Chapter 7 Reasoning 130 Chapter 7 Reasoning 7.1 Introduction.............................................................................................................. 131 7.2 Reasoning Strategies for the Sinking Domain ......................................................... 131 7.2.1 Experiential Reasoning ...................................................................................... 133 7.2.2 Common-sense Reasoning ................................................................................. 135 7.2.3 Scientific Reasoning .......................................................................................... 138 7.2.4 BSL Reasoning .................................................................................................. 143 7.2.5 Reasoning strategy switch.................................................................................. 145 7.2.6 Summary................................................................................................................ 148 7.3 Reasoning Strategies for the Floating Domain ........................................................ 148 7.3.1 Fused versus isolated aggregate causal model for the sinking and floating domain................................................................................................................ 148 7.3.2 Aggregated bugged student causal model for the sinking and floating domain .... 153 7.4 Levels of Precision for Causal Reasoning ............................................................... 153 7.5 Conclusions.............................................................................................................. 155 Figure 7.1: Overview of the reasoning strategies ....................................................................... 133 Figure 7.2: Interaction between Experiential Reasoning, prior knowledge and BSL ................ 134 Figure 7.3: Interaction between Common-sense Reasoning, prior knowledge and BSL ........... 136 Figure 7.4: Interaction between Scientific Reasoning with general physics rules, prior knowledge, and BSL .................................................................................................................. 139 Figure 7.5: Interaction between Scientific Reasoning with derived rules prior knowledge, and BSL.......................................................................................................... 142 Figure 7.6: Interaction between BSL Reasoning, prior knowledge, and BSL ............................ 144 Figure 7.7: Aggregated bugged student causal model for sinking and floating ......................... 152 Table 7.1: Definitions of various reasoning strategies ............................................................... 131 Table 7.2: Definitions of various interactional links .................................................................. 132 Table 7.3: A comparison between the source and target situations............................................ 135 Table 7.4: A comparison between student common-sense reasoning model and a correct model ........................................................................................................................... 137 Table 7.5: Correct general physics rules .................................................................................... 138 Table 7.6: Switch of reasoning strategies ................................................................................... 147 Table 7.7: A summary of rules and BSL ‘couples’ for the sinking domain ........................ 149-150 Table 7.8: Levels of precision for qualitative reasoning ................................................... 156-1567
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Chapter 7 Reasoning 130
Chapter 7Reasoning
7.1 Introduction.............................................................................................................. 1317.2 Reasoning Strategies for the Sinking Domain......................................................... 131
7.3 Reasoning Strategies for the Floating Domain ........................................................ 148 7.3.1 Fused versus isolated aggregate causal model for the sinking and floating domain................................................................................................................ 1487.3.2 Aggregated bugged student causal model for the sinking and floating domain.... 153
7.4 Levels of Precision for Causal Reasoning ............................................................... 1537.5 Conclusions.............................................................................................................. 155
Figure 7.1: Overview of the reasoning strategies....................................................................... 133Figure 7.2: Interaction between Experiential Reasoning, prior knowledge and BSL................ 134Figure 7.3: Interaction between Common-sense Reasoning, prior knowledge and BSL ........... 136Figure 7.4: Interaction between Scientific Reasoning with general physics rules, priorknowledge, and BSL .................................................................................................................. 139Figure 7.5: Interaction between Scientific Reasoning with derived rulesprior knowledge, and BSL.......................................................................................................... 142Figure 7.6: Interaction between BSL Reasoning, prior knowledge, and BSL............................ 144Figure 7.7: Aggregated bugged student causal model for sinking and floating ......................... 152
Table 7.1: Definitions of various reasoning strategies ............................................................... 131Table 7.2: Definitions of various interactional links .................................................................. 132Table 7.3: A comparison between the source and target situations............................................ 135Table 7.4: A comparison between student common-sense reasoning model anda correct model ........................................................................................................................... 137Table 7.5: Correct general physics rules .................................................................................... 138Table 7.6: Switch of reasoning strategies................................................................................... 147Table 7.7: A summary of rules and BSL ‘couples’ for the sinking domain........................149-150Table 7.8: Levels of precision for qualitative reasoning ...................................................156-1567
Chapter 7 Reasoning 131
Chapter 7
Reasoning
7.1 Introduction
The Articulation-cum-Reflection Tool provided by the BSL System assumes the auxiliary role of
a reasoning tool as well as an object of reasoning when students were requested to solve for B, S,
and L whilst exploring the tasks in Stage 2 of the system. Thus, it primarily aims to facilitate
students’ reasoning and evoke their prior physics, as well as their common-sense knowledge. We
investigated how students reason by examining the reasoning strategies employed in the midst of
the problem solving process and the influence of prior physics knowledge on their reasoning.
Prior physics knowledge, in the context of this research, encompasses physics concepts,
quantitative relationships between concepts or laws, which are acquired from past formal
classroom instruction. Next, the patterns for the interaction between the exposed reasoning
strategies are abstracted, followed by drawing up an aggregated student causal model4 of
buoyancy, from which they derived their causal reasoning. Undeniably, the qualitative nature of
the BSL System compels the students to reason qualitatively and it is the interest of this research
to examine the level of precision of their qualitative reasoning.
7.2 Reasoning Strategies for the Sinking Domain
The four main categories of reasoning strategies that emerge from the transcripts for sinking
domain are Experiential Reasoning, Scientific Reasoning, Common-sense Reasoning, and BSL
Reasoning. The definition of each reasoning strategy is given in Table 7.1.
Term Definition
Experiential Reasoning Reasoning with knowledge acquired from past experiences includingeveryday or laboratory experiences. It could be in the form of a concretesituation or example.
Scientific Reasoning Reasoning with physical laws which encompasses general physics rules,derived physics rules which could be erroneous, or mathematicalexpressions.
