Cognitive load issues in teaching and learning mathematics

Slava Kalyuga

Review of CLT principles

Reducing cognitive load in mathematics instruction

Learner prior knowledge and instructional guidance

Responding to alternative approaches

Implications

Outline

Constructing mental representations of a situation or task

Long-Term Memory Knowledge base

Working Memory

Sensory Memory: Incoming information

385674 + 938475 = ?

How many windows are in your house?

Working memory

CIABBCABCJVCVCR

CIA BBC ABC JVC VCR

Working memory

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

WM and LTM: Role of knowledge in cognition

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

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)

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

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

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

Imagination effect

simple materials (expert learners): Imagination was better than study

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

Imagination effect

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

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

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

40

30

10

-10

-20

-30

-40

20

B

Y

Follow the numbered steps

-X X

workpiece

A

10 20 30 40-10-20-30-40

40

30

10

-10

-20

-30

-40

20

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

Chandler and Sweller, 1992

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?

Split-attention effect

Ward & Sweller (1990)

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

Instructional implications

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)

Reducing cognitive load in mathematics instruction

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

Learning from user manuals(Sweller & Chandler, 1994; Chandler & Sweller, 1996):

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

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

Instructional implications

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

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

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)

Transient Information Effect

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

Transient information effect with 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

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)

Improving the effectiveness of animations

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

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.

Conventional: find a value for angle X

Goal-free: find the values of as many angles as possible

Goal free effect: cognitive resources are directed to problem states and their associated moves

Reducing cognitive load in mathematics instruction

Suitable for problems that have a limited search space

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

The goal free effect

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.

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

Worked example effect

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.

Limitations of worked examples

Completion problems, faded examples

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

Learner prior knowledge and instructional guidance

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

Adaptive learning environments

Problem solving:

Novices:search-based

Experts:rapid retrieval and application of schemas

Rapid diagnostic assessment of learner expertise

x = - 4/5

Solve for x: 5x = - 4

5x/5 = - 4/5

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!

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

Responding to alternative approaches

Evidence from CLTChih-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 .75Problem Solving + principle guidance

.18 .31

Problem Solving +principle guidance + reflection

.97 1.15

Worked Example (with principle guidance)

.55 .74

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

Page 1 Page 2

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

Evidence from within CLTV. 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

Evidence from within CLTV. 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

Is it possible to reconcile these alternative results with CLT?

Do we need to revise some basic approaches in CLT?

Responding to alternative approaches

Solve for x: 5x = - 4

5x/5 = - 4/5

x = - 4/5

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

Responding to alternative approaches

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

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

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)

Practical implications

s.kalyuga@unsw.edu.au

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