Educational Objectives For Problem Solving Skills

1. Educational Objectives For Problem Solving Skills David Sands and Tina Overton Dept. Physical Sciences

2. We need: To identify problem-solving processes Expert-novice comparisons To teach these processes Field dependence - independence Classify these processes A taxonomy - Marzano

3. Why a new taxonomy? Bloom’s taxonomy is about 60 years old Confuses knowledge with processes “The new taxonomy avoids this … by postulating three domains of knowledge that are operated on by three systems of thought and their component elements” – Marzano and Kendall, 2008 Thought systems: self system metacognitive cognitive – 4 components

5. Problem solving “Whenever there is a gap between where you are now and where you want to be, and you don’t know how to find a way to cross that gap, you have a problem.” Hayes, J. The complete problem solver, 1980 “Problem solving is what you do when you don’t know what to do.” Wheatley, G H. Problem solving in school mathematics, 1984 (MEPS Technical Report, School of Mathematics and Science Centre, Purdue University)

6. Can problem-solving be analyzed? Up until the 1970s problem-solving was general Around 1980 the importance of domain-specific knowledge was recognised. Two very commonly cited papers: Expert and Novice Performance in Solving Physics Problems, Jill Larkin, John McDermott, Dorothea P Simon, Herbert A Simon, Science 208 1980 1335-1342 Categorization and Representation of Physics Problems by Experts and Novices, Micheline T H Chi, Paul J Feltovich, Robert Glaser, Cognitive Science 5 1981 121-152

7. Larkin et al: • Computer modelling of problem-solving: • cognition is an analogue of a computer programme • PS framed in terms of productions resulting from condition-action pairs; if x then do y. • Students - direct syntactic translations, experts - make representations • (surface features vs. deeper structure) • Pencil and paper representations augment working memory • Distinguish between internal and external representations • concentrate on internal • (Nersessian postulates that internal and external representations form a coupled system which facilitate analogical reasoning) • Essential conclusion; novices use means-end analysis, experts use a forward looking approach.

8. Chi, Feltovich and Glaser • asked novices and experts to categorize end-of-chapter physics problems • novices concentrate on surface features, eg. Common components, such as an inclined plane • experts concentrate on the deep structure, eg the physical principles behind the problem, such as conservation of energy or momentum. • Production rules are also mentioned; experts work forward from principles, novices search for a solution. • Clearly parallels the work of Larkin et al. • Representations also mentioned

9. Glaser emphasises the role of knowledge; Robert Glaser, Education and Thinking: the role of knowledge, American Psychologist, 39(2) 1984 93-104 “… the possession and utilization of an organized body of conceptual and procedural knowledge, and a major component of thinking is seen to be the possession of accessible and usable knowledge.” (cf. Marzano) “… knowledge of novices is organised around the literal objects explicitly given in a problem statement. Experts’ knowledge, on the other hand, is organized around principles and abstractions that subsume these objects.”

10. Glaser on representations “We define a problem representation as a cognitive structure … that is constructed by a solver on the basis of domain-related knowledge and its organization. At the initial stage of problem analysis the problem solver attempts to ‘understand ‘ the problem by constructing an initial problem representation”. The use of representations constitutes part of the thinking!

11. Representations A recurring, and perhaps dominant, theme in expert-novice literature. George M Bodner and Daniel S Domin, Mental Models: The role of representations in problem solving in chemistry University Chemistry Education, 4(1) 2000 p24-30

12. Stieff, for example, reports that experts use a number of strategies to solve organic chemistry problems. These include: [M Stieff and and Sonali Raje, Expertise & Spatial Reasoning in Advanced Scientific problem Solving, Proc 8th Internantional conference on the Learning Sciences, Utrecht, NL, 2008, p366-373] • initial visualization to understand the problem • visualizing • reasoning through images (generating new structures through the mental use of) • as well as a host of analytical strategies.

