Cognitive load issues in teaching and learning mathematics

1. Cognitive load issues in teaching and learning mathematics Slava Kalyuga

2. Outline • Review of CLT principles • Reducing cognitive load in mathematics instruction • Learner prior knowledge and instructional guidance • Responding to alternative approaches • Implications

3.  

4. Working memory 385674 + 938475 = ? How many windows are in your house?

5. Working memory CIABBCABCJVCVCR CIA BBC ABC JVC VCR

6. WM and LTM: Role of knowledge in cognition • Why chess grandmasters always beat weekend players? (De Groot, 1946/1965, Chase & Simon, 1973) • Knowledge of large numbers of different game configurations held in LTM dramatically altered the characteristics of WM. Similar mechanisms for all high-level cognitive skills (e.g., reading) • LTM: not a passive store, it is actively used in most of cognitive processes (learning, problem solving, thinking) • WM is very limited when dealing with novel information, but has no known limits when dealing with information that has been organized and stored in LTM as schemas

7. Why learning could be difficult? • High element interactivity => high intrinsic/relevant cognitive load • b is larger than c, a is larger than b. Which is the largest? • Instructional design => high extraneous/ wasteful cognitive load • unnecessary search processes • redundant information • unnecessary inferences when information is not provided explicitly

8. Managing intrinsic load Appropriately segmenting and sequencing tasks from simple to complex Simplifying tasks by omitting some of the interacting elements initially Getting familiar with separated elements (e.g., variables) first – pre-training Rote learning Initially presenting complex material as isolated elements allows to process them serially, rather than simultaneously (isolated-interactive elements effect - Pollock et al., 2002)

9. Cooper, Tindall-Ford, Chandler, and Sweller (2001): Instruction on how to use a spreadsheet application Imagining procedures and concepts (mental practice) vs simple study of procedures Imagination effect

12. Imagining procedures or concepts enhances learning compared to repeatedly studying materials (but: only for more knowledgeable learners) The effect depends on the learners’ prior knowledge level. Imagination effect

13. Imagination effect Ginns et al. (2003) Complex materials (novice learners): study was better than imagination

14. Imagination effect simple materials (expert learners): Imaginationwas better than study

15. Imagination effect Leahy & Sweller (2005) Phase 1 (novices) vs Phase 2 (experts) As learners’ levels of expertise increased, the advantage switched from studying to imagining examples

16. Reducing extraneous cognitive load in mathematics instruction Split-attention effect Split attention situations: learners have to mentally integrate multiple sources of information and this integration overburdens limited working memory capacity

17. Split-attention situation • diagrams accompanied by textual statements • neither text nor diagrams are intelligible in isolation • understanding requires searching and matching elements from the text to the appropriate entities on the diagram and their mental integration • applies to any two or more interdependent sources of information (text and text, text and tables, etc.) • Split attention effect: physically integrating corresponding sources of information within instruction may reduce extraneous cognitive load

18. Y 40 A 30 20 B 10 X -X -40 -30 -20 -10 10 20 30 40 -10 -20 -30 workpiece -40 -Y We assume that the tool is located at the origin. Firstly, we have to instruct the machine to quickly go to the point A. The NC command for a quick movement without cutting is G00 and is denoted with a broken line. We also have to instruct the machine where to go. Point A has the absolute position (20, 30). The NC command for a movement to the point A is X20 Y30. The complete command for this movement is therefore G00 X20 Y30. A straight line cut from A to B is required. The NC command for a straight line cut is G01 and is denoted by an unbroken line. We now have to instruct the machine to cut to point B. To achieve this the NC command for the point B is required. The NC command for point B is X-20 Y10. The complete command for this movement is G01X-20 Y10. The NC com- mand to return the tool back to the origin is simply G00 X0 Y0. This completes the NC program code for this job.

