References 1

Building a bridge between school and university- critical issues concerning interactive applets

Timo Ehmke (Kiel / GER) Lenni Haapasalo (Joensuu, FIN)Martti E. Pesonen (Joensuu, FIN)

NBE ’05, Rovaniemi, Finland

Based on the project

From Visual Animations to Mental Models in Mathematics Concept Formation

(sponcored by DAAD and the Academy of Finland)

Martti E. Pesonen (Department of Mathematics, University of Joensuu, FIN)

Lenni Haapasalo (Department of Applied Education, Joensuu, FIN)

Timo Ehmke (Leibniz Institute for Science Education / IPN, University of Kiel, GER)

15.09. '05NBE '05 Ehmke, Haapasalo & Pesonen

References 1References 1 Haapasalo, L. & Kadijevich, Dj. (2000). Two Types of Mathematical

Knowledge and Their Relation. Journal für Mathematikdidaktik 21 (2), 139-157.

Haapasalo, L. (2003). The Conflict between Conceptual and Procedural Knowledge: Should We Need to Understand in Order to Be Able to Do, or vice versa? In L. Haapasalo & K. Sormunen (eds.) Towards Meaningful Mathematics and Science Education.

Kadijevich, Dj. & Haapasalo, L. (2001). Linking Procedural and Conceptual Mathematical Knowledge through CAL. Journal of Computer Assisted Learning 17 (2), 156-165.

Kadijevich, Dj. (2004) Improving mathematics education: neglected topics and further research directions. University of Joensuu. Publications in Education 101.


References 2References 2 Pesonen, M., Haapasalo, L. & Lehtola, H.

(2002) Looking at Function Concept through Interactive Animations. The Teaching of Mathematics 5 (1), 37-45.

Pesonen, M., Ehmke, T. & Haapasalo, L. (2005). Solving Mathematical Problems with Dynamic Sketches: a Study on Binary Operations. To appear in the Proceedings of ProMath 2004 (Lahti, Finland).

http://www.joensuu.fi/mathematics/MathDistEdu/MAA2001/index.html

http://www.joensuu.fi/mathematics/MathDistEdu/MAA2001/index.html


References 3References 3 Sierpinska, A., Dreyfus, T., & Hillel, J. (1999).

Evaluation of a Teaching Design in Linear Algebra: the Case of Linear Transformations. Recherche en Didactique des Mathématiques, 19 (1), 7-40.

Tall, D. & Bakar, M. 1991. Students’ Mental Prototypes for Functions and Graphs. Downloadable on Internet at

http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1991f-mdnor-function-pme.pdf

Tall, D. 1992. The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof. In D. Grouws (ed.), Handbook of research on mathematics teaching and learning. NY: MacMillan, 495-511.




... References 3... References 3 Vinner, S. & Dreyfus, T. (1989). Images and

Definitions for the Concept of Function. Journal for Research in Mathematics Education 20 (4), pp. 356-366.

Vinner, S. (1991). The Role of Definitions in Teaching and Learning. In D. Tall (ed.): Advanced mathematical thinking (pp. 65-81). Dordrecht: Kluwer.

Holton, D. (2001) The Teaching and Learning of Mathematics at University Level. An ICMI Study. Dordrecht: Kluwer.


BackgroundBackground • Mathematics is considered as organized body of knowledge.

• Students are largely passive, practicing old, clearly formulated, and unambiguous questions for timed examinations. •Theory is abstract and depends on an unfamiliar language.

These features leave students dispirited and bored, and their performance in more advanced courses is poor because the foundations are weak.

The assessment is reduced to bookwork and stereotyped questions, to be remembered without becoming a vital part of the student. (Joint European Project MODEM; http://www.joensuu.fi/lenni/modem.html

http://www.joensuu.fi/lenni/modem.html








School vs. UniversitySchool vs. University• The main problem: how students could

develop their procedural school thinking towards abstract conceptual thinking?

• Neglected topics: • promoting the human face of mathematics • relating procedural and conceptual knowledge• utilizing mathematical modelling in a humanistic,

technologically-supported way• promoting technology-based learning through

multimedia design and on-line collaboration


AimsAims / 1 / 1st st stepstep To generate hypotheses, what special

benefits do the dynamic interactions offer and what new types of difficulties in conceptual thinking arise.

What advantages are there in manual dragging by the students (within the applets) and what in automatic animation?

How students use the tracing function and what significance do the given hints have?


AimsAims / 2 / 2nd nd stepstep To analyse whether different representations

(symbolic, verbal, graphic) given through interactive applets) lead to different test performance.

To consider possible explanations to these difficulties (e.g. why conceptually identical but functionally slightly different implementations lead to diverging interpretations).