13. M Suwa and B Tversky: “External representations contribute to the dynamic construction of ideas” M Hegarty, B Meyer, H Narayanan, Diagrammatic Representation and Inference, Proc. Inference 2002 Lecture Note ins Artificial Intelligence series, Springer p341-343 • Free working memory • Cue retrieval from long term memory • Allow perceptual judgements about spatial relations • Allow the generation of new ideas

14. Nersessian has integrated these ideas into a theory of MODEL BASED REASONING in which the construction and manipulation of iconic, analogical mental models is considered as a form of reasoning in its own right. (Nancy J Nersessian, Creating Scientific Concepts, Bradford books 2008) Internal – external representations form a coupled system which facilitates model-based reasoning!

15. Example, Stieff and Raje (2008): 10 experts, 7 tasks each Take out Basic Recall and we are left with visualization and diagrammatic reasoning as the principal mechanisms of problem solving.

16. Summary Types of knowledge needed to solve problems: • Declarative knowledge; domain specific content, facts, laws, principles • Procedural knowledge; production rules/sequences, how to. • Conceptual knowledge; knowing why, understanding, schemata • Strategic knowledge; knowing when, where; heuristics In addition • Need experience in constructing, interpreting, manipulating representations for analogical or diagrammatic reasoning

17. Two Questions: Can we teach content knowledge adequately? Yes, but not easily Can we teach problem-solving? No definitive answer: probably, but definitely not as an adjunct to other activities; has to be integrated.

18. Field Dependence/Independence Summarised by Witkin et al Field-Dependent and Field-Independent Cognitive Styles and Their Educational Implications Author(s): H. A. Witkin, C. A. Moore, D. R. Goodenough, P. W. Cox Source: Review of Educational Research, Vol. 47, No. 1 (Winter, 1977), pp. 1-64 FD most commonly characterised as an inability to disembed information from its context (field) Also involves an inability to see order in, or to superimpose order on, a field – FDs takes their cues from others. Continuum: extremes possible, should be viewed as tendencies

19. Implications for problem-solving • Well documented that FD students struggle; • cannot disembed the problem from its context • working memory saturated from irrelevant perceptual material Deeper implication: students might not acquire sufficient content knowledge. Witkin discusses some educational implications, but two examples from science will illustrate the point

20. Strawitz Barbara M Strawitz, Cognitive Style and the acquisition and transfer of the ability to control variables, J. Res. Sci. Teach. 21(2) 1984 133-141, Barbara M Strawitz, Cognitive Style and the effects of two instructional treatments on the acquisition and transfer of the ability to control variables: a longitudinal study, J. Res. Sci. Teach. 21(8) 1984 833-841]. Teaching young teenagers the principles of scientific experimentation, ie. controlling variables in a well defined experimental task. FI students pick up these skills. FD students unable to develop these skills unless instructed in an appropriate way. With the right approach FD students are just as effective as FI students

21. Antonietti and Gioletta Allessandro Antonietta and Maria Alfonsa Gioletta, Individual differences in analogical problem solving, Person. individ. Diff. 18(5) 1985 611-619, Students were given either text-only or text-and-visual instructional material which they had to use to solve a problem. FI students: “The same diagrams were finally synthesized into one model” “It was then easier to identify the existing relationships and isolate relevant information despite its complexity” FD students: “Stated that too many cause-and-effect relationships were depicted in the diagrams and described in the text, making it difficult to extract what was relevant and solve the problem”

22. Recently FDI has been redefined as ability rather than cognitive style. Ability is distinct from general intelligence. Field dependence–independence as visuospatial and executive functioning in working memory: implications for instructional systems design and research Kent A. Rittschof, Education Tech Research Dev DOI 10.1007/s11423-008-9093-6

23. Conclusions • Expert – novice studies have elucidated the nature of problem-solving within specific domains; the role of recall, representations, analogical reasoning • Marzano’s taxonomy allows an objective to be associated with each task • Content knowledge and problem-solving should be taught in an integrated manner

24. Illustration Raluca Teodorescu, Cornelius Bennhold, and Jerry Feldman, of George Washington University, have constructed a taxonomy of introductory physics problems (TIPP) based on Marzano’s taxonomy. This is used to teach and assess problem solving in a reformed physics class at GW The following, taken from Jerry Feldmann’s presentation at GIREP-EPEC 2009, summarises their problem-solving strategy

25. Thanks to Jerry Feldman and Raluca Teodorescu, GW University

26. Fin. Thank you for your time.