Algebra and the
Mathematical Practices
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Warm-Up
What is the sum of 37 and 28?
Do it mentally and write down your answer.
Warm-Up
What is the sum of 37 and 28?
How did you solve it?
Warm-Up
What is the sum of 37 and 28?
What does this have to do with
Mathematical Practices and Algebra?
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Common Core State Standards for MATHEMATICS
What is Algebra?
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What is Algebra?
The traditional image of algebra, based in more than a
century of school algebra, is one of simplifying
algebraic expressions, solving equations,
learning the rules for manipulating symbols
– the algebra that almost everyone, it seems, loves to hate.
School algebra has traditionally been taught and learned as
a set of procedures disconnected both from
other mathematical knowledge and from
students' real worlds.
From Teaching and Learning a New Algebra by James J. Kaput
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What is Algebra?
In traditional algebra students memorize procedures
that they know only as operations on strings of symbols,
solve artificial problems that bear no meaning to
their lives, and are graded not on understanding of the
mathematical concepts and reasoning involved, but on
their ability to produce the right symbol
string—answers about which they have no reason to
reflect and that they found (or as likely guessed) using
strategies they have no need to articulate.
From Teaching and Learning a New Algebra by James J. Kaput
What is Algebra?
Worst of all, their experiences in algebra too often
drive them away from mathematics
before they have experienced not only their own
ability to construct mathematical knowledge and to
make it their own, but, more importantly, to
understand its importance and usefulness to their
own lives.
From Teaching and Learning a New Algebra by James J. Kaput
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What is Algebra?
Although algebra has historically served as a
gateway to higher mathematics,
the gateway has been closed for many
students in the United States, who are
shunted into academic and career dead ends
as a result.
From Teaching and Learning a New Algebra by James J. Kaput
What is Algebra? Really…
Algebraic reasoning in its many forms, and the use
of algebraic representations such as graphs, tables,
spreadsheets and traditional formulas, are
among the most powerful intellectual
tools that our civilization has
developed. Without some form of symbolic
algebra, there could be no higher mathematics
and no quantitative science, hence no technology
and modern life as we know them.
From Teaching and Learning a New Algebra by James J. Kaput
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Our Challenge…
Our challenge then is to find ways to make the
power of algebra (indeed, all mathematics)
available to all students - to find ways of
teaching that create classroom
environments that allow students to
learn with understanding.
From Teaching and Learning a New Algebra by James J. Kaput
What we know…
What we already know about algebra teaching and learning:
• begin early in part, by building on students’ informal
knowledge
• integrate the learning of algebra with the learning of other
subject matter by extending and applying mathematical
knowledge
• include the several different forms of algebraic thinking by
applying mathematical knowledge
• build on students' naturally occurring linguistic and cognitive
powers encouraging them at the same time to reflect on what
they learn and to articulate what they know
• encourage active learning and the construction of
relationships that puts a premium on sense-making and
understanding.
From Teaching and Learning a New Algebra by James J. Kaput
What does the Common Core have to
say about algebra?
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One of the common domains in grades
K through 5 is
Operations and Algebraic Thinking.
Why are these two concepts linked?
Big Ideas of Early Algebra
Equivalence and Equations
“Equals” means equivalent sets rather than a place to
write an answer. Simple real-world problems with
unknowns can be represented as equations. Equations
remain true (balanced) if the same change occurs to each
side. Unknowns can be found using the balance strategy.
Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.
Big Ideas of Early Algebra
Patterns and Functions
Operations almost always change an original number to a
new number. Simple real-world problems with variables
can be represented as “change situations”. “Backtracking”
or reversing a change can be used to find unknowns.
Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.
Big Ideas of Early Algebra
Properties
Arithmetic properties apply. The commutative law and
associative law apply to addition and multiplication but
not to subtraction and division. Addition and subtraction
are inverse operations, and multiplication and division are
inverse operations. Adding or subtracting zero, and
multiplying or dividing by 1, leaves the original number
unchanged. In certain circumstances, multiplication and
division distribute over addition and subtraction.
Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.
Big Ideas of Early Algebra
Representations
Different representations (e.g. graphs, tables of values,
equations, drawings, everyday language) help with
identifying trends and finding and interpreting solutions to
real-world problems.
Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.
Operations and Algebraic Thinking.
Why are these two concepts linked?
