{{:2018:mathematics:mathematical-thinking-banner-790x50.jpg|Mathematical Thinking Banner}}
====== TEACHER-ONLY WORKSHOPS - Mathematical Thinking ======
=== Private Universe in Mathematics - Harvard-Smithsonian PD Videos & Activities ===
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.

=== Intended Outcomes - Research Philosophy & Student Perceptions (7min) === 
{{ youtube>Vgbo91z1u6U?640x360 |Research Philosophy, Process and Intended Outcomes (7min)}}

=== Video 1: Research, Philosophy, Process + Outcomes + Perceptions (7min) ===
  * IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:[[http://viewpure.com/ECg6jNWIqBs|Project Philosophy and Intended Outcomes(7min)]]

=== TEACHER RESOURCES - The Private Universe in Mathematics ===
TEACHER WORKSHEETS: The [[:2018:mathematics:teacher-workshops:answers-to-problems:home|ANSWERS_TO PROBLEMS]]  web page provides a simpler teaching guide, with explanations provided on an activity-by-activity basis.

STUDENT WORKSHEETS: The [[:2018:mathematics:student-problems:home|STUDENT-PROBLEMS]] web page contains the list of activities and associated instructions that can be handed out to students.

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=== MATHEMATICAL THINKING WORKSHOP - FROM KINDY TO HIGH SCHOOL - A VIDEO SUMMARY ===
**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**

{{ youtube>oaO9JL-hbGo?640x360 |2J Get Started with Mathematical Thinking (3min)}}

=== Video 2. 2J Getting Started with Mathematical Thinking (3min) ===
  * IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:[[http://viewpure.com/oaO9JL-hbGo|2J Get Started with Mathematical Thinking (3min)]]

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{{ youtube>8yXVmcx1rYQ?640x360 |Teacher Feedback - 2D & 2S First Session -  Mathematical Thinking (3min)}}

=== Video 3. Teacher Feedback - 2D & 2S First Session -  Mathematical Thinking (4min) ===
  * IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:[[http://viewpure.com/8yXVmcx1rYQ|2D + 2S Get Started with Mathematical Thinking (3min)]]

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=== Long-term Research Strategy & Formal Maths Instruction - Teacher Perceptions ===
{{ youtube>u4MIr7egrbY?640x360 |Long-term Outcomes: Teacher Perceptions (5min)}}

=== Video 4. Long-term Outcomes -Teacher Perceptions (5min) ===
  * IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:[[http://viewpure.com/u4MIr7egrbY|Outcomes: Teacher Perceptions  (3min)]]

What do teachers say: Is there any [[https://youtu.be/u4MIr7egrbY|evidence]] that these activities promote improvement in Mathematical Thinking and/or cognitive development for K-12 students?

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=== Formulating general solutions before being introduced the to the formal rules ===
{{ youtube>JjW2SJ8fyXs?640x360 |Solving problems before being introduced to the formal rules (7min)}}

=== Video 5. Formulating general solutions before being introduced to the formal rules (7min) ===
  * IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:[[http://viewpure.com/JjW2SJ8fyXs|Problem Solving (7min)]]

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

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=== 2D Mathematical Thinking ===
{{ youtube>OeMVf7kJNYs?640x360 |2D Mathematical Thinking}}

=== Video 6. 2D Mathematical Thinking ===
  * IF UNABLE TO ACCESS YOUTUBE VIDEO, TRY:[[http://viewpure.com/OeMVf7kJNYs|2D Mathematical Thinking]]

====== Teaching Practice - Video Feedback: ======
[[https://www.teachingchannel.org/videos/improve-teaching-with-video|{{ :teaching:teaching-practice:video-teaching-practice-640x360.jpg?640x360 |Using Video To Improve Teaching Practice}}]]

** Video 1. [[https://www.teachingchannel.org/videos/improve-teaching-with-video|Using Video To Improve Teaching Practice]] **


