Andrew Wiles and Fermat's Last Theorem

Italian Algebraists and the Solution to the Cubic

Motivating the Mind to Think Mathematically

Design class and curriculum to encourage critical thinking

Class

1) Elevate your view of the student

2) Teach Socratically.

3) Leverage student knowledge and skills to encourage self-discovery of new ideas.

4) Establish class traditions.

Make the students do the thinking.

How much "telling" are we doing?

“The understanding is not a vessel which must be filled, but firewood, which needs to be kindled; and love of learning and love of truth are what should kindle it.”

- Plutarch

Algebra students should derive the quadratic formula.

Geometry students should demonstrate the Pythagorean Theorem.

Curriculum

Textbooks should serve as a guide to scope and sequence and a source for basic problem sets.

Don’t be afraid to design more creative problems that encourage critical thinking or to present ideas by more engaging methods.

(more on this to follow)

"We hold that the child's mind is no mere sac to hold ideas; but is rather, if the figure may be allowed, a spiritual organism, with an appetite for all knowledge." - Charlotte Mason

"What if education, including higher education, is not primarily about the absorption of ideas and information, but about the

formation

of hearts and desires?" - James K.A. Smith

First and foremost, retire your search for the "magic curriculum."

"Riddle of the Week"

Probability Games (Monty Hall Problem)

"Quotable/Notable/Lexiconical"

Develop better problems and questions that exhort students to think critically

Give more take-home tests to allow space for critical thinking outside of time pressure

Require students to explain their solution method in words

Give test problems that contain no numbers (Dan Meyer type problems)

Let students prepare a lesson plan and teach class

Training the Mind to Problem Solve Efficiently and Creatively

Encourage patient, thoughtful problem solving

Help students organize and develop a logical thought process

Don't hand out "free formulas"

(or, "Let's name the animals, not just count them")

Whenever possible, make students "purchase" formulas by walking through the derivation, mathematically or geometrically (e.g., area of a circle, volume of a pyramid, the quadratic formula, the Pythagorean Theorem, etc. . )

When understanding works, throw the formula in the trash (e.g., percent markup, interest, volume of a cylinder)

Developing an Appetite for

Problem Solving

Exemplifying the Characteristics of

the Mathematical Mind

Incorporate math riddles that "trick" the student into thinking critically

Provide ahead of time the bonus question for your next test

Why not? Students will do just about anything for a bonus, but it’s often just the A students that even have the time (or energy) left at the end of a test to tackle a bonus, and at that point you are simply widening the gap between your higher scores and your lower scores.

Advocate for equal opportunity critical thinking!

Live and model a life of curiosity and wonder

Is it obvious to our students that our interest in mathematical thinking goes beyond the textbook?

"To make this transition [from a focus on mastering procedures to 'mathematical thinking'] easier, students need to be introduced to 'mathematical thinking' as quickly as possible, including reinforcement, in their K-12 math experiences. The majority of K-12 textbooks on the market do little of that." - James Nickel

Coordinate with other disciplines to make cross-curricular activities and assessments

Share with your students real mathematical problems from your daily life

What do I know? What am I asked for? What could I infer from what I know? What could I calculate from what I know?

Do I know any relationships between the info that I have and the info that I need? Can I translate the question into an algebraic equation?

Is this the BEST way to approach this problem? Is there a more efficient, more elegant path? (

correct answer vs. beautiful solution

)

Give fewer but more complex, multi-step problems that integrate multiple skills and ideas (avoid "skill silos")

Give more take-home tests that require sustained, engaged thought (don't be constrained by the classroom clock)

Leave out pertinent information in a word problem or riddle

Research projects on famous Christian mathematicians/scientists (math/science/bible/history)

Math poem contests (math/English)

Bring an oscilloscope to music class (math/music/science)

**Training the Mathematical Mind**

"Five sailors arrive at a deserted island that has only coconuts and one monkey. The sailors collect all the coconuts into one big pile and agree to divide up the coconuts into equal shares the next morning. However during the night each sailor wakes up one at a time afraid to trust the others and decides to take his share secretly. So each sailor takes 1/5 of the coconuts and hides it. Each time there is one coconut left over and the sailor gives that to the monkey. In the morning they divide what is left of the pile into equal shares and there is still one coconut left for the monkey.

How many coconuts were in the original pile?"

Martin Gardner Monkey & Coconuts Problem

0,1,1,2,3,5,8,13,21,34,55,89,144...

Jason Faulkner

Heritage Preparatory School

Atlanta, GA

Brett Edwards

Atlanta Classical Christian Academy

**2013 ACCS Conference, Atlanta, GA**

**June 20, 2013**

Emphasis on Quality of Problems vs. Quantity of Problems

Temptation for the math teacher is to do an example of every potential problem variation.

Create problems that allow a student to use their creativity in finding a solution.

Ultimate goal is to develop students that can solve problems independently, not process imitators

Tell Stories about History's Greatest Problem Solvers

Start from a Christian Worldview

Transformation

vs. information

". . . how we think about distinctly Christian education [should] not be primarily a matter of sorting out which Christian ideas to drop into eager and willing mind-receptacles; rather, it [should] become a matter of thinking about how a Christian education shapes us, forms us, molds us to be a certain kind of people

whose hearts and passions and desires are aimed at the kingdom of God

. And that will require sustained attention to the practices that effect such transformation." - James Smith

We are not filling minds, we are transforming souls.

Renewal

of the mind

"Do not conform to the pattern of this world, but

be transformed by the renewing of your mind

. Then you will be able to test and approve what God's will is--his good, pleasing and perfect will." Romans 12:2

What does it look like to exhort our students towards mind renewal?

Do we call on the help of the Holy Spirit in this process?

"Paul has hereby created the context for the key command which sets all of his ethics apart from any suggestion of 'spontaneity,' as though once you were in Christ, indwelt by the Spirit, all you had to do was let the new life 'come naturally.' No: the mind must be transformed, so that you can

think out for yourself, weigh up and consider

, what God's will actually is. Unless the mind is fully involved, not only are you not growing up as a fully (and fully integrated) human being; you are not engaging in virtue at all." - N.T. Wright

Cultivation

of wisdom and virtue

Mind renewal is essential to the cultivation of wisdom and virtue.

This cultivation and transformation is a process, a "long obedience in the same direction."

Revelation

of God's nature

"For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse." Romans 1:20

Do we model for our students delight in the revelatory beauty and order of mathematics?

Math is

God's way of

Revealing His hand.

Through equations and fractions,

Numbers and symbols - even the grains of sand.

Complex and yet full of order, His creation is

Pointing us to the truth that the world makes sense, and all control is His.

"I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future." - Andrew Wiles

Do we leverage their own ?

sensus divinitatis

"There is within the human mind, and indeed by natural instinct, an awareness of divinity." - John Calvin

Designing Assessments That Truly Test the Mathematical Mind

John Milton Gregory's Four Layers of Knowledge

Assess Weekly and Cumulatively

Quiz on Weaknesses

Critical Concepts

Recognition

Recall

Detail

Vivid

Knowledge

"We know a fact faintly to recognize when we see it."

"We can recall a fact or describe it generally."

"We know the fact, truth, or concept so that we can readily explain, prove and illustrate it."

“Mounting to the highest grade of knowledge, we may so know and vividly see a truth in its deeper significance and wider relations that its importance, grandeur, or beauty impresses and inspires us.”