NCNC - breakout I - What is Mathematics?

NCNC II

1st Breakout session
Dana Johnson - What is Mathematics?

Handout - powerpoint

Ask your students. What is mathematics? What does a mathematician do?

grade school image of mathematics is strictly numbers - computation

Mathematics - study of patterns in number and space

linear path -- learn to take the test -
arithmetic - algebra - geometry - etc.
"calculus rules the world"
really calc is useful tool
all kinds of other things

Mathematicians -
-recognize and describe patterns
-write their own questions about relationships in patterns
-prove it (deductive reasoning)
-create models that are abstractions of real world problems
-apply models to real world

Inductive reasoning-
general conclusion from a limited set of observations
 - particular case to generalization

Deductive reasoning -
using logic to draw conclusions from statements that we accept to be true

most disciplines do not have the ability to use deductive reasoning to prove something to be true
Math usually produces results of absolute truth and is often used as a tool in other disciplines.
Not doing a lot of deductive reasoning in our schools with regular kids and need to do more.

exercise with triangular numbers

using manipulatives is just good teaching
some kids have it etched in their minds after one look, other kids need the spatial representation
learning style issue mostly - NOT gifted issue

concrete pattern,
expanded pattern beyond what is given
generalize pattern
extended patterning

Deductive reasoning is a powerful tool. We need to highlight that for students, especially as they get to middle school.

The thing about mathematics is you never know when it's going to be useful. A lot of mathematical theory figured out a hundred years ago is now being directly applied in science.

exercise with Goldback's Conjecture

Let your students play with numbers and figure out their own approaches to big difficult problems. Do not let them get the idea that every math problem should be something you can solve in just a minute or so.

Process of Doing Mathematics -
study a lot about the field
think of a new question or problem
work on the problem
communicate the result
think of new questions that are extensions of the knowledge

Linda Sheffield's Model of problem solving
star shape - points are relate, create, investigate, evaluate, communicate
can start at any of the points and move to any of the others

(passed around a book by Sheffield- Extending the Challenge in Mathematics)

Where do we do this in our curriculum?
science more than in math

get away from rote processes into truly mathematical processes
- not an argument against memorizing table, but that should not be the primary focus of math classes

Habits of mind of mathematicians
curiousity
creativity
precision
tenacity
skepticism
collaboration

Teachers need to provide more problems that are open-ended, interesting, have possibilities for extension
Inquiry based teaching and learning is essential
emphasis on concepts
problem solving process is important
assign some problems that will not be solved overnight
expect students to pose problems
expect clear and precise explanations
it's ok to tackle problems in class that the teacher does not already know the answer to
keep a math journal
give choices in problems
ask students to make presentations about their approach to problems

Ask the right questions -
What is going on here?
What patterns do you notice?
Is this always true?
How do you know?
Can you state a generalization?
Can you think of another question?
Is there another way to do this?
How do you know that is right?

Impediments -
shortage of knowledgeable, confident teachers
need materials that are a rich source of problems
internet can be a help in locating rich problems but students can use it as a source for finding other people's solutions rather than working on it themselves
students are often used to being spoon-fed and it is also easier to teach that way

To Thales the primary question was not what do we know, but how do we know it. - Aristotle