Common-sense Reasoning Reasoning based on common-sense which could be derived byobserving the simulated laboratory model in the BSL System.
BSL Reasoning Reasoning based on relationships among B, S and L
Table 7.1: Definitions of various reasoning strategies
4 ‘Aggregated student causal model’ refers to the combined student causal models of all the participants inthe research
Chapter 7 Reasoning 132
In this chapter, four different types of links are used to represent the interactions between each
reasoning style, the constellations of prior physics knowledge and BSL. They are: conceptual
link, non-causal link, causal link, and inferred causal link. The definition of each type of link is
given in Table 7.2.
Term Definition
Conceptual link A link which binds two entities together basing on an existing underlyingsimilar concept or principle which is inferred by the researcher.
In other words, when a student reasons, some form of relevant priorphysics knowledge is invoked. It is the interest of the researcher to inferwhat prior knowledge has been abstracted and thus ‘establish’ a conceptuallink between the utterance and this abstracted knowledge.
Non-causal link A link between two entities by virtue of an existing non-causal relationshipbetween them. In other words, a change in one entity does not affect theother and vice versa
Causal link A link between two entities by virtue of an existing cause effectrelationship between them (explicitly articulated by students)
Inferred causal link A causal link which is inferred by the researcher (not explicitly articulatedby students)
Table 7.2: Definitions of various interactional links
Causal link, in this research, could either be quantitative relationships or qualitative causal
relationships. The former comprises mathematical expressions or specific values while the latter
encompass three different types of qualitative relationships namely:
Positive proportionality
Two entities are known to assume a positive proportionality relationship when a change in one
causes change in the other with the same direction (de Koning, 1997).
Negative proportionality
Two entities are known to assume a negative proportionality relationship when a change in one
causes change in the other with an opposite direction (de Koning, 1997).
Neutral proportionality
Two entities are known to assume a neutral proportionality relationship when the directions of
change in both the entities are not specified.
An overview of the interactions between each reasoning style, relevant prior knowledge and BSL
is illustrated in Figure 7.1.
Chapter 7 Reasoning 133
Figure 7.1: Overview of the reasoning strategies
7.2.1 Experiential Reasoning
As shown in Figure 7.2, Experiential Reasoning encompasses reasoning with previously
experienced situations or concrete examples. Reasoning with concrete examples is basically a
form of inductive reasoning where generalisations are drawn from specific examples. The
following excerpt exemplifies reasoning with a concrete example.
Excerpt 1: Plastic and steelS4: Increase in density…E: hmm…of the bodyS4: Density...that means …the weight …er…weight is actually…that means you are changing the…this parameter…S4: That means you are using steel or copper. Now you are increasing… Can’t increase anddecrease…S4: I increase so the…weight…oh yea, the weight should increase that means this will go down,isn’t it?E: OkS4: Yea you use plastic and steelS4: String increase...in fact three increases…
Based on the above excerpt, the example of plastic and steel was opted for instead of the prior
example steel and copper possibly due to a more salient difference in their relative densities.
Prior physics knowledge
Quantitative relationships betweenconcepts or physics lawsConcepts
Experiential Reasoning
ConcreteSituations
ConcreteExamples
Key
Conceptual link Non-causal link Causal link
S
Forces
Objects ofreasoning
B
Common sense ReasoningObjects of model
LiquidString
Body
Scientific Reasoning
GeneralPhysicsRules
DerivedRules
Equations
BSL reasoning
Liquid ForceString Force
Body Force
L
Chapter 7 Reasoning 134
However, it portrays an implicit kind of inductive reasoning. Thus, an attempt is made to
rephrase and fill the gaps in the utterance so that it assumes an explicit form inductive reasoning
which is demonstrated below:
Premise 1: Steel is denser than plasticPremise 2: Steel is heavier than plasticConclusion: Density of body increases and volume is constant, therefore weight (B) increases.
Figure 7.2: Interaction between Experiential Reasoning, prior knowledge and BSL
As shown in Figure 7.2, another component of Experiential Reasoning is reasoning with
knowledge drawn from previously encountered situations. Mayer (1992) defines analogical
reasoning as abstracting a solution strategy from a previous problem and applying it to a new
Inferred physicsknowledge
Concepts
Prior physics knowledge
Concepts
Experiential ReasoningConcrete Situations
Concrete Examples
Forces
Inertia
B S
Splash
Hydrodynamics/aerodynamics
Drag force
Shape ofobject
L
Work done againstdrag force
Quantitative relationshipsbetween concepts or laws
Hooke’s law
KeyConceptual link
(Student,Task)
Causal link
Inferred causal linkRate of liquid
displacement orvelocity of body
Relativedensities
Push rod and boxinto water (S4,T3.1)
Pull body fromthe deep (S5, T1)
Mercury andwater
(S1, S2, T9)
Splash and size ofpool (S4,T8.2)
Elongation ofspring
Push and size ofobject(S4,T3.1)
Steel andcopper
Plasticand steel(S1,T4)
(S,T)
Chapter 7 Reasoning 135
problem which is to be solved. This definition is mentioned in Chapter 2. However, in our case,
seemingly there is no such abstracted solution strategy and neither is there a related previous
problem but merely a past experience. Nevertheless, this facet of experiential reasoning could be
tied loosely to analogical reasoning because both belong to the similarity-based type of
reasoning. The principle for a successful analogical transfer between the two problems as being
recognition, abstraction, and mapping (Mayer, 1992) has been discussed in Chapter 2 too. It
provides a framework for examining students’ reasoning with previously experienced concrete
situations. For this purpose, a parallel comparison between the source and target situations is
drawn for two instances of concrete situation: pull body from the deep and push and size of
object. The results of the comparison are tabulated in Table 7.3.
