On this page: An overview of the Mathematical Thinking Project, including a list of example activities and links to supporting teaching resources (for teachers only). Following students from Kindy through high school and beyond. The longest and most detailed study ever undertaken of how children develop mathematical thinking.

• IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:Project Philosophy and Intended Outcomes(7min)

TEACHER WORKSHEETS: The ANSWERS_TO PROBLEMS web page provides a simpler teaching guide, with explanations provided on an activity-by-activity basis.

STUDENT WORKSHEETS: The STUDENT-PROBLEMS web page contains the list of activities and associated instructions that can be handed out to students.

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

• IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:2J Get Started with Mathematical Thinking (3min)

• IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:2D + 2S Get Started with Mathematical Thinking (3min)

• IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:Outcomes: Teacher Perceptions (3min)

What do teachers say: Is there any evidence that these activities promote improvement in Mathematical Thinking and/or cognitive development for K-12 students?

• IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:Problem Solving (7min)

Can students solve problems without having been introduced to formal rules:

• IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:2D Mathematical Thinking

Video 1. Using Video To Improve Teaching Practice

• TRAINS
• SHIRTS-AND-PANTS
• TOWERS-FOUR-HIGH
• CUPS-BOWLS-PLATES
• TOWER-OF-HANOI
• WORLD-SERIES

REMINDER: Teachers should not need to suggest strategies or provide answers

• STEAMpunks 2017 Workshops - Introduction
• STEAMpunks 2017 Workshops - Outcomes
• STEAMpunks 2017 Workshop Part 1
• STEAMpunks 2017 Workshop Part 2
• STEAMpunks 2017 Maths Workshop - Shirts & Pants
• Maths Case studies - K-6 Teachers

• PUP Workshop overview - Descriptions & Structure
• Mathematical thinking PUP Maths workshop video resources:
  1. Workshop 1 (1hr) - Problems & Possibilities
  2. Workshop 2 (1hr) - Proof-making
  3. Workshop 3 (1hr) - Developing Notations
  4. Workshop 4 (1hr) - Being a Mathematician
  5. Workshop 5 (1hr) - Student collaboration
  6. Workshop 6 (1hr) - Deep understanding
• Mathematical Patterns Teachers Lab
• Cases versus Induction Theories of Mathematical Learning
• Unifix blocks, local purchase 1,000 blocks = A$150 - Purchase via Kesco
• Cuisenaire rods Purchase large set

1. Private Universe Project In Mathematics Written overview - Workshop components, activities, materials and timelines.
2. Private Universe Project Mathematics Workshops - Overview and links for teachers and administrators.
3. Courses for Teachers Learning Math - Data analysis, statistics and probability.
   1. Mathematics Workshop Towers Four High
   2. Video Workshop Sessions Workshop videos and transcripts
   3. Written Support Materials Private Universe Project in Maths

• Teacher Notes (PDF) PUP Workshop 1
• Mathematical thinking Workshop 1 (1hr)

An unprecedented long-term study followed the development of mathematical thinking in a randomly selected group of students for 12 years.

Research is showing that children begin their schooling with some surprising mathematics abilities.

Is there a way to keep this interest alive, and these abilities blossoming, all the way through high school and beyond?

In an overview of the study, we look at some of the conditions that made their math achievement possible. Go to this unit. ¹⁾

• Video http://www.learner.org/vod/vod_window.html?pid=1356
• Transcript - http://www.learner.org/workshops/pupmath/workshops/wk1trans.html
• Resources: Workshop1 (10 pages .pdf)
• 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

• Teacher Notes (PDF) PUP Workshop 2
• Proof-making Workshop 2 (1hr)

• On-Screen Math Activities - Towers Build all possible towers that are five (or four, or three, or n) cubes high by selecting from plastic cubes in two colors. Provide a convincing argument that all possible arrangements have been found.
  1. Focus Question - We have seen teachers presenting a number of carefully constructed arguments for finding all of the combinations of towers four-high, when selecting from two colors. Which arguments are convincing? Why?
  2. Focus Question - What are some similarities and/or differences in the mathematical reasoning by the teachers and the students that you observed?

Does it make sense to argue about mathematics? Can kids learn mathematics by debating and convincing each other? A long-term study shows that this can be an effective tool for learning.

Proof making is one of the key ideas in mathematics. Looking at teachers and students grappling with the same probability problem, we see how two kinds of proof — proof by cases and proof by induction — naturally grow out of the need to justify and convince others. Go to this unit. ²⁾

• Video - http://www.learner.org/vod/vod_window.html?pid=1358
• Transcript: http://www.learner.org/workshops/pupmath/workshops/wk2trans.html
• Resources: Workshop1 (10 pages .pdf)
• Support - Proof strategies (pages 4-6) - http://www.learner.org/workshops/pupmath/support/maher98.pdf
  1. Random building
  2. Recognise relationships (such as opposite coloured cubes in corresponding positions)
  3. Early checking characterised by recognition of 'duplicates'
  4. Patterns and relationships built on recognition of sets and opposites

• Teacher Notes (PDF) PUP Workshop 3
• Developing Notations Workshop 3 (1hr)

We learn how to foster and appreciate students' notations for their richness and creativity, and we look at some of the possibilities that early work on problems that engage students in creating notation systems might open up for students as they move on toward algebra.