====== MATHEMATICAL THINKING WORKSHOP RESOURCES ======

=== There are simplified teacher guides for a number example activities ===
  * [[:2018:mathematics:teacher-workshops:trains:home|TRAINS]]
  * [[:2018:mathematics:teacher-workshops:shirts-and-pants:home|SHIRTS-AND-PANTS]]
  * [[:2018:mathematics:teacher-workshops:towers-four-high:home|TOWERS-FOUR-HIGH]]
  * [[:2018:mathematics:teacher-workshops:cups-bowls-plates:home|CUPS-BOWLS-PLATES]]
  * [[:2018:mathematics:teacher-workshops:tower-of-hanoi:home|TOWER-OF-HANOI]]
  * [[:2018:mathematics:teacher-workshops:world-series:home|WORLD-SERIES]]

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=== Some videos that were prepared for STEAMpunks 2017 Workshops ===
REMINDER: **Teachers should not need to suggest strategies or provide answers**
  * [[https://youtu.be/EYvos73Ux4k|STEAMpunks 2017 Workshops - Introduction]]
  * [[https://youtu.be/Vgbo91z1u6U|STEAMpunks 2017 Workshops - Outcomes]]
  * [[https://youtu.be/xdlgwVEDmB8|STEAMpunks 2017 Workshop Part 1]]
  * [[https://youtu.be/ECg6jNWIqBs|STEAMpunks 2017 Workshop Part 2]]
  * [[https://youtu.be/WJPDjOWmlZE|STEAMpunks 2017 Maths Workshop - Shirts & Pants]]
  * [[http://www.learner.org/vod/vod_window.html?pid=1695|Maths Case studies - K-6 Teachers]]

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====== THE ORIGINAL HARVARD-SMITHSONIAN WORKSHOP RESOURCES ======
  * PUP Workshop overview [[http://www.learner.org/workshops/pupmath/workshops/descriptions.html| - Descriptions & Structure]]
  * Mathematical thinking [[http://www.learner.org/resources/series120.html|PUP Maths workshop video resources:]]
    - [[http://www.learner.org/vod/vod_window.html?pid=1356|Workshop 1 (1hr) ]] - Problems & Possibilities 
    - [[http://www.learner.org/vod/vod_window.html?pid=1358|Workshop 2 (1hr) ]] - Proof-making
    - [[http://www.learner.org/vod/vod_window.html?pid=1359|Workshop 3 (1hr) ]] - Developing Notations
    - [[http://www.learner.org/vod/vod_window.html?pid=1360|Workshop 4 (1hr) ]] - Being a Mathematician
    - [[http://www.learner.org/vod/vod_window.html?pid=1361|Workshop 5 (1hr) ]] - Student collaboration
    - [[http://www.learner.org/vod/vod_window.html?pid=1362|Workshop 6 (1hr) ]] - Deep understanding
  * Mathematical Patterns [[http://www.learner.org/teacherslab/math/patterns/|Teachers Lab]]
  * Cases versus Induction [[http://www.learner.org/workshops/pupmath/support/mahermartino96.pdf|Theories of Mathematical Learning]]
  * Unifix blocks, local purchase 1,000 blocks = A$150 - [[http://www.kesco.com.au/catalogue?catalogue=KESCO&category=KE-UNIFIX-MATHS-CUBES|Purchase via Kesco]]
  * Cuisenaire rods [[http://www.teachersuperstore.com.au/numicon-cuisenaire-rods-large-set|Purchase large set]]

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=== Research References ===
  -  Private Universe Project In Mathematics [[http://www.learner.org/workshops/pupmath/support/pupmintro.pdf|Written overview]] - Workshop components, activities, materials and timelines.
  - Private Universe Project [[http://www.learner.org/workshops/pupmath/about/overview.html|Mathematics Workshops]] - Overview and links for teachers and administrators.
  - Courses for Teachers [[http://www.learner.org/resources/series158.html#|Learning Math]] - Data analysis, statistics and probability.
    - Mathematics Workshop [[http://www.learner.org/vod/vod_window.html?pid=1356|Towers Four High]]
    - Video Workshop Sessions [[http://www.learner.org/workshops/pupmath/workshops/descriptions.html#|Workshop videos and transcripts]]
    - Written Support Materials [[http://www.learner.org/workshops/pupmath/support/index.html|Private Universe Project in Maths]]

