Chapter 7 Reasoning 7.3 Reasoning Strategies for the ... file12/04/2011 · Chapter 7 Reasoning 131...

Chapter 7 Reasoning 130

Chapter 7Reasoning

7.1 Introduction.............................................................................................................. 1317.2 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.................................................................................. 1457.2.6 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................................................................................................................ 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

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


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.


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


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


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)


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.

Concretesituation

Pre-requisitesfor successful

transfer

Dimensions for transfer

Past situation(Source situation)

New problem(Target situation)

Recognition Domain Hydrodynamics/aerodynamics

Hydrostatics

Functional role ofS

Pull PreventAbstraction

Manipulation ofvariable

Increase depth ofsubmergence

Increase depth ofsubmergence

pull body fromthe deep

Mapping Causal relationshipfor depth ofsubmergence

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.


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


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


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’.


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.


Figure 7.4: Interaction between Scientific Reasoning with general physics rules, prior knowledge, and BSL


General physics rules

Body Liquid


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


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


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.


Figure 7.5: Interaction between Scientific Reasoning with derived rules,prior knowledge, and BSL



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


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’.


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


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.


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


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


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


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

S1 SB SB BSL SL BSL BSL BSL BSL BSL BSLS2 BSL depth SL BSL BSL BSL BSLS3 depth volume density and

volumeS4 SB SB SB SB SB SBS5 independent

of materialBSL BSL SB SB BSL BSL BSL

S6 SB BSL same volume BSL same body SL SL

Table 7.7: A summary of rules and BSL ‘couples’ for the sinking domain (to be continued)


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 T9equilibrium volume of

immersionS7 depth

depth

same volume SB SB SB

SB

density andvolume

BSL BSL

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


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.


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


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...


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


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

nature.


Task 1 Task 2 Task 3.1 Task 3.2 Task 4 Task 5.1 Task 5.2 Task 6Level 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3

B × × × × × × × × × ×S × × × × × × × × × ×

S1

L × × × × × × × × × × × × × ×B × × × × × × × ×S × × × × × × × ×

S2

L × × × × × × × ×B × × × × × × × ×S × × × × × × × ×

S3

L × × × × × × × ×B × × × × × × × × × ×S × × × × × × × ×

S4

L × × × × × × × × ×B × × × × × × × ×S × × × × × × × ×

S5

L × × × × × × × × × ×B × × × × × × × ×S × × × × × × × ×

S6

L × × × × × × × × × ×B × × × × × × × ×S × × × × × × × ×

S7

L × × × × × × × ×B × × × × × × × ×S × × × × × × × ×

S8

L × × × × × × × ×B × × × × × × × ×S × × × × × × × ×

S9

L × × × × × × × ×Table 7.8: Levels of precision for qualitative reasoning (to be continued)


Task 7.1 Task 7.2 Task 7.3 Task 8.1 Task 8.2 Task 8.3 Task 9Level 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3 0 1 2.1 2.2 3

B × × × × × × ×S × × × × × × ×

S1

L × × × × × × ×B × × × × × × × ×S × × × × × × ×

S2

L × × × × × × ×B × × × × × × ×S × × × × × × ×

S3

L × × × × × × ×B × × × × × × ×S × × × × × × ×

S4

L × × × × × × × ×B × × × × × × ×S × × × × × × ×

S5

L × × × × × × ×B × × × × × × ×S × × × × × × ×

S6

L × × × × × × ×B × × × × ×S × × × × ×

S7

L × × × × ×B × × × × × × ×S × × × × × × ×

S8

L × × × × × × ×B × × × × × × ×S × × × × × × ×

S9

L × × × × × × ×Note: missing data for S7: Tasks 8.2 and 8.3× indicates an occurrence

Table 7.8: Levels of precision for qualitative reasoning

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