**Author details**

Bing H. Ngu\* and Huy P. Phan

\*Address all correspondence to: bngu@une.edu.au

The University of New England, Armidale, Australia

## **References**

[1] Ngu BH, Phan HP. Comparing balance and inverse methods on learning conceptual and procedural knowledge in equation solving: A cognitive load perspective. Pedagogies: An International Journal. 2016;**11**(1):63-83

[12] Ayres P. Impact of reducing intrinsic cognitive load on learning in a mathematical

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

http://dx.doi.org/10.5772/intechopen.70568

303

[13] Vygotsky L. Mind in Society: The Development of Higher Psychological Processes.

[14] Bandura A. Self-Efficacy: The Exercise of Control. New York: W. H. Freeman & Co.; 1997

[15] Bandura A. Self-efficacy: Toward a unifying theory of behavioral change. Psychological

[16] Snyder CR. Conceptualizing, measuring, and nurturing hope. Journal of Counseling

[17] Snyder CR. Hope, goal blocking thoughts, and test-relayed anxieties. Psychological

[19] Daulta MSN. Impact of home environment on the scholastic achievement of children.

[20] DeBacker TK, Nelson RM. Variations on an expectancy-value model of motivation in

[21] Fenollar P, Román S, Cuestas PJ. University students' academic performance: An integrative conceptual framework and empirical analysis. British Journal of Educational

[22] Martin AJ, Marsh HW. Academic resilience and its psychological and educational correlates: A construct validity approach. Psychology in the Schools. 2006;**43**(3):267-281

[23] Chouinard R, Karsenti T, Roy N. Relations among competence beliefs, utility value, achievement goals, and effort in mathematics. British Journal of Educational Psychology.

[24] Dupeyrat C, Mariné C. Implicit theories of intelligence, goal orientation, cognitive engagement, and achievement: A test of Dweck's model with returning to school adults.

[25] Phan HP, Ngu BH. Introducing the concept of optimized functioning: Establishing evidence for further consideration. The International Journal of Pedagogy and Curriculum.

[26] Phan HP. Maximizing academic success: Introducing the concept of optimized function-

[27] Fraillon J. Measuring student well-being in the context of Australian schooling: Discussion Paper. The Australian Council for Research: Carlton South, Victoria; 2004

[18] Franken RE. Human Motivation. 6th ed. Thomson Wadsworth: Belmont, CA; 2007

science. Contemporary Educational Psychology. 1999;**24**(2):71-94

Contemporary Educational Psychology. 2005;**30**(1):43-59

domain. Applied Cognitive Psychology. 2006;**20**(3):287-298

Cambridge, MA: Harvard University Press; 1978

Review. 1977;**84**(2):191-215

Reports. 1999;**84**:206-208

and Development. 1995;**73**:355-360

Psychology. 2007;**77**(Pt 4):873-891

ing. Education. 2015;**135**(4):439-456

2007;**77**(3):501-517

2015;**22**(4):1-19

Journal of Human Ecology. 2008;**23**(1):75-77


[12] Ayres P. Impact of reducing intrinsic cognitive load on learning in a mathematical domain. Applied Cognitive Psychology. 2006;**20**(3):287-298

**Author details**

**References**

**55**:5-14

in Psychology

Bing H. Ngu\* and Huy P. Phan

\*Address all correspondence to: bngu@une.edu.au

The University of New England, Armidale, Australia

302 New Pedagogical Challenges in the 21st Century - Contributions of Research in Education

An International Journal. 2016;**11**(1):63-83

DOI: 10.1007/978-1-4419-8126-4

[1] Ngu BH, Phan HP. Comparing balance and inverse methods on learning conceptual and procedural knowledge in equation solving: A cognitive load perspective. Pedagogies:

[2] Ngu BH, Phan HP. Unpacking the complexity of linear equations from a cognitive load

[3] Phan HP, Ngu BH, Williams A. Introducing the concept of optimal best: Theoretical and

[4] Phan HP, Ngu BH, & Yeung AS. (in press). Achieving Optimal Best: Instructional Efficiency and the Use of Cognitive Load Theory in Mathematical Problem Solving.

[5] Sweller J, Ayres P, Kalyuga S. Cognitive Load Theory. New York, NY: Springer; 2011.

