Researchers (professors at the Rutgers University Graduate School of Education) conducted all of the documented problem-solving sessions with the students; every sessions was video-recorded with one or more cameras for later analysis. Over two thousand videos were recorded and analysed.

Example videos are provided to illustrate how data were collected and analysed, how students worked on fundamental problems and how this work led to exceptional growth in the students’ mathematical understanding.

By videotaping children as they worked together on mathematical tasks over long periods of time, the researchers were able to trace the origin and development of their mathematical ideas, observing what children said to one another and showed to one another.

The videos and transcripts were used to study the meanings that children gave to mathematical situations and to note the different representations they made public.

Researchers observed, described, and coded the videotape data, and they kept written and electronic files of the emerging theoretic, analytic, and interpretative ideas about the students’ mathematical behaviours.

A detailed analysis of data made it possible to trace the origin and evolution of children’s arguments. How children expressed their ideas through spoken and written language, through the physical models they built, through the drawings and diagrams they made, and through the mathematical notations they invented.

• IF UNABLE TO ACCESS YOUTUBE VIDEO (requires Flash) TRY:Dan Meyer

Mathematics arose from the need to count, measure, and calculate, but the discipline evolved to include abstraction, logical reasoning, and the search for and analysis of patterns.

Good mathematical problems are therefore those which give rise to the need for abstraction, systematization, and pattern recognition. A focus of the study was therefore to select problems that would give rise to these needs and on solving problems that were not part of the regular curriculum (because it was important for the students to come to the problems fresh, without pre-taught algorithms).

For those reasons, the major strand of the longitudinal study consisted of problems in combinatorics, because in working on these problems, students can find the need to organize their work systematically, look for patterns, and generalize their findings; in ways that are outside the regular elementary school curriculum and therefore unfamiliar to students.

Students worked on a variety of combanitorics problems. For example, the shirts and pants problem, which asked how many different outfits can be made from a given number of clothing items?

Many St2 and St3 students drew pictures of outfits; some drew lines between shirts and jeans, and others made lists of outfits.

Notational choices influenced the way they reasoned about the data.

For example, Stephanie used “blue-white” to stand for the white shirt/blue jeans outfit, and also for the blue shirt/white jeans outfit. Contextual issues also played a role in the problem solving. For example, Dana discarded the white jeans/yellow shirt outfit on grounds that the resulting outfit did not match and was thus not fashionable! That different students got different answers was not problematic for the children; many students seemed comfortable with the notion that answers varied between three and seven outfits.

Students willingly shared their interpretations and strategies and talked to each other about their findings. Over time, when the children were again presented with this problem, they did not remember how they had solved the problem earlier, nor did they remember their earlier answers.

Of particular interest is that evidence of further elaboration of earlier strategies emerged. Students used and built on strategies of their partners. For example, Stephanie indicated different outfits by drawing lines between drawings of shirts and jeans, as Dana had done earlier.

Much later, student-developed techniques for checking and for keeping track, such as controlling for variables, were complete. Earlier ideas and strategies were refined to produce complete, elegant solutions rather quickly. Students built on their heuristics to solve more complex extensions of the problem to include belts and hats as parts of outfits.

Over the months, there is evidence of durable learning. Also, there is evidence that children learned from each other as revealed in elements of one another’s strategies reappearing in their second attempt at solving the problems.

More importantly, perhaps, is that each student incorporated strategies in unique ways.

For example, Michael’s use of lines and notation to show combinations differed significantly from the way that Stephanie and Dana used lines and notation. This suggests the importance of encouraging the use of students’ personal representations in building solutions.

These videos also demonstrate some benefits of group work. The contribution of each student to the cumulative body of knowledge enables students to arrive at their own solutions while simultaneously benefiting from the knowledge of others.

What was especially significant for the researchers was the evidence of how students built on earlier ideas and, without intervention or approval from researchers and continued their problem solving, driven by earlier heuristics and sense making to produce correct solutions that they could confidently justify.

