Mathematics is the study of quantity, structure, space and change ...
Mathematics is the study of quantity, structure, space and change ...
Mathematics is the study of quantity, structure, space and change ...
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<strong>Ma<strong>the</strong>matics</strong> <strong>is</strong> <strong>the</strong> <strong>study</strong> <strong>of</strong> <strong>quantity</strong>,<br />
<strong>structure</strong>, <strong>space</strong> <strong>and</strong> <strong>change</strong>. Ma<strong>the</strong>maticians<br />
seek out patterns formulate new conjectures<br />
<strong>and</strong> establ<strong>is</strong>h truth by rigorous deduction<br />
from appropriately chosen axioms <strong>and</strong><br />
definitions.
National Council <strong>of</strong> Teachers <strong>of</strong> <strong>Ma<strong>the</strong>matics</strong><br />
(NCTM)<br />
Momentum started in early 1980s<br />
1989 NCTM Curriculum <strong>and</strong> Evaluation<br />
St<strong>and</strong>ards for School <strong>Ma<strong>the</strong>matics</strong><br />
Principles <strong>and</strong> St<strong>and</strong>ards for School<br />
<strong>Ma<strong>the</strong>matics</strong> (2000)<br />
NCTM Leadership
Equity-expectations <strong>and</strong> support for all<br />
Curriculum-you are major part <strong>of</strong> curriculum<br />
Teaching-effectiveness requires<br />
Learning-students learn w/ underst<strong>and</strong>ing<br />
Assessment-support student <strong>and</strong> teacher<br />
Technology-influences <strong>and</strong> enhances student<br />
learning
Numbers <strong>and</strong> Operations<br />
Algebra<br />
Geometry<br />
Measurement<br />
Data Analys<strong>is</strong> <strong>and</strong> Probability
Problem Solving<br />
Reasoning <strong>and</strong> Pro<strong>of</strong><br />
Communication<br />
Connections<br />
Representation
Curriculum focal points are important<br />
ma<strong>the</strong>matical topics for each grade level,<br />
pre-K–8. These areas <strong>of</strong> instructional<br />
emphas<strong>is</strong> can serve as organizing <strong>structure</strong>s<br />
for curriculum design <strong>and</strong> instruction at <strong>and</strong><br />
across grade levels. The topics are central to<br />
ma<strong>the</strong>matics: <strong>the</strong>y convey knowledge <strong>and</strong><br />
skills that are essential to educated citizens,<br />
<strong>and</strong> <strong>the</strong>y provide <strong>the</strong> foundations for fur<strong>the</strong>r<br />
ma<strong>the</strong>matical learning.
Away From:!<br />
Classrooms as collections<br />
<strong>of</strong> individuals!<br />
Teacher as “sole authority”<br />
for right answers!<br />
Mere memorization <strong>of</strong><br />
procedures!<br />
Emphas<strong>is</strong> on mechan<strong>is</strong>tic<br />
finding <strong>of</strong> answers!<br />
Treating ma<strong>the</strong>matics as a<br />
body <strong>of</strong> <strong>is</strong>olated concepts<br />
<strong>and</strong> procedures!<br />
Toward:!<br />
Classrooms as<br />
ma<strong>the</strong>matics communities!<br />
Logic <strong>and</strong> ma<strong>the</strong>matical<br />
evidence as verification!<br />
Ma<strong>the</strong>matical reasoning"<br />
Conjecturing, inventing,<br />
<strong>and</strong> problem solving!<br />
Connecting ma<strong>the</strong>matics,<br />
its ideas, <strong>and</strong> applications!
The term “st<strong>and</strong>ards” was popularized by<br />
NCTM in 1989<br />
Grade-by-grade objectives vary state to state<br />
Testing programs associated with st<strong>and</strong>ards<br />
put pressure on teachers <strong>and</strong> students to<br />
perform so that test scores are ra<strong>is</strong>ed at all<br />
costs
Your Responsibilities!<br />
■ What does it mean to DO ma<strong>the</strong>matics?!<br />
■ How does a person go about LEARNING<br />
ma<strong>the</strong>matics?!<br />
■ As a future elementary teacher, how<br />
should you go about teaching<br />
ma<strong>the</strong>matics through problem solving?!
Doing ma<strong>the</strong>matics means !<br />
generating strategies for solving problems!<br />
applying those approaches !<br />
seeing if <strong>the</strong>y lead to solutions !<br />
checking to see if your answers make<br />
sense.!
Creation <strong>of</strong> Knowledge<br />
◦ Not empty vessels<br />
Assimilation <strong>and</strong> Accommodation<br />
Reflective Thought
Learning depends on:<br />
Learner (working in h<strong>is</strong> or her ZPD)<br />
Social interactions<br />
Culture
Build new knowledge from prior knowledge<br />
Provide opportunities to talk about<br />
ma<strong>the</strong>matics<br />
Build in opportunities for reflective thought<br />
Encourage multiple approaches<br />
Treat errors as opportunities for learning<br />
Scaffold new content<br />
Honor diversity
■ Underst<strong>and</strong>ing <strong>is</strong> <strong>the</strong> measure <strong>of</strong> quality <strong>and</strong> <strong>quantity</strong> "<br />
<strong>of</strong> connections between new ideas <strong>and</strong> ex<strong>is</strong>ting ideas !<br />
■<br />
■<br />
Knowing ≠ Underst<strong>and</strong>ing (students may know something<br />
about fractions, for example, but not underst<strong>and</strong> <strong>the</strong>m)!<br />
Continuum <strong>of</strong> underst<strong>and</strong>ing!<br />
! ! Relational ! ! ! Instrumental!<br />
! Underst<strong>and</strong>ing! Underst<strong>and</strong>ing!<br />
! ! What to do <strong>and</strong> Why! Just do it!
■ The need to replace <strong>the</strong> question<br />
“Does <strong>the</strong> student know it?” with <strong>the</strong><br />
question “How does <strong>the</strong> student<br />
underst<strong>and</strong> it?”<br />
■ Early number concepts!<br />
■ Computation!
Effective new concepts <strong>and</strong><br />
procedures<br />
Less to remember<br />
Increased retention <strong>and</strong> recall<br />
Enhanced problem-solving<br />
abilities<br />
Improved attitudes <strong>and</strong> beliefs
Change language