Learning Environment that
Empower Mathematics Learners
Dr Cynthia Seto
Primary Mathematics Chapter
Academy of Singapore Teachers
Outline of Workshop
• Learning Environment
• Motivation and Effort
– Cultivating a Growth Mindset
• Social Space
– Establishing Norms for Mathematics Learning
– Making Thinking Visible
– Encouraging Discussions with Justifications
Classroom Learning Environment
The learning environment refers to the social,
psychological and pedagogical contexts in
which learning occurs and which affects student
achievement and attitudes.
(Fraser, 1998)
Learning Environment
Social
Context
Psychological
Context
Lewinian Formula
B = f (P,E)
B (Behaviour)
P (Person)
E (Environment)
(Lewin, 1936)
Pedagogical
Context
Learning Environment
Murray (1938) proposed a need–press model to
describe an individual’s personal needs and
environmental press. He defined needs as the
specific, innate and personal requirements of an
individual, such as personal goals.
B = f (P, E)
• An individual’s need to achieve these goals, or drive to
attain them, is also a factor in an individual’s personality.
• Environmental factors (beyond an individual’s control)
can either enhance or inhibit an individual’s achievement
of personal needs and goals.
Mathematics Framework 2013
Source: MOE. (2012, p. 16).
Learning Experiences
MOE (2012, p. 22) emphasised that
“Learning experiences must include opportunities
where students discover mathematical results on
their own, …, must be given opportunities to
work together on a problem and present their
ideas using appropriate mathematical language
and methods.”
Learning Experiences
Learning experiences refer to ‘the interaction
between the learner and the external conditions
in the environment to which he can react
(Tyler, 1949, p. 63).
Game 1 : Take 1 – 4 coins
Start with 2 players and a pile of 20 coins.
Rule:
1. Each player takes turn to take 1, 2, 3 or 4
coins from the pile.
2. The player who takes the last coin(s) wins the
game.
Play as many rounds as you can to determine the
strategy you will use to win the game.
Adapted from https://plus.maths.org/content/play-win-nim
Game 2 : Take 1 – 5 coins
Start with 2 players and a pile of 20 coins.
Rule:
1. Each player takes turn to take 1, 2, 3, 4 or 5
coins from the pile.
2. The player who takes the last coin(s) wins the
game.
Play as many rounds as you can to determine the
strategy you will use to win the game.
Adapted from https://plus.maths.org/content/play-win-nim
What is your motivation
to play the games?
Motivation
Motivation is a force that energises, sustains,
and directs behaviour towards a goal.
(Eggen & Kauchak, 1992)
Goal
– Provides the impetus for and direction for action
Action
– Entails effort: persistence in order to sustain an
activity for a long period of time
Goal Setting
Motivational benefits of goals depend on 3 properties.
Proximity
• Promote self-efficacy
• Enhance skill acquisition
(Bandura & Schunk, 1981)
Specificity
“You can remember the 4
times-table!” as compared to
“Do your best!”
Difficulty
Compared with easier goals,
difficult goals raised children’s
motivation during arithmetic
instruction (Schunk, 1991)
Growth vs Fixed Mindsets
Believe that they can learn
anything if they put in the
work, practice and effort
to learn it (Dweck, 2006)
Assume that they
cannot increase their
knowledge and skills in
any meaningful way
Growth vs Fixed Mindset
Growth Mindset
• Intelligence can be developed.
Fixed Mindset
• Intelligence is static.
• Has the desire to learn and
• Has the desire to look smart
display the tendency to
and display the tendency to
 Embrace challenges
 Avoid challenges
 Persevere despite obstacles
 Give up easily
 See effort as path to mastery
 See effort as fruitless
 Learn from feedback
 Ignore feedback
 Be inspired by others’ success
 Be threatened by others’
success
Mathematics Framework
How do we provide opportunities for students to
build their mathematical content knowledge
while developing their ability to use
mathematical processes with fluency?
The Social Space in our Classroom
Social space refers to the interactions teachers
provide to facilitate students to learn
mathematics (Fraser, 2012; OWP/P Architects et
al., 2010; Fraser, 1994).
3 considerations:
• Establishing Norms for Mathematics Learning
• Making Thinking Visible
• Encouraging Discussions with Justifications
Norms for Mathematics Learning
Everyone can
learn and
achieve a level
of mastery in
mathematics.
Mistakes are
opportunities
for learning.
Questions are
important to
help us learn.
Paired Convesation Protocol: Why
• Making Thinking Visible
• Encouraging Discussions with Justifications
Paired Conversation Protocol: How
1. Paraphrase and clarify problem for one another
–
–
–
–
what is asked
What is given
What happens
What the units are etc
2. Estimate the answer
– Justify your own estimate
– Compare the solutions such as drawings, diagrams,
calculations
Adapted from Zwiers, O’Hara, & Pritchard (2014)
Paired Conversation Protocol
3. Discuss which method you would recommend
for problems like this. Why?
4. Discuss connections between the two methods.
How do they relate?
5. Summarise how to solve problems like this; use
this problem as an example
6. Co-create a similar problem, solve it and share
with others
Adapted from Zwiers, O’Hara, & Pritchard (2014)
A Word Problem
Mrs Tan is four times as old as her son.
Five years ago, the sum of their ages was 60.
What is Mrs Tan’s present age?
How do we engage our students in
discussions with justifications?
Activity 1
• In pairs, have a role-play based on Handout 1.
• Discuss
– which method would you recommend for problems
like this. Why?
– connections between the 2 methods
• Generate a summary for how to solve problems
like this.
• Co-create a similar problem, write it, and
solve it (then share with others)
Activity 2
• In pairs, name yourself as Student 1 or
Student 2
• Use the ‘Paired Conversation Protocol’ to
discuss your solution with your partner.
I never teach my pupils.
I only attempt to provide the conditions
in which they can learn.
Albert Einstein
References
1.
2.
3.
4.
5.
6.
7.
8.
Bandura, A., & Schunk, D. H. (1981). Cultivating competence, self-efficacy
mechanisms governing the motivational effects of goal systems. Journal of
Personality and Social Psychology, 41, 586 – 598.
Dweck, C. S. (2006). Mindset: The new psychology of success. New York:
Random House.
Fraser, B. J. (1994). Research on classroom and school climate. In D.L. Gabel
(Ed.), Handbook of research on science teaching and learning (pp. 493 –
541). New York: Macmillan.
Fraser, B. J. (1998). Classroom environment instruments: Development,
validity and applications. Learning Environments Research, 1, 7–33.
Fraser, S. (2012). Authentic childhood. To ON: Nelson Education.
Lewin, K. (1936). Principles of topological psychology. New York: McGraw
MOE. (2012). Primary Mathematics Teaching and Learning Syllabus.
Ministry of Education: Singapore.
Owp / P Architects, VS Furniture, & Bruce Mau Design. (2010). The third
teacher: 79 ways you can use design to transform teaching and learning.
New York, NY. Abrams.
References
9.
Schunk, D. H. (1991). Self-efficacy and academic motivation. Educational
Psychologist, 26, 207–231.
10. Tyler, R. W. (1949). Basic principles of curriculum and instruction. USA: The
University of Chicago Press.
11. Zwiers, O’Hara, & Pritchard (2014). Common Core Standards in diverse
classrooms: Essential practices for developing academic language and
disciplinary literacy. Stenhouse: ALDNetwork.org
Retrieved on May 20, 2015 from https://www.scoe.org/files/Math-Paired
CC-Protocol.pdf