[6] Sweller J. Human cognitive architecture: Why some instructional procedures work and others do not. In: Harris K, Graham S, Urdan T, editors. APA Educational Psychology Handbook. Washington, D.C.: American Psychological Association; 2012. p. 295-325 [7] Seligman M, Csíkszentmihályi M. Positive psychology. American Psychologist. 2000;

[8] Seligman M. Flourish: Positive psychology and positive interventions. In: The Tanner

[9] Ryan RM, Deci EL. On happiness and human potentials: A review of research on hedonic

[10] Phan HP, & Ngu BH. (2017). Positive psychology: The use of the Framework of Achievement Bests to facilitate personal flourishing. In A. A. V. Boas (Ed.), *Well-being* 

[11] Phan HP, Ngu BH, Wang H-W, Shih J-H, Shi S-Y, & Lin R-Y. (In press-2018). Fostering well-being experiences: Guidance from the Framework of Achievement Bests. Frontiers

*and Quality of Life* (pp. 19-33). Rijeka, Croatia: Intech: Open Science|Open Minds

Lectures on Human Values. MI, USA: The University of Michigan; 2010

and eudemonic well-being. Annual Review of Psychology. 2001;**52**:141-166

theory perspective. Educational Psychology Review. 2016;**28**:95-118

methodological contributions. Education. 2016;**136**(3):312-322

Educational Psychology Review. DOI: 10.1007/s10648-016-9373-3


[28] Phan HP, & Ngu BH. (In press-2018). Expanding on the theoretical concept of 'Optimization' for effective learning: Establishing empirical evidence from an Eastern sociocultural context. In A. D. Liem & T. S. Hong (Eds.), Student motivation, engagement, and growth: Asian insights. Singapore: Routledge

[43] Sweller J. Element interactivity and intrinsic, extraneous, and germane cognitive load.

Achievement Bests Framework, Cognitive Load Theory, and Equation Solving

http://dx.doi.org/10.5772/intechopen.70568

305

[44] Chen O, Kalyuga S, & Sweller J. (2017). The Expertise Reversal Effect is a Variant of the More General Element Interactivity Effect. Educational Psychology Review, **29**(2), 393-405.

[45] Sweller J. Cognitive load theory, learning difficulty, and instructional design. Learning

[46] Vincent J et al. MathsWorld. Australian Curriculum ed. Vol. 8. South Yarra, VIC:

[47] Cai J et al. The development of students' algebraic thinking in earlier grades: A crosscultural comparative perspective. ZDM—The International Journal on Mathematics

[48] Ngu BH, Chung SF, Yeung AS. Cognitive load in algebra: Element interactivity in solv-

[49] Ngu BH, Phan HP, Yeung AS, & Chung SF. (in-press). Managing element interactivity in equation solving. Educational Psychology Review. DOI: 10.1007/s10648-016-9397-8 [50] Ayres PL. Systematic mathematical errors and cognitive load. Contemporary Educational

[51] Linchevski L, Williams J. Using intuition from everyday life in 'filling' the gap in children's extension of their number concept to include the negative numbers. Educational

[52] Cramer K, Wyberg T. Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning. 2009;**11**(4):226-257

[53] Parker M, Leinhardt G. Percent: A privileged proportion. Review of Educational

[54] Ngu BH, & Phan HP. Will learning to solve one-step equations pose a challenge to 8th grade students? International Journal of Mathematical Education in Science and Tech-

[55] Hiebert J, Wearne D. Instruction, understanding, and skill in multidigit addition and

[56] Rittle-Johnson B, Alibali MW. Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology. 1999;**91**(1):175-189 [57] Rittle-Johnson B, Siegler RS, Alibali MW. Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology.

[58] McNeil NM et al. Middle-school students' understanding of the equal sign: The books

nology. 2017;**48**(6):876-894. DOI: 10.1080/0020739X.2017.1293856

they read can't help. Cognition and Instruction. 2006;**24**(3):367-385

subtraction. Cognition and Instruction. 1996;**14**(3):251-283

Educational Psychology Review. 2010;**22**(2):123-138

ing equations. Educational Psychology. 2015;**35**(3):271-293

DOI: 10.1007/s10648-016-9359-1

and Instruction. 1994;**4**(4):295-312

Macmillan; 2012

Education. 2005;**37**:5-15

Psychology. 2001;**26**(2):227-248

Research. 1995;**65**(4):421-481

2001;**93**(2):346-362

Studies in Mathematics. 1999;**39**(1/3):131-147


[43] Sweller J. Element interactivity and intrinsic, extraneous, and germane cognitive load. Educational Psychology Review. 2010;**22**(2):123-138