When depth ofsubmergenceincreases, work doneagainst drag forceincreases (inferred byresearcher)
When depth ofsubmergenceincreases, S increases
Recognition Domain Dynamics HydrostaticsAbstraction Type of force Push Support/push
push and sizeof object
Mapping Causal relationshipfor size of body
A bigger body needsa greater push
A bigger surface willresult in a greaterpush from L
Table 7.3: A comparison between the source and target situations
Based on the first situation in Table 7.3, pull body from the deep, it can be seen that the first
condition itself, Recognition, is not satisfied due to an incompatible domain where one situation
involves motion while the other is otherwise. Consequently, this results in a misappropriated
Mapping despite having two similar features for Abstraction. The consequence of such a
misappropriated Mapping is that previous knowledge is wrongly applied to the target situation.
Such similar explanation could be accorded to the second example of push and size of object too.
7.2.2 Common-sense Reasoning
Common-sense reasoning is demonstrated when students apply one or more common-sense
rules. As shown in Figure 7.3, the two abstracted categories of common-sense rules relate to a
change in the body or liquid attribute.
Chapter 7 Reasoning 136
Some examples of student conclusions that can be drawn from Figure 7.3 are shown below.
Conclusion 1A change in a body attribute causally effects a change in B.
Conclusion 2A change in the position of a body does not causally effect B, S, and L
Figure 7.3: Interaction between Common-sense Reasoning, prior knowledge and BSL
Prior physics knowledge
Common-sense Reasoning
LiquidBody
Change in attribute
Forces
Concept
Quantitative relationshipsbetween concepts or laws
Key
Conceptual link
Non-causal link
Causal link
Boat
Slantedposition
UnstableLighter
Float
Floating
No changein attribute
Change in attribute
Hollow Position
LB S
Very smallliquid
column
Partialimmersion
Equivalentto no liquidL=0, L≈ 0
Anexternalforce ispresent
No changein attribute
Chapter 7 Reasoning 137
In Table 7.4, a comparison between an aggregated student common-sense reasoning model and a
correct model is drawn to expose the inadequacies and inconsistencies in the inferences students
make.
Category Student’s repertoire of rules Correct rulesRule 1: If true then change B Rule 1: If true and change is ρo or
volume of body then change BRule 2: If true and change isposition of body then constantB, constant S, and constant L
Rule 2: If true and change isposition of body with conditionthat body remains fully submergedthen constant B, constant S, andconstant L
Change in body attribute
Rule 3: If false then constant B,and constant S
Rule 3: If false then constant B,and if change in liquid attribute isnot ρl then constant S and constantL
Change in liquid attribute Rule 4: If false then constant L Rule 4: If false and if change inbody attribute is neither change involume of body nor volume ofimmersion then constant L
Note: Words in italics mean inadequacy in student’s ruleTable 7.4: A comparison between student common-sense
reasoning model and a correct model
A conclusion that could be derived from Table 7.4 is that the common-sense rules used by the
students are oversimplified rules. They appear to be true at the surface level but are, in actual
fact, incomplete when taking into consideration the conditions for these rules to hold.
In this part of the analysis, it is found that students made erroneous assumptions when solving B,
S and L. A noteworthy excerpt for S4 while executing Task 4, State is as follows:
Excerpt 2: Hollow implies floatingS4: er….er…this is a hollow oneE: hmm…hmm..S4: There is a…a space here…there is a…what do you call it? The…the…gas…whatever…the…E: Whatever… Ok…S4: There is a space here, there is a gap so that the liquid force has to…. against the. the… whatdo you call it? …. The space…the force…the force…do you know when there is a space, thisobject tend to float, isn’t it? The liquid force will have to be a bit strong to prevent this objectfrom going up
S4 assumes that a hollow object has the tendency to float though it clearly contradicts with the
model which depicts a fully submerged and static body. Figure 7.3 depicts some other examples
of erroneous assumptions made. They are: incorrect association between a slanted position and
instability, an external force acting on a partially submerged body, and a very small liquid
column which is just enough to cover a body, is inferred to be negligible. This finding seems to
confirm one of Henle’s (1978) stated reasons for faulty reasoning as being ‘the import of
additional and unwarranted factual assumptions into it’.
Chapter 7 Reasoning 138
7.2.3 Scientific Reasoning
In Chapter 2, the term scientific reasoning indicates the discovery of rules or laws through
hypotheses and experiments. Here, the three categories of rules coded are general physics rules,
derived physics rules, and quantitative rules in the form of equations. General physics rules refer
to commonly known rules. On the other hand, derived physics rules are rules abstracted from
students’ prior knowledge and transformed to fit the goals of a seemingly novel situation of the
system. Reasoning with both general and derived rules appears to be a form of deductive
reasoning. We examine how these rules are applied, and also, the basis for the derivation of
these rules.
Table 7.5 lists some of the general physics rules which involve only one parameter change at a
time. In the later part of this section, these rules are employed to evaluate some of the students’
applied general physics rules.
Rule No RuleRule 1 If density changes and volume is constant, then mass changes in the same directionRule 2 If density is constant and volume is constant, then mass is constantRule 3 If volume (dimension) changes and density of body is constant, then mass changes in
the same directionRule 4 If mass changes and g is constant, then weight (B) changes in the same directionRule 5 If volume of immersion changes and density of liquid is constant and g is constant,
then upthrust (L) changes in the same direction and tension (S) changes in theopposite direction
Rule 6 If volume of immersion is constant and density of liquid is constant and g is constant,then upthrust (L) is constant and tension (S) is constant
Rule 8 If density of liquid changes and volume of immersion is constant and g is constant,then upthrust (L) changes in the same direction and tension (S) changes in theopposite direction
Note: words in italic refer to premises which are generally omitted by studentsTable 7.5: Correct general physics rules
i. General physics rules and equations
In Figure 7.4, it can be seen that the commonly applied general physics rules relate to volume of
immersion, dimensions, density, mass and weight. These rules are grounded either on
Archimedes Principle or several physics equations. They are the density, weight or force
equations.
