sentences whilst remembering the last word of
each sentence (i.e., reading span) or performing simple arithmetic whilst
memorizing letters (i.e., operation span) (Unsworth & Engle, 2005). Gf is
most often evaluated by applying a Raven’s progressive matrices test which
comes in a few different levels of difficulty: colored, standard, and advanced.
This test is a non-verbal problem solving test where the participant has to
choose the correct tile that will complete a three-by-three matrix (see Figure
3). Raven’s progressive matrices are supposed to be independent of the
participant’s language and culture but an increase in scores has been observed
over time (e.g., Brouwers, Van de Vijver, & Van Hemert, 2009; Raven, 2000;
Wongupparaj, Kumari, & Morris, 2015).
19
Figure 3: Example of a task similar to those used in Raven’s
Progressive Matrices.
4.2 Observing brain activity
Like all other organs in our bodies, the brain will develop during childhood
and adolescence. Therefore, some cognitive processes that are trivial to an
adult might be impossible to perform for a child. For example, the concept of
time is very hard to grasp for a child while an adult finds it mostly
unproblematic.
Brain imaging techniques can provide support for hypotheses about
educational issues (De Smedt et al., 2010). For instance, in study 3 (in the
present thesis) we showed that different mathematical practice provided, not
only behavioral difference, but also long lasting neural differences in the brain
(Wirebring et al., 2015a). These results supplied evidence that the hypotheses
of which brain regions that became more or less active correspond to the
previous behavioral data and thus strengthen the conclusion drawn about the
behavioral results. By using brain-imaging techniques, it is also possible to
detect cognitive processes that are not manifested in observable behavior. It
has even been shown that the brain activity can be prognostic of future
behavior. Wirebring et al. (2015b) showed that word pairs that were retrieved
but subsequently forgotten were characterized by lower brain activity than
word pairs that were retrieved and remembered. Hence, brain-imaging
studies can provide information on how the brain process information that is
impossible to obtain in behavioral studies. By combining brain-imaging and
20
behavioral studies, we can more effectively evaluate different methods of
learning.
Techniques have been developed to study brain activation and in later years
this has been important for cognitive psychologists since the complex nature
of the brain earlier only could be studied in special cases where injuries or
illness did disrupt the normal brain function. Electro-encephalogram (EEG),
a non-invasive technique that measures electric brain activity via electrodes
attached to the scalp, has been available since the mid 20th century but this
technique has a disadvantage, its lack of spatial resolution. Signals are quickly
detected but it is much harder to locate their origin within the brain. To study
neurological processes that correlates to cognitive activities like language or
mathematics there is a need to increase this spatial resolution. Functional
Magnetic Resonance Imaging (fMRI) is a comparatively new technique of
registering brain activation. In contrast to classic EEG, fMRI has a high spatial
resolution but a somewhat lower temporal resolution. This is due to the
biological and physical processes that constitute the signal source during
fMRI. This is quite complex, but in the following section I will try to explain it
without going into the deeper physics of it.
4.2.1 The technique behind an fMRI image
Every voxel3 in a typical fMRI-image is about 3x3x3 mm (i.e., about 1/500 of
a teaspoon). fMRI depicts the brain (or any part of the body) in several “slices”
(usually 27). The scanner makes a pass over the brain in about 2 second and
this is then repeated several times with several stimuli to get a reliable picture
(movie). fMRI utilizes magnetic properties of hemoglobin molecules in the
blood to measure blood flow as this indicates that there is activity.
In the strong magnetic field (1,5-7T, an ordinary refrigerator magnet
produces about 0,005T) within the scanner the hydrogen (nuclei) in the body
will be positioned in line with the magnetic field. In the scanner there are
emitters that send out radio waves in specific frequencies that will excite
hydrogen atoms within water molecules. The hydrogen absorbs the radio wave
energy and this will change its orientation to one angled to the magnetic field.
When the radio pulse subsides the excited hydrogen return to the original
orientation and release the excess energy as an energy pulse back to the
detectors.
To be able to discern between the different parts of the “brain slice” the
scanner is equipped with gradient coils that induce gradient magnetic fields
that utilizes this sensitivity of inhomogeneous magnetic fields. This has the
effect that the hydrogen atoms will orient differently depending on where in
the slice they are located. This will give rise to a slightly diverse timing in the
3 A voxel is the smallest visible element in a depicted volume. Comparable to pixels when dealing with digital
photography.
21
return signal that is correlated to the gradient fields. And by this it is possible
to determine a three-dimensional coordinate of the return signal, i.e., in which
voxel the activation took place.
As the brain activates a certain area the consumption of oxygen will increase
and the blood flow will need to increase accordingly. The body does this by
enlarging the capillary vessels in the active area so that they will let more
oxygenated blood through. The ratio of oxygenated hemoglobin will actually
rise to a higher level than normal and this provides us with an opportunity to
distinguish the active areas from the surrounding, non-active, tissue. As the
de-oxygenated hemoglobin molecule is paramagnetic it distorts the return
signal from the tissue but when the concentration of deoxyhemoglobin
decreases, as the concentration of oxyhemoglobin is rising, the signal will
become stronger. This contrast between in deoxy- and oxyhemoglobin gives
us the possibility to detect a useful signal (often called a BOLD-signal). For a
more comprehensive description of fMRI see for example Huettel (2014).
4.2.2 A downside to fMRI
A weakness of this imaging technique is that the fMRI-scanner gives us an
indirect image of neural activation. This means that we can’t be absolutely
sure that the increase of blood flow indicates that the main activity is depicted.
The scanner can’t distinguish between neuron clusters forwarding
information and clusters that does the actual processing of the information.
This makes it even more important to use a good experiment design and
relevant contrasts (e.g., comparisons between a) reading and finding spelling
errors and b) reading and solving mathematics tasks) to make sure that what
is being measured is the active brain area (Huettel, 2014).
4.2.3 fMRI and Mathematics
There has been quite extensive research in cognitive neuroscience about
which brain areas that are active when mathematics is involved. However,
most of the studies are done concerning quite simple mathematical concepts,
e.g., addition, subtraction or one-digit multiplication. This is partly because
the fMRI-technique is comparatively new but also due to the fact that the
technique places restrictions on the experiment design. Time constraints are
a reason that more complex mathematical procedures have not been studied
in fMRI. A long exposure time would decrease the signal-to-noise ratio and
give few results or results that are difficult to interpret. There is also a problem
with complex tasks that would demand pen and paper or similar tools to be
solvable, since physical movement or speech also generate brain activity. This
would add activation patterns to the result and can sometimes be hard to
distinguish from the pattern that you are really interested in (Huettel, 2014).
