Join The Parade, New South Wales - Ph:+61-2-1234-5678

Mathematical Thinking Banner

NESA Mathematics K-10


Working Mathematically - Students...

Number & Algebra - Students...

Measurement and Geometry - Students...

Statistics and Probability - Students...


Number & Algebra ...

Measurement & Geometry ...

Statistics & Probability ...

Work Samples ...


  • 'A schoolgirl making the decision to not study maths, as early as high school, is unknowingly precluding herself from access to the jobs with the highest growth rates and the highest future earning potential.' ABC News 06/02/2018 1)
  • 'Students taking advanced maths in Year 12 at its lowest-ever level - A new report shows that students are increasingly turning away from more challenging mathematics subjects in the last years of high school. The number of those studying advanced mathematics in year 12 is at its lowest level in two decades – and the biggest drop-off is amongst girls. ABC News 08/03/2019 2)
  • 'Quite simply, if a learner’s thinking is superficial, biased, weak, lacking depth or breadth then his learning will be superficial, biased, weak, and lacking depth or breadth. Thinking and learning are inextricably linked' - Lane Clark 3)
  • 'It is not what you cover, it is what you uncover that is most important.' Prof. Walter Lewin 4)

Video 1: Five Principles of Extraordinary Math Teaching (13min)

Five principles to encourage mathematical thinking:

  1. Start with a question (or a misconception).
  2. Students need time to struggle (make effort).
  3. You are not the answer key (do not give answers)
  4. Say 'yes' to student ideas (it's not the same as saying 'you are correct or wrong' - what if 2 + 2 = 12 and we explore the consequences?)
  5. Play ('the highest form of research' - N. V. Scarfe) 5)

A Private Universe In Mathematics - Project Overview

The workshops are the outcome of unique, longitudinal classroom research (over twenty continuous years) by educators from universities and institutions including Rutgers, Harvard, MIT, The Smithsonian Center for Astrophysics and Annenberg Media.

This is the longest study of its type ever undertaken.

The research follows an 'ordinary' group of learners from first grade through high school and beyond, concentrating on their work on a set of mathematical (especially combinatorics) tasks.

The data drawn from this research, illustrates and explains how teachers and student groups work collectively to build mathematical ideas over time.

The classroom videos and journal entries show how reasoning that takes the form of proof can evolves and can be promoted from early childhood.

Assessment example Print out a class list with at least one column for each week/lesson (or create an Excel or Google sheet). During each session, allocate a numeric value (say 1, 2 or 3), to refelect the level of understanding that a student may demonstrate in that session. At the end of the assessment period, sum the totals for each student and normalise the result. Ascribe a comment for each student, based on their score. Cross-check by browsing through each student journal/notebook as an alternative/supplementary method of evaluation.

Video 2: Introduction - A Private Universe in Mathematics (3min)

The study has produced over 4,500 hours of video, detailed transcripts and related on-line resources and investigates how students’ reasoning evolved from elementary and high school years to college and beyond.

These resources are an invaluable tool not only for primary- and secondary-school teachers of science or mathematics, but also for anyone involved in teacher preparation.

MATHEMATICAL THINKING - K-6 + High School & Beyond

Video 3: The Research - A Private Universe in Mathematics (7min)

The theme of these Workshops is “how to develop authentic mathematical thinking by and for those who may have little mathematical training and/or confidence.

This is done by blending long-standing teaching research and effective strategies within the context of new technologies”.

Both teachers and learners are encouraged to develop new metaphors for their roles in the classroom. Students are urged to see themselves as something other than “sponges” soaking up the teacher's information.


Video: How Mathematics Helps Us Understand 'The Tragedy of the Commons' (5min)

EXTENSION: More Challenging Maths Activities:


Find out more about the Harvard-Smithsonian Center for Astrophysics here: www.cfa.harvard.edu

  1. Learn more about Annenberg Media and browse the resources and workshops they offer to teachers: www.learner.org
  2. To watch A Private Universe video: www.learner.org/resources/series28.html
  3. To watch Minds of Our Own video: www.learner.org/resources/series26.html
  4. The Private Universe in science project videos: www.learner.org/catalog/extras/puptwsup.html
  5. To watch the Private Universe in Science workshop videos: www.learner.org/resources/series29.html
  6. The Private Universe in mathematics project videos, see: www.learner.org/resources/series120.html
  7. The Private Universe in mathematics project K-12 License
2018/mathematics/home.txt · Last modified: 25/06/2019/ 19:29 by