In a way that comes naturally to them, students like to represent their ideas symbolically, showing that algebraic thinking must begin early. In the, 'shirts and pants' task, the origins of algebraic thinking, generalised solutions and controlling for variables is all there. The notion of extending that particular solution structure to more complicated problems like cups, bowls, and plates provides opportunities for children to develop their thinking in ways that the study considered to be very, very important.

Each team will need to have/draw, 15 of each item in the specified colours (below) to build place settings.

Students keep a record of their solution(s), including their justifications/proof, to share during the next workshop.

Students should share and compare their solution(s) and justification(s) with others in their group.

REMINDER Students naturally develop their own strategies, with guidance from a facilitator who who shows that teachers should not need to suggest strategies or provide answers

Record the date and the title 'Cups, Bowls and Plates' on a clean page in your journal (write your name at the top of the page if you are sharing or not using your own journal).

Pretend that there is a birthday party in your class today. It’s your job to set the places with cups, bowls, and plates.

• The cups and bowls are blue or yellow.
• The plates are either blue, yellow, or orange.

1. Is it possible for 10 children at the party to each have a different

combination of cup, bowl and plate? Write down your prediction.

1. Is it possible for 15 children at the party each to have a different combination of cup, bowl and plate? Write down your prediction.

Compare and justify your solutions with the group.

What did you used to think?

What do you think now?

Reflect on whether this is a problem you might use with your students.

How do you think that your students would solve this problem?

Reflect on whether this is a problem you might use with your students.

How do you think that your students would solve this problem?

How far can children’s natural curiosity and interest in math take them? How do students do mathematics before they are taught procedures for solving problems?

One of the most important conditions of the long-ter mathematics study was that students were invited to work together to solve problems. By sharing and justifying their ideas, students are able to clarify their own thinking. Collaborative work thus becomes the vehicle for advancing each individual student’s ideas. This activity focuses on how teachers and students can foster thoughtful mathematics through collaboration and building a learning community.

• In mathematics, how do we make visible an idea, or keep track of a line of thought?

From kindergarten to calculus, mathematics involves notations, symbols as surrogates for abstract ideas.

Often, the goal of math education is to give students the standard notation: the written language of symbols and equations that is commonly accepted in the math profession. But, how often have you seen students come up with unique and surprising ways to express a mathematical idea? It happens more often than many teachers realise.

In this exercise, we are giving students a chance to create their own notations and explore if this can help advance their learning. Notation

The simple question, “Can you convince me?” is key to mathematical success. This program introduces the idea of proof as one of the key ideas in mathematics. Delving into the mathematics of the 'Cups, Bowls & Plates' problem, we’ll see how two kinds of proof (proof by cases and proof by induction), naturally grow out of the need to justify and convince others. The teacher plays a critical role by using several kinds of questions to help students move toward successful justification of their answers.

• Go to Student //Cups, Bowls & Plates// problem page

Students keep a record of their solution(s), including their justifications/proof, to share later.

Students should share and compare their solution(s) and justification(s) with others in their group/class.

• What did you used to think?
• What do you think now?