Most current researchers agree that the algebraic
reasoning most appropriate for elementary school
typically falls into one of two subcategories:
Generalized Arithmetic
and
Functions.
Early Algebra and Mathematics Specialists by M.K. MURRAY
Generalized Arithmetic
The reasoning that occurs as students recognize
patterns that emerge during their study of the
four basic operations, and to the claims they make
and later justify, and eventually express with
symbolic notation.
Early Algebra and Mathematics Specialists by M.K. MURRAY
Generalized Arithmetic
For example, a student solving the problem 37 +
28 may take 3 from the 28 and add it to 37; the
resulting problem becomes 40 + 25. At first, the
student may state a generalization of what he
notices as with words: When you take an amount
from one addend and add the same amount to
the other addend, you still get the same total
when you add them together.‖ This serves as the
basis for the symbolic expression of the
relationship, (a+b) = (a+c) + (b-c).
Early Algebra and Mathematics Specialists by M.K. MURRAY
Generalized Arithmetic
7 + 5 = 12
7 + 6 = ___
7 + 5 = 12
8 + 5 = ___
9 + 4 = 13
9 + 5 = ___
9 + 4 = 13
10 + 4 = ___
What do you notice?
Make a statement about this.
How would you convince someone this is true?
Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter
Generalized Arithmetic
7 - 5 = 12
7 - 6 = ___
7 - 5 = 12
8 - 5 = ___
9-4=5
9 - 5 = ___
9-4=5
10 - 4 = ___
Does this work for subtraction?
Is there another rule?
Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Common Core State Standards for MATHEMATICS
What Mathematical Practices are we using?
Generalized Arithmetic
7 x 5 = 35
7 x 6 = ___
7 x 5 = 35
8 x 5 = ___
9 x 4 = 36
9 x 5 = ___
9 x 4 = 36
10 x 4 = ___
How is multiplication different from addition?
In a multiplication problem, if you add 1 to a
factor, I think this will happen to the product…
Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter
Generalized Arithmetic
7 x 5 = 35
7 x 6 = ___
7 x 5 = 35
8 x 5 = ___
9 x 4 = 36
9 x 5 = ___
9 x 4 = 36
10 x 4 = ___
Choose one of the original equations and either
draw a picture, make an array, or write a story
for the original equation. Then change it just
enough to match the new equations.
Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter
Generalized Arithmetic
What are some of the generalizations we made?
Articulate them.
Keeping symbols connected to representations
is the key.
Early Algebra and the Common Core: What Do Teachers Need to Know? A lecture by Susan Jo Russell and Deborah Schifter
Functions
Refers to the generalization of numeric patterns.
Such patterns often arise from contextual
situations, and may be represented with pictures,
number lines, function tables, symbolic notation,
and graphs.
Early Algebra and Mathematics Specialists by M.K. MURRAY
Functions
For example, six pennies are added to a jar every
day and the children analyze the growth.
Early Algebra and Mathematics Specialists by M.K. MURRAY
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Pool Border
How many different
ways can you find to
count the border tiles
of an 8 x 8 pool
without counting them
one at a time?
Pool Border
Another approach to the Border Problem is to build a
series of pools in steps, each with one more tile on
the side, and then find a way to generalize the
number of border tiles on an n x n pool.
Adapted from Lappan, Mundy and Phillips, “Experiences with Patterning” in Algebraic Thinking: Grades K-12; NCTM, 1999.
Operations and Algebraic Thinking.
Why are these two concepts linked?
Operations and Algebraic Thinking
K
• Understand addition as putting together and adding to, and understand subtraction as
taking apart and taking from.
1
• Represent and solve problems involving addition and subtraction.
• Understand and apply properties of operations and the relationship between addition
and subtraction.
• Add and subtract within 20.
• Work with addition and subtraction equations.
2
• Represent and solve problems involving addition and subtraction.
• Add and subtract within 20.
• Work with equal groups of objects to gain foundations for multiplication.
3
• Represent and solve problems involving multiplication and division.
• Understand properties of multiplication and the relationship between multiplication
and division.
• Multiply and divide within 100.
• Solve problems involving the four operations, and identify and explain patterns in
arithmetic.
Common Core State Standards for MATHEMATICS
Operations and Algebraic Thinking
4
• Use the four operations with whole numbers to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.
5
• Write and interpret numerical expressions.
• Analyze patterns and relationships.
Common Core State Standards for MATHEMATICS
Closing
and
Reflections