The researchers classified student forms of reasoning into four categories:

1. justification by cases,
2. inductive argument,
3. elimination argument
4. analytic method (use of formulas)

The list (below) illustrate some ways that students used justification by cases. Many students proceed by selecting one or more colours/items and listing all possibilities for that colour/item combination in some organised fashion. They then argue from symmetry that all/any other cases would be the same/different (by reference to concepts such as 'pairs', 'duplicates' and/or 'opposites').

For example:

• Some students used specific codes, such as 'R', 'Y' and 'B' to represent colours (such as Red, Yellow and Blue).
• Other students used generalises codes such as 1, 0, and X to indicate the three colors, and they focused on the placement of the duplicate colour, using 1 to indicate the colour that was duplicated and X and O to indicate the other two colours.
• By comparison, students in later Stages who are introduced to these same problems for the first time, generally jump straight to an abstract 'formula' in their search for 'the correct answer. They are reluctant to make estimates and/or explain their reasoning.

A detailed synopsis and list of resources used in the initial research phase of this project (1989 -2001) is available to view on-line.

There is a collection of videos and resources developed as workshops for teachers completed prior to commencement of the research project. These videos explore the reasons why teaching science is so difficult and offers practical advice to help you teach more effectively. These videos were recorded in the 1990's but remain equally relevant today.

Each of the above programs focus 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. ¹⁾

Each week, one person should be responsible for facilitating the Site Investigations. The facilitator does not need to be the AP/leader. Participants may choose to rotate the role of facilitator on a weekly basis. The facilitator should be responsible for organising materials and documenting each workshop session (see 'Record your discussions' below).

You should keep an eye on the clock to ensure 'teacher talk' is minimised and that students have at least ten minutes at the close of each session to reflect and annotate their journals.

Be sure to review each workshop before proceeding to the next.

The lesson/activity sequence is flexible, depending only on factors such as student capabilities and the weather on the day.

A few of the workshops do require special materials - although most of these can be replaced by those created from trash, through the use of a little craft/ingenuity.

Each participant/student is encouraged to keep a journal; to keep track of predictions, strategies, working-out, reactions to readings and videotapes, to collect and reflect on data, and to record teaching ideas for yourself. Take every opportunity to explicitly:

1. Write down a prediction
2. Record working strategies, notes, drawings
3. At the end of each session, reflect and write 'what I used to think' and 'what I think now'

We recommend that you make an audiotape recording/video of all workshops/discussions. A simple Go-Pro or similar camera is sufficient. Videos/audiotape are low maintenance, cheap and easy to use. The content is a gold mine for use as evidence/assessment.

You will need paper and 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.

The mathematical term 'combinatorics' is mentioned throughout the project to describe particular kinds of problems. Combinatorics problems involve the mathematics of systematic counting based on strategies, such as pattern recognition and grouping.

These ideas are important for children as they develop operational skills with whole numbers, basic understanding of probability and discrete mathematics, algebraic concepts such as variables, and overall ideas about justification and generalisation.

Make sure that you retain documentation; journals, videos and related works.

On starting your first activity, ask each student to write down and answer (simple YES or NO), these two questions:

1. Am I good at maths? YES/NO
2. Do I like maths? YES/NO

Develop the habit of asking “What did I used to think?” and “What do I think now.”

After a few lessons, ask the students to again write down answers to these two questions:

1. Am I good at maths? YES/NO
2. Do I like maths? YES/NO

shirts and c

Listed here are the major combinatorics problems the students encountered from elementary school through high school (for students and teachers), along with brief discussions of solutions (for teachers only). Each problem is capable of being presented in a variety of ways, for example, by simply adding to the variety or number of items in a given problem.

PROBLEM - Solve the problem below for yourself and develop a way to convince others that you have found all possible outfits:

• 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 he make?

Share and compare your solution and justification with others in your group.

SOLUTION - He can make six different outfits; each of the two pairs of pants can be matched with each of the three shirts.