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=== Workshop 1. Following Children's Ideas in Mathematics ===
  * Teacher Notes (PDF) [[http://www.learner.org/workshops/pupmath/support/pupm1.pdf|PUP Workshop 1]]
  * Mathematical thinking [[http://www.learner.org/vod/vod_window.html?pid=1356|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. [[http://www.learner.org/workshops/pupmath/workshops/wk1trans.html|Go to this unit]]. ((VOD WS1 (1hr) http://www.learner.org/vod/vod_window.html?pid=1356 ))
  * Video http://www.learner.org/vod/vod_window.html?pid=1356
  * Transcript - http://www.learner.org/workshops/pupmath/workshops/wk1trans.html
  * Resources: [[http://www.learner.org/workshops/pupmath/support/pupm1.pdf|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

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=== Workshop 2. Are You Convinced? ===
  * Teacher Notes (PDF) [[http://www.learner.org/workshops/pupmath/support/pupm2.pdf|PUP Workshop 2]]
  * Proof-making [[http://www.learner.org/vod/vod_window.html?pid=1358|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.
    - 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?
    - 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. [[http://www.learner.org/workshops/pupmath/workshops/wk2trans.html|Go to this unit]]. ((VOD WS2 (1hr) http://www.learner.org/vod/vod_window.html?pid=1358 ))
  * Video - http://www.learner.org/vod/vod_window.html?pid=1358
  * Transcript: http://www.learner.org/workshops/pupmath/workshops/wk2trans.html
  * Resources: [[http://www.learner.org/workshops/pupmath/support/pupm2.pdf|Workshop1 (10 pages .pdf)]]
  * Support - Proof strategies (pages 4-6) - http://www.learner.org/workshops/pupmath/support/maher98.pdf
    - Random building
    - Recognise relationships (such as opposite coloured cubes in corresponding positions)
    - Early checking characterised by recognition of 'duplicates'
    - Patterns and relationships built on recognition of sets and opposites

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=== Workshop 3. Inventing Notations ===
  * Teacher Notes (PDF) [[http://www.learner.org/workshops/pupmath/support/pupm3.pdf|PUP Workshop 3]]
  * Developing Notations [[http://www.learner.org/vod/vod_window.html?pid=1359|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? [[http://www.learner.org/workshops/pupmath/workshops/wk3trans.html|Go to this unit]]. ((VOD WS1 (1hr) http://www.learner.org/vod/vod_window.html?pid=1359 ))
  * Video http://www.learner.org/vod/vod_window.html?pid=1359
  * Transcript - http://www.learner.org/workshops/pupmath/workshops/wk3trans.html
  * Resources: [[http://www.learner.org/workshops/pupmath/support/pupm3.pdf|Workshop1 (8 pages .pdf)]]

  * Notation examples
    - Worksheet - {{ :2018:mathematics:pup-mathematics:ws1-02.pdf |Shirts & Cups - More Strategies}}
    - Worksheet - {{ :2018:mathematics:pup-mathematics:ws2-01.pdf |Towers - Patterns & Cases}}
    - Worksheet - {{ :2018:mathematics:pup-mathematics:ws2-02.pdf |Towers - Doubling & Problems}}
    - Worksheet - {{ :2018:mathematics:pup-mathematics:ws2-03.pdf |Towers - Example Student Strategy}}

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=== Workshop 4. Thinking Like a Mathematician ===
  * Teacher Notes (PDF) [[http://www.learner.org/workshops/pupmath/support/pupm4.pdf|PUP Workshop 4]]
  * Problem solving & Being a mathematician (Fern) [[http://www.learner.org/vod/vod_window.html?pid=1360|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?
[[http://www.learner.org/workshops/pupmath/workshops/wk4trans.html|Go to this unit]]. ((VOD WS1 (1hr) http://www.learner.org/vod/vod_window.html?pid=1360 ))
  * Video http://www.learner.org/vod/vod_window.html?pid=1360
  * Resources: [[http://www.learner.org/workshops/pupmath/support/pupm4.pdf|Workshop1 (8 pages .pdf)]]
  * Towers of Hanoi {{:2018:mathematics:pup-mathematics:ws4-01.pdf|Strategies 1}}

