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A Mathematician’s Lament

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A Mathematician’s Lament

Inquiry Based Learning

Pete Anderson

pganders@henrico.k12.va.us

Teacher of Mathematics

Glen Allen High School

HCPS

This workshop is entirely inspired by this text by Paul Lockhart.Free: http://www.maa.org/devlin/lockhartslament.pdfPurchase:http://www.amazon.com

Web Site: http://gahs-math.wikispaces.com

- What does it mean to “teach”?
- Modeling
- Problem Solving Techniques
- How to Break the Cycle
- Redefine the meaning of “Word Problem”
- What have you done lately?

“I had this one kid, though, man was she sensational! Her sheets were impeccable – every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician one day.”

“I just use the Paint-by-Numbers kits supplied by the school board.

Sadly, our present system of mathematics education is precisely this kind of nightmare.”

Read on

“The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, ‘math class is stupid and boring,’ and they are right.”

“Simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”

- Triangle in a Box
- http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Triangle_in_Box.html
- Two Segments – Four Sided Shape
- http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/TwoSegments_FourSidedShape.html
- Point on a Grid
- http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Level1.html
- Line on a Grid
- http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Level2.html
- Parabola
- http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Level3.html
- Angle of Incidence & Reflection
- Sketchpad file

“Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck.”

- Solve an Easier Related Problem
- Physical Representations
- Work Backwards
- Venn Diagrams
- Algebra
- Finite Differences
- Other Ways to Organize Information
- Other Ways to Change Focus
- Other Forms of Spatial Organization

- Draw a Diagram
- Systematic Lists
- Eliminate Possibilities
- Matrix Logic
- Look for a Pattern
- Guess and Check
- Sub-problems
- Unit Analysis

- Problem Solving Strategies: Crossing the River With Dogs and Other Mathematical Adventures(ISBN: 1559530685 / 1-55953-068-5 )
- Available at AbeBooks.com for as little as $1 (used).
- GAHS Math text samples

Crossing the River ...

- Level 1 – with a wolf, goat, cat, and cabbage
- Level 2 – with Missionaries and Cannibals
- Level 3 – with Dogs
In groups of ___, you have __ minutes to solve the problem.

- Sailor Cat needs to bring a wolf, a goat, and a cabbage across the river. The boat is tiny and can only carry one passenger at a time. If he leaves the wolf and the goat alone together, the wolf will eat the goat. If he leaves the goat and the cabbage alone together, the goat will eat the cabbage. How can he bring all three safely across the river?
- Online version
- Flipchart

- Three missionaries and three cannibals must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are missionaries present on the bank, they cannot be outnumbered by cannibals (if they were, the cannibals would eat the missionaries.) The boat cannot cross the river by itself with no people on board. How can they all cross the river alive?
- Online version
- Flipchart

- The situation: The five members of a family and their five dogs (each family member owned one of the dogs) were hiking when they encountered a river to cross. They rented a boat that could hold three living things: people or dogs. Unfortunately, the dogs were temperamental. Each was comfortable only with its owner and could not be near another person, not even momentarily, unless its owner was present. The dogs could be with other dogs, however, without their owners. The crossing would have been impossible except that Lisa’s dog had attended a first-rate obedience school and knew how to operate the boat. No other dogs were that well educated.
- The task: Explain in detail how do the five family members and their dogs cross the river?

- What skills were required?
- What previous knowledge was required?
- Is there more than one solution? Is one solution better than another?

Original| Mine| HCPS – draft w/ template

- Anno’s Hat Tricks
- Oh Crap. It’s Trash Day!
- The Locker Problem
- anything from “The Canterbury Puzzles” – 1908

Rubrics:

Original| Mine| HCPS – draft w/ template

“By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject.”

Completing Exercises vsSolving Problems

“… a self-perpetuation monster: students learn about math [from a traditional text and methodology] from their teachers, and teachers learn about it from their teachers.”

“… Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions.”

- Problem Level
- Unit Level
- Nine Weeks Level

- PBL
- Flipped Classroom
- Backwards Design

lesson plan

Define: Write a list of questions but do not answer them yet. “What do you need to do or know in order to solve this problem?”

Design: Create a working sketch and equation that will solve the

problem.

Do: Gather information needed to substitute into the equation

and solve.

Debrief: Turn in your work with the grade you believe you have

earned. Discuss with the teacher why you deserve this

grade.

Sample lesson plan with and with PBL approach

http://geometryonline.pbworks.com/PBL

Teaching Content and Problem-Solving Skills

by Ted McCain

Amazon.com

A Four Step Process

- Understanding the Problem
- Devising a Plan
- Carrying Out the Plan
- Looking Back
- Intro to Text
- Full Text
- Amazon.com

- What do you have to teach?
- What do you want to teach?
- Why not do both?
The flipped classroom is a model of teaching in which a student’s homework is the traditional lecture viewed outside of class on a vodcast. Then class time is spent on inquiry-based learning which would include what would traditionally be viewed as a student’s homework assignment.