The possible outfits are:

• blue pants/white shirt
• blue pants/blue shirt
• blue pants/yellow shirt
• white pants/white shirt
• white pants/blue shirt
• white pants/yellow shirt

PROBLEM: Suppose Stephen was given an extra pair of pants, a black pair. How many different outfits can he now make?

SOLUTION: He can make 12 different outfits (by simple multiplication, the number of shirts (3) times number of pants (4).

Imagine giving this problem to students that you teach.

1. Do you think it is an appropriate problem for your students?
2. How do you think your students would solve and justify the problem?

PROBLEM - Solve the problem below for yourself and develop a way to convince others that you have found all possible outfits (that there are no more and no less).

Share and compare your solution and justification with others in your group.

Solve the following extensions to the Shirts and Pants problem and then share your solutions.

1. Remember that Stephen has a white shirt, a blue shirt, and a yellow shirt.
2. He has a pair of blue jeans and a pair of white jeans.
3. Stephen has now been given a brown belt and a black belt.
4. His mother has told Steven that he must always wear one of his belts, to stop his pants falling down.
5. How many different outfits can he make now?

PROBLEM - REVERSIBILITY -Stephen's brother (Billy-bob) has exactly 16 different outfits.

• Decide how this might be possible.
• Specify what pieces of clothing Billy-bob might have, so that he can make exactly 16 different outfits?

Share and compare your solution and justification with others in your group.

SOLUTION

1. Shirts & Pants - Solutions
2. Shirts & Pants - More Strategies

MATERIALS: Each group is given a total of 100 Unifix cubes comprised of two colours (50 cubes of each colour = total 100 cubes).

PROBLEM: Work together and make as many different towers four cubes tall as is possible when selecting from two colours.

See if you and your partner(s) can plan a good way to find all the towers four cubes tall.

SOLUTION: At each position in the tower, there are two colour choices. Therefore, there are 2⁴ = 2×2×2×2 = 16 possible towers that are four cubes tall.

This can be generalized to an n-tall tower with two colours to choose from; there are 2 × 2 × 2. . . × 2 = 2ⁿ possible towers that are n cubes tall, when there are two colours to choose from.

This can also be generalized to an n-tall tower with m colours to choose from; there are m × m × m. . . × m = mⁿ possible towers that are n cubes tall with m colours to choose from.

In the following discussions, we will call the first generalization (the n-tall tower with two colours) the towers problem, and we will call the second generalization (the n-tall tower with m colours) the generalized tower problem.

MATERIALS: Each group has the same number of cubes as previous activity; 100 Unifix cubes comprised of two colours (50 cubes of each colour = total 100 cubes).

Write down your estimate before you start to build towers.

If you use all the brick that you used to build the four-tall towers with two colours, how many 3-tall towers could you make when selecting from the same cubes with two colours.

Write down your estimate before you start to build towers.

• Starting with the same number of cubes, could you build more three-tall towers than four-tall towers, or less three-tall towers than four-tall towers?

Develop a way to convince others that you have found all possible combinations and have included no 'duplicates'.

MATERIALS: Each group has the same number of cubes as previous activity, 100 Unifix cubes comprised of two colours (50 cubes of each colour = total 100 cubes).

Write down your estimate before you start to build towers.

PROBLEM: Work together and make as many different towers five cubes tall as is possible when selecting from two colours.

See if you and your partner(s) can plan a good way to find all the towers four cubes tall.

Develop a way to convince others that you have found all possible outfits and have included no 'duplicates'.

NOTE FOR TEACHERS: This final part only requires students to 'estimate' the number of cubes required. They do not need to build the towers 5-high. Building towers 5-hight is listed as a separate activity (it requires more than the total number of cubes allocated for the 4-high activity).

FOR STUDENTS: AFTER YOU HAVE COMPLETED THE ABOVE CHALLENGE: What is the total number of cubes of each of the two colours needed to build all possible combinations of towers 5-high.

Write down “Total number of cubes for towers 5-tall +”, then write down the number that you estimate, and draw a circle around it.

Develop a way to convince others that you have found all possible towers, and have included no 'duplicates'.

Write down your answer and draw a square around it.

