Problem Posing and Problem Solving: What do non-routine, ill-posed mathematics projects look like?

Introductions
Course overview

Course philosophy
Assignments
Spreadsheets

King's chessboard: How much rice is that?

Differentiating Instruction with Marbles: Is This Algebra or What? (Available on the Blackboard within the Course Documents; if you have trouble accessing it, email me. I can send it as an attachment.)

Teachers Explore Linear and Exponential Growth: Spreadsheets as Cognitive Tools (Available on the Blackboard within the Course Documents; if you have trouble accessing it, email me. I can send it as an attachment.)

Reflective pods: REFLECTION #1

a.  Group name

b.  What did you learn from the article Four Kids with Marbles (Differentiating Instruction with Marbles: Is This Algebra or What?)

Process Standards: p. 52 – 67
Problem Solving: p. 334 - 342

• Evidence: Why? How do you know? What does that mean? Can you explain more? Can you provide some details/examples?
• Self-regulated Learning
• Metacognitive reflection
• reflection in action and after action
• scaffolding
• open-ended problems
• multiple representations
• multiple strategies
• differentiating instruction

06/07

Teachers Explore Linear and Exponential Growth: Spreadsheets as Cognitive Tool (available in the Course Documents on the Blackboard)

GENERAL

• teaching for understanding

• formative and summative assessment

• cognitive tools

• modeling: linear, exponential, quadratic

• multiple representations <=> multiple intelligences and learning modes

• it is ok to make a mistake - we are all learners

• changing "a mile wide, an inch deep" curriculum

• inquiry environment

SPECIFIC

• chart and visualizing

• technology integration

• examples: double, triple, ..., half, gossip, chain letters,

• Exponential decay: paper example, radioactive decay

• Comparing linear and exponential growth - slopes

• meaning of large numbers

• stories

• humor

• zooming/changing the domain

• open-ended problems/stories

Developing first problem set

Spreadsheets mathematics; Grade book

ONLINE

Grade sheets and self-evaluations

Problem set due by MIDNIGHT

Next problem set

• Which one holds more?
• Different ways of finding volume
• Developing volume formula
  • rectangular box: How many cubes in my box?
    • fill the box
    • cover the bottom -
    • not enough to cover the bottom - only as much as the longest side
  • cylinder: How many cubes in my jar?
• Concepts that we know:
  • measurement
  • area
  • estimation

Virtual Manipulatives: http://nlvm.usu.edu/en/nav/vlibrary.html

http://matti.usu.edu/nlvm/nav/frames_asid_273_g_3_t_4.html

• K-2 students explore similarities and differences among two-dimensional shapes.
• 3-5 identify characteristics of various shapes/quadrilaterals.
• 6-8 examine and make generalizations about properties of particular shapes/quadrilaterals.
• 9-12 develop logical arguments to justify conjectures about particular polygons

Rhombus problem

PS2 - sharing ideas

HOMEWORK:

• Reasoning and proof: p. 342 - 348 
• Bring paper copy of your PS2

REFLECTION #5.
In preparation for the discussion about Reasoning and Proof Standard, in your first part of the reflection, reflect on your existing knowledge and experience with reasoning and proof. Your second part of reflection should be written based on careful reading (textbook, p. 342-348). Instead of just summarizing what you read, design a higher-order level questions to ask your classmates. Describe why you think that question will help us learn more about Reasoning and Proof Standard.

I will expect every group to facilitate a class discussion guided by their questions.

Multiple Representations: Open-ended, rich problems 
Reasoning and Proof: Inductive and deductive reasoning; What constitutes a mathematical proof at different levels of understanding?

READINGS DISCUSSION:  Reasoning and proof: p. 342 - 348 

PROBLEM SET 2:
- feedback from colleagues
- naming problems to help scaffolding

DYNAMIC GEOMETRY
- drag-function
- triangle properties

http://matti.usu.edu/nlvm/nav/frames_asid_273_g_3_t_4.html

PS2 HINT:
(1) Technology integration is your choice, more open this time.
(2) Technology integration can be part of the challenge or any other problem; just make sure it is interactive and describe how it is interactive. If you are using a website, make sure that you write your problem and explanation in such a way that I do not have to go to a website to check it. I usually do go and check it, but I still expect good amount of details in your paper.