The details of the student articulated general physics rules are found in Appendix M. Here, we
present a summary of the findings. General rules have been employed more frequently for B than
for S or L. Seemingly, S9 did not apply any general rule for S and L at all. Tasks that incur a
higher frequency use of general rules for B are Task 4 (Density of Body), Task 5.1 (Width of
Body), and Task 5.2 (Height of Body) while for L, it is Task 9 (Density of Liquid). Student S6
was the only participant who related L to volume of displaced liquid while S3 was the only
student to relate L to pressure difference, though incompletely.
Chapter 7 Reasoning 139
Figure 7.4: Interaction between Scientific Reasoning with general physics rules, prior knowledge, and BSL
Scientific Reasoning
General physics rules
Body Liquid
Prior physics knowledge
Quantitative relationships between concepts or lawsEquations
WidthVolume ofdisplaced
liquid
Density of body
Volume
Height
Mass
ForcesS LB
Key for symbolsρ - densityg - acceleration due to gravityh - depthm - massV - volumeA - horizontal cross-sectional areaF - forceP - pressureW - weight
Key for notations
Density ofliquid
Archimedes’ Principle
VolumeMass
Weight
Conceptual link
Causal link
Inferred causal link
Non-causal link
W=mgρ=m/V F=ma
Volumeof immersion
Pressuredifference
Upthrust
Weight
P=F/A
Chapter 7 Reasoning 140
Some of the applied rules are either insufficient or erroneous when compared with the correct
general physics rules listed in Table 7.5. Two instances are used to provide evidence of
insufficient rules. Rules in bold print are student erroneous rules while underlined premises are
necessary conditions.
Sufficiency of applied ruleRule 2: If density is constant and volume is constant, then mass is constantRule 4: If mass changes and g is constant, then weight (B) changes in the same directionCombined correct rule for B: If density of body is constant and volume of body is constant andg is constant, then mass and weight (B) are constant
Insufficient rule 1 with only one necessary premiseS2 and S9: If density of body is constant, then B is constant
Insufficient rule 2 with only two necessary premisesS3 and S8: If density of body is constant and volume is constant then B is constant
In the above example, the three necessary conditions for a sufficient rule in Task 7.1 (Shape:
Cone) have been underlined. However, the rules employed by S2, S3, S8 and S9 are considered
insufficient because they do not fulfil these necessary conditions.
Erroneous applied ruleRule 3: If volume (dimension) changes and density is constant, then mass changes in the samedirectionRule 4: If mass changes and g is constant, then weight (B) changes in the same directionCombined correct rule for B: If volume (dimension) of body changes and density of body isconstant and g is constant, then mass and weight (B) changes in the same direction
First erroneous applied ruleS7: If volume increases, density increases
Erroneous modified ruleS7: If width increases then volume increases
If volume increases then mass increases If mass increases then density decreases If density decreases then weight (B) decreases
Second erroneous applied ruleS8: If volume increases then density increases
If density increases then mass increases If mass increases then (B) increases
The above example is for Task 5.1 (Width of Body). Students S7 and S8 were the only two
participants who mistakenly asserted that the density of body is not constant and this accounts
for the three erroneous applied rules. Both of them demonstrated a chained type causal reasoning
with flawed intermediate rules which could lead to a correct conclusion.
ii. Derived rules and equations
As depicted in Figure 7.5, the categories of derived rules coded are centre of gravity rule,
equilibrium rule, static rule, depth rule, and surface area rule. Student perception of surface area
is varied. Generally, it is viewed as the surface area of the bottom of the body. However, some
Chapter 7 Reasoning 141
regarded it as the area of the side surface or bottom and side surfaces. Figure 7.5 also shows that
conflicting rules have been applied for the width, height and volume of liquid column as well as
depth of the body. In this part of the analysis, the bases for the derivation of these rules are
examined and discussed. They are as follows:
Schema-based
Centre of gravity or its abbreviated form cg, is defined simply as the point in an object at which
its weight acts. In Figure 7.5, cg is considered as an entity for the schema of weight. Thus, the
derivation of the cg rule, which encompasses stability and position of cg, and distance from cg,
is schema-based.
Model-based
BSL as depicted in the forces model are in equilibrium and this prompted the erroneous derived
equilibrium rule which states that all the three said forces will remain constant when the
equilibrium condition is true. Theoretically, when BSL are in equilibrium, it means that the
resultant of these forces is zero and the object they act on, which is the block, is in a static
condition. Such a condition is also known as static equilibrium. However, as shown in the
Extended Articulation list in Table 6.7, the terms, static and equilibrium are not used together but
interchangeably. An offshoot of the static rule is the no friction rule which could be inferred as a
rule applied to a situation where no work is done against friction and thus resulting in a constant
L.
Abstraction from formula
In theory, L originates from the difference between the upward and downward resultant forces
due to the liquid pressure which act on a fully or partially submerged body The liquid pressure
assumes two different algebraic equations forms: P=hρg and P=F/A. Here, ‘h’ is depth and thus,
probably, an origin for the derived depth rule while the ‘A’ in the second formula is cross-
sectional area of the object which is perpendicular to the line of action of L. However, as seen in
Figure 7.5, the ‘A’ which gives rise to the surface area rule is mutated in many forms.