22
The studies that have been made (e.g., Dehaene, Piazza, Pinel, & Cohen,
2003; Delazer et al., 2003; Houde, Rossi, Lubin, & Joliot, 2010; Ischebeck,
Zamarian, Egger, Schocke, & Delazer, 2007; Wirebring et al., 2015a;
Zamarian, Ischebeck, & Delazer, 2009) have all identified interesting neural
networks that are in use while making calculations, solving novel tasks, or
recalling from memory. Some of these networks are common with other
subjects as well (e.g., connections to long-term memory or processing of
complex tasks) and some are more specifically linked to mathematics (e.g.,
abstract number sense or calculations). In the following section I will describe
the main neural networks and their connection to mathematics based on their
function.
4.2.4 Brain functions connected to mathematics
There are some basic functions that are in use when a person is doing
calculations or more complex mathematics. There are rules or principles that
we have learned some time ago that need to be retrieved from memory. We
have to make simple calculations using for example addition or multiplication.
While we do this there is a need for our working memory to briefly store and
retrieve information that is being processed. If the task at hand is novel or
complex we need to figure out how to handle this new situation and how to
use our prior knowledge to solve the task. There might also be visual
representations to consider or to manipulate in some way. All this is handled
by the brain although in different areas. The brain is a very complex and
interconnected organ and most processes activate large portions of the brain.
4.2.4.1 Memory retrieval
Retrieval from long-term memory is important for all tasks, mathematical or
not. Important ideas, rules, or rote learned knowledge can be found here and
can be more or less easy to retrieve. It is fairly simple to recall rote-learned
information (e.g., the multiplication table) which can be useful to relieve our
working memory by automatization. However, since rote-learned information
does not carry information on underlying ideas or basic concepts it is not
always obvious which rote-learned knowledge that should be used. For
example, I still recall a few German words that were supposed to control direct
object (durch, für, gegen, ohne, um), but since I cannot remember what a
direct object is I have little use of them. As discussed previously, knowledge
that is achieved with some effort will be easier to recall (e.g., Bjork & Bjork,
2011; Jonsson et al., 2016). For example, Wirebring et al. (2015a) (study 3 in
this thesis) showed that students that learned mathematics by a given solution
method had to work harder to recall this method than students that
constructed the solution method themselves. The former group also showed a
higher activation was in the Angular Gyrus (AG), an area related to for
23
example reading, mathematics, memory retrieval and social cognition
(Seghier, 2013).
Dehaene et al. (2003) argue that the angular gyrus is active during
mathematics since verbal or linguistic properties are the basis of arithmetic
tasks. They also propose that rote learned addition and multiplication are
stored in verbal memory in the same way that grammatical rules can be
remembered as a ditty. This view is shared with Ischebeck et al. (2007) as they
observe that the angular gyrus and temporal lobes are activated when
retrieving rote-learned mathematics, i.e., simple addition and the
multiplication table. Perhaps this activation of the angular gyrus is a marker
for memory retrieval (Grabner et al., 2009) or maybe it is an indication that
the trained “knowledge” is manifested verbally (Delazer et al., 2003; Zamarian
et al., 2009). The AG has been found to be active in non-mathematical tasks
as well and this could indicate the verbal connection.
4.2.4.2 Novel and complex tasks
When a person encounters a novel or complex task, working memory will be
activated to a higher degree. There is also need for memory retrieval and
structuring of information and prior knowledge. Much if this work is done in
the frontal lobe of the brain although there are also other parts of the brain
that seem to have a part in manipulation of information in working memory
(Koenigs, Barbey, Postle, & Grafman, 2009). Studies have shown that higher
complexity yields more activation of the Prefrontal Cortex, an area connected
to problem solving and working memory. This area is more active during
childhood then during later years (Houde et al., 2010; Zamarian et al., 2009).
This might not be unexpected since more tasks are novel as you are younger.
Complex tasks often have a visual component as well, either as a sketch,
diagram or graph or as information which induce the solver to make mental
pictures. The visuo-spatial processing is conducted in the Posterior Superior
Parietal Lobule, further back in the brain (Dehaene et al., 2003). This
indicates that a truly complex mathematics task will be hard to study in fMRI
since there are so many areas active at the same time. There is also the
temporal difficulty mentioned previously, that a lasting task will decrease
signal-to-noise ratio and give results that are harder to analyze.
4.2.4.3 Calculation and number sense
Most mathematics, at least in pre-university schooling, include numbers and
calculations with numbers to some degree. Some of the rote-learned
computations can be retrieved from long-term memory without the need for
processing, but mental arithmetic is still needed to solve even the simpler
tasks. Much calculation is conducted in working memory but there is also an
area that seem to be activated only during mathematics, the Horizontal Intra-
Parietal Sulcus. Dehaene et al. (2003) observed that this area was not
24
activated by words in general but by number words, which led them to
conclude that this area was the most mathematically specific of the three they
studied. The horizontal intra-parietal sulcus is proposed to code the abstract
meaning of numbers and is activated by mental arithmetic, e.g., subtraction.
4.3 Summary
As mathematics is activating a large portion of the brain, most studies have to
be designed to pinpoint specific brain functions or type of mathematics.
Additionally, increased time for reflection would add interference to the fMRI
signals, and so would also aid in form of pen and paper, calculators or asking
questions do. Put together, examining mathematics with fMRI is not an easy
task but with considerable thought on experiment design it still is possible to
distinguish brain activation connected to mathematics from other cognitive
tasks (e.g., reading). One such experiment is reported in study 3 in this thesis.
The previous sections have given us an overview over the fMRI-technique
and results from some previous studies connecting brain activity to the
learning of mathematics. There are some benefits in using this kind of
methodology since fMRI can give a deeper insight into the brain processes that
govern our behavior. The fMRI-technique can also help to sort out processes
that behavioral studies will have a harder time to pick out. For example, in
study 3 we used fMRI to help with the explanation to the significant advantage
of CMR-practice over AR-practice in study 1. In study 3 we could see that the
CMR-group activated brain areas connected to memory retrieval processes in
a significantly lesser degree than the AR-group did. The AR-group also had
higher activation in areas connected to working memory. Together with the
results from study 1 this could explain how AR-practice differs from CMR-
practice. Students that practice with CMR seems to have an easier access to
the practiced solution methods and easier to apply them than the AR-students
will have. This was of course implicated by the behavioral results in study 1 as
well, but the question about if the difference in test results in study 1 could
have been due to a higher activation for CMR-students in other areas where
complex tasks are processed. However, study 3 showed no other areas within
the mathematics network where the CMR-students had higher activation
levels than the AR-students. Therefore, the reason for the significant
advantage of CMR over AR is somehow connected to the deeper encoding and
ease of retrieval of CMR-practiced solution methods. This result could maybe
have been found out in a series of behavioral studies as well but the use of
fMRI gave us this result in a single experiment. The fMRI-experiment also
gave us implications in what to pin-point in coming studies, namely, the
reason for the deeper encoding of CMR-practiced solution methods.
25
5 Method
When the project ‘Learning mathematics by Imitative and Creative Reasoning’
began the project group was formed by researchers from mathematics
education, cognitive psychology, and cognitive neuro-science. One of the
benefits of this constellation was that there were different methodologies that
met and, as all involved were interested to learn from each other, there has
been both interesting discussion and education during the design processes.
