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The *Private Universe Project in Mathematics* is a collaboration between multiple research institutions, including *Harvard-Smithsonian Center for Astrophysics*, *MIT*, Rutgers SUNJ and *Annenberg Media* ^{1)}.

It focuses on the development and teaching of K-12 mathematical thinking.

There are on-line videos, workshop guides and related resources. ^{2)}.

This is the longest evidence-based study of its type ever undertaken.

The video series follows a group of children - beginning in first grade and continuing through graduation from high school - to investigate how students build mathematical ideas over time.

This research documents how children formulate extraordinarily interesting and complex mathematical ideas, even at a very young age.

The *Private Universe Project in Mathematics* demonstrates and honours the power and sophistication of these ideas, and explores how mathematics teaching can be structured to resonate with children's sophisticated thinking.

The workshops offer a rare opportunity to follow the mathematical development of one group of students throughout grades 1-12, and to observe teachers in the process of redefining what mathematics is for themselves and for their students. ^{3)} ^{4)}

**“It is safer to accept any chance that offers itself, and extemporize a procedure to fit it, than to get a good plan matured, and wait for a chance of using it.”** - Thomas Hardy

Some of the best known mathematicians/mathematical discoveries, such as *Pascal's Triangle*, are the result of seemingly superficial pursuit. It’s interesting, for example, that a 17th century professional gambler’s unexpected losses provided the starting point for one of the most important branches of mathematics - the theory of probability. ^{5)}

This project begins by engaging 30 teachers in a two-week professional development workshop. Despite clear goals having been designed for this initial work… We hear how, as a result of Dr. Maher’s professional development work with teachers, the project was modified, developed and extended.

For those reasons, the study was undertaken as a 'research program' rather than a 'teaching program'. The researchers worked alongside teachers and requested that teachers did not provide students with explicit strategies or answers.

Archival footage shows how the students worked toward a solution over time by devising their own way of representing and solving mathematical problems.

**Instead of adapting or replacing the way that maths was otherwise taught in the participating schools, all classroom activities were carried out in parallel with the 'regular' curriculum.**

SYNOPSIS

**For most primary school mathematical assessment tasks, the numerical value of the answer is simply marked as right or wrong, and each student's mathematical ability is assessed accordingly.**

**For the Mathematical Thinking Project, the answers are of minor importance. Teachers can continue with their 'normal' mathematical teaching program. The Mathematical Thinking project is not intended to replace or be merged into the schools current day-to-day teaching of mathematics.**

**The Mathematical Thinking project was simply designed to discover how far children's natural curiosity and interest in mathematics takes them, whilst observing how students work towards a solution over time by devising their own way of representing and solving problems.**

- Start with a question (or a misconception).
- Allow students time to struggle (ensure students invest effort)
- The teacher is not
*the answer key*(be less helpful - do not give answers / strategies) - 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?)
- Pose problems that encourage play ('Play is the highest form of research' - A. Einstein)

Mathematical thinking is focused on promoting and developing the natural, deep, mathematical and problem solving abilities of children as then progress from primary school and beyond.

The *Mathematical Thinking* project was intended to see how far children's natural curiosity and interest in maths could take them, without them having been explicitly taught procedures for solving mathematical problems.

The activities are designed to help uncover and stimulate development of each student's own mathematical ideas and strategies; To help make their understanding and thinking more visible and powerful through the collaboration and communication required to convince others as they develop their solution(s) to each problem.

**The end-goal of each activity is to encourage a deep understanding and consequent ability to generalise strategies to solve real-world problems in a real-world context.**

- Students take 'ownership' of each problem.
- It is essential that teachers do not offer strategies or solutions, but students are encouraged to copy, discuss, share their own strategies and solutions and convince their peers.
- There are no 'right/wrong' strategies and/or 'right/wrong' answers but students must convince the teacher/others by providing a convincing arguments (proof).

**Q:** Steven's mother says he must wear a different outfit to school each day.

- He has only one red, one white, and one blue shirt.
- He one pair of blue jeans and one pair of white jeans.
- How many different outfits (pants and shirts) can he make.
- Convince us that you have found all possible outfits and there are no more or less outfits.