IngredientsIngredients mathematical: the concept definitions pedagogical: framework of concept

building technical: dynamic Java applets,

WebCT test tools

Example of a dynamic applet

http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/SavonlinnaLETTET2005/LeTTET_Figure1JSP.html

http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/


Features of the interactive tasksFeatures of the interactive tasks dragging points by mouse automatic animation/movement dynamic change in the figure tracing of depending points hints and links (text) hints as guiding objects in the figure response analysis (in Geometria applet)


Interactive Graphical Representations (IGR)Interactive Graphical Representations (IGR)


Interactive Graphical Representations (IGR)Interactive Graphical Representations (IGR)

http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/RovaniemiNBE2005/index.html

http://www.joensuu.fi/mathematics/MathDistEdu/








Theoretical backgroundTheoretical background Interplay between conceptual (C) and

procedural knowledge (P) (cf. Ref #1)

Multiple representations of concept attributes (cf. Ref #1)

Interactive Graphical Representations (IGR) (cf. Ref #2 - #4)


Interplay between Interplay between PP and and CC Procedural knowledge (P) denotes dynamic and

successful utilization of particular rules, algorithms or procedures within relevant representation forms. This usually requires not only knowledge of the objects being utilized, but also the knowledge of format and syntax for the representational system(s) expressing them.

Conceptual knowledge (C) denotes knowledge of and a “skilful, conscious drive” along particular (semantic) networks, the elements of which can be concepts, rules (algorithms, procedures, etc.), and even problems (a solved problem may introduce a new concept or rule) given in various representation forms.


Developmental approachDevelopmental approach assumes that P enables C development. The term reflects the philogenetic and

ontogenetic nature of knowledge.

..


Educational approachEducational approach is based on the assumtion that P depends on C. The term refers to educational needs,

typically requiring a large body of knowledge to enable transfer.

..


Which one of the situations represents conceptual or/and procedural knowledge?


Utilising MODEM theory…Utilising MODEM theory…

… emphasis being on (D), (I) , and (P)


Multiple representations Multiple representations of concept attributesof concept attributes.

http://www.joensuu.fi/lenni/programs.html









MethodsMethodsThe focus: to concentrate on students’ difficulties to utilize sketches that contain special technical or mathematical features. Cognitive findings were represented just for considering possible explanations to these difficulties.


StudyStudy #1#1

First semester Introductory Mathematics (N = 42)

a 2-hour exercise sessions in 2 groups interactive sketches to introduce the function concept answers were sent directly to the teacher students’ actions were recorded by a screen capturer the material was analyzed with qualitative methods.


Second semester Linear Algebra (N = 82)

Test items were posed to the students using WebCT focus on an exceptionally poorly solved problem

containing an IGR in the plane (52 students) a query soon afterwards asking about reasons for poor

performance (43 answers) An open-ended feedback question expressions

interpreted and classified

Study #2Study #2


ResultsResultsby Study #1 ....by Study #1 ....

.... and Study #2.... and Study #2


Study #1 results: drag/animateStudy #1 results: drag/animate

dragging is very popular throughout the tests... ... and in some problems it is crucial dragging is useful when studying what happens in

special places, and when controlling values animation is useful in getting students’ attention to

special situations most students use animations when it is helpful or

necessary

Q: What advantages are there in manual dragging, what in automatic animation?


Example of tracingExample of tracing


one half of the students used tracing if available tracing facility was not well guided, 2/3 did not

clear the traces messy figure students with totally wrong ideas did not use

tracing

Q: What can be said about tracing?

Study #1 results: tracingStudy #1 results: tracing


Study #1 conclusionsStudy #1 conclusions

Hints must be offered only when crucial; students stop using hints as soon they find them useless.

Link to formal definition is practically useless ...

... because of students’ pure C understanding (cf. the concept image vs. concept definition in Vinner 1991)

The role of the applet hints?


Study #2 results:Study #2 results:Students’ explanationsStudents’ explanations

0 10 20 30 40 50 60 70 80 90 100

Saw that not draggable

Did not notice

Hard to say

a) u moved, result visible and moved

b) u moved and result visible

c) u, v and result visible

d) confused, answered positive

Error in the figure

%

http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/RovaniemiNBE2005/NBE05_Figure1JSP.html

QuickTime™ jaTIFF (LZW) pakkauksen purkuohjelma

tarvitaan elokuvan katselemiseen.