Within the studies that this thesis is based upon there are many different
methods used and the following sections will address them and connect them
to both mathematics and the object of study in each study. As a member of the
project group my work has comprised task design, experiment design, data
collection, data analysis, and writing. The task- and experiment design phases
have taken quite a lot of time during the start of the project and later on, data
collection and analysis took their time as well. In the beginning of the project
extensive piloting was done, since the interventions (and partially also the
analysis method) were completely new, to secure that the students were able
to solve the tasks and that the tasks did promote the desired reasoning.
Therefore, the articles were not submitted until later on in my doctoral
studies. In the following sections the design process will be elaborated to
describe the different development stages involved.
5.1 Task design
In all studies mathematics tasks are important. In three of the studies (1, 2,
and 3) the tasks were designed by researchers to be as tightly connected to
specific reasoning types as possible. This was important since these studies
focused the outcome of specific reasoning. This will be addressed further in
section 5.1.1. In the last study the tasks we study as part of the students
reasoning, are designed by the teachers as part of their ordinary preparation
for the lessons. These tasks were not as strictly formulated as the tasks in study
1-3 and therefore not as easily distinguished as to what type of reasoning they
would promote. The teacher-designed tasks are discussed further in section
5.1.2.
5.1.1 Design by researchers
The tasks in studies 1-3 were designed by researchers to promote either AR or
CMR. The goal was that both these versions of the tasks should have the same
target knowledge (i.e., a solution method in the shape of a formula). The AR-
tasks would need to explicitly give the solution method while the CMR-tasks
required the students to be able to construct the solution method by
themselves. To increase the likelihood of this we designed the CMR-tasks with
three sub-tasks as elaborated below. The design would be as similar as
26
possible to eliminate that eventual layout differences would influence the
results.
The design of the tasks was an extensive process that took place during a
couple of years where a lot of different tasks were tried out and adjusted to be
tried again, through several pilot-studies. The main reasons for the extended
design process was the need to find tasks that a) could reach the same target
knowledge either via AR or CMR, b) were not familiar to the students so that
they already knew the solution method in advance c) not too difficult to solve
by CMR and d) not so easy that solving by CMR provided no challenge. This
left us a comparatively small window of implementation where the tasks were
just hard enough to be solved by CMR, without the students falling into an
AR-mode of reasoning because of familiarity with the tasks.
The first pilot-study was a think-aloud study with four students, where we
video-recorded their work. The analysis indicated that our hypothesis
concerning the importance of CMR held but also that the initial tasks were to
extensive to be used in a large-scale study. Therefore, the tasks were adjusted
and new tasks were constructed for a second pilot-study. This time two classes
were involved and the students solved several multiple-choice tasks
individually on a computer, with an observer seated next to them. The task
solving process was recorded and after all tasks were solved the observer and
the student went through the recording and discussed difficulties and the
choices made by the students4. Analysis showed that some of the tasks were
not suitable. These tasks were based on fictitious mathematics which did not
engage the students. The third pilot-study tested some of the old tasks and
some new tasks and this time the students (two classes) both practiced and
were tested with a computer where the software also recorded their answers.
The data was purely quantitative and comprised answers and solution times.
After this trial, smaller changes were made to the tasks and instruction and
then we regarded the tasks ready for the larger data-collection.
Basically, students that are solving AR-tasks are presented with a formula
and the task leaves it up to the students to decide whether or not to think about
the mathematical properties behind the formula (Figure 4). It is not necessary
to do so to solve the task but it is possible. During the first two pilot-trials we
observed a few AR-students that applied CMR in the first couple of AR-tasks
but this was not common practice. One of these students explained afterwards
that he wanted to check if we tried to fool him with an erroneous formula, but
when the first formula checked out he trusted the tasks and continued without
controlling the properties.
4 Some results from this pilot-study was analyzed and reported in (Liljekvist, Lithner, Norqvist, & Jonsson,
2014).
27
Figure 4: Example of an AR-task
In one of the studies (study 2) XAR-tasks were used. They were constructed
by reuse of the AR-tasks but with an additional explanation of why the given
solution method works. In some cases, an extra picture was added to clarify
the explanation further (see Figure 5). The given explanations where
controlled by four experienced teachers before the study to see if the
explanations were reasonable in comparison to explanations in textbooks.
Figure 5: Example of a XAR-task
Students that solve CMR-tasks will not have a choice. They are forced to
consider the mathematical properties in order to be able to solve the tasks
(unless they are guessing, and strict guessing can almost never yield a correct
answer). The tasks were designed to give them a first task that could be solved
by observing and mentally extending the accompanying figure a few steps, a
28
second task that forced them to consider a more generalized idea and, a third
task that asked for a formalized algebraic expression (Figure 6).
Figure 6: Example of a CMR-task (top), second CMR-task
(middle), and last CMR-task (bottom).
29
5.1.2 Authentic, design by teachers
In study 4 we observed students’ work with authentic teacher made tasks5.
These tasks were not primarily designed to capture all aspects of CMR but
were rather chosen as tasks with potential for CMR from a larger number of
tasks that the teacher planned to use. We judged the task to have potential for
CMR based on the novelty aspect in mind.
Two of the tasks were problem solving tasks with geometrical focus (i.e.,
optimizing volume of a cylinder and calculating area of a trapezoid). Both
tasks had a clear goal and a given start, but little instruction on how to reach
the goal. Both these tasks were classified as CMR-tasks. The teacher explained
that the purpose of the task was for the students to practice problem solving.
The other task in this study had clear instructions on how to proceed until the
last subtask which had less instruction and was judged to have CMR-potential.
The teachers ambition with the task was to let the students build on previous
knowledge about differentiation to find a new method of differentiation
connected to rational functions (i.e., the quotient rule).
5.2 Data collection
The included studies report on three experimental studies (study 1-3) and one
observational study (study 4). In the following sections I will describe the
methods used in these four studies to present similarities and differences
between them regarding research design and sample. All participants in all
studies were given and signed a written informed consent.
5.2.1 Behavioral-experiments
The experiment design in two of the studies (study 1 and 2) addressed
differences in learning depending on the type of reasoning that the practice
tasks promoted (i.e., AR, CMR, and in study 2 also XAR). In both experiments
the sample comprised natural-science students from Swedish upper-
secondary school (16-17 year olds). In study 1 we had 91 participants and in
study 2 there were 104 participants, which we met for three sessions that took
about 45 minutes each.
During the first session the students took two cognitive tests, Raven’s
Progressive Matrices and Operation Span, to measure fluid intelligence and
working memory respectively. The students also provided background
information (i.e., mathematics grade, age, and gender). These measures were
then used to match the participants into two (study 1) or three (study 2)
similar groups for the following session.
The second session consisted of a practice session where the students
solved practice tasks via a computer program. The software saved their
5 See study 4 for the complete tasks.
30
progress (i.e., solution time and answers) on a server. Within each of the
practice groups students solved either AR-, CMR-, or XAR6-tasks. The AR-
and XAR-groups solved a total of 70 sub-tasks while the CMR-group solved
42 sub-tasks, due to the additional time needed to complete the CMR-tasks.
