PEMSTL Mid-winter

InstituteJanuary 10, 2009

Place the numbers 1-8 in the rectangles so that no two consecutive numbers are next to each other horizontally, vertically, or diagonally.

Powerful Ideas in

Mathematics

COMPOSITION

In order to do mathematics, the human mind must compose units, which are countable objects , and the conception of units must be flexible. The act of composing units if referred to as composition

 

COMPOSITION

•Counting

•Units of measure

•Clustering units

•What constitutes a shape?

•Operations – adding, multiplying

Activity: Geometric Composition

DECOMPOSITION

In order to do mathematics, the human mind must be able to decompose units into smaller pieces, The act of forming smaller pieces from units is referred to as decomposition.

 

DECOMPOSITION•Break into parts

•Fractions

•Decimals

•Ratio

•Percent

•Measurement

•Operations – subtracting, dividingDiscuss how to complete the following problem:

56-29

RELATIONSHIPS

In order to do mathematics, the human mind must perceive of relationships between units and/or partitions of units as entities that can be studied, describes, and manipulated.

 

RELATIONSHIPS•Between numbers and sets of

numbers

•Between shapes and parts of shapes

•Ratio

•Proportion

•Scale

•Statistics

•Probability

•Functions

I was traveling to Logan. I passed a sign that said Logan was 48 miles away. My speedometer read 70 mph. My car typically gets 24 mpg. What relationships can be discussed?

REPRESENTATION

In order to do mathematics, the human mind must conceive of ways to represent abstractions with some form of symbols that can be manipulated and upon which operations can be carried out in proxy.

 

REPRESENTATION•Written symbol or drawing stands

for an idea

•Numerals

•Symbols of language that makes sense

“It is never enough in mathematics to simply learn the symbols and the rules that govern their use. The symbols are only representations for complex ideas, and it takes time and effort to fully explore those complex ideas.” Schwartz, 2008

CONTEXT

In order for mathematics to be meaningful, it helps to have a context in which the mathematical ideas reside. In most cases, real-world applications for mathematics provide the necessary context.

CONTEXT•See the real-world practical

context from which mathematical abstractions are derived

•Problem-based learning

•Why should students invest the effort to learn?

•Teachers need to know the use of the mathematics they are teachingDiscuss: How can you make learning multiplication and division facts relevant to students?

NUMBER SYSTEMS

Number Systems•Unury system – one mark for each

number

•Mayan - base 20

•Sign value notation

• Egyptian – base 10, hieroglyphics

• Roman numerals

•Place value notation

• Hindu-Arabic

Bases•Binary – 0 and 1, based on vacuum tubes

being open or closed

•5 (quinary)

•8 (octal) – used by the Yuki tribe of Northern California as they counted the spaces between fingers

•10 (decimal)

•12 (duodecimal) – dozen, gross, 24 hours in a day

•20 (vigisimal) – Mayan, central & western Africa

•60 (sexagesimal) – Sumeria, MesopotamiaActivity: Chip Trading

Your Number System