Study #2 results: Study #2 results: GeneralGeneral opinions of the testsopinions of the tests

4

35

55

1 1 2

25

59

16

24

59

4 7

010203040506070

significantlypositive

somewhatpositive

neutral somewhatnegative

significantlynegative

non-response

Functions Internal Binary Operations External Binary Operations


... in more ... in more

detaildetail


... expressions... expressions Tasks suitable for testing mastery of the function concept. (girl, 90 %) Hard to get information out of applets and to understand… … but they are nice, different from ordinary exercises. (girl, 50 %) Especially the figure-based tasks are difficult, because nothing alike

was done before. (boy, 38 %) Some problems easy, some not. Especially the problems concerning

two variable functions were not easy. (boy, 75 %) Problems were difficult, since the concepts are just sought. Training,

training! (girl, 34 %) Terrible tasks, even many of the questions are too difficult. (boy, first

trial, 40 %) Well, it was moderately easy on my second trial. Many problems

were similar. (the same boy, second trial, 95 %)


Defects in metacognitive thinkingDefects in metacognitive thinking(cf. Haapasalo & Siekkinen in this NBE)(cf. Haapasalo & Siekkinen in this NBE)

experts’ vs. novices’ strategies essential vs. irrelevant elements & actions easily too many dimensions:

mathematical, technical, observational example: one variable ignored

dynamical picture

http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/SavonlinnaLETTET2005/LeTTET_Figure4JSP.html


Technical problems Technical problems

conflicts in using e.g. Javascript in the questions and orientation module in WebCT

browser problems with Java

browser problems with mathematical fonts


Advantages of interactive appletsAdvantages of interactive applets

students become engaged with the content and the problem setting

students get a ”feeling” of the relation between the given parameters

dynamic pictures offer new possibilities to solve problems (e.g. trace or use scaling)

automatic response analysis provides feedback and ”learning when doing”


new kind of representation form is unfamiliar for many students

computer activities are time consuming problems in embedding to traditional

curriculum problems in measuring the results students are conservative in new situations

Disadvantages of interactive appletsDisadvantages of interactive applets


The need of pedagogical tutoringThe need of pedagogical tutoring Concerning teacher’s tutorial measures: a) face-to-face tutoring is best for metacognitive

defects, at least for less experienced students b) for technical guidance also audio solutions

should be taken into account.

Concerning appropriate pedagogical framework:

It is the students’ social constructions that lead to a viable definition for the concept (- ideal case!)


Example Example (from Haapasalo & Siekkinen in this NBE)(from Haapasalo & Siekkinen in this NBE)


Novice learner (“Alien”)Novice learner (“Alien”)(cf. Haapasalo & Siekkinen in this NBE)(cf. Haapasalo & Siekkinen in this NBE)


Expert learner Expert learner (cf. Haapasalo & Siekkinen in this NBE)(cf. Haapasalo & Siekkinen in this NBE)


Advantages of the Advantages of the WebCT…WebCT… questions can be authored using plain text style or html

code (mathematics, pictures, applets) easy to use for the students quizzes can be corrected automatically, or at least by

making minor revisions data can be examined, manipulated and stored in many

ways after submitting the quiz the students can see the whole

worksheet equipped with their own answers, together with the correct answers, and comments written by the teacher

… … might be objectivist / behaviorist loaded.might be objectivist / behaviorist loaded.


Disdvantages of the Disdvantages of the WebCT…WebCT… technical solutions can become expensive the lack of support for (higher) mathematics not easy to use for the authors, e.g. navigation is

complicated and running slowly it is not possible to correct all the answers to a

certain problem manually in a row the assessment and teacher’s comments cannot

be seen before answering all the questions Therefore the test system cannot be used

efficiently for “exam as a learning tool”…can be fatal regarding constructivism.


Conclusions (1/3)Conclusions (1/3) To shift from paper and pencil work towards

technology-based interactive learning, an adequate pedagogical theory is needed.

Applets alone are not a big step to shift procedural school teaching to the university mathematics aiming for conceptual understanding.

More or less systematic pedagogical models connected to an appropriate use of technology can help us to achieve both of these goals.


Conclusions (2/3)Conclusions (2/3) Interactive applets can be used not only for

learning but also for assessment and for increasing new kinds of complexity for the content.

Simultaneous activation of P and C allows the teacher to be freed from the worry about the order in which student’s mental models develop when interpreting, transforming and modelling mathematical objects.


Conclusions (3/3)Conclusions (3/3) University mathematics can be learned outside

institutions by utilising web-based activities.

Most students’ difficulties appear in the steps of mathematising and interpreting. To validate this result, the correlation between test performance in IGR vs. paper-and-pencil problems are to be examined.

The on-going research in the DAAD project will focus on qualitative research of students’ thinking processes.


IBMT principleIBMT principle ((IInteraction nteraction BBetween etween MMathematics and athematics and TTechnology)echnology) by Kadijevich, Haapasalo & Hvorecky (2004):by Kadijevich, Haapasalo & Hvorecky (2004):

“When using mathematics, don’t forget available tool(s); when utilising tools, don’t forget the underlying mathematics.” QuickTime™ ja

TIFF (LZW) pakkauksen purkuohjelmatarvitaan elokuvan katselemiseen.

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