PROBLEM - Solve the problem below for yourself and develop a way to convince others that you have found all possible combinations of cups, bowls and plates and have no 'duplicates':

Pretend that there is a birthday/class 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?
2. 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.

SOLUTION - Each of the two cup choices can be matched with each of the two bowl choices, and each cup-bowl pair can be matched with any of the three different plate choices. Therefore, there are 2 x 2 × 3 =12 possibilities.

Therefore,

1. yes, it is possible for ten children at the party each to have a different combination of cup, bowl, and plate.
2. no, it is not possible for fifteen children at the party each to have a different combination of cup, bowl, and plate.

Reflect on whether this is a problem you might use with your students. How do you think that your students would solve this problem?

1. Shirts & Cups - More Strategies
2. Towers - Patterns & Cases
3. Towers - Doubling & Problems
4. Towers - Example Student Strategy

PROBLEM: This Saturday there will be a 500 metre relay race at the school.

• Each team that participates in the race must have a different uniform (a uniform consists of a solid coloured shirt and a solid colored pair of shorts).
• The colors available for shirts are yellow, orange, blue, or red.
• The colors for shorts are brown, green, purple, or white.

How many different relay teams can participate in the race?

SOLUTION: There are four choices for shirts and four choices for shorts, so there are 4×4=16 ways to make uniforms. Therefore, sixteen different relay teams can participate.

MATERIALS: Your group has two colours of Unifix cubes.

PROBLEM: Work together and make as many different towers five cubes tall as is possible when selecting from two colours. See if you and your partner can plan a good way to find all the towers five cubes tall.

SOLUTION: There are 2⁵ = 32 possible unique towers five cubes tall.

PROBLEM: Your group has three colours of Unifix cubes. Work together and make as many different towers four cubes tall as is possible when selecting from three colours. See if you and your partner can plan a good way to find all the towers four cubes tall.

Since there are three choices for each of four positions, there are 3⁴ = 81 possible towers that are four cubes tall when selecting from three colors.

PROBLEM: Consider the pizza problem, focusing on the number of pizza combinations that can be made when selecting from among five different toppings.

SOLUTION: There are 2⁵ = 32 different pizzas.

You have been invited to participate in a TV Quiz Show and the opportunity to win a vacation to Disney World.

The game is played by:

1. choosing one of four possibilities for winning and…
2. then picking a tower out of a covered box.

If the tower pick draw out of the covered box matches your choice, you win.

You are told that the box contains all possible towers that are three tall that can be built when you select from cubes of two colours, red, and yellow.

You are given the following possibilities for a winning tower:

• All cubes are exactly the same colour.
• There is only one red cube.
• Exactly two cubes are red.
• At least two cubes are yellow.

Which choice would you make and why would this choice be better than any of the others?

SOLUTION: In order to decide which is the best choice, we need to find the probability of each choice. The total number of 3-tall towers is 2³ = 8.

The probabilities are:

• All cubes are exactly the same color: There are two ways (all red or all

yellow). The probability is 2 ÷ 8 = 0.25.

• There is only one red cube: There are three ways; the red cube can be on the

top, in the middle, or on the bottom. The probability is 3 ÷ 8 = 0.375.

• Exactly two cubes are red: This is the same as saying exactly one cube is

yellow. The probability is the same as for exactly one red cube: 3 ÷ 8 = 0.375.

• At least two cubes are yellow: This is equivalent to saying that either exactly two cubes are yellow or exactly three cubes are yellow. As discussed above, the probability that exactly two cubes are yellow (the same as the probability that exactly two cubes are red) is 0.375. Since there is only one way for exactly three cubes to be yellow, that probability is 1 ÷ 8 = 0.125. The probability of either event is therefore 0.375 + 0.125 = 0.5 (We can use addition because the two events are mutually exclusive.)

Therefore, “At least two cubes are yellow” is the most likely event.

Assuming you won, you can play again for the Grand Prize which means you can take a friend to Disney World… But now your box has all possible towers that are four tall (built by selecting from the two colours yellow and red). You are to select from the same four possibilities for a winning tower.