Details of the student derived rules are shown in Appendix N. Some of the conclusions that
could be drawn are: derived rules are used more frequently for L than for S or B, and types of
derived rules applied for S resemble that of B except for the cg rule. This could be due to a
strong association between tension and weight for a suspended object. The derived rules used for
L are predominantly area and depth rules. The depth rule results in conflicting conclusions but
are applied for B, S and L. Students S1, S2 and S5 even applied the depth rule for Task 8.3,
which increases the height of the liquid column.
Chapter 7 Reasoning 142
Figure 7.5: Interaction between Scientific Reasoning with derived rules,prior knowledge, and BSL
Prior physics knowledge
Scientific Reasoning
Derived rules
EquationsConcepts
Forces
Key for symbolsρ - densityg - acceleration due togravityh - depthA - cross-sectional areaF - forceP - pressure
Key for notations
Conceptual link
Causal link
Non-causal link
Center of gravity(rule)
Distance fromcg
P=F/AP=hρgEquilibrium of forcesCentre of gravity
Heightof body
Widthof body
Volume of immersionWeight Density of body
Static (rule)
No friction
Equilibrium (rule)
Width ofliquid column
Height of liquidcolumn
Volume of liquidcolumn
B LS
Depth(rule)
Surface area (rule)Bottom surface areaSide surface areaBottom and side surface areas
Chapter 7 Reasoning 143
The surface area rule, though it appears in various forms, yields very consistent conclusions for
L too. Both S4 and S9 had the most number of varying area rules for L which thus suggests a
prevailing fuzzy schema for L. As for S5, in Task 6, he demonstrated the use of an elastic area
rule for L, which was stretched beyond the normal consistent area rule so that it would appear to
fit into a solution which he appeared to be more certain of. The following excerpt exemplifies
such a phenomenon.
Note: Bold italics in brackets refer to comments
Excerpt 3: Elastic ruleE: Why should this one (referred to L) increase?S5: That’s my problem now because I can’t give a proper reason…E: So you feel it should increase…S5: This one as well should increase (referred to L) if I use my guessing, should increaseE: Why is it so? When you guess, surely you base it on some kind of reasonS5: Yea, because of the area here (left and right vertical area, and bottom area) but then Ialways think that this area (vertical area) does not make much difference… probably it is true
Task 8.1 which portrays a situation with a very small liquid column that is just sufficient to cover
the object, appears to be quite a problematic task viewing the conflicting causal relationships
given by the students.
7.2.4 BSL Reasoning
In Figure 7.6, the scientific terms weight, tension and upthrust which correspond to BSL, are
familiar terms. However, as previously mentioned in Chapter 6, L is misconstrued as force due
to pressure. The rules used for BSL reasoning are the BSL rule and ‘couples’5. The BSL rule
considers the equilibrium state of the model where magnitude of B is perceived as the sum of the
magnitude for both S and L, or S is viewed as the resultant of B and L. The ‘couples’ as depicted
in Figure 7.6 are SB, LB, BS, LS, BL and SL, means that the first entity is causally dependent on
the second entity. In other words, the first entity is considered as a dependent variable while the
second entity is otherwise.
The details for the usage of BSL Reasoning strategy are found in Appendix O. A general
observation that can be made is that it is applied predominantly for S and rarely for both B and
L. The content analysis apparently shows that student S3 was neither a BSL rule nor ‘couple’
reasoner. Students S1, S2 and S5 appeared to be very consistent BSL rule users. However, the
rest of the students seemed to be predominantly BSL ‘couple’ users. S1 seemed to apply the
BSL rule almost in the middle part of the exploration while for students S7 and S8, it was
towards the end of the exploration of sinking situations for Stage 2 of the system. This suggests
5 Examples of ‘couples’ are SB, LB, BS, LS, BL, and SL ‘couples’ where the first entity in each ‘couple’is causally dependent on the second entity in the ‘couple’.
Chapter 7 Reasoning 144
an initial incomplete rule is honed when used over an extended time. The discussion of the
application of the aforementioned ‘couples’ is described very briefly.
Note: In a ‘couple’, the first entity is causally dependent on the second entity.E.g. SB ‘couple’ - S is causally dependent on B
Figure 7.6: Interaction between BSL Reasoning, prior knowledge, and BSL
SB ‘couple’
Based on the information given in Appendix O, the SB ‘couple’ is predominantly perceived as a
positive proportionality relationship and this ‘couple’ holds only when L is constant. Once again,
it suggests a strong dependency of S, tension, on B, which is weight and this confirms the earlier
observation made about S and B. The SB ‘couple’ is switched to the SL ‘couple’ or BSL rule
after Task 6, Volume of Immersion, and this could be due to the influence of some past
knowledge about buoyancy, though incomplete.
LB ‘couple’
For the sinking domain, this ‘couple’ is wrong as both L and B are independent variables and L
will not in any way causally depend on B. This ‘couple’ is rarely used for the sinking domain
and is applied only at the beginning of the exploration. Possibly, at this stage of exploration,
some of the students were still in midst of searching for a more consistent and logical rule. On
Prior physics knowledgeConcepts
BSL Reasoning
Forces
B S L
Equilibrium offorces
Tension UpthrustWeight
BSL
Key
Force due topressure
Conceptual link
Causal link
Liquid ForceString ForceBody Force
BS BLSB SLLSLB
Chapter 7 Reasoning 145
the other hand, the ‘couple’ is correct for the floating domain since B is an independent variable
while L is otherwise. A few students used this ‘couple’ for the floating domain of the Questions
Stage.
BS ‘couple’
Just like the LB ‘couple’, the BS ‘couple’ is not correct logically because S is a dependent
variable while B is otherwise and this suggests why it is used only once in Task 1 (Depth of
Submergence).
LS ‘couple’
LS ‘couple’ is also considered logically wrong due to the same reasons accorded for the BS
‘couple’ and once again, this suggests why it is used sparingly.