(NOTE: If you are not familiar with the term “zone of proximal development” please find out; get into agreement with your group about this concept, before proceeding to discuss the reflective question.)

1. Geometry Concept Map
2. Geometric Transformations
3. GSP: Triangle Properties

• Sum of angles in a triangle
• Sum of exterior angles of a triangle
• Perpendicular bisectors
• Medians

4. PS3: Congruence OR Similarity OR Pythagorean Theorem
     PS4: Concepts subsumed under the Pascal's Triangle - I will provide more information after you finish PS3

Reflection #7:
Each group member will write an essay (250-300 words) on the assigned topic and post it for other members to be able to read it. It is expected that at least two resources be used and referenced. Your essay should include the following:
a. Define your topic.
b. What are some examples of real-life applications?
Your second reflection: reflect on your group members’ essays.  Topics are assigned in the order listed below. The order of the group members should correspond with alphabetical order by last name.

Triangle Sum: interior angle, measuring an angle, sum of  the angles in a triangle

Exterior Angles: exterior angle, measuring an angle, exterior angles property

Perpendicular Bisectors:
Angle Bisectors -
Medians -
Altitudes -

PS#3 has to include interactive dynamic geometry

Reflection #8: Integrating mathematics and art

How can integrating art with mathematics increase students’ understanding of mathematics concepts? Provide some specific examples of mathematical concepts explored through art activities. Please use and reference resources.  

ONLINE

Working on a problem set #3: Congruence OR Similarity OR Pythagorean Theorem

Reflection #9: 1st entry: Select one main characteristic of each process standard (in your opinion) and right a couple of sentences about it. Make sure to select a different characteristics than people that posted their first entry before you.
2nd entry: For your second entry, pick standard in order according to your order in the group: Problem solving, Reasoning and Proof, Representations and Connections. Look what others wrote about that standard and based on that and your own understanding of that standard write an essay about it (approximately 350 words).

• Tossing a coin: H or T
• Tossing 2, 3, ..; assume that order does matter
• Developing Pascal's triangle
• In how many different ways can you order three color: Amber, Blue and Coral
• How many different outfits can you have if you have 3 t-shirts, 4 pairs of jeans and 2 vests
   

1. Permutations of unlike objects

(a) In how many ways can you order a group of 5 students?

 P(5) = 5x4x3x2x1= 5! (Factorial symbol is just a symbol used to simplify our writing.

 So,

 P(5) = 5x4x3x2x1= 120

 (b) In how many ways can you order a group of 5 students?

 Permutations are arrangements in which order is important.

2. Permutations of a subset of a set (objects in a set)

In how many ways can you select a group of 4 students in a class of 20 students?

 ₂₀P₄ = 20x19x18x17 = 20!/16! = 20!/(20-4)!   

 _nP_k = n!/(n-k)!

 What is _nP_n ?

_nP_n = n!/(n-n)!= n!/0! = n!

(Because mathematicians agreed that 0! = 1; this is one of the strange examples that does not make much sense but it makes everything work)

 So,

0! = 1

1! = 1

2! = 2x1 = 2

3! = 3x2x1 = 6

…

n! = nx(n-1)x(n-2)x…x3x2x1 = …

3. Permutations involving like objects

(a) You have 2 t-shirts and one pair of jeans. In how many different ways can you order these 3 objects? (Think of these objects on one shelf, next to each other.)

 P(3) =3!           

(b) If I had 3 jeans and 7 t-shirt that I wanted to lay them next to each other on a long shelf, I would want to permute 17 objects in all possible ways. Therefore,

P(10) = 10! = 10x9x8x7x…x3x2x1 = 3 628 800

 (c) You have 2 t-shirts of the same color and one pair of jeans. In how many ways can you order these 3 objects?

                 3!/2!      

                    In general,  n!/k! (where k is set of objects in which order does not matter)

4. Combinations = arrangements of objects in which the order makes no difference

(a) In how many ways can you select a pair out of 3 students?