Which choice would you make this time and why would this choice be better than any of the others?

The total number of four-tall towers is 2⁴ = 16. The probabilities are:

• All cubes are exactly the same colour: There are two ways (all red or all yellow). The probability is 2 ÷ 16 = 0.125
• There is only one red cube: There are four ways; the red cube can be on the top, second from the top, second from the bottom, or on the bottom. The probability is 4 ÷ 16 = 0.25
• Exactly two cubes are red: The number of ways to accomplish this is C(4, 2) = 6. The probability is therefore 6 ÷ 16 = 0.375
• At least two cubes are yellow: This means that exactly two cubes are yellow, exactly three cubes are yellow, or exactly four cubes are yellow. As discussed above, the probability that exactly two cubes are yellow (the same as the probability that exactly two cubes are red) is 6÷16 = 0.375. The probability that exactly three cubes are yellow is the same as the probability that one cube is red: 4 ÷ 16 = 0.25. Since there is one way for exactly four cubes to be yellow, that probability is 1 ÷ 16 = 0.0625. The probability of any one of the three events is therefore 0.375 + 0.25 + 0.0625 = 0.6875.

Therefore, “At least two cubes are yellow” is the most likely event.

PROBLEM: A local pizza shop has asked us to help them design a form to keep track of certain pizza sales. Their standard “plain” pizza contains cheese. On this cheese pizza, one or two toppings could be added to either half of the plain pizza or the whole pie.

How many choices do customers have if they could choose from two different toppings (sausage and pepperoni) that could be placed on either the whole pizza or half of a cheese pizza?

List all possibilities. Show your plan for determining these choices.

Convince us that you have accounted for all possibilities and that there could be no more.

With two topping choices, there are four possibilities for the first half pizza, because each topping can be either on or off that half of the pizza. The four choices are: plain (sausage off, pepperoni off), sausage (sausage on, pepperoni off), pepperoni (sausage off, pepperoni on), and sausage/pepperoni (sausage on, pepperoni on). Consider each of the four possibilities in turn.

1. Case 1: Plain. There are four possibilities for the other half of the pizza, the four listed above (plain, sausage, pepperoni, and sausage/pepperoni).
2. Case 2: Sausage. There are three possibilities for the other half of the pizza: sausage, pepperoni, and sausage/pepperoni. (We omit plain, because we already accounted for the plain-sausage pizza in Case 1.)
3. Case 3: Pepperoni. There are two possibilities remaining for the other half of the pizza: pepperoni and sausage/pepperoni. (Plain and sausage are already accounted for.)
4. Case 4: Sausage/pepperoni. There is only one possibility left for the other half of the pizza; that is sausage/pepperoni.

There are 4 + 3 + 2 + 1 = 10 possible pizzas with halves.

A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a cheese pizza with tomato sauce.

A customer can then select from the following toppings: peppers, sausage, mushrooms, and pepperoni.

How many different choices for pizza does a customer have?

List all the possible choices.

Find a way to convince each other that you have accounted for all possible choices.

SOLUTION: There are 2⁴ (2 × 2 × 2 × 2) = 16 possible pizzas.

The pizza shop was so pleased with your help on the first problem that they have asked us to continue our work.

Remember that they offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms, and pepperoni.

The pizza shop now wants to offer a choice of crusts:

• regular (thin) or…
• Sicilian (thick).

How many choices for pizza does a customer have?

List all the possible choices.

Find a way to convince each other that you have accounted for all possible choices.

SOLUTION: Each of the 16 four-topping pizzas has two choices of crust, so there are 32 pizzas.

At customer request, the pizza shop has agreed to fill orders with different choices for each half of a pizza.

Remember that the shop offer a cheese pizza with tomato sauce.

A customer can then select from the following toppings: peppers, sausage, mushroom, and pep- peroni. There is a choice of crusts: regular (thin) and Sicilian (thick). How many different choices for pizza does a customer have? List all the possible choices. Find a way to convince each other than you have accounted for all possible choices.

