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Training the Mathematical Mind
Transcript of Training the Mathematical Mind
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
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.”
Algebra students should derive the quadratic formula.
Geometry students should demonstrate the Pythagorean Theorem.
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
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)
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
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
Heritage Preparatory School
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
". . . 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.
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
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."
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?
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 ?
"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
"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.”