BL ‘couple’
Based on the definition of B in the system, the BL ‘couple’ is also logically wrong due to the
same reasons given in the LB ‘couple’ (for sinking domain). However, a predominantly negative
proportionality relationship between B and L suggests that B is perceived as apparent weight and
not the actual weight.
SL ‘couple’
It is already mentioned that SL is applied only after Task 6 (Volume of Immersion). The SL
‘couple’ is often perceived as a negative proportionality relationship. However, such a causal
relationship will only be true if B is constant.
7.2.5 Reasoning strategy switch
A sample coded data for the switch of reasoning strategies is shown in Appendix P. The dots and
the arrows in Table 7.6 illustrate the type of reasoning and switch of strategy that occurred when
the students articulated their causal explanation. Both these representations are placed in a
sequence which reveals the sequence of events that occurred. As shown in Table 7.6, the
students predominantly applied Scientific Reasoning strategy. The next frequently used strategy
is the BSL Reasoning strategy. Experiential Reasoning and Common-sense Reasoning strategies
are very rarely employed. The pattern of switch that occurs most frequently seems to be the
switch from Scientific Reasoning to BSL Reasoning and very seldom is BSL Reasoning used to
start off a task. This supports the finding for the preceding section which states that BSL
Reasoning is used predominantly for S and rarely for B, or L. Thus, S being the dependent
variable has to be solved after B and L if this correct order of event is observed.
Chapter 7 Reasoning 146
Task 1 Task 2 Task 3.1 Task 3.2 Task 4 Task 5.1 Task 5.2 Task 6NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL
S1
• • • • •
••
• • • •
•
S2
• •
• • •
• •
• •
• •
S3 • ••
• •
S4
• • •
•
•
• • •
S5 • • • •
• • •
S6 •• •
••
S7
• •
• •
• •••
• • •
• •
•
Key Table 7.6: Switch of reasoning strategies (to be continued)• No switch of strategy Switch is from left to right other Switch is from right to left
Chapter 7 Reasoning 147
S8 ••
• • •
S9 • • •
••
•• •
• • •
Task 7.1 Task 7.2 Task 7.3 Task 8.1 Task 8.2 Task 8.3 Task 9NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL NO ER CR SR BSL
S1 • •• •
• •
•
S2 ••
• • •
S3 • • • • • • •S4 • •
S5 • • •
•
S6
• • •
• • • •
S7 • • • •
S8 • •S9 •
• •
••
•
• • •
Table 7.6: Switch of reasoning strategies
Chapter 7 Reasoning 148
7.2.6 Summary
Most scientific and some common-sense rules coupled with BSL rules used for reasoning about
BSL in the sinking domain are summarised and tabulated in Table 7.7. In conclusion, general
rules are used most frequently for B while for S, it is BSL rule or ‘couples’, and for L, it is
derived rules. Possibly, this is due to the fact that B is the most familiar and straightforward of
all. As for L, the main reason for a recurrent use of derived rules might be attributed to students’
incomplete conception of L. S is perceived as a resultant force or strongly associated with its two
counterparts, and this might explain the predominant use of BSL rule or ‘couples’ for S.
In summary, the concrete examples used for the Experiential Reasoning are more apt than the
concrete situations due to lack of transfer skills. As for Common-sense Reasoning, most of the
rules used are incomplete and oversimplified though valid conclusions could be attained at times.
General rules are most frequently applied for B while derived rules are largely used for L. The
derived rules for L are more varied and numerous compared to B, thus suggesting a more fuzzy
understanding of L than B. In addition most of the derived rules for L appear to be incomplete
abstractions from the formula for pressure which thus often result in an invalid conclusion. As
for S, BSL Reasoning is the main strategy used, probably due to the fact that it is perceived as the
dependent variable by most of the students.
7.3 Reasoning Strategies for the Floating Domain
The reasoning strategies applied for the floating domain are similar to the four previously
mentioned ones: Experiential Reasoning, Common-sense Reasoning, Scientific Reasoning, and
BSL Reasoning. The similarities and differences in the application of these reasoning strategies
for the sinking and floating domain are discussed in the next sub-section.