With the support from the district, teachers take the first steps towards implementing a more thoughtful approach to mathematics. What effect will this have on math scores? Go to this unit. ³⁾

• Video http://www.learner.org/vod/vod_window.html?pid=1359
• Transcript - http://www.learner.org/workshops/pupmath/workshops/wk3trans.html
• Resources: Workshop1 (8 pages .pdf)

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

• Teacher Notes (PDF) PUP Workshop 4
• Problem solving & Being a mathematician (Fern) Workshop 4 (1hr)

What does a mathematician do? What does it mean to “think like a mathematician”? This program parallels what a mathematician does in real-life with the creative thinking of students.

Now in high school, the students take a fresh look at thinking they had done in the early years of the long-term study. What is the process through which students make connections between seemingly unrelated ideas in math? Go to this unit. ⁴⁾

• Video http://www.learner.org/vod/vod_window.html?pid=1360
• Resources: Workshop1 (8 pages .pdf)
• Towers of Hanoi Strategies 1

ADVANCED TOWERS ACTIVITIES: Choosing from two colours, RED and YELLOW, how many total combinations exist for towers 5 tall that each contain 4 red?

* Teacher Notes (PDF)PUP Workshop 5 * Teacher Notes Workshop 5 * Self-directed collaborative leaarning. Kindy, through HS & beyond. Workshop 5 (1hr)

One of the strands of the Rutgers long-term study was to find out how useful ideas spread through a community of learners and evolve over time. Here, the focus is in on the teacher's role in fostering thoughtful mathematics.

Two teachers, one highly experienced and one just beginning, are creating a community of learners. How does this re-define what it means “to teach”? Go to this unit. ⁵⁾

• Muybridge cat photos: http://www.learner.org/workshops/pupmath/support/catrun.pdf
• Video http://www.learner.org/vod/vod_window.html?pid=1361
• Resources: Workshop1 (14 pages .pdf)
• Pascal's Triangle handout (pdf)

1. Worksheet - Cuisenaire® Rod Activities

TRAINS ACTIVITY:

• Trains - Kindy Students arrange shorter rods end-to-end to match the length of a given longer rod.
• Trains Year 2 Students try to find all possible ways to arrange shorter rods end-to-end to match the length of a given rod. They count the number of possibilities and compare results.
• Towers (Fourth Grade) Students try to find out how many different towers four blocks high they can build by selecting from blocks of two colours.

Cuisenaire® Rods are used in the kindergarten and second-grade activities. The traditional set of rods that students use is designed in 1cm increments, starting with white as 1cm. If possible, use rods that are proportionally larger than the traditional set of rods (easier to differentiate relative sizes.

• There are 10 rods in each set.
• Each rod has a permanent colour name but has deliberately not been given a permanent number name. For example, the length of the dark green rod might be called 'four' in one activity and 'one' in another.
• “Trains” can be constructed by placing rods together.
• Trains may be multiples of the same rod,or a mix of different rods.

The children construct trains to aid them in finding solutions to the given problems.

PROBLEM:

1. How many different ways can we make dark green? (Kindergarten)
2. What are all the different ways that we can make a train equal to the length of one magenta rod? (note: Cuisenaire® refers to this rod as 'purple').

An extension problem:

1. Can you work out how to find the answer to a similar question for a rod of any length?

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.

On-Screen Math Activities

• Building Pascal’s Triangle. A researcher (Robert Speiser) probes Stephanie’s understanding of the relationship of the numbers in row n of Pascal’s Triangle to towers n high when choosing from two colours.
• World Series Problem. Two evenly matched teams play a series of games in which the first team to win four games wins the series. What is the probability that the series will be decided in
  • four games?
  • five games?
  • six games?
  • or seven games?
• “n choose r” Students derive the formula for determining the number of ways that a subset of r objects can be selected from a total of n objects.

Focus Question - How do the Pizza problems, Towers problems, and World Series problem relate to Pascal’s Triangle?

Refer to the Pascal’s Triangle Worksheet below:

• Can you model Pascal’s Triangle with block towers?
• How does the doubling rule work?
• Can you explain how and/or why the addition rule works?

Patterns provide much of the backbone and motivation in mathematical problem solving. This is true for children as well as for professional mathematicians. In the video you will see Stephanie's exploration of patterns leading to a comparison between Pascal’s Triangle and the Towers Problem.

In the early grades, children were seen on video establishing such patterns to help them determine how many different towers there are and to try to convince themselves and others when they believe they have found all of the possibilities. In the video clips of the children in the early grades you can see several patterns emerging.