This is what I haveto teach.

- My Lesson Plan
- My Vodcast– Geometry
- My Youtube Channel

a.k.a. Understanding by Design

Stage 1 starts with educators identifying the desired results of their students by establishing the overall goal of the lessons by using content standards, common core or state standards. In addition, UbD's stage 1 defines "Students will understand that..." and lists essential questions that will guide the learner to understanding. Stage 1 also focuses on identifying "what students will know" and most importantly "what students will be able to do".

Stage 2 focuses on evidence of learning by assessment. Teachers plan performance tasks and evidence of understanding. Performance tasks determine what the students will demonstrate in the unit and what evidence will prove their understanding. This can include self-reflections and self-assessments on learning.

Lastly, stage 3 lists the learning activities that will lead students to your desired results.

Geometry: Unit One

- What are the Standards of Learning
- Unpack the Standards using Bloom's
- Essential Questions
- How and why do we communicate through measure?
- How and why do we mathematically describe naturally occurring phenomenon?
- When, if ever, is a mathematical solution practical?

- Performance Based Assessment - "It's in the Hole!"
- Directions
- Sample
- Rubric
- Photos

- Activity/Lesson Plan #4
- Activity/Lesson Plan #3
- Activity/Lesson Plan #2 | Henrico 21 Entry
- Activity/Lesson Plan #1
- Plan a schedule for the marking period and fill in with needed traditional material

- Advanced Concepts in Creating and Reviewing Units
- by Grant Wiggins and Jay McTighe, 2012.
- Publisher: Association for Supervision and Curriculum Development (ASCD), Alexandria, VA
- 8 1/2" x 11," 134 pages with downloadable resources available online.
- ISBN-13: 978-1-4166-1409-8
- Amazon.com

“In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide.”

- Lack of initiative.
- Lack of perseverance.
- Lack of retention.
- Aversion to word problems.
- Eagerness for formula.

Dan Meyers

part one

part two

time = 11:39

- Use multimedia.
- Encourage student intuition.
- Ask the shortest question you can.
- Let students build the problem.
- Be less helpful.

Dan Meyers

“We are encouraged to forego the difficult task of making decisions based on our individual wisdom and conscience, and to ‘get with the program.’ It is simply the path of least resistance:”

TEXTBOOK PUBLISHERS : TEACHERS ::

- pharmaceutical companies : doctors
- record companies : disk jockeys
- corporations : congressmen
- all of the about

A megaphone is formed by taking a cone with a radius of 20 centimeters and an altitude of 60 centimeters and cutting off the tip. The cut is made along a plane that is perpendicular to the axis of the cone and intersects the axis 12 centimeters from the vertex. What is the lateral area of the megaphone? Round your answer to the nearest hundredth.

How much money can I make by buying whole cantaloupe, chopping it up into pieces, repackage it, and sell it back to Martins?

As manager of a locally owned grocery, you currently sell whole cantaloupe melons but are considering packaging your own fresh cut (chunked) cantaloupe. Your study of situations reveals that the average melon has a 7 inch diameter. To chunk the melon, you must remove the seeds (2 inch diameter) and the outer peel which is ½ inch thick.

Part One: What is the volume of the original (whole) melon?

Part Two: What is the volume of the melon after removing the peel and seeds?

- Parallel Lines
- Quadrilaterals
- Right Triangles
- Circles
- Coordinate Geometry

“Now I’m not saying that math teachers need to be professional mathematicians – far from it. But shouldn’t they at least understand what mathematics is, be good at it, and enjoy doing it?”

Is this an exercise or problem?

The BWW Method

by

Pete Anderson

and

Darren Steadman

Durer: Melancholia

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

sketchpad

Santa Croce Cathedral – Florence, Italy

They are very easy.

- If a woman had never read a novel or written anything since college, would she be a good English teacher?
- If a man had never gone to a concert, played an instrument, or sang, would he be a good music teacher?
- If a woman had never gone to a museum, painted a picture, played with clay, or drawn, would she be a good art teacher?
I told you they were easy!

Peter Flom

Yahoo Contributor Network

- If you were forced to take 12 years of music theory before you were permitted to pick up an instrument, would you do so?
- If you were not allowed to paint until you could name 100 types of brushes and distinguish 200 pigments, would you paint?
- If you had to learn all about iambic pentameter before you were permitted to write a poem, would you write one?
OF COURSE NOT!

Peter Flom

Yahoo Contributor Network

“If your idea is that learning is for turning children into economic units that are going to have value in some global competition then this current education system is what you want and you’re welcome to it.”

“If however you view learning as the pursuit of beauty and knowledge and understanding for its own sake so that we can live richer and happier lives then that’s what I’m all about.”

Paul Lockhart