7.3.1 Fused versus isolated aggregate causal model for the sinking and floating domain
Detailed information pertaining to four common tasks in the sinking and floating scenarios found
in Stage 2 of the BSL System are shown in Appendix Q. It can be concluded that three students
S1, S3, and S6 demonstrated a clear separate conception for the sinking and floating phenomena
when they maintained that S=0 or S remain constant at 0, for all the above mentioned floating
situations. This first category of reasoners are known as isolated reasoners who view sinking and
floating phenomena as distinctively different. As a matter of fact, in the initial stage, S3 did not
belong to this category and the details of such a change are discussed in Chapter 8. As for the
second category of reasoners, they are called fused reasoners who perceive the sinking and
floating domain as one and the rest of the six students fall under this category. This is further
Chapter 7 Reasoning 149
A summary of rules and BSL ‘couples’ for Body Force in sinking scenario (for each task and student)Student T1 T2 T3.1 T3.2 T4 T5.1 T5.2 T6 T7.1 T7.2 T7.3 T8.1 T8.2 T8.3 T9
depth BSLS1BS BL
width height BL density &volume
density
density andvolume
S2 mass mass mass
mass
density densityW=mg
W=mg density &volume
density Imply g mass
S3 depth same body density volume density andvolume
width,volume,density
heightvolumedensity
upthrust Imply depthS4 depth density
mass weight BL
cg cg cg upthrust cg
cg
BL
S5 depth density volume position density andvolume
same body
density andvolume
width BSLS6 BL
weight weight same cuboid
cg BS
equilibrium density width,volume,density
heightvolumedensity
S7 depth
same volume
position volume
weight mass mass
density andvolume
density volume,density
volumemass
S8 same object
mass mass weight
density andvolume
density densityS9 depth area areaweight
width area volume ofimmersion
density BLarea
volume volume volume
A summary of rules and BSL ‘couples’ for String Force in sinking scenario (for each task and student)Student T1 T2 T3.1 T3.2 T4 T5.1 T5.2 T6 T7.1 T7.2 T7.3 T8.1 T8.2 T8.3 T9
S8 SL BSL SB SB SB BSL SL SL BSL SLS9 depth static area SB SB area area SL volume volume volume SL
A summary of rules and BSL ‘couples’ for Liquid Force in sinking scenario (for each task and student)Student T1 T2 T3.1 T3.2 T4 T5.1 T5.2 T6 T7.1 T7.2 T7.3 T8.1 T8.2 T8.3 T9
static heightS1 BSL area areaBSL
area area volume ofimmersion
area volumedepthheightS2 depth area area area area area volume of
immersionarea aerodynamic
shapearea area
depthdensity
S3 depth pressure only positionchanges
density andvolume
LBS4 depth area
depth
areaP=F/A
areaP=F/A
area area volume length static F=maρ=m/vdensity
S5 depth independentof material
area area density area area area area area volume width depth density
density volume ofimmersion
S6 depth Volume ofdisplacement
area area
depth
area area
depth
area area volume
volumemass
densityvolumemass
S7 gravity force equilibrium volume
W=mg
volume density andvolume
F=ma
density andvolume
static area staticS8no friction BSL
areano friction
area area area area area area volume area area density
static LB areaS9depth
LBBSL
LB LSLS
area area volume volume volume area
Table 7.7: A summary of rules and BSL ‘couples’ for the sinking domain
Chapter 7 Reasoning 151
confirmed by their BSL solutions in the task where the ellipse S was not placed at the origin of
the one-dimensional graph.
Student S2 applied almost the same rules for both the sinking and floating tasks while most of
the rules used by S4, S5, and S7 were mostly similar. Student S9 demonstrated the least number
of similar rules. Unfortunately, a comparison of the rules for S8 cannot be made due to the loss
of audio-taped data. Once again, the change of the rules are further described in Chapter 8.
For the task on Density of Liquid, students S2, S5 and S7 applied a LS negative proportional rule
which yields a decreasing S when L increases. However, in the interface, the starting value for S
has already been fixed at zero and consequently, the S end value selected for the graph falls into
the negative region. This means that S will be a downward force, which is an unrealistic situation
in the simulated model. Thus, it is evident that occasionally rules are applied without regarding
the feasibility of its outcome.
Isolated reasoners only used LB or BL ‘couples’ for the floating situations. When LB ‘couple’ is
used, B is considered as an independent variable while L is regarded as a dependent one. As for
the BL ‘couple’, it is the reverse. Though the latter ‘couple’ is not logically correct, it still leads
to a valid conclusion. Here, we examine how these rules are applied under two differing
conditions where an attribute of the body or liquid is manipulated.
Manipulation of an attribute of the body
Whilst executing task on density of body, both S1 and S6 reasoned about B first, followed by
using BL ‘couple’ for L, which results in a valid conclusion. As for S3, she reasoned about L
first, followed using LB ‘couple’ for B, which also yields the same result.
Manipulation of an attribute of the liquid
As for the task on Density of Liquid, both S1 and S6 solved for L first, followed by applying LB
‘couple’ for B, which produces an invalid outcome. However, S6 reasoned that the block
remained the same and subsequently, altered his answer for B leaving L untouched. Student S3
demonstrated complete reasoning for L when she mentioned that the causal effect of an increased
density of liquid is nullified when the body rises a little and thus decreasing its immersed volume.
The above descriptions suggest that students do not have a correct perspective of the roles of B
and L in a floating scenario. In actual fact, B is the independent variable while L is a dependent
one. On the contrary, some might think that the two roles can be swapped depending on the type
of change that occurs.
Chapter 7 Reasoning 152
Figure 7.7: Aggregated bugged student causal model for sinking and floating
Mass ofdisplaced liquid
Densityof liquid
Verysmall
column
Static
Equilibriumstate
Depth ofsubmergence
Surfacearea
Weight
Position of cg
Width of body
Volume ofbody
Mass of body
Key
+
+
+
+
+
+
+
+
+-
+
+
+
+
~
+
+
+ -, =0, ≈0 -
S+
-+
-
S
F+
FloatingS=B-LB=L
∴ S=0
SinkingS=B-L
B-L+ + -
F~
F+
S+
F+ S-
F-
F+S+
Conceptual link
Various causal links
Non-causal link
positive proportionality
negative proportionality
neutral proportionality
equal
+
Weight ofdisplaced liquid
Depth ofimmersion
+
-
Volume ofimmersion
Volume ofdisplaced liquid
B
++
~ ~
+
+- ++-
Density ofbody
Height of column
Width of columnVolume of column +
= =
S
S+
S -
F -
Sinking
Floating
Erroneous conception
Variables
Height ofbody
Buoyant force
L
B L
upthrust
Chapter 7 Reasoning 153
7.3.2 Aggregated bugged student causal model for the sinking and floating domain
Figure 7.7 depicts a general representation of the bugged conceptual model that the students
applied when they reasoned about BSL in both the sinking and floating situations. The text boxes
in Figure 7.7 that are black in colour represent students’ bugged concepts in the buoyancy
domain. They are surface area, equilibrium state of body, depth of submergence, position of cg,
extreme size of the liquid column, and the static state of the body. On the other hand, bugged
causal relationships are represented by black proportionality boxes. For example as shown in
Figure 7.7, the variable Depth of Submergence is wrongly regarded as having a positive or
negative proportionality causal link with B, S and L.