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

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=== Workshop 5. Building on Useful Ideas ===
 * Teacher Notes (PDF)[[http://www.learner.org/workshops/pupmath/support/pupm5.pdf|PUP Workshop 5]]
 * Teacher Notes {{ :2018:mathematics:pup-mathematics:ws5-01.pdf |Workshop 5}}
 * Self-directed collaborative leaarning. Kindy, through HS & beyond. [[http://www.learner.org/vod/vod_window.html?pid=1361|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"?
[[http://www.learner.org/workshops/pupmath/workshops/wk5trans.html|Go to this unit]]. ((VOD WS1 (1hr) http://www.learner.org/vod/vod_window.html?pid=1361 ))
  * Muybridge cat photos:  http://www.learner.org/workshops/pupmath/support/catrun.pdf
  * Video http://www.learner.org/vod/vod_window.html?pid=1361
  * Resources: [[http://www.learner.org/workshops/pupmath/support/pupm4.pdf|Workshop1 (14 pages .pdf)]]
  * Pascal's Triangle [[http://www.learner.org/workshops/pupmath/support/index.html|handout (pdf)]]
  - Worksheet - {{ :2018:mathematics:pup-mathematics:ws5-01.pdf |Cuisenaire® Rod Activities}}

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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.

{{ :2018:mathematics:combinatorics:cuisinaire-blocks-01.png |Cuisenaire blocks}}

  * 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:
  - How many different ways can we make dark green? (Kindergarten)
  - 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:
  - Can you work out how to find the answer to a similar question for a rod of any length?

=== Combinations versus Permutations ===
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:
  - vanilla
  - chocolate
  - pistachio
  - boysenberry
  - cherry
  - 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.

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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.

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=== 5.4 Pascal’s Triangle ===
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.

{{ :2018:mathematics:teacher-answers:pascals-triangle-animated-200.gif|pascals triangle animated}}
^Colours ^Height of the tower = n ^^^^^^
|colour1 |  0 |      1 |      2 |      3 |  ...  |  n |
|colour2 |  n |  n - 1 |  n - 2 |  n - 3 |  ...  |  0 |
^Towers  ^Height of the tower     ^^^^^^
|1-tall  |  1 |        |        |        |  ...  |  1 |
|2-tall  |  1 |      2 |      3 |        |  ...  |  1 |
|3-tall  |  1 |      3 |      6 |        |  ...  |  1 |
|4-tall  |  1 |      4 |      4 |      4 |  ...  |  1 |

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{{:2018:mathematics:teacher-answers:pascals-triangle-300.png |pascals triangle}}{{ :2018:mathematics:teacher-answers:pascals-triangle-coefficient.png|pascals triangle coefficient}}
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Source [[https://en.wikipedia.org/wiki/Pascal%27s_triangle|Wikipedia - Pascal's Triangle (left image) and '' n choose k '' coefficient (right image)]]
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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 [[https://www.dcode.fr/pascal-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 [[http://mathforum.org/dr.math/faq/faq.comb.perm.html|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.

| 1 + 1              | =  2              | = 2<sup>1</sup>  |  =  2|
| 1 + 2 + 1          | =  2 x 2          | = 2<sup>2</sup>  |  =  4|
| 1 + 3 + 3 + 1      | =  2 x 2 x 2      | = 2<sup>3</sup>  |  =  8|
| 1 + 4 + 6 + 4  + 1 | =  2 x 2 x 2 x 2  | = 2<sup>4</sup>  |  = 16|

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

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=== Workshop 6. Possibilities of Real-Life Problems ===
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 |

MATERIALS: At least two copies of [[http://www.learner.org/workshops/pupmath/support/catrun.pdf|Muybridge's cat photographs]] on 11x17 paper and on transparencies, metric rulers (clear plastic ones work best), graph paper, a calculator (graphing calculator if possible), and pens or markers for preparing solutions to the problems. You will also want to have an overhead projector, blank transparencies and pens for sharing solutions. [[http://www.learner.org/workshops/pupmath/workshops/wk6trans.html|Go to this unit]]. ((VOD WS1 (1hr) http://www.learner.org/vod/vod_window.html?pid=1362 ))
  * Video http://www.learner.org/vod/vod_window.html?pid=1362
  * Resources: [[http://www.learner.org/workshops/pupmath/support/pupm6.pdf|Workshop1 (6 pages .pdf)]]

