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- This is very much a work in progress (November 2017)

How might children solve problems that use mathematical ideas, before the procedures to solve them are formally introduced in school. And/or, will a more hands-on teaching framework help an otherwise 'average', range of students who have become disengaged from, or are having difficulty progressing with maths.

From the analysis of the problem solving of 150 students in a variety of settings from high-school to graduate study, four major forms of reasoning evolved: (1) Justification by Cases, (2) Inductive Argument, (3) Elimination Argument, and (4) Analytic Method (use of formulas.) The predominant method for students was reasoning by cases where they used the heuristic of controlling for variables or a recursive argument. Only graduate students and one senior undergraduate student 1 correctly used analytical methods.^{1)}

Sanchez and Sacristan concluded from studying the written work of students that students are not accustomed to expressing mathematical ideas, and offer as an explanation that the emphasis mainly is on producing correct solutions (2003). One consequence is that students develop the belief that all problems can be solved in a short amount of time and they will not persist if a problem cannot be solved quickly.

This workshop series for teachers explores why teaching science is so difficult and offers practical advice for teaching it more effectively. Each program focuses on one theme and one content area and uses specific examples to show how students' preconceived ideas can create critical barriers to learning. Education experts review classroom strategies and results and recommend new ways to involve students and approach difficult topics.

LINKS (For facilitator only)

- Purchase the EBook of the full study - US$149 - https://www.springer.com/us/book/9789400706149 - 2011 edition covers 20 years longitudinal study - Also at Walmart US$164

- Home Page - Private Universe in Mathematics - http://www.learner.org/workshops/pupmath/index.html
- Series overview, including a workshop 'organiser' - http://www.learner.org/workshops/pupmath/about/overview.html
- Synopses (all workshops) - http://www.learner.org/workshops/pupmath/about/synopses.html
- Support materials for each workshop - http://www.learner.org/workshops/pupmath/support/index.html
- Detailed workshop structure & support materials required -http://www.learner.org/workshops/pupmath/about/structure.html
- Full workshop 1 instruction sheet - http://www.learner.org/workshops/pupmath/support/pupm1.pdf
- Official PUP Annenberg Brochure - http://www.learner.org/workshops/pupmath/pdf/PUPmath.pdf

**Paper, pens or markers** - for preparing solutions to the problems. If you are a large group, you may want to have an overhead projector, blank transparencies and pens for participants to use for sharing solutions.

**Unifix® or other snap cubes** - Each participant will need about 100 cubes (50 of each of two colors) to complete the homework assignment for Workshop Two. Although not essential, sharing and discussing solutions will be much easier if everyone is using the same two colors. If this is impossible, sets of cubes need to be made up for each participant to use with two colors that can be designated as “light cubes” and “dark cubes” when their solutions are discussed. If Unifix® cubes are not available, use the “cut-out cubes” sheet included with the packet at the end of Workshop 1. You can also use the online version in the Towers Problem Web activity.

Unifix® or other snap cubes, or cut out cubes (see above)

Try the following activities BEFORE watching the workshop video.

Solve the problem below for yourself and develop a way to convince others that your solution is correct.

Stephen has

- a white shirt, a blue shirt, and a yellow shirt.
- a pair of blue jeans and a pair of white jeans.

YOUR TASK: Work out how many different outfits can Stephen make?

- Share and compare your solution and justification with others in your group.
- Imagine giving this problem to students that you teach.
- Do you think it is an appropriate problem for your students?
- How do you think your students would solve and justify the problem?

Solve the following extensions to the Shirts and Pants problem.
** 1. Add a new clothing item** Remember that Stephen has

- a white shirt, a blue shirt, and a yellow shirt.
- a pair of blue jeans and a pair of white jeans.
- Stephen ALSO now has a brown belt and a black belt.

YOUR TASK: Work out how many different outfits can Stephen make now?

** 2. Reversibility** Mario has exactly 16 different outfits.

YOUR TASK: Decide how this might be possible. Specify what pieces of clothing Mario might use to make these 16 outfits.

- Share your solutions with the group.

Solve the following problem for yourself and develop a way to convince others that your solution is correct.