After practice the students had encountered 14 different tasks with different
solution methods that we later tested for during the last session.
A week after the second session we met the students for the last time while
they took the test. They were tested on all 14 solution methods with three test-
tasks for each method. The first test-task explicitly asked them for the
algebraic expression they used during practice. The time on this task was
restricted to 30 seconds to restrict (re)construction of forgotten information.
We hypothesized that the students that did not remember the formula could
perhaps remember the solution idea they used during practice, therefore the
second test-task asked for a numerical answer and was also restricted to 30
seconds for the same reasons as the first test-task. The third test-task asked
for the same numerical answer as the second but with no time restrictions to
allow for eventual (re)construction. During the test-session the software again
recorded solution times and answers.
The AR-practice and the test tasks were designed to be as similar to
textbook tasks and teacher made tests as possible, asking for solution methods
that were practiced during the second session and for the use of these
methods. This was done to try to ensure that the test did not especially benefit
the CMR-students. A follow-up study have also shown that the reason that
CMR seems to be more effective is not connected to any similarity between
practice and test tasks (Jonsson et al., 2016).
5.2.2 fMRI-experiment
The fMRI-study (paper 3) focused on the eventual difference in brain activity
during the test, after practicing via AR or CMR. In this study 40 of the 72
participants were upper-secondary students (18-20 yo) while the rest were
first year university students (18-22 yo). Practice was again made at a
computer that recorded practice data. The practice groups were matched into
two similar groups based on mathematics grade, gender, and their score on
Raven’s matrices. The test session was done individually in an fMRI-scanner
which put some restrictions on the experiment design.
As the fMRI-scanner registers all brain activity it would be very hard to
distinguish the mathematics from eventual activity pertaining motoric
functions, speech etc. Hence the test was done with multiple choice questions
where the participant left his/her answers via a response-pad where each
finger corresponded to an answer. Because of this the first test-task (i.e.,
recalling the formula) was removed and the second and third test-tasks were
6 Only used in the study of paper 2.
31
adjusted to different numerical questions with the same time restriction (30
seconds to read the task + 6 seconds to choose an answer). After each test-task
there was a baseline task to control for perception, attention, and reading. This
task asked the participants to identify if there were any spelling errors in a text
with the same layout as the mathematics tasks. This base-line task was then
used to single out the mathematics specific processes from for example,
reading and visual processing.
5.2.3 Observations
The observational study (paper 4) had a completely different approach. We
started out by constructing an observational instrument (i.e., a document that
let us structure our notes). This was piloted in an upper-secondary class
followed by a rudimentary analysis process which led us to reconsider some
parts of the observational instrument. We also realized the need for pre- and
post-interviews with the teacher and post-interviews with the student groups
that we observed. We also added a curriculum reading log in which the teacher
elaborated on the materials used in preparation and what would be used
during class. Another pilot study was made to try out the new tools, also in an
upper-secondary class. This time we saw that the tools worked out so we went
on by trying to find teachers that wanted to participate in the study.
We visited four teachers at a Swedish upper-secondary school. The teachers
all agreed to participate but since only two of them had planned to use tasks
with potential for CMR during their lessons we included these two teachers in
the study. Within each lesson we randomly selected two student groups to
observe while they solved the mentioned tasks. This was done to be able to
have a closer look at how the reasoning sequences unfolded and how their
progression through the task was connected to their reasoning.
We used observational notes, audio-recordings, and pictures to capture the
discussion and work of the students. We also gathered information on the
class and the intention for the lessons from the teachers and had a post
interview with the students to entangle some critical moments that we
observed. Altogether we observed four student groups (three pairs and one
triplet).
5.3 Methods of analyses
The method of analysis differed a bit between the four studies since study 1-3
used quantitative data and study 4 relied on qualitative data. In the following
sections I will describe the analysis methods and how they differed from each
other.
32
5.3.1 Statistical
All results in studies 1-3 were quantitative and therefore analyzed statistically.
Study 1 and 2 only used behavioral data and this was quantified by the
software that we used to collect it. The same was true for study 3 but here
additional fMRI-data was analyzed. The behavioral data was first scanned for
outliers. After this a statistical analysis was chosen and performed depending
on the nature of the data. In study 1 we chose to perform an analysis of
variance with practice group (AR or CMR) as the fixed factor and the test score
as the dependent variable. We also performed a linear regression analyses to
evaluate the impact of the different measured factors (practice result,
mathematics grade, cognitive index and gender) on the test score. Study 2 was
analyzed with a multiple analysis of co-variance since there were two
dependent variables (practice score and test score) and three practice groups,
and since the matching of cognitive proficiency between the groups got
disrupted by drop-outs it was controlled for as a co-variate. There were also
linear regression analyses made that evaluated the predictors of the test score,
similarly to the first study. In the third study the behavioral data was analyzed
with an analysis of variance where the dependent variable was evaluated with
practice group as the fixed factor. The fMRI-data was analyzed in several steps
to sort out the activation depending on the practice condition. First the fMRI-
images were adjusted for time and spatial differences, secondly the
mathematics condition was compared to a reading condition to delimit a
mathematics network and finally the activity within the mathematics network
was analyzed to distinguish any differences between the two practice groups.
In study 1 and 2 all statistical analyzes were conducted in SPSS7 and in study
3 the analysis was done in SPM88.
5.3.2 Qualitative analysis
As the data in study 4 was of the qualitative kind (e.g., transcripts, notes, and
pictures) we needed to find a different way to analyze it and this took quite
some time. The audio recordings were transcribed and we started to identify
moments when the reasoning sequence took a new direction. These vertices
were then categorized in regards to the reason for the change in solution
method. We also controlled if the next vertex indicated a progress in the task
solving process to be able to distinguish if the solution method brought the
students closer to the solution or not. The edges between the vertices were
then analyzed according to which type of reasoning dominated each edge.
When we had the vertices, edges, and all classifications ready, we visualized
the complete reasoning sequence in a graph where the x-axis represented the
turns in the students’ discussion and the y-axis represented progression
7 Statistical Package for the Social Sciences, version 22 and 23.
8 Statistical Parametric Mapping, version 8, which is an add-on to Matlab.
33
towards the answer. The graphs were then analyzed to see if we could find
patters and co-occurrences that were connected to the task design (AR or
CMR), the reasoning of the student group, and the student groups motivation
and persistence. A much more elaborate description of the methodology can
be found in the included manuscript (i.e., study 4).
34
6 Summary of the articles/Result
In this section I will give a short summary of each of the included studies. I
will also present an additional result from studies 1-3 that was not reported in
the articles. The additional result concerns information about the practice
sessions and can serve as a comparison to the test results that are reported in
the included studies.
6.1 Study 1 – Learning mathematics through algorithmic
and creative reasoning
The aim of this study was to investigate the learning effects of practicing
mathematical tasks through AR and CMR on task-solving performance.