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=== References ===
Find out more about the Harvard-Smithsonian Center for Astrophysics here: www.cfa.harvard.edu
  - Learn more about Annenberg Media and browse the resources and workshops they offer to teachers: www.learner.org
  - To watch //A Private Universe// video: www.learner.org/resources/series28.html
  - To watch //Minds of Our Own// video: www.learner.org/resources/series26.html
  - The //Private Universe in science// project videos: www.learner.org/catalog/extras/puptwsup.html
  - To watch the //Private Universe in Science// workshop videos: www.learner.org/resources/series29.html
  - The //Private Universe in mathematics// project videos, see: www.learner.org/resources/series120.html

  - The Patterns in Mathematics teacher’s lab can be found here: www.learner.org/teacherslab/math/patterns
  - 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

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====== GUIDING QUESTIONS FOR TEACHERS (REFLECTION) ======
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 [[http://www.learner.org/workshops/pupmath/support/pupm7.pdf|Workshop 7]] is designed help teachers design //ADVANCED// activities.

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**Philosophy**
  - What similarities and/or connections did you notice between the parts of the video involving the pre-K children and those focusing on students in elementary and secondary grades?
  - What similarities and differences for solving the Towers problems did you observe across the various segments: (a) for the children and the adults and (b) for the children over time?
  - What issues may have come up for the teacher/researchers working with the children?

**Students**
  - How do students use notation as a problem-solving tool?
  - When and how do we see students connecting meaning to the symbols that they use?
  - How might the students’ early building be reflected in the work they did in later investigations?
    - Is it a direct application of ideas built in early grades?
    - Is it in the way they reason and demand careful explanation?
    - Is it in the way they rethink, reformulate, and then rebuild what they have done before?
    - Is it in the way they listen to each other?
  - What might you say about the “flow of ideas” as the students study and work together?

**Teachers**
  - What part does the teacher/researcher play in the building of students’ mathematical ideas?
  - How might the task design contribute to:
    - the building of ideas by learners?
    - the teacher’s role in fostering the building and sharing of ideas?
  - What ideas and issues raised by the video relate to local, regional, or national standards?
  - What implications do you see for your own teaching? 

**Think about & discuss:**
  - something you will change in the way you introduce the next unit you are going to teach. 
  - what you will do to get your students to work together, to share their solutions, and to justify those solutions to one another.
  - how you will work together with one of your colleagues who has also participated in the workshop. 
  - how you will share your work with other colleagues and your principal?

++++

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=== WHAT IS PROOF? ===
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. ((https://en.wikipedia.org/wiki/Heuristic))

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====== TEACHER WORKSHOP REFERENCES ======
=== Source: The Original Harvard-Smithsonian Teacher Workshop Videos ===
  * The original Workshop descriptions and video summaries are [[http://www.learner.org/workshops/pupmath/support/pupmintro.pdf|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.
  - What does it mean to be a teacher of mathematics? 
  - What is the connection between learning and teaching?

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Central to each research session is a 60-minute video show children and teachers engaged in authentic mathematical activity and discussion.

These  episodes  come  from  a  variety  of sources in diverse school communities and across grade levels from pre-kindergarten through grade 12. The episodes and accompanying narratives in each videotape focus on: 
  - students and teachers actively engaged in doing mathematics;
  - conditions that encourage meaningful mathematical activity; and
  - implications for learning, teaching, and assessment.

These materials and activities have been developed in long-term research programs about mathematical thinking that share certain presuppositions about learning and teaching.

Key to this perspective is that knowledge and competence develop most effectively in situations where students, frequently working with others, work on challenging problems, discuss various strategies, argue about conflicting ideas, and regularly present justifications for their solutions to each other and to the entire class.

The role of the teacher includes selecting and posing the problems, then questioning, listening, and facilitating class-room discourse, usually without direct procedural instruction.

Students frequently worked together in small groups on meaningful problem activities. One classroom of children was followed from first through third grades by regularly videotaping small-group problem-solving sessions, whole class discussions, and individual task-based interviews.

From 1991 to the present, the research team has continued documenting and studying the thinking of a focus group of these children through grade 12. The videotapes from these later years document the students as they participated in problem-based activities developed by the university researchers in classroom lessons, after-school sessions, a two-week summer institute, and individual and small-group task-based interviews.