As shown in Figure 7.7, the principle concept of buoyancy which is the weight of displaced
liquid has been entirely excluded in the students causal explanation whilst exploring BSL
System. As for the floating domain, some of the students utilised the two critical concepts for the
floating domain which are S equal 0 and LB positive proportional ‘couple’ for solving the tasks
in Stage two of the system.
7.4 Levels of Precision for Causal Reasoning
The levels of precision for hypothesis (Ploetzner & Spada; 1992) which have been discussed in
Chapter 2, are adapted for the purpose of this analysis. In the BSL System, students use
qualitative graphs to represent their solutions. Consequently, this could prompt them to consider
the relative gradients for the BSL graphs or relative changes in BSL in their reasoning as well.
Such a kind of reasoning could be regarded as qualitative reasoning with a second order or level
of precision. Thus, the level two precision is subdivided into two categories, primarily, to cater
for such a type of more precise causal relationship. However, for the purpose of this analysis, the
evaluation of the levels of precision for students’ causal reasoning will only be confined to
whatever was articulated and also for sinking tasks with two-dimensional graphs. Level 0 is
provided to make room for instances with no articulated relationship. The modified version of
the levels is as follows and excerpts will be used to exemplify every level except for level 0:
Level 0: No articulated relationship
Level 1: GeneralA change in one entity effects a change in another entity with no mention of the its direction ofchange
Excerpt 4: Task 5.1-Width of BodyE: Why is it so? Explain one by oneS6: er…these two (referring to S and L)… just affected by body…body forces, isn’t it?E: Is that what you think?S6: Yea,…mostly affected by the weight of the thing...
Chapter 7 Reasoning 154
Level 2.1: Qualitative relational of the first order of precisionA change in one entity effects a change or no change in another entity with a mention of itsdirection of change
Excerpt 5: Task 6-Volume of immersionS7 Ok, liquid force, force in the liquid which supports…ah haE: Do you understand?S7: ah haE: Please explainS7: Liquid force is higher because the…er…the density of water is higher…higher …
Level 2.2: Qualitative Relational of the second order of precisionA change in one entity effects a change in another entity with a mention of its direction of changeand also its relative qualitative amount of changeorA change in one entity effects a change in more than one other entities with a mention of theirrespective directions of change and also their relative qualitative amount/s of change
Excerpt 6: Task 5.2-Height of BodyE: Increase the heightS1: Ok, increase the height, this (referring to B) will increase proportionately….now surfacecontact will still be the same. I am sure it will be constant or decrease a bit.orExcerpt 7: Task 5.1-Width of BodyE: What are you thinking now?S6: …er…think this one (referring to S) will be…just slightly increase….the liquid force…verymuch…increase very much…and this one (referring to B)…same as the…red one, what isit?….the string forceE: Why is it so? Explain one by oneS6: er…these two… just affected by body…body forces, isn’t it?E: Is that what you think?S6:Yea,…mostly affected by the weight of the thing...so…increasing the width, will increase theweight by just small fraction but the surface area down here is going to be much greater(referring to the bottom of the block)…since we increase the width so the liquid force will bemuch greater
Level 3: Quantitative relationalA change in one entity effects a change in another entity with a mention of its direction of changeand also its relative quantitative amount of changeorTwo or more entities are considered quantitatively relational if they are related through a physicallaw or some form of mathematical expression
Excerpt 8: Problem 1 of Problem Solving StageS6: String force…huh…laughedS6: When width increase, area increase, and volume increase. If width increase by 1…if width is4, bottom area will be 16, if 4,5, bottom area will be 20 and 20…4,4,4…16,4 is 64…20,4 is 80,so the volume will be much bigger. I have to see the immersed volume first…or the height of thebody…S6: If this is increased, the liquid force increases and string force decreases…orExcerpt 9: Task 5.1-Width of BodyS4: er….this is a small areaE: ah…haS4: We need less force….to make it stableE: hmm…hmmS4: Bigger area, less pressure, isn’t?S4: Smaller area, big pressureE: hmm…hmm
Chapter 7 Reasoning 155
S4: Pressure is force per unit area. Yea think so… because this is small surface of area. So weneed less force compared to this one
Results shown in Table 7.8 reveal that the level of precision for the students’ causal reasoning
seems to be predominantly at level 2.1. Only S1, S2, and S3 demonstrated level 2.2 of causal
reasoning. Its low frequency of occurrence suggests that the students’ verbal qualitative
reasoning is not precise enough even though linear qualitative graphs in the interface could
afford a higher level of precision. Five of the students, S1, S2, S4, S6, and S7, reasoned
quantitatively, with some of them using physics formulas that have been mentioned in Chapter 6.
7.5 Conclusions
In conclusion the students employed four reasoning strategies: Experiential Reasoning,
Common-sense Reasoning, Scientific Reasoning and BSL Reasoning whilst exploring in the
computer-based environment. The most frequently applied strategy is Scientific Reasoning
which could be further sub-divided into reasoning with general, derived rules or equations.
General rules are most frequently used for B, while most derived rules are related to L.
Equations are rarely employed in reasoning. The BSL Reasoning strategy is most frequently for
S. Prior knowledge plays a crucial role in facilitating students’ qualitative reasoning about BSL.
Nonetheless, the results in this research show that often, an incomplete reasoning with past
knowledge leads to erroneous rules that in turn yield invalid or even infeasible conclusions.
Also, the data analysis reveals that there are two categories of buoyancy reasoners, one being the
fused reasoners and the other, the isolated reasoners. Most of the students reason qualitatively
with the first order precision while some move beyond this level, reasoning causally with a
precision which is qualitative relational of the second order precision or quantitative relational in