**A:** The teachers were surprised to find that students came up with a very wide range of answers for the number of possible Shirts & Pants outfits that Steven could wear; For example:

- He can wear 5 different outfits.
- He can wear 4 different outfits.
- He can wear 2 different outfits.
- He can wear 6 different outfits.
- He can wear 6 different outfits.
- He can wear 3 different outfits.
- He can wear 24 different outfits.

Instead of responding that students obtained a 'wrong' answer, the teachers probed deeper to help uncover the students' hidden thinking; For example:

**“You say you could find five (or whatever number) different outfits. How can you convince us that there are no more or less than five?”**

- He has six but there are only five school days, so he can only wear five different outfits.
- White and yellow do not 'go together', so he can only wear four outfits.
- Both pants would be dirty after two days, so he could only wear two outfits.
- He can wear six outfits: Wp-Rs Wp-Ws Wp-Bs Bp-Rs Bp-Ws Bp-Bs.
- I drew each of the shirts and jeans. I drew arrows to join them up and counted six outfits.
- I used colours to draw all of the outfits and then I counted them - He can wear six different outfits.
- His mother could buy him another pair of jeans and then he could wear three different outfits.
- He can cut the sleeves and legs and sew them together in different ways to get more outfits, so I think he could wear maybe 24 outfits.

NOTE: All of the above are actual, un-edited sample answers supplied by 2018 Stage1 students.

In general, Stage1 students have only minimal interest in what may be the 'right' answer.

When explaining their strategies and answers over time, students quickly and spontaneously start to use terms such as *pattern*, *opposite*, *match*, *combination*, *test*, *estimate*, *predict* and *proof*.

As a teacher, consider each of the above answers…

**How might you adapt your teaching to help each of the students develop a deeper understanding of the problem, or encourage them to investigate alternative solutions if they say that they are stuck?**

As student progress through the school system, this becomes more difficult. At we found that Stage 3 students are generally only interested in finding the 'right' answer; They are more interested in finding the right values to plug into the right formula than in understanding and then investigating alternative solutions to the underlying problem.

Work recorded is student journals and multi-media recordings provides evidence of students' ability to confidently build, modify, generalise and explain their strategies.

This evidence provides teachers with an easy and authentic basis for student assessment.

Despite the problem being posed in a highly abstract format, the students ( 2018 Year 1 and Year 2) developed five different and effective solutions.

Interestingly (and despite having been taught some multiplication strategies in their conventional day-to-day maths learning), all of the solutions were based on some kind of 'Direct counting' strategy and a 'proof by cases'; where physical materials are used to model the problem and the objects are simply counted without any obvious reference to the multiplicative structure.

**As a teacher, why is this interesting? Could/should this feedback influence the way you teach multiplication in your next conventional maths lesson?**

Conventional maths teaching is focused on student acquisition of knowledge - The memorisation and practice of explicit strategies, where 'knowledge' is assessed by testing a student's ability to plug a set of given values into one or more previously prescribed formuli.

See NESA K-10 Curriculum St1, St2, St3.

PROBLEM: A man has three Shirts and two pairs of Pants. How many different things does he have?

STRATEGY: From NESA St1 Multiplication and Division, Students 'Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031)“.

SOLUTION:

- Using addition by groups/array, he can have 1 row of 3 things plus 1 row of 2 things.
- 3 + 2 = 5

PROBLEM: A man has three pairs of sneakers and two hats.

- How many different combinations of sneakers and hats can he make?
- Show how you solved this problem.

STRATEGY: HINT: Refer to 'Sample Question - B' for a worked example (repeated on next two lines):

- Q: A man has three pairs of sneakers and two hats. How many combinations does he have?
- A: 3 x 2 = 6

SOLUTION: 6

- I worked out the answer this way. I substituted 'pants' and 'shirts' in my question for the 'sneakers' and 'belts' in the 'Sample Question - B': I worked out the answer 3 x 2 = 6

Of more than 100 Stage 1 students participating in the 'Shirts & Pants' problem in 2017-18, not one student demonstrated that they had used a group/array strategy to solve, calculate, or to prove their answer. Where physical materials were used to model the problem, the objects were simply counted without any obvious reference to the multiplicative structure.

By contrast, the presentation of problems in purely abstract form were at first converted by the students to various forms of physical representation, with some students progressing towards more sophisticated, algebraic-style notation, multiplicative strategies.

Students were most motivated by solving the problem and explaining their strategies. Few of the students were concerned with, or motivated by 'the answer'

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