   Permutations               Combinations

AB                               AB

BA

AC                               AC

CA

BC                               BC

CB

 3x2 is all possible pairs

Since order does not matter we have to divide by 2 and we can write that as (3x2)/2 = 3!/2!

 (b) One triple out of 5 students?

 5x4x3 if the order does matter (Just for fun, I can write that as 5!/2!)

 BUT, order does not matter, so I have to divide by 3! (because there are 3! possible ways in which I can order three students)

 (5!/2!) 3! = 5!/3!2!

 (b) How many groups of 4 students can you have if your class has 20 students? (Notice, order does not matter; so, these are called combinations “of 20 elements in groups of 4)

                  If order matters, we know we are talking about permutations (see problem #2)

 ₂₀P₄ = 20!/(20-4)! = 20!/16! =

 = 20x19x18x17x16x15x14x…x3x2x1/16x15x14x…x3x2x1

 = 20x19x18x17x16 = 1 860 480

 BUT, order does not matter, so we have to divide with 4! (which also can be written as P(4) or  ₄P₄)

SO, we have

 ₂₀C₄ = ₂₀P₄ / ₄P₄ = [20!/(20-4)!]/4! = 20!/4!16! = 1 860 480/4! = 1 860 480/24 = 77520

 (c) To generalize, If we have n people for a party and k tables, in how many different ways can we sit these people?

 _nC_k = _nP_k / _kP_k = [n!/(n-k)!]/k! = n!/k!(n-k)! (combinations of n elements in groups of k; REMEMBER - combinations mean that order does not matter)

5. (a) Calculate

₅C₀ =

₅C₁ =

₅C₂ =

₅C₃ =

₅C₄ =

₅C₅ =

 (b)  Is there a row in Pascal’s triangle that matches these numbers?

(c)   Remember, these numbers are coefficient of the expression (x + y)⁵. How can you write that down? Think different representations of this expression using what you know about combinations (case (a)) and what you know about Pascal’s triangle (case (b)). 

6. TRY THIS: How many (a) different gender (b) same gender, pairs can you have if your class has 12 girls and 8 boys?

7. TRY THIS:
   Observe a row in Pascal's Triangle. What happens if we first subtract and than add and continue alternating like that in a horizontal direction? What if we first add and than subtract and continue alternating like that in a horizontal direction? Can you generalize one of these cases for any row? Can you prove that?

       (Problem #7 is a challenging problem that does not have an obvious real-life connection for you at this point. It is a significant property of combinations that is represented with Pascal's triangle and it can be used in probability calculations.  If you decide to use this problem as your challenging problems, I will not take points of for the fact that it is not real-life related).

  ANNOUNCEMENT:

(1) Problem set 4. Select either permutations or combinations for your concept in the problem set 4. Combinations are a little bit more complex than permutations, but it might be easier to scaffold them. 

 Reflection #10:

    Entry 1. Share one thing from your self-evaluation with your group.
    Entry 2. Reflect on your group writings.
                

Communications: Language of Mathematics

• Development of representations for the concept permutations/combinations - all problems focusing on the same concept; scaffolding
• Where to put vocabulary - after the problems are designed and solved, skim over and decide if the place where you want your vocabulary is appropriate
• Metacognitive reflection - compare your metacognitive thinking in this problem set to your metacognitive thinking during developing PS 1
• Technology integration - technology tool is your choice - just make sure to include and describe interactivity; yes, you van integrate technology in more than one problem

Reflection #11:

Entry 1. Your assignment for today is to write an essay on the topic selected from the PPT: "Mathematics around us." Before doing that, as part of this entry, decide with your group members who is writing on which topic from this presentation.

Mathematics around us

Entry 2. Essay on the topic you selected in agreement with your group members. Provide at least one reference/resource in addition to the PPT (250 - 300 words).

ONLINE CLASS

June 2006

These pages are always under construction: I am trying to keep them up-to-date with my activities :) Questions and/or comments are welcome!

Maintained by:  Mara Alagic
Mathematics Education
Curriculum & Instruction Department
Wichita State University
Wichita, Kansas  67260-0028