Source Wikipedia - Pascal's Triangle (left image) and '' n choose k '' coefficient (right image)

The Co-efficient Relationship: A second useful application of Pascal's triangle is in the calculation of combinations. For example, the number of combinations of n things taken k at a time (called n choose k ) can be found. Rather than performing the calculation, one can simply look up the appropriate entry in the triangle. Provided we have the first row and the first entry in a row numbered 0, the answer will be located at entry 'k' in row 'n'.

PLAYERS: For example, suppose a basketball team has 10 players and wants to know how many ways there are of selecting 8.

• The answer is entry 8 in row 10, which is 45; that is, 10 choose 8 is 45
• HINT: Use an on-line Pascal's Triangle calculator

SOCKS: What if you have five socks - how many different ways can you choose two objects from a set of five objects?

• Find 'place 2 in row 5' = 10 ways. Because of this choosing property, the binomial coefficient [5:2] is usually read “five choose two.”
• The probability of choosing one particular combination of two socks is 1/10. For more about permutations and combinations, see the Dr. Math FAQ.

The Summing Relationship: Given that we know a row in Pascal’s Triangle, the entries in the next row can be found in the following manner:

• The first number in the new row will be 1.
• The second number in the new row will be the sum of the 1st and 2nd numbers in the previous row.
• The third number in the new row will be the sum of the 2nd and 3rd numbers in the previous row.
• And finally the last number in the new row will be 1.

The mathematics involved in Pascal's triangle forms an important starting point for the branch of mathematics known as combinatorics.

Find the pattern represented in the triangle.

Another relationship among the numbers in Pascal’s triangle fits with the children’s earlier discovery that as the height of the towers increase by 1 block, the number of different possible towers doubles.

If you sum the numbers in any row of Pascal’s Triangle, you will observe that those sums double (column on right-hand-side) as you progress down the rows

PROBLEMS

1. Can you model Pascal’s triangle with block towers.
2. How and why does the doubling rule work?
3. Can you explain how and why the addition rule works?

Students come up with a surprising array of strategies and representations to build their understanding of a real-life calculus problem — before they have ever taken calculus.

Students or professional mathematicians — both go through the same processes of “doing mathematics” when confronted with real-life problems. How can teachers help students uncover the beauty of mathematics?

More about cats and other problems

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

1. The Patterns in Mathematics teacher’s lab can be found here: www.learner.org/teacherslab/math/patterns
2. Access the A Private Universe online teacher’s lab here: www.learner.org/teacherslab/pup

STEAMpunks Science Workshop(s) video flyer - https://youtu.be/sCswReodSTk

• 'Are you convinced' workshop
  • Block 4 high both for teachers & for students including deep analysis
  • Organising by cases (trial + error) versus organising by induction (predicting the unknown)
  • http://www.learner.org/vod/vod_window.html?pid=1358
• From six minutes in - good overview of constructivism, problems, assessment - http://www.learner.org/vod/vod_window.html?pid=91
• Adapt lessons to address misconceptions and poor content, re-construct and re-frame conventional questions (as per Dan Meyer)
• Make an explicit commitment to discover student ideas and remedy at least one student misconception at the start of each topic
• Implement pre and post semester/unit quizzes
• Ask 'what I used to think' and 'what I think now' at the end of each session

You have completed a series of six video workshops in which you observed teachers and stu- dents of all ages working on a variety of mathematical problems, and have worked on the same problems yourself. Many of these investigations may have looked different from the mathematics often seen in classrooms.

• Resource Workshop 7 is designed help teachers design ADVANCED activities.

Read more ...

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 judgement, guesstimate, stereotyping, profiling, or common sense. ⁷⁾

• The original Workshop descriptions and video summaries are here

• The workshops are designed to model classroom activities and providing an interactive forum for teachers, administrators, and other interested adults to explore issues about teaching and learning mathematics.

1. What does it mean to be a teacher of mathematics?
2. What is the connection between learning and teaching?

Read more...

Due to copyright issues, not all of the articles for these workshops are available online. However, the sources of all the publications are listed where possible. Links to PDF files are listed for those available for download. ⁹⁾

For FREE ACCESS to many resources, check out The Internet Archive - a 501©(3) non-profit, who provide free access to researchers, historians, scholars, the print disabled, and the general public. Our mission is to provide Universal Access to All Knowledge.

The Universal History of Numbers: From Prehistory to the Invention of the Computer - A riveting history of counting and calculating from the time of the cave dwellers to the late twentieth century, The Universal History of Numbers is the first complete account of the invention and evolution of numbers the world over. As different cultures around the globe struggled with problems of harvests, constructing buildings, educating their citizens, and exploring the wonders of science, each civilization created its own unique and wonderful

• Available from The Internet Archive
• For old web content such as old, 1996 and later DSE web content, visit the Internet Archive WayBackMachine

1. All
2. Cups, Bowls & Plates
3. Shirts & Pants
4. Towers of Four
5. Tower Of Hanoi