19. Chandler and Sweller, 1992 Follow the numbered steps Y 7A straight line cut from A to B is required 6The complete command for this movement is therefore G00 X20 Y30 8 The NC command for a straight line cut is G01 and is denoted by an unbroken line 5Point A has the absolute position (20, 30). The NC command for a movement to the point A is X 20 Y30 40 A 9We now have to instruct the machine to cut to the point B 30 4We also have to instruct the machine where to go 10 To achieve this the NC command for the point B is required. The NC command for the point B is X-20 Y10 3The NC command for a quick movement without cutting is G00 and is denoted with a broken line 20 10 B 2Firstly, we have to instruct the machine to quickly go to the point A 11The complete command for this movement is G01 X-20 Y10 -40 -30 -20 -10 10 20 30 40 -X X -10 12The NC command to return the tool back to the origin is simply G00 X0 Y0 1We assume that the tool is located at the origin -20 -30 workpiece -40 -Y

20. Split-attention effect A car moving from rest reaches a speed of 20 m/s after 10 seconds. What is the acceleration of the car? u = 0 m/s v = 20 m/s t = 10 s v = u + at a = (v - u)/t a = (20 - 0)/10 a = 2 m/s² A car moving from rest (u) reaches a speed of 20 m/s (v) after 10 seconds (t): [v = u + at, a = (v - u)/t = (20 - 0)/10 = 2 m/s²]. What is the acceleration of the car? Ward & Sweller (1990)

21. Instructional implications • Multiple representations (text, pictures, video, etc.), online nonlinear(‘hypertext’) environments may cause split attention • Integrate interdependent sources (e.g., the text into the graphic) • Avoid covering or separating information that must be integrated for learning • Design space for guidance or feedback close to problem statements, both being visible

22. Reducing cognitive load in mathematics instruction Redundancy effect: if a source of information (textual or graphical) is intelligible on its own, then any additional redundant sources of information should be removed rather than integrated (e.g. pie-charts)

23. Learning from user manuals(Sweller & Chandler, 1994; Chandler & Sweller, 1996): • Mentally integrating information from the manual and hardware (e.g., computer screen and keyboard): split-attention and redundancy situations • 1st group: manual (split-source) plus hardware - conventional format • 2nd group: integrated manual plus hardware • 3rd group: integrated manual only • The 3rd group was superior in both written and practical skills: the hardware (e.g., lab equipment) appeared to be redundant

24. Instructional implications • Temporarily eliminate the computer during the initial instructional period • replace computer with diagrammatic representations of the screen and keyboard • integrate segments of textual instructions at their appropriate locations on the diagram • Alternatively, eliminate the manual and place everything on the screen (computer-based training) in an integrated format

25. Instructional implications • Avoid redundant graphics, stories, and lengthy text (e.g., additional concrete materials in mathematical word problems) • No split-attention and redundancy effects were demonstrated in areas of low element interactivity • Repetition is not redundancy! • General rule: integrate if sources of referring information are unintelligible in isolation, but eliminate if they are intelligible in isolation

26. Transient Information Effect • Decline in learning due to transient information (e.g., spoken words, animation frames) disappearing before the learner has time to adequately process it • Related to two technology-generated procedures that transform permanent into transient information: • transforming written information into spoken information (modality effect: advantages of using both –visual and auditory – channels of WM to effectively extend WM capacity) • transforming static graphical information into dynamic animated information

27. Transient Information Effect Leahy & Sweller (2011): Primary school children studied how to read temperature/time graphs using lengthy segments of verbal information: written text superior to spoken information

28. Transient Information Effect • When the same material was divided into smaller chunks - a modality effect was obtained (audio/visual information superior to visual only) • The shorter spoken text reduced the influence of transience; learners could remember the shorter spoken text when processing the diagrams • Written information is permanent (no transiency) • The transient information effect does not apply to low element interactivity or biologically primary information (e.g., lengthy conversations, movies)

29. Transient information effect with animations • As animation frames roll from one to another, visual information disappears from sight. If information from previous frames is needed to understand later frames, then a transient information effect occurs • Animations without learner control cannot be revisited, unlike static diagrams that are constantly accessible

30. Improving the effectiveness of animations • Reducing extraneous cognitive load (e.g., split-attention, redundancy effects) • Allowing learner control. Slowing or stopping the flow of information that has to be simultaneously processed reduces cognitive load • However, complete control of an animation may only benefit learners if they have the necessary monitoring skills .