We let 131 upper secondary students from the natural-science program in
Sweden practice on solution methods to 14 different tasks, either by AR or
CMR. The two groups were matched on cognitive prerequisites, mathematics
grade and gender. The AR-group got five practice tasks where a solution
method (i.e., a formula) was presented and this was repeated for all 14 solution
methods. The CMR-group got three tasks with increasing difficulty for each
solution method where the last task was to construct a mathematical formula
that described the sought after relation. One week later we gave both groups
the same test on the 14 solution methods. The test asked three questions for
each solution method. The first asked for the formula and was restricted to 30
seconds. This was to ensure that they had to remember the formula and not
have time to (re)construct it. The second question asked for a numerical
answer and was also restricted to 30 seconds, again to ensure that no
(re)construction would take place. The third question was the same as the
second but now with a 300 second time limit to give enough time for eventual
(re)construction of a forgotten formula.
After excluding participants from the sample due to attrition and in a few
cases to prevent ceiling effects we were left with 91 students (48 AR and 43
CMR). The result showed that the CMR-group significantly outperformed the
AR-group on the test, both on the composite level (see Figure 7) and on the
three different tasks. A closer study of the data also revealed that high
cognitive capacity was more important for the AR-group to perform well
during the post-test than for the CMR-group. This is contrary to common
beliefs as many regard problem solving and similar tasks as “only for the high
achievers”.
In the discussion we try to find reasons to why CMR seems to be more
efficient. One reason that comes up is that the CMR-group has to struggle with
mathematical properties of the task to find a solution to do this. The AR-
students get to see and use the correct formula repeatedly but do not have to
struggle at all. The positive struggle that the CMR-group is subjected to might
be one reason to why they perform better. The theory of didactical situations
35
also suggest that you learn better by constructing the solution compared to
imitating a given method. Thus, CMR will produce better opportunities to
understand and learn mathematics.
Figure 7: Practice and test scores on composite
level as retrieved from Jonsson et al. (2014).
6.2 Study 2 – The affect of explanations on mathematical
reasoning tasks
The aim of this study was to see if the addition of an explanation of the
mathematical principle behind the formula in the AR-task would enhance the
efficiency of the AR-tasks.
104 upper secondary students from the natural-science program in Sweden
were recruited to participate in the study. The method was similar to study 1
and the AR and CMR-tasks were the same but another group, XAR, was
added. The XAR-group got the same tasks as the AR-group but with an
additional explanation that briefly but carefully described the principle behind
the given formula. In other words, while the AR-tasks included information
about how to solve the task the XAR-tasks also included mathematical
arguments clarifying why the suggested method was correct.
All three groups were matched on cognitive capacity (i.e., Ravens matrices
and operation span), mathematics grade and gender. One week after the
practice session the students took the same test as in the previous study. The
results indicate that the added explanation did not give any significant effect
on performance compared to the AR-group. The tendency from the earlier
study that the CMR-group outperformed the AR-group was still there but with
fewer participants it did not show up as significant. Compared to the XAR-
group though, the CMR-group performed significantly better (see Figure 8).
36
Cognitive capacity was still more important for the AR-groups than for the
CMR-group but the added explanation seemed to lessen this effect slightly for
the XAR-group.
In the discussion the non-effect from the added explanation is discussed in
terms of how Brousseau’s theory of Didactical Situations (1997) actually
predicts this result. The importance of struggle again comes into play and the
eventual lack of engagement in the explanation is also hypothesized to be an
explanation.
Figure 8: Practice and test scores as retrieved from Norqvist
(2016).
6.3 Study 3 – Learning mathematics without a suggested
method: Durable effects on performance and brain
activity
The aim of this study was to replicate parts of the first study and add the
perspective of brain activity. Hypothetically, the performance would be
similar to the first study but since the test was made in an fMRI-scanner with
multiple-choice questions we could not be sure. Another hypothesis was that
the CMR-group would show less activity in the left angular gyrus since study
1 had shown that CMR-practice yielded a better recollection of the solution
methods than AR-practice.
73 students, 40 from the third year at upper secondary school and 33 first
year engineering students were recruited to participate. All were right-handed
and had normal or corrected-to-normal vision. The participants were divided
into two matched groups, AR and CMR, based on cognitive prerequisites,
mathematics grade and gender. The students practiced on nine different task
types with different solution methods and the practice method was the same
as in study 1. Six days after practice the students took a test while in an fMRI-
37
scanner. This test consisted of one multiple-choice for each solution method.
Between each mathematical test task there was a baseline task, checking for
spelling errors, that served as a contrast in the later analysis.
The result showed that the CMR-group outperformed the AR-group on the
test (see Figure 9A). It was also apparent that the AR-group had a higher
activation of the left angular gyrus (see Figure 9B) and the left precentral
cortex. The results also show that the right superior parietal cortex is
important for mathematical performance.
A
B
Figure 9: (A) Mathematics test scores for the two groups. (B) Difference in activation
between groups in Left Angular Gyrus, as retrieved from Wirebring et al. (2015)
We concluded that the CMR-group again outperformed the AR-group on
the post test. We also discussed the two indicated areas where the AR-group
had a higher activation rate than the CMR-group. The higher activity in the
angular gyrus indicates that the AR-students have to work harder to retrieve
the formula from memory. The difference in the precentral cortex indicates
that there is more stress on working memory in the AR-group.
6.4 Study 4 – Unraveling students’ reasoning: analyzing
small-group discussions during task solving
The aim of this study was to examine students’ reasoning to see how the
reasoning sequence would unfold in actual classroom situations. We were also
interested in how students’ reasoning would influence their progression
towards a solution.
We visited two classrooms in an upper secondary school and observed two
student groups in each classroom for the time it took them to complete a task,
constructed and presented to them by the teacher. One of the four student
groups did not complete the given task during the observed lesson. Initial
38
analysis showed that there were two interesting dimensions to regard, group
characteristics (i.e., the student-group’s motivation and persistence) and task
design (i.e., AR or CMR). After transcribing the audio-recordings we have
segmented them into sections by utilizing Lithner’s (2008) framework of
mathematical reasoning. A moment when the students’ reasoning took a new
trajectory was called a vertex and the segment between two such vertices was
called an edge (Lithner, 2008). After the classification we compared the
vertices and determined how they compared to each other regarding progress
towards the answer (i.e., return to previous ideas – step down, no progression
– horizontal edge, or progress – rising edge or step up). The edges were then
categorized according to the students’ reasoning (i.e., either CMR or AR). We
then visualized the students’ reasoning in graphs (see Figure 10) and analyzed
the patterns, the amount of rising edges and types of reasoning, as well as how
the group characteristics and task design would influence reasoning and
progress.
Figure 10: Example of visualization of reasoning sequence as retrieved from Van
Steenbrugge & Norqvist (2016). Blue indicates AR, Yellow CMR, and Black un-characterized
reasoning (all other markings are explained in the included paper (Study 4)).