[28] Phan HP, & Ngu BH. (In press-2018). Expanding on the theoretical concept of 'Optimization' for effective learning: Establishing empirical evidence from an Eastern sociocultural context. In A. D. Liem & T. S. Hong (Eds.), Student motivation, engage-

[29] Phan HP, Ngu BH, Williams A. The Optimal Outcomes Questionnaire. Armidale,

[30] Kieran C. The learning and teaching of school algebra. In: Grouws D, editor. Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan; 1992.

[31] Stacey K, MacGregor M. Learning the algebraic method of solving problems. The Journal

[32] Carpenter TP, Franke M, Levi L. Thinking Mathematically: Integrating Arithmetic and

[33] Ngu BH, Yeung AS, Tobias S. Cognitive load in percentage change problems: Unitary, pictorial, and equation approaches to instruction. Instructional Science. 2014;**42**(5):685-713

[34] Ngu BH, Yeung AS. Fostering analogical transfer: The multiple components approach to algebra word problem solving in a chemistry context. Contemporary Educational

[35] Ngu BH, Yeung AS. Algebra word problem solving approaches in a chemistry context: Equation worked examples versus text editing. The Journal of Mathematical Behavior.

[36] Ngu BH, Yeung AS, Phan HP. Constructing a coherent problem model to facilitate algebra problem solving in a chemistry context. International Journal of Mathematical

[37] Hegarty M, Mayer RE, Monk CA. Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational

[38] Mayer RE. Mathematical ability. In: Sternberg RJ, editor. Human Abilities: An Information-Processing Approach. New York, NY: Freeman; 1985. pp. 127-150

[39] Miller GA. The magical number seven, plus or minus two: Some limits on our capacity

[40] Sweller J. Implications of cognitive load theory for multimedia learning. In: Mayer RE, editor. Cambridge handbook of multimedia learning. New York: Cambridge University

[41] Ngu BH et al. Reducing intrinsic cognitive load in percentage change problems: The

[42] Jelsma O, van Merriënboer JJ. The ADAPT design model: Towards instructional control

for processing information. Psychological Review. 1956;**63**(2):81-97

equation approach. Learning and Individual Differences. 2016;**51**:81-90

of transfer. Instructional Science. 1990;**19**(2):89-120

Algebra in Elementary School. Portsmouth, NH: Heinemann; 2003

Education in Science and Technology. 2015;**46**(3):388-403

ment, and growth: Asian insights. Singapore: Routledge

304 New Pedagogical Challenges in the 21st Century - Contributions of Research in Education

Australia: The University of New England; 2015

of Mathematical Behavior. 1999;**18**(2):149-167

Psychology. 2012;**37**(1):14-32

Psychology. 1995;**87**(1):18-32

Press; 2005. p. 19-30

2013;**32**(2):197-208

p. 390-419


[59] Atkinson RK et al. Learning from examples: Instructional principles from the worked

[60] Renkl A. Learning mathematics from worked-out examples: Analyzing and fostering self-explanations. European Journal of Psychology of Education. 1999;**14**(4):477-488

[61] Paas F, Van Gog T. Optimising worked example instruction: Different ways to increase

[62] van Gog T et al. Instructional design for advanced learners: Establishing connections between the theoretical frameworks of cognitive load and deliberate practice.

[63] Huk T, Ludwigs S. Combining cognitive and affective support in order to promote

[64] Rey GD, Buchwald F. The expertise reversal effect: Cognitive load and motivational explanations. Journal of Experimental Psychology: Applied. 2011;**17**(1):33-48

[65] Ashcraft MH, Kirk EP. The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology: General. 2001;**130**(2):224-237

[66] Fraser K et al. The emotional and cognitive impact of unexpected simulated patient

death: A randomized controlled trial. CHEST Journal. 2014;**145**(5):958-963

examples research. Review of Educational Research. 2000;**70**(2):181-214

germane cognitive load. Learning and Instruction. 2006;**16**(2):87-91

Educational Technology Research and Development. 2005;**53**(3):73-81

learning. Learning and Instruction. 2009;**19**(6):495-505

306 New Pedagogical Challenges in the 21st Century - Contributions of Research in Education