31. Improving the effectiveness of animations Segmenting. As with speech, short sequences may not cause transience problems and be superior to the equivalent static graphics. The length of animations could be managed by the use of segmentation Segmenting may be unnecessary for higher knowledge learners (prior knowledge can reduce number of interacting elements) .

32. Learning Human Movement or Motor Skills Animations could be more effective than static diagrams if they involve learning about perceptual-motor knowledge. Wong et al. (2009); Ayres et al. (2009): making origami shapes, tying knots, solving puzzle-rings; Arguel & Jamet (2009): teaching first-aid techniques Learners observing animations performed better and found the task easier than those studying a series of static key frames

33. Reducing cognitive load in mathematics instruction Problem-solving as an instructional method is associated with a significant extraneous cognitive load: • Means-ends analysis - defining differences between problem states; finding moves to reduce those differences; considering sub-goals, etc.

34. Reducing cognitive load in mathematics instruction Goal free effect: cognitive resources are directed to problem states and their associated moves Conventional: find a value for angle X Goal-free: find the values of as many angles as possible Suitable for problems that have a limited search space

35. The goal free effect • Traditional problems: Calculate the value of the parameter X. Evidence: students continued to use the means-ends strategy on post-instruction test problems • Goal-free (nonspecific goal) problems: Calculate the values of as many parameters as you can Evidence of acquired schemas: students worked forward on post-instruction test problems

36. Limitations Goal-free technique may not be appropriate under conditions where a very large number of moves can be generated. Goal-free technique is effective for problems that have a limited search space. In areas of high search space worked examples could be used.

37. Worked example effect • Worked example: a problem statement followed by all the appropriate steps to solution • Studying worked examples requires the learner to attend only to each problem state and its associated move (Sweller and Cooper, 1985) • Zhu and Simon (1987): a class learning by examples covered the 3-year curriculum in algebra and geometry in 2 years at a higher level of performance • Example-problem pairs could be more motivating than studying worked examples alone

38. Limitations of worked examples Worked examples are most effective for novice learners Worked examples may not be effective for learners who already acquired problem-solving schemas in the domain (expertise reversal effect – see the next lecture). When a worked example is structured in a way that produces high extraneous cognitive load, the benefit is reduced.

39. Completion problems, faded examples

40. Learner prior knowledge and instructional guidance • Expertise reversal effect: instructional designs or procedures that are effective for novices may be ineffective for more expert learners, and vice versa (Kalyuga, 2007) • Novice learners may benefit most from well guided low-paced instructional procedures, while more knowledgeable learners may benefit more from minimally guided forms of instruction

41. Adaptive learning environments Dynamic (real-time) tailoring of instructional methods and formats to levels of learner expertise. How to measure levels of learner expertise rapidly, in real time? Cognitive diagnostic assessment

42. Problem solving:Novices:search-based Experts:rapid retrieval and application of schemas Rapid diagnostic assessment of learner expertise

43. Solve for x: 5x = - 4 5x/5 = - 4/5 x = - 4/5

44. Rapid diagnostic approach • Presenting learners with a task for a limited time and asking them to indicate their first step towards solution • Skipping intermediate steps reflects a higher level of proficiency: the learner has corresponding operations automated or is able to perform them mentally • Less could be more!

46. Responding to alternative approaches • CLT: explicit instruction prior to problem solving (worked example effect) for novice learners • Productive failure/preparation for future learning /invention learning: benefits of initial problem solving activities prior to explicit instruction – especially for conceptual learning/far transfer/delayed tests • Kapur, 2008; Schwartz & Bransford, 1998; Schwartz & Martin, 2004) • Kapur & Bielaczyc, 2012; Schwartz, Chase, Oppezzo, & Chin, 2011; DeCaro & Rittle-Johnson, 2012; Loibl & Rummel, 2014

47. Evidence from CLT Chih-Yi Hsu (thesis): delayed (one week) transfer posttest: p < .05

48. The sample of principle-based worked example (Hsu et al., 2015) Page 1 Page 2

49. V. Likourezos (thesis) Task: Construct a perpendicular to a line from a point off the line using a pair of compasses and a straight edge

50. Evidence from within CLT V. Likourezos (thesis): delayed transfer posttest: n.s.