Chapter 13 - Promoting Algebraic Reasoning
Strands of Algebraic Reasoning
Structure in the Number System: Connecting Number & Algebra
Generalization with Number Combinations
Generalization with Place Value
Generalization with Algorithms
Meaningful Use of Symbols
Meaning of Equal Sign
Relational Thinking
The Meaning of Variables
Variables Used as Unknown Values
Variables Used as Quantities That Vary
Structure in the Number System
Making Sense of Properties
Making & Justifying Conjectures
Patterns & Functions
Growing Patterns
Repeating Patterns
Functional Thinking
Common Misconceptions with Algebraic Reasoning
Used to generalize arithmetic
Study of structures in number system, including those arising in arithmetic (algebra)
Study of patterns, relations and functions
Process of mathematical modeling, including the meaningful use of symbols
Decompose and composing numbers
Notice generalized concepts by adding one lily pad, minus one rock
Helps with finding all of the possible combinations for given numbers
Children may wonder if "1 up & 1 down" works for really big numbers
fundamental to mental math is generalizing place-value concepts
Hundreds chart is useful tool for children to realize relationship of tens and ones
Moves on the hundreds chart can be seen with arrows, my use count-by-ones approach
Pick a number, skip count by different values, find a two-skip count pattern
"when will this be true?" "Why does this work?"
Choose specific numbers that will elicit certain strategies
EX: 504-198 will elicit strategy of using a close of benchmark to make problem easier
A discussion to appreciate the relationship between numbers can make problems easier to solve
Strategy of making 10 can be applied to any of the basic addition facts
One of the most important symbols in mathematics
US textbooks rarely explain the relationship of equivalence
Most children see the = means "the answer is" not that it means it's equivalent
they wonder why the numbers don't have to be identical to be equal
When students are older, if they don't understand = they struggle with all other algebra
Conceptualizing = as a balance
Children think about the equal sign in 3 ways
- operational view: = means do something
- relational-computational: relation between answers to calculations
- relational-structural: relationships between two sides of equal sign rather than actually computing the amounts
Boxes or letters can be used in open sentences for missing numbers
Same number in different places represents it occurring in the same place
Don't ask to solve the problem but instead write an equation
More difficult for students & not explicit in curriculum
Explain the variable stands for the number of because children can confuse variable with a label
Important to let children work through the process of determining problems are the same/different based on contextual situation
Slight changes to arithmetic problem can open more opportunities to examine math ideas at hand
"What do you notice?"
Have students generalize the concept without use of numbers
Most common form of justification for young children is use of examples
Some students will be set with 1/2 examples while others may need multiple examples to understand
Children can use physical materials to justify reasoning behind the conjecture
Fair Shares for Two
pattern where core repeats - red/blue is core, continually repeats the pattern. red/blue/red/blue. Always fully repeated & never only partially shown
Help children identify core of the pattern
Focus on children finding AB patterns around them in the world (open/close door, day/night, light/dark)
Progression from step to step, sequences
Children identify core but also look for generalization of relationship that will explain how core is changing
Fairly straightforward instruction that includes visuals
Using a function machine to show the relationship in an equation
Easy, medium, hard
design functions that are appropriate for children
Use calculator to have children become comfortable with skip counting
Child thinks = means to do something
When checking "does this always work?" Child only checks a few examples
Child overgeneralizes +1/-1 in addition & subtraction
Overgeneralizes that all patterns repeat
Confuses labels as variables & vice versa