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MAT 3030 Test 1
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Key Concepts:
Terms in this set (27)
Instrumental Understanding
doing something without understanding
Relational Understanding
knowing what to do and why
Tools vs. Manipulatives
A tool is any object, picture, or drawing that can be used to explore a concept. Manipulatives are physical objects that students and teachers can use to illustrate and discover mathematical concepts. A tool is used to visualize a mathematical concept and only your mind and impose the mathematical relationship on the object. A manipulative can be a testing ground for emerging ideas.
Constructivism
theory based on the idea that knowledge is learned by the "knower"
Schema
making inferences on what to expect from past experiences or knowledge
Sociocultural Theory
learning built from the social interactions and communication between people in the culture in which they live
Zone of Proximal Development
the area that challenges the student and promotes growth but is not too difficult or too easy
Subitize
seeing a set of objects and knowing immediately how many it is. Example: 5 on a dice, you would automatically know it was 5 without individually counting 5 dots
Cardinality
the ability to recognize that things have 1 to 1 correspondence. This is one computer and one water bottle
Learning Trajectory
group of instructional tasks corresponding with the level of knowledge
Number Sense
the ability to think flexibly about numbers including various ways to represent and use numbers
Procedural Knowledge
how to complete an algorithm or procedure
Conceptual Knowledge
connected knowledge; mental connections among mathematical facts, procedures, and ideas.
Counting on
the ability to start counting from a given number other than one
Composing vs. Decomposing Numbers
putting numbers together vs. taking them apart
Base-ten Grouping
count by groups of tens and ones; 53 = 5 sets of 10s and 3 singles
Unitary Grouping
count by ones
Equivalent Grouping
non-standard base-ten; before counting students would trade and then count 10, 20, 30....
Standard Language vs. Base-ten Language
fifty three vs. five tens and three singles
Five Levels of Understanding Place Value
1) single numeral;
2) position names;
3) face value;
4) transition to place value;
5) full understanding
Compatible numbers
numbers that easily combine to equal benchmark numbers
Direct Modeling
involves the use of manipulatives or drawings along with counting to directly represent the meaning of an operation or story problem
Invented Strategies
any strategy other than the standard algorithm or does not involve the use of physical materials or counting by ones
Standard Algorithms
1) knowing the step by step procedure and how it is performed;
2) knowing why the algorithm works and how it can be applied;
3) knowing when an algorithm should be used and comparing the value and usefulness of different algorithms.
Invented Strategies vs. Standard Algorithms
1. Invented strategies are number oriented rather than digit oriented.
2. Invented strategies are left-handed rather than right handed.
3. Invented strategies are a range of flexible options rather than 'one right way.'
Benefits of Invented Strategies
Students make fewer errors;
less reteaching is required; students develop number sense;
the basis for mental computation and estimation;
flexible methods are often faster than the standard algorithm;
algorithm invention is a significantly important process of doing mathematics;
serve students well on standardized tests;
Computational Estimation
using higher level thinking to flexibly and quickly produce an approximate result that will work or the situation.
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