SOLUTION: The first half of the pizza can have 2⁴ = 16 possible topping configurations, as described above. Consider each of those configurations in turn.

Following the procedure described above for the two-topping half-pizza problem, we find that there are 16 + 15 + 14 +. . . + 3 + 2 + 1 possible pizzas; this sum is given by 16 × 17 ÷ 2. Since each pizza can have a thick or thin crust, we multiply by 2.

Therefore, the number of possible pizzas is 16 × 17 ÷ 2 × 2 = 272.

What's the Difference? In English we use the word 'combination' loosely, without thinking if the order of things is important. In other words: * 'My fruit salad is a combination of apples, grapes and bananas' We don't care what order the fruits are in, they could also be “bananas, grapes and apples” or “grapes, apples and bananas”, its the same fruit salad.

• 'The combination to the safe is 472'. Now we do care about the order; '724' won't work, nor will '247'. It has to be exactly 472 (in that order).

So, in Mathematics we use more precise language:

• When the order doesn't matter, it is a Combination.
• When the order does matter it is a Permutation.

A 'combination lock' is really a “permutation lock'

BOWLS & CONES - On-Screen Math Activities (Ice Cream Problems)

• Bowls: There are six flavours of ice cream. If the ice cream is served in bowls that can hold up to six scoops, how many different ways can the ice cream be served?
• Cones: In a variation of the problem, the ice cream can be served in cones stacked up to

four scoops high. Given that the order of stacking matters, how many different cones could be served?

PROBLEM: The new pizza shop has been doing a lot of business. The owner thinks that it has been so hot this season that he would like to open up an ice cream shop next door.

He plans to start out with a small freezer and sell only six flavours of ice cream:

1. vanilla
2. chocolate
3. pistachio
4. boysenberry
5. cherry
6. butter pecan.

BOWLS: The cones that were ordered did not arrive in time for the grand opening so all the ice cream was served in bowls.

• How many choices for bowls of ice cream does the customer have?
• Find a way to convince each other that you have accounted for all possibilities.

CONES: The cones were delivered later in the week.

The owner soon discovered that people are fussy about the order in which the scoops are stacked. on the cone. One customer said “After all eating chocolate then vanilla is a different taste than eating vanilla then chocolate.”

The owner also quickly discovered that she couldn’t stack more than four scoops in a cone.

How many choices for ice cream cones does a customer have?

Find a way to convince each other that you have accounted for all possibilities.

PROBLEM I: How many feet and toes are there? Here is the the TEACHER ANSWER Video

PROBLEM II: How many different two-digit numbers can be made from the digits 1, 2, 3, and 4?

Each of four cards is labelled with a different numeral: 1, 2, 3, and 4.

How many different two-digit numbers can be made by choosing any two of them?

SOLUTION:

• Counting I: Assuming that you are not permitted to reuse digits, there are four

choices for the first digit and three choices for the second digit, giving 4 x 3 = 12 two-digit numbers:

• These two-digit numbers are 12, 13, 14, 21, 23, 24, 31, 32, 34, 41, 42, and 43

• Counting II: There are four choices for the first digit and four choices for the second digit. This makes 4⁴ = 16 different two-digit numbers:
• These two-digit numbers are 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, and 44

This is a variation of the Towers (Pascal's Triangle) problems.

Using Powell’s et al. (2003) notation to denote coordinates on the taxicab grid, (n,r) indicates a point n blocks away from the taxi stand and r blocks to the right. So the blue dot is at (5,1), the red dot is at (7,4), and the green dot is at (10,6).

Taking the shortest route means going in two directions only (down and to the right).

Finding the number of shortest paths from the taxi stand (0,0) to any point (n,r) involves the number of ways to select r segments of one kind of movement in a path that includes two kinds of movements; i.e., the number of shortest paths to (n,r) is C(n,r). For the specific cases given above, the shortest paths are:

1. Blue: C(5,1) = 5
2. Red: C(7,4) = 35
3. Green: C(10,6) = 210