The result showed that task design is important for which reasoning the
students will use. Although an AR-task does not exclude CMR, it only occurs
in our data if the students have difficulties and strive to handle them by
themselves. We also observed that group characteristics were important for
the chosen reasoning type. Student groups that were less motivated and less
persistent were more prone to giving up and using AR than the more
motivated and more persistent student groups.
39
6.5 Additional result
In study 1-3 we also gathered data on the practice session. However, we did
not report or discuss this extensively in the studies. What is obvious is that
CMR-practice is much more taxing on the cognitive abilities than AR-practice.
This becomes evident if we observe which variables that predicts the practice
result. For AR-practice none of the included variables (i.e., gender, cognitive
proficiency index, mathematics grade, practice time) are predictive of the
practice result while the CMR-practice result is highly dependent on cognitive
capacity and mathematics grade (see Table 1). This is consistent throughout
study 1 and 2 but not apparent in study 3, maybe because of the smaller
sample9. This is almost the opposite to the result from the test where the AR-
groups test result was predicted by cognitive measures and to some degree
mathematics grade and gender while the CMR-groups test result was
predicted by their practice result (see Table 2). In the tables below the XAR-
condition from study 2 is merged with the AR-condition since they performed
similarly.
Table&1&
Regression)Analysis)Summary)For)Variables)Predicting)Practice)Result&
&
AR&
&
CMR&
Variables&
B)
))
SE)B)
))
ß)
))
B)
))
SE)B)
))
ß)
Study&1&
&
&&
&
&&
&
&
&
&&
&
&&
&
&&Cognitive&index&
.031&
&
.014&
&
.349*&
&
.079&
&
.047&
&
.249&
&&Mathematics&grade&
.004&
&
.004&
&
.151&
&
.031&
&
.011&
&
.418**&
&&Gender&
.022&
&
.023&
&
.131&
&
.060&
&
.058&
&
.135&
Study&2&
&
&
&
&
&
&
&
&
&
&
&
&&Cognitive&index&
.008&
&
.008&
&
.120&
&
.128&
&
.033&
&
.432***&
&&Mathematics&grade&
.006&
&
.002&
&
.375**&
&
.041&
&
.008&
&
.555***&
&&Gender&
.013&
&
.012&
&
.120&
&
-.068&
&
.053&
&
-.144&
Study&3†&
&
&
&
&
&
&
&
&
&
&
&
&&Raven’s&matrices&
-.895&
&
1.825&
&
-.129&
&
2.088&
&
3.847&
&
.126&
&&Mathematics&grade&
-.344&
&
.343&
&
-.272&
&
.986&
&
.801&
&
.288&
&&Gender&
-4.794&
&&
3.408&
&&
-.353&
&&
3.063&
&&
7.789&
&&
.089&
*p<.001,&**p<.01,&***p<.05.&
†&includes&one&of&the&sample&groups&(students).&
9 Only data from one of the sample groups were available from study 3 at the time of writing this thesis.
40
Table&2&
Regression)Analysis)Summary)For)Variables)Predicting)Test)Result&
&
AR&
&
CMR&
Variables&
))))B)
))
SE)B)
))
ß)
&&
))B)
))
SE)B)
))
ß)
Study&1&
&
&&
&
&&
&
&
&
&&
&
&&
&
&&Cognitive&index&
.151&
&
.047&
&
.492**&
&
-.003&
&
.039&
&
-.010&
&&Mathematics&grade&
.013&
&
.013&
&
.155&
&
.004&
&
.010&
&
.046&
&&Practice&score&
-.027&
&
.472&
&
-.008&
&
.897&
&
.131&
&
.811***&
&&Gender&
-.005&
&
.072&
&
-.009&
&
-.034&
&
.048&
&
-.069&
Study&2&
&
&
&
&
&
&
&
&
&
&
&
&&Cognitive&index&
.059&
&
.018&
&
.282**&
&
.032&
&
.024&
&
.172&
&&Mathematics&grade&
.030&
&
.005&
&
.588***&
&
.010&
&
.007&
&
.219&
&&Practice&score&
-.060&
&
.286&
&
-.019&
&
.382&
&
.110&
&
.597**&
&&Gender&
.117&
&
.028&
&
.342***&
&
.018&
&
.032&
&
.060&
Study&3†&
&
&
&
&
&
&
&
&
&
&
&
&&Raven’s&matrices&
9.046&
&
3.806&
&
.472*&
&
2.071&
&
4.074&
&
.102&
&&Mathematics&grade&
1.726&
&
.733&
&
.494*&
&
.830&
&
.872&
&
.198&
&&Practice&score&
1.087&
&
.534&
&
.534&
&
.607&
&
.253&
&
.497*&
&&Gender&
17.363&
&&
7.502&
&&
.463*&
&&
2.825&
&&
8.159&
&&
.067&
*p<.05.&**p<.01.&***p<.001.&
†&includes&one&of&the&sample&groups&(students).&
41
7 Discussion
In the included studies there are some coherence in results that could be
noted. First, studies 1-3 show that CMR is more effective than (X)AR10
regarding both memory retrieval and re-construction of practiced solution
methods. Second, studies 1-3 also show that practicing by (X)AR gives rise to
higher taxation on cognitive abilities during the test situation than CMR-
practice does. This is confirmed by the neuro-cognitive data in study 3, which
shows that AR is more neurologically demanding during the test situation.
Third, the additional data shows that CMR-practice will however tax cognitive
proficiency during practice while (X)AR-practice does not to the same extent.
Fourth, study 4 confirms that if CMR is preferred, the task design is very
important. Even though CMR is not excluded when (X)AR-information is
available, it is much less likely to occur, especially if the students lack
motivation and persistence. Bear in mind that the discussion is limited to the
results in the four included studies and the additional result in the previous
chapter, hence the word “shows” is used to mean in the context of these
studies.
During the remainder of this chapter these results will be discussed in
relation to the research questions (Sections 7.1-7.3) as well as in relation to
limitations, generalizability, and implications for further studies and teaching
practice (Section 7.4)
7.1 How will the task design influence students’ solutions
process, mathematical reasoning, and brain activity?
From a theoretical point of view both Brousseau (1997) and Lithner (2008)
argue that tasks that contain prescriptive solution methods will most likely be
solved with said methods. And according to Brousseau (1997), this will not
lead to the creation of any new knowledge. Tasks should instead be designed
to promote the construction of new knowledge by emphasizing important
concepts or mathematical properties and at the same time not give away the
entire solution method. This is basically what happens in tasks with potential
for CMR. In study 4, it was clear that students preferred to use AR as far as
possible and as long as the task was designed with given methods these were
used. The lack of effect from the provided explanations in study 2 can also be
seen as a sign of that students prefer clean AR and do not bother with
redundant information. However, in a few instances in study 4 students did
use CMR when solving AR-tasks. Common for these instances was that the
students had difficulties in solving the task with the given method, which led
them to discuss the mathematical properties that was important and decide
on how to move on.
