1. Cognitive load issues in teaching and learning mathematics
Slava Kalyuga
Physics perspective: evidence is a king. Some theories miserably fail, some spectacularly advance; some perpetuate either by being modified or by outlining their clear boundaries. Direct instruction vs. inquiry learning. This talk – about only one specific controversy.

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
= ?
How many windows are in your house?

5. Working memory
CIABBCABCJVCVCR
CIA BBC ABC JVC VCR

6. Role of knowledge in cognition
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. Imagination effect 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

12. The effect depends on the learners’ prior knowledge level.
Imagination effect
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.

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

14. Imagination effect
simple materials (expert learners): Imagination was 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
Split-attention situation

18. We assume that the tool is located at the origin
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 G01 X-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. -Y X -X Y
workpiece A 10 20 30 40
-10
-20
-30
-40 B

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

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

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
Conditions
(preceding common example-based explicit instruction)
Total scores M SD
Problem Solving only
.19
.75
Problem Solving + principle guidance
.18
.31
Problem Solving +principle guidance + reflection
.97
1.15
Worked Example (with principle guidance)
.55
.74

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.
Conditions
(preceding common example-based explicit instruction)
Test scores M SD
Problem Solving only
21.71
7.03
Problem Solving + guidance
20.54
5.95
Worked example
22.42
5.71

51. Evidence from within CLT
V. Likourezos (thesis): cognitive efficiency
(posttest scores/cognitive load): p<0.005
Conditions
(preceding common example-based explicit instruction)
Scores M SD
Problem Solving only
32.74
16.66
Problem Solving + guidance*
24.86
10.04
Worked example*
42.53
17.85

52. Responding to alternative approaches
Is it possible to reconcile these alternative results with CLT?
Do we need to revise some basic approaches in CLT?
Involved in CLT for 20 years; recommendations are highly intuitive! Polya; WE with self explanations.
Failed attempts helped to appreciate the elegance of the standard solution

53. Responding to alternative approaches
Solve for x:
5x = - 4
5x/5 = - 4/5
x = - 4/5
(“benefit of doubt”)

54. Responding to alternative approaches
Traditional CLT: acquisition of domain-specific schemas - the only stated goal!
Complex learning task may involve different instructional goals associated with specific activities
Specific instructional goals of cognitive activities need to become an attribute of CLT

55. Specifying instructional goals
Generation/exploration phase prior to explicit instruction:
“pre-instruction” goals: e.g., activating relevant learner prior knowledge;
exploring initial ideas potentially related to the critical conceptual features (to be taught later);
enhancing learner awareness of problem situations
High levels of cognitive load might not interfere with achieving some of these goals

56. Implications
Moving away from the explicit instruction – limited guidance dilemma in complex learning
contrasting explicit and limited-guidance instruction is unjustified for complex learning environments
variety of activities with different goals: methods with various levels of guidance co-exist
Research should control not only levels of learner prior knowledge, but also instructional goals of the corresponding learning activities

57. Practical implications
Failure to establish universal instructional approaches (e.g., explicit or limited guidance).
Cheng’s (2014): analysis of mathematics instructional practices in high-achieving Asian educational systems based on TIMSS (Trends in International Mathematics and Science Study) data. No stable patterns were detected.
CLT: different factors need to be considered, especially levels of prior knowledge (and goals)

58.
Sweller, J., Ayres, P., & Kalyuga, S. (2011). Cognitive Load Theory. New York: Springer (250 p.)
Kalyuga, S. (2015). Instructional Guidance: A Cognitive Load Perspective. Charlotte, NC USA: IAP– Information Age Publishing