In the workshop videotapes, participants will see some of the same children solving mathematical problems at various grade levels over the years. In addition, the activities they engage in have  been used in other communities and at other grade levels.

Workshop participants will explore particular questions about learning and teaching mathematics based on the shared experience of watching the videotapes.
Key questions are:
  - How do children (and adults) learn mathematics?
  - How do children (and adults) learn to communicate about mathematics and to explain
and justify solutions to problems? 
  - What conditions—environments, activities, and interactions—are most helpful in facilitating this development?

About the Workshop
  - What does it mean to be a teacher of mathematics? 
  - What is the connection between learning and teaching?

Each videotape includes episodes of children engaged in mathematical problem-solving.

The goal is for participants to become able to recognize what is mathematical in students’ activity by attending very closely to what they do and say. As they observe, study, and discuss what they see on the tape from the perspective of the questions listed above, participants will gain insights about learning and teaching.

Participants are encouraged to select and use appropriate problems with their own students and read further about the learning and teaching of these ideas.
  * Source ((Workshop Intro. http://www.learner.org/workshops/pupmath/support/pupmintro.pdf))

**Example excerpts from video content...**

NARRATOR: During the summer, Arthur begins the project by engaging 30 teachers in a two-week professional development workshop. Arthur has defined clear goals for this initial work...

CAROLYN MAHER: Okay, let's get a piece of a paper and write down what you're saying and see if you all agree. I think Jeff hasn't been with us for a while and he doesn't know what we're talking about. But let's take one at a time.

CAROLYN MAHER: There are a couple of ways of approaching the problem. There's the notion of you can start with towers one tall, each colour. And once you have the tower - let's say red and blue - you start with a red tower and now you're making a two tall tower. Well, you can either put a red on the top or a blue on the top. So, from that one, you get two. Likewise for the tower with the blue on the bottom. You could either put a red on the top or a blue on the top. From that one, two.

So, from the two towers - one tall - you generate four towers two tall. Each of those towers you can choose to either put a red on the top or a blue on the top, red on the top or blue on the top. From the eight, two more, two more, two more, two more. You could begin to generalize this idea and, later on, you can come up with a nice way of expressing your justification, which leads you to a kind of way of reasoning that we call inductive reasoning.

NARRATOR: For about half an hour, the students shared their different approaches. When it was Stephanie's turn, she presented a version of Proof by Cases to justify her solution for the number of towers three high.

NARRATOR: We have seen fourth graders naturally develop two mathematically sound methods of proof-making:

  - By induction; building taller towers from shorter towers, one at a time, and using known results about the shorter towers to derive results about the taller ones;
  - By cases; organising towers of a given size into different cases which could be studied separately.

What are some of the similarities and/or differences in the mathematical reasoning by teachers and students that you may have observed in this program?

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=== Related Resources ===
Due to copyright issues, not all of the articles for these workshops are available online. However, the [[http://www.learner.org/workshops/pupmath/support/index.html|sources of all the publications are listed where possible]]. Links to PDF files are listed for those available for download. ((http://www.learner.org/workshops/pupmath/support/index.html ))

For FREE ACCESS to many resources, check out [[https://archive.org/about/|The Internet Archive]] - a 501(c)(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 [[https://archive.org/about/|The Internet Archive]]
  * For old web content [[https://web.archive.org/web/19970628173208/http://www.dse.nsw.edu.au:80/index.html|such as old, 1996 and later DSE web content]], visit the [[https://web.archive.org/|Internet Archive WayBackMachine]]

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=== Videos ===
  - **[[http://steampunks.org/doku.php?id=workshop:maths:home|All]]**
  - **[[http://steampunks.org/doku.php?id=workshop:maths:shirts:home|Cups, Bowls & Plates]]**
  - **[[http://steampunks.org/doku.php?id=workshop:maths:shirts:home|Shirts & Pants]]**
  - **[[http://steampunks.org/doku.php?id=workshop:maths:towers:home|Towers of Four]]**
  - **[[http://steampunks.org/doku.php?id=workshop:maths:hanoi:home|Tower Of Hanoi]]**

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=== References ===