Math Teaching at Gann

To top it all off...

• This method doesn't work for all learners
• It promotes performance over mastery

"Children with a mastery orientation have learning goals – they are concerned with increasing their competence and abilities while mastering new tasks over time. Conversely, children with a performance orientation have performance goals – they are concerned with eliciting positive judgments about their work...There is strong evidence that a mastery orientation can boost children’s academic performance, in the short- and long-term."

Summary of the work of Dweck and Leggett (1988) on Goal Orientation. Quoted from http://www.tauckfamilyfoundation.org/outcomes/child-outcomes/mastery-orientation

Geometric Sequences

The following is a Geometric Sequence because the way to get from one term to the next is to multiply the previous term by something rather than to add (or subtract).

Can we apply what we just did on the previous example?

Arithmetic Sequences and Fractions

What if we make 10 the 1st output of some sequence?

If 10n = 10, then how do we figure out what fraction of the way there 2 is? Or 4?

Results of Learning Math in this Way

More Formally...

"Constructivism focuses our attention on how people learn. It suggests that math knowledge results from people forming models in response to the questions and challenges that come from actively engaging math problems and environments - not from simply taking in information, nor as merely the blossoming of an innate gift. The challenge in teaching is to create experiences that engage the student and support his or her own explanation, evaluation, communication, and application of the mathematical models needed to make sense of these experiences."

• Lack of engagement in the learing process
• Lack of enjoyment with mathematics
• Lack of being able to explain it to others
• Lack of wanting to learn more and/or apply it outside of school

http://mathforum.org/mathed/constructivism.html

Motivating Questions

How did you learn math?

• By rote memorization?

• By being told "how" but not "why"?

• By the "drill and kill?" method? (when done correctly, this actually is not such a bad thing)

What does 10(1/5) equal?

What does 32^(1/5) equal?

What frustrated you about learning math?

Example from Algebra 2

Complete the missing spaces in the pattern: 0, 2, ____, 6 _____, 10, _____ , 14, _____, 18....

The series of numbers above defines an Arithmetic Sequence. As the change is the result of adding the same change consistently, we can write a rule for the series based on the location of each number.

Textbook Choices Influence Teaching Philosophies

• Our text, Discovering Geometry, replicates this sort of learning to an extent

• It focuses on experimentation and discovery, using inductive reasoning to arrive at rules (conjectures)

• Downside: formal proof of these rules is not always given and is up to the teacher to supplement

• Upside: 1) Students own more of their learning 2) this style of learning can be applied to future courses

Making the Ancient Modern

Well , sort of.

Ancient Greek Mathematics focused on Geometry. Euclid wrote down his Elements around 300 B.C. Building up the ideas in Geometry is to consistently and routinely investigate and deduce new properties based on previously known & proven ones. This is an excellent way to learn math.

Furthermore, a final upside to this is the affect on math learning of girls in the classroom.

Research concludes that girls learn better in group situations and when they can make more meaning of what that are learning.

By making meaning for our students, we give them the tools to understand how mathematical principle work and can be applied to other concepts down the road. Furthermore, it takes away the need for pure memorization as students can reconstruct where a fact came from (in most cases).

(more about that can be found in this talk given at my college reunion: http://palmer.wellesley.edu/~ofernand/Palmer_Wellesley/Presentation.pdf)