10 (X)AR denotes both AR and XAR
42
The student-group’s motivation and persistence also seemed to be
important factors for CMR to occur. One of the groups in study 4 was less
motivated and less persistent than the others and when they met a CMR-task
they never chose to reason creatively. Their reasoning was based upon
formerly known methods and when they did not work the students started
exploring methods by random. This could also be due to what Bjork and Bjork
(2011) calls undesirable difficulties (e.g., an unbridgeable gap between prior
knowledge and task requirements). The extra effort that can be so beneficial
for mathematics learning (e.g., Bjork & Bjork, 2011; Hiebert & Grouws, 2007)
and which is needed to perform CMR (Jonsson et al., 2016) will of course
require tasks that are within reach for the students’ resources and heuristics
(Schoenfeld, 1985).
By design, tasks that focus the use of a given solution method will promote
AR. CMR is not excluded from this type of task but is much less likely to occur
than AR. Providing an explanation as was done in study 2 (i.e., an explanation
to why a solution method works and is valid), does not seem to increase test
scores, although the explanation is focusing the important mathematical
properties. The explanation will not induce the productive struggle that CMR
does by definition, and this might be why XAR-information does not improve
test results compared to AR.
CMR-practice also seems, according to the results in study 3, to contribute
to less effortful memory retrieval and less brain activity during the test-
situation. The students that practiced by AR had a significantly higher
activation in a part of the brain that is connected to verbal memory retrieval,
the left Angular Gyrus (e.g., Delazer et al., 2003; Ischebeck et al., 2007;
Seghier, 2013). This would indicate that the solution methods were encoded
as a string of words, much like an automated multiplication table, rather than
as mathematical relations or properties. As will be discussed in the next
section, this lack in retention of mathematical relations or properties can
affect the test situation by inducing higher strain on cognitive abilities. That
our results showed no areas where brain activity was higher for the group that
practiced with CMR-tasks than with the AR-tasks confirms that retrieval and
eventual reconstruction during the test was less effortful for the CMR-group.
Although this thesis indicates that promoting CMR is preferable, AR can
sometimes be the appropriate choice. There are some concepts or properties
that are difficult to grasp or construct for students until later on in their
schooling, for example that if you multiply two negative numbers the answer
will be positive. There is also need for automatization of procedures to be able
to focus one’s cognitive abilities to the difficult parts of a problem. It would be
impractical if every little calculation would need the full attention of the
brain’s mathematics network. The problem is therefore not that AR occurs or
that rote-learning is common, but that these types of non-reflective learning
is too common. When learning occurs without connection to mathematical
43
properties or concepts it becomes fragmented, hard to remember, and as
Boaler (1998) saw irrelevant.
7.2 How will students’ cognitive variation affect their
solution rate and does task design matter?
In studies 1-3 cognition plays a significant role in the test-situation, and to a
larger extent if the students have practiced by AR or XAR. The CMR-students
do not seem to be as influenced by cognitive proficiency during the test
situation, but rather by the extent of how well they have performed during
practice. This conclusion was also confirmed by fMRI in study 3, where AR
showed to be more demanding during the test situation. However, as the
additional results show, CMR is taxing high on cognition during practice while
(X)AR does not. So, in a sense, test-results for CMR-students are also a
dependent on cognitive abilities, although in an indirect way.
In our studies students that practiced by CMR performed better than the
(X)AR-students during the test. One could argue that this difference is due to
a performance boost of the capable and high achieving students that would
benefit most from creative reasoning. However, in study 1 the largest
difference in test result, between the two practice groups, can be found in the
lowest cognitive tertile.
The struggle provided by the CMR-tasks could be one reason to why these
tasks render higher test results, even though many students did not solve all
practice tasks. As was suggested by Hiebert and Grouws (2007), discussed in
studies 1-3, and later on supported by Jonsson et al. (2016), effortful struggle
with important mathematical concepts and properties is essential for
mathematical understanding. Jonsson et al. (2016) also concluded that task
design is vital for what level of effort students will allocate to the solving
process. Furthermore, Bjork and Bjork (2011) points out that practicing with
desirable difficulties does not only increase test results but also improve long-
turn retention and transfer of skills. The importance of devoting effort to a
task was also argued for by Brousseau (1997) in the theory of didactical
situations, where he suggests that the teacher should delegate the
responsibility of the task solution to the students within the a-didactical
situation. Thus, there are both theoretical and empirical arguments that CMR
may be considered in the teaching practice.
7.3 How can these results influence teaching practice?
The result that (X)AR-practice puts less strain on cognition than CMR-
practice, and that the opposite is true during tests is maybe not surprising but
can be rather alarming. If practice in the classroom does not prepare the
students to cope with the coming test situation, then what we test is not
whether or not students have learned mathematics, but rather the students’
44
cognitive abilities. This could be deceitful for both students and teachers. If
the teacher observes that the students can solve most tasks they may believe
that the students are learning. If the difference between the practice- and test
scores from study 1-3 would be seen as indications of what could happen in a
classroom, there could be a significant drop in performance from practice to
test. Of course the experiment scores are somewhat an extreme case of this
but, from discussions with teachers from lower secondary school to university,
it does not seem uncommon that they meet students that are either
underestimating or overestimating their abilities, especially when it comes to
clearing or failing tests. Bear in mind that the students practiced by
themselves without any help from teacher or peers. This means that all
difficulties that the students met during their practice had to be either
overcome by themselves or left behind. What the result would be if the
students would have received help is a question left to another study, but from
the results of Stein et al. (1996) we can assume that the results might be even
better for the CMR-group, provided that the help was of good quality (i.e.,
addressed conceptual hurdles) that did not collapse the didactical situation.
Since much lesson time is devoted to solving tasks (Mullis et al., 2012;
Wakefield, 2006), some of these problems could be solved by presenting the
students with more well designed and mathematically challenging tasks that
focus creative reasoning and mathematical properties rather than
unconnected procedures. Constructing and designing new tasks can however
be tedious work, but many textbook tasks have all components needed. The
problem is often that there is too much information for the tasks to have
potential for CMR. The following example from an upper-secondary textbook
can illustrate this:
422. In a rectangle the sides are 8 cm and 6 cm long. The long sides will
be shortened by 30% and the short sides will be increased by 25%.
a) How long are the sides in the new rectangle?
b) What is the area of the new rectangle?
c) What is the percentual difference between the areas of the old and new
rectangles?
To construct a task with higher potential for CMR we could just remove
some of the explicit sub steps and thereby leave it up to the student to consider
which steps that are needed to solve the tasks (provided that the student does
not know any solution methods by heart). Hence, we would end up with this
task:
422. In a rectangle the sides are 8 cm and 6 cm long. The long sides will
be shortened by 30% and the short sides will be increased by 25%. What is
the percentual difference between the areas of the old and new rectangles?
45
We could even remove the lengths so that the students would have to make
an assumption about how long the sides are, or try to make a more general
solution.
