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A Mathematician’s Lament. Inquiry Based Learning. Pete Anderson [email protected] Teacher of Mathematics Glen Allen High School HCPS.

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A mathematician s lament

A Mathematician’s Lament

Inquiry Based Learning

Pete Anderson

[email protected]

Teacher of Mathematics

Glen Allen High School


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

Workshop topics
Workshop Topics Lockhart.

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?

B ottom of page 1
b Lockhart.ottom of page 1

“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.”

Bottom of page 2
Bottom of page 2 Lockhart.

“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

Top of page 3
Top of page 3 Lockhart.

“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.”

Bottom of page 3
Bottom of page 3 Lockhart.

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

Modeling with technology
Modeling with Technology Lockhart.

  • 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

Bottom of page 4
Bottom of page 4 Lockhart.

“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.”

Problem solving techniques
Problem Solving Techniques Lockhart.

  • 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

A text by ted herr and ken johnson
A text by Ted Herr and Ken Johnson Lockhart.

  • 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

Let s solve a problem
Let’s Solve a Problem Lockhart.

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.

Level 1 wolf goat cat and cabbage
Level 1: Wolf, Goat, Cat and Cabbage Lockhart.

  • 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

Level 2 missionaries and cannibals
Level 2: Missionaries and Cannibals Lockhart.

  • 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

Level 3 crossing the river with dogs
Level 3: Crossing the River with Dogs Lockhart.

  • 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?

How did you solve the problem
How did you solve the problem? Lockhart.

  • What skills were required?

  • What previous knowledge was required?

  • Is there more than one solution? Is one solution better than another?

Rubrics drive instruction
Rubrics Drive Instruction Lockhart.

Original | Mine | HCPS – draft w/ template

Shall we do another
Shall we do another? Lockhart.

  • Anno’s Hat Tricks

  • Oh Crap. It’s Trash Day!

  • The Locker Problem

  • anything from “The Canterbury Puzzles” – 1908


Original | Mine | HCPS – draft w/ template

Middle of page 5
middle of page 5 Lockhart.

“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.”

Break Lockhart.

Turn evil into good1
Turn Evil into Good Lockhart.

Completing Exercises vsSolving Problems

Top of page 6
Top of page 6 Lockhart.

“… 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.”

Middle of page 6
Middle of page 6 Lockhart.

“… 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.”

How to break the cycle
How to Break the Cycle Lockhart.

  • Problem Level

  • Unit Level

  • Nine Weeks Level

  • PBL

  • Flipped Classroom

  • Backwards Design

Original lesson plan area circumference
Original Lesson Plan Lockhart.Area & Circumference

lesson plan

The 4 ds to problem solving
The 4 Ds to Problem Solving Lockhart.

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


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


Pbl lesson plan area circumference
PBL Lesson Lockhart.Plan Area & Circumference

Sample lesson plan with and with PBL approach


Teaching for tomorrow
Teaching for Tomorrow Lockhart.

Teaching Content and Problem-Solving Skills

by Ted McCain


How to solve it g polys
How To Solve It – G. Polys Lockhart.

A Four Step Process

  • Understanding the Problem

  • Devising a Plan

  • Carrying Out the Plan

  • Looking Back

  • Intro to Text

  • Full Text

  • Amazon.com

The flipped classroom
The Flipped Classroom Lockhart.

  • 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.

Traditional pacing guide
Traditional Pacing Guide Lockhart.

This is what I haveto teach.

The flipped classroom1
The Flipped Classroom Lockhart.

  • My Lesson Plan

  • My Vodcast– Geometry

  • My Youtube Channel

Backwards design
Backwards Design Lockhart.

a.k.a. Understanding by Design

Backwards design1
Backwards Design Lockhart.

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".

Backwards design2
Backwards Design Lockhart.

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.

Backwards design3
Backwards Design Lockhart.

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

Example of the ubd approach
Example of the Lockhart.UbD Approach

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 

End result
End Result Lockhart.

Understanding by design
Understanding by Design Lockhart.

  • 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

Who are simplicio and salviati
Who are Lockhart.Simplicio and Salviati?

Top of page 9
Top of page 9 Lockhart.

“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.”

Students come pre installed with viruses
Students come pre-installed with Lockhart.viruses

  • Lack of initiative.

  • Lack of perseverance.

  • Lack of retention.

  • Aversion to word problems.

  • Eagerness for formula.

Dan Meyers

Dan meyer math class needs a makeover
Dan Lockhart.Meyer Math class needs a makeover

part one

part two

time = 11:39

Redefine the meaning of word problem
Redefine the meaning of “Word Problem” Lockhart.

  • Use multimedia.

  • Encourage student intuition.

  • Ask the shortest question you can.

  • Let students build the problem.

  • Be less helpful.

Dan Meyers

Bottom of page 10
Bottom of page 10 Lockhart.

“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:”


  • pharmaceutical companies : doctors

  • record companies : disk jockeys

  • corporations : congressmen

  • all of the about

Make this more interesting
Make this more interesting Lockhart.

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.

I m going to make some dough with this cantaloupe
I’m going to make some dough with this cantaloupe! Lockhart.

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

More information
More information Lockhart.

As a word problem
As a “word problem” Lockhart.

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?

Shall we do another1
Shall we do another? Lockhart.

  • Parallel Lines

  • Quadrilaterals

  • Right Triangles

  • Circles

  • Coordinate Geometry

Middle of page 11
Middle of page 11 Lockhart.

“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?”

What have you done lately
What have you done lately? Lockhart.

Is this an exercise or problem?

Magic squares
Magic Squares Lockhart.

The BWW Method


Pete Anderson


Darren Steadman

Durer: Melancholia

Squaring the circle
Squaring the Circle Lockhart.

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.


Galileo s tomb
Galileo’s Tomb Lockhart.

Santa Croce Cathedral – Florence, Italy

Let me ask you some questions
Let me ask you some questions. Lockhart.

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

Here are some more
Here are some more Lockhart.:

  • 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?


Peter Flom

Yahoo Contributor Network

What is learning for
What is learning for? Lockhart.

“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