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.

YOUR TASKS:

- Is it possible for 10 children at the party to each have a different combination of cup, bowl, and plate?
- Is it possible for 15 children at the party each to have a different combination of cup, bowl, and plate?

- Compare and justify your solutions with the group.
- Reflect on whether this is a problem you might use with your students.
- How do you think that your students would solve this problem?

Stephen has a white shirt, a blue shirt, and a yellow shirt. He has a pair of blue jeans and a pair of white jeans.

- How many different outfits can Stephen make?

There is to be a birthday party for one of the students in class. There are blue and yellow cups, blue and yellow bowls, and blue, yellow, and orange plates.

- Can 10 children have a different combination of cups, bowls, and plates?

We’ve just seen the teacher/researchers repeating the same, or similar, problems three times over the course of almost a year. What can we say about the changes in the students’ methods over time?

Build all possible towers that are five (or four, or three, or n) cubes high by selecting from plastic cubes in two colours. Provide a convincing argument that all possible arrangements have been found.

We’ve had a chance to look at an overview of the Rutgers’ long-term study.

- In what ways are the students in the study similar to your students?
- How are they different?

An unprecedented long-term study conducted by Rutgers University followed the development of mathematical thinking in a randomly selected group of students for 12 years - from 1st grade through high school - with surprising results. In an overview of the study, we look at some of the conditions that made their math achievement possible. Go to this unit.

- Support materials - http://www.learner.org/workshops/pupmath/support/index.html
- PDF Handout and activities description - http://www.learner.org/workshops/pupmath/support/pupm1.pdf
- Workshop 1 transcript: http://www.learner.org/workshops/pupmath/workshops/wk1trans.html

In the second grade, the researchers introduced the problem: Shirts and Pants.

Stephen has:

- a white shirt, a blue shirt, and a yellow shirt.
- a pair of blue jeans and a pair of white jeans.

How many different outfits can he make?

Let's see how students approach this problem.

Some students solved the problem by drawing lines between drawings. One student drew lines between the words. This is interesting because last time, when he came up with a completely different answer.

Since the students were able to come up with a way to solve this problem on their own, would they be able to use this strategy to solve a more complex problem?

To test this, the researchers presented an extension of the Shirts and Pants problem.

Called “Cups, Bowls, and Plates,” the problem adds another choice to be considered.

Pretend that today is somebody's birthday in your class. It's just 'finger food', so we're not going to have forks and knives today. Your job is to make different combinations…

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.

Is it possible for 10 children at the party each to have a different combination of cup, bowl, and plate?“

The next day, researchers interviewed students to explore their thinking in more depth. After reviewing the previous work with Shirts and Pants, researchers asked students to justify their total number of combinations - 12. The student first used the same strategy that they had used with Shirts and Pants - linking choices by drawing lines, then counting the lines… Then had an idea for a different way to solve the problem: multiplication!

Students seem to like to represent their ideas symbolically. Algebraic thinking must begin early.

In the first task, shirts and pants:

- the origins of algebraic thinking are there.
- the notion of the generalized solution is there.
- the notion of controlling for variables is there.
- the notion of extending that particular solution structure to more complicated problems like cups, bowls, and plates is there.

So that's very, very important.

Right after the shirts and pants activity, the Rutgers team introduced a new challenge: Towers. Students were asked to select from stacking cubes of two colors and assemble ìtowersî of a given height. This problem is from a branch of discrete mathematics called combinatorics- which is usually taught in high school or college as part of probability.

Towers = Combanatorials and probability

“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

A heuristic technique (/hjuːˈrɪstɪk/; Ancient Greek: εὑρίσκω, “find” or “discover”), often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, guesstimate, stereotyping, profiling, or common sense. ^{2)}

- Detailed synopsis and list of resources - http://www.learner.org/workshops/privuniv/intro.html

This innovative workshop for teachers explores the reasons why teaching science is so difficult and offers practical advice to help you teach more effectively. Each program focuses on one theme and one content area and uses specific examples to show how students' preconceived ideas can create critical barriers to learning. Education experts also review classroom strategies and results and recommend new ways to involve students and approach difficult topics.