By removing the prescriptive parts of the task it becomes more effortful and
creative. Now each student can have an idea on how the task could be solved
and a discussion between classmates could focus on the mathematical
properties rather than on implementation of known procedures. The teacher
also has the opportunity to easily adjust the tasks to individual students by
removing more or less of the guiding steps, and by helping students with
conceptual problems. This adjustment takes much less time than having to
construct the task from scratch and though the original task probably will be
solved faster there will be less struggle with important mathematics. This
might be one way of utilizing the CMR-idea as a way to construct tasks that
are more effortful for the students but still concerns the content that the
textbook prescribes.
It is also necessary to consider the result that AR-practice (as discussed
under 7.2) is more cognitively demanding during test than during practice. If
students fulfill the learning goals by rote learning and if these learning goals
were best assessed by giving the students cognitively demanding test tasks, all
would be well. However, as many studies show, this is not the case. If we want
students to become proficient in mathematics, rote learning alone will not be
enough (e.g., Bjork & Bjork, 2011; Hiebert, 2003; Schoenfeld, 1985).
Personally I do not think that this is fair to the students. If students are
supposed to solve mathematical problems and reason creatively they must get
a chance to practice these competencies as well as the necessary procedures.
Studies 1-3 are made in an experimental setting and the classroom
environment is of course much more complex. However, the studies of this
thesis combined with many other studies in mathematics education give us
reasonable evidence that task design and students reasoning are important
aspects to consider when discussing students’ mathematical learning.
In classroom practice the teacher could address the question about task
appropriateness by choosing tasks that will fit into the curriculum and
presenting them at a moment where the students can solve the task but do not
know a simple procedure to do so. The teacher also has the possibility to
present new content in such a way that the students has to consider
mathematical properties instead of being served pre-defined procedures.
7.4 Limitations and implications for further studies
The included studies give indications on how to take the idea of creative
reasoning into a classroom study. The involved teachers would have to be
either educated in the different types of reasoning to be able to work as
46
creatively as possible, or scripted as to what help and which response they
would give the students. The teacher introductions and the tasks that the class
would be working with have to a larger extent to be designed with CMR in
mind to engage as many students as possible in this type of reasoning. It would
also be good for a study if it could encompass a complete section of the subject
knowledge that the class should learn that year (e.g., percentages or linear
equations).
Designing tasks that would fit the rather small window comprising what the
students know and what they are about to learn has also been a difficulty that
we have encountered. As have been addressed earlier, CMR-tasks need to be
solvable but not by any previously known solution method, and this have been
a challenge for the research group during the task design process since we
have little direct evidence of what the students’ prior knowledge were. We
chose not to test the students’ prior knowledge since a test of the particular
solution methods in the experiment would have influenced both the practice
and post-test. Therefore, we decided to rely on more indirect sources (i.e.,
which tasks the students had met in the textbook and their mathematics
grade) as a measure of prior knowledge. It would have been of interest for the
tasks to address topics in the current curriculum and as we only visited the
classrooms thrice it was hard to find tasks that fit.
It would also be interesting to meet with textbook authors and discuss the
development and testing of a new type of textbook that could put more focus
on creative reasoning from the beginning of each chapter. Such a textbook
could be evaluated by comparing observations of students’ work with it and
with one of the common textbooks, as well as with pre- and post-tests of
knowledge. However, a new textbook alone is insufficient unless teaching is
adopted to match the intentions of the new material.
There are also many adjustments that could be made to the existing studies
to examine how, for example, repeated practice, delayed tests, or different
student groups would influence the results. In fact, one such study has already
been made by one of our pre-service teachers, where the object was to see if a
more heterogeneous and slightly younger group of students would yield the
same result as study 1. The result showed that the same pattern occurs during
practice and also the importance of the need to think through the tasks for this
group of students (Wikman, 2015).
Since studies 1-3 were pseudo-experiments, we tried to reduce complexity
as much as possible to be able to study the influence of the tasks rather that of
something else. This meant that information to the students were scripted and
quite vague when it came to the the purpose of the different sessions. The fact
that the students did not know that they were going to be tested could also
have been important for their test performance. Students could have
concentrated harder to remember the solution methods and maybe even tried
to rehearse them before the test. Then again, it was not a high-stake test so the
47
students might not have bothered to engage in rehearsing since they have
plenty of other school-related and higher-stake-tasks to attend to.
Letting students talk to each other while solving the tasks could also be
interesting to study. From a result perspective this could increase the CMR-
students practice performance and according to the results in study 1-3, the
test scores. On the other hand, the help the students give each other could be
of an AR-type which might lessen the focus on important mathematical
properties and hence decrease the test-scores. It would also be interesting to
observe the student discussions, the questions and help they give each other,
both in the CMR-group and the AR-group, to notice if the task design will be
important for the quality of these discussions.
Another interesting but maybe impractical study would be to make an
fMRI-study of the practice situation. It would be very interesting to see how
the CMR-practice would compare to the AR-practice on a brain activation
level. Alas, at the moment I have no inkling on how such a study would be
designed or which parts of the practice situation that could be or should be
observed. Since the fMRI-environment puts restraints on the experiment
design a lengthy creative reasoning sequence would be hard to observe, with
all the noise that would infiltrate the data. Maybe some parts of the reasoning
process could be focused or maybe some sort of multiple choice questions
could lessen the time needed to construct a solution method. Maybe the
productive struggle could be isolated and observed but as I said, at this
moment I am just speculating. This kind of study could however give us deeper
insight into why CMR-practice seems to be so effective compared to AR-
practice and which brain processes that are active in sustaining this difference.
7.5 Conclusion
The studies included in this thesis confirm that task design is important for
what type of reasoning the students will engage in, and that the reasoning is
subsequently important for what is learned, just as Brousseau (1997) and
Lithner (2008) argued. The included studies also suggest that creative
reasoning is not only beneficial for the high achievers but equally (if not more)
fruitful for cognitively less proficient students. As a complement, the
additional result indicate that CMR-practice is highly taxing on cognition
while during the test practice performance becomes more important.
Conversely, practicing by AR seems to be quite effortless while the test seems
to place a higher demand on cognition. The results of study 4 also informed
us that CMR is important for progression through demanding tasks and that
the reasoning also is dependent on student characteristics, such as motivation
and perseverance along with subject knowledge.
These findings are of importance for both teacher practice and textbook
design. It is essential that students are well prepared for coming tests as well
48
as for future life and none of these will benefit from knowing a lot of superficial
procedures. Even if creative mathematical reasoning always can be used to
understand concepts and mathematical properties, task solvers may rather
use an available algorithm to solve a task as rapidly as possible. If students are
taught mathematics mostly based upon becoming proficient in using different
procedures, they may be even less likely to engage in creative reasoning if it is
not required from the task (Schoenfeld, 1985). Therefore, including more
creative mathematically founded reasoning tasks, and teaching that supports
students’ work with such tasks, is important if we want students to grow up to
become mathematically literate.
49
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