**I am so very pleased to welcome Dr. Shari Beck of Teaching to Make a Difference to Minds in Bloom. Her in-depth post on Developing Algebraic Thinking shows you exactly how to apply the standards to each elementary grade level.**

Algebraic Thinking is part of one of the major
content strands included in the Common Core State Standards for Mathematics,
and a vertical progression of algebraic concepts can be seen in the specified
standards from one elementary grade level to the next. Beginning in Kindergarten, students are asked
to find missing numbers that sum with another number to make 10. In Grades 1-2, students must find unknown
whole numbers in all positions of addition and subtraction problems. The objective of finding missing unknowns is
extended to include multiplication and division in Grade 3 where students also
begin searching for arithmetic patterns.
By Grade 4, students are required to solve multistep word problems where
they begin to represent unknown quantities in an equation with a letter. In Grade 5, a focus on patterns is included
by having students identify apparent relationships between the corresponding
terms of two numerical patterns.
Students are actually required to write ordered pairs of numbers and
graph these ordered pairs in a coordinate plane.

The idea of teaching algebra to young learners
can be an intimidating thought for some, especially for teachers who may have
struggled with an Algebra class in high school or college. While some may think that algebraic thinking
requires higher order thinking skills and may not be feasible for all students,
the developers of the Common Core integrated this type of thought process into
the standards with the idea that students should be exposed to algebraic
thought beginning as young as Kindergarten.
Teachers can accomplish the goal of each standard by using various
representations in the classroom, and this type of approach can foster creative
and critical thought processes. When
teachers are able to provide problems that actually relate to the real world of
the student and have those students relate ideas among the various
representations, the development of algebraic thinking with learners at all
elementary levels can occur and build a foundation for the learning of algebra
in future grade levels.

The concept of algebraic thinking can be
taught through the use of five representations which include:

- the concrete representation
- the verbal representation
- a representation by a table
- representation by a graph
- representation by an algebraic equation

Posing problems in a real world format and requiring students to work through the various representations helps develop the objectives of the strand of algebraic thinking in the Common Core Standards.

The level of difficulty with the pattern in a
given algebraic thinking problem can be increased as students progress and as
the grade level increases. The teacher
can first model each of the five representations for the students and then
allow students to creatively and critically develop their own representations
with future problems. Once students are
efficient and taking a given problem and developing the representations for it,
the teacher can then simply give a pattern and ask the students to creatively
write a problem to fit it along with critically thinking about the development
of the five representations. To further
explain the five representations, the following example problem, which would be
appropriate for students as young as Kindergarten, will be used.

As a reward for good behavior in class, the teacher hands out stickers. For each day that Ben is good, he earns 2 stickers. If Ben currently has 4 stickers, how many more days must he have good behavior to have a total of 10 stickers?

For the concrete representation of this
problem, a teacher can use basic centimeter cubes (or any other object which
can be laid out in groups). A student begins by laying out 2 objects to represent
the number of stickers that Ben has received after the first day that he had
good behavior. Then, the student lays
out 4 objects in a separate pile to represent the total number of stickers that
Ben had received after his second day of good behavior. Thus, the second pile represents two days of
good behavior. Next, the student lays
out 6 objects in a third pile to represent the total number of stickers that
Ben had received after his third day of good behavior. At this point, the teacher can remind the
students of the posed question of “…how many more days must he have good
behavior to have a total of 10 stickers?”
Students continue laying out the next pile in the pattern until a total
of 10 objects are in the pile. For the
Kindergarten standard which focuses on algebraic thinking, students are being
required to find the number that makes 10.
First and second graders are finding the unknown where 6 + ? = 10. Third graders are being required to identify
an arithmetic pattern where there is a common difference from one pile of
objects to the next. All of these tasks
relate to the algebraic thinking standards for the specified grade levels.

With the verbal representation, students are
asked to provide a description of the pattern by discussing how the number of
stickers is changing from one day to the next.
For example, a student may say that “Ben has 2 stickers after his first
day of good behavior. Ben has 4 stickers
after his second day of good behavior.
Ben has 6 stickers after his third day of good behavior, and his total
number of stickers is increasing by 2 each time. In order for Ben to have 10 total stickers,
he has to earn 4 more stickers. This would
require that he has had good behavior for a total of 5 days.” In grade levels where students are able to
write out the description, the teacher should require that it be written. Younger students may simply orally give the
description of the pattern.

When a student represents the problem with a
table, two columns are used with a label of total days of good behavior and
number of stickers. Students are asked
to complete the table based on either the concrete representation or their
verbal description. As students progress
with the concept of algebraic thought, the teacher can require that they
determine the labels of the columns in the table. Students in Grade K-5 should be able to reach
this third level of representation for the problem. The remaining two representations may only be
posed in Grades 4 and above, depending on the level of the students. Advance students in Grades K-3 can be
challenged with the remaining two representations in an effort to further
develop their critical thought processes.

The student will use the ordered pairs from
the table to plot the points in order to represent the problem with a
graph. Be sure that the students are
labeling the axes as appropriate. In
this case, the horizontal axis would be labeled based on the first column of the
table (total days of good behavior), and the second column would be labeled
based on the second column of the table (total number of stickers).

Using the table and the graph, students can
then be required to find a pattern in the number of total stickers. When the student realizes that the number of
stickers is twice that of the total number of days of good behavior, the
student can represent this expression as 2*D (or whatever letter they choose to
select for total number of days. Then,
students can choose a letter to represent the total number of stickers, such as
S, so that an equation can be formed as S=2*D.
While the “letters” are not yet referred to as “variables”, students are
laying the foundation for solving algebraic problems in higher grade levels
that involve variables.

The idea of using these five representations
to develop algebraic thinking in Grades K-5 can be developed through the use of
a graphic organizer such as the one shown below.

A detailed
explanation of how to use this type of organizer in class can actually be
downloaded for free from my TPT store.
You can download a copy of Developing
Algebraic Thinking Through Five Representations by simply clicking on the link.

Dr. Shari Beck is currently a mathematics
professor at Navarro College where she has taught for the past 12 years. Her career in education spans the last 20
years where she has a B.S. and M.S. in Mathematics and a Ph.D. in Curriculum
and Instruction. She enjoys working with
preservice teachers focusing on EC-Grade 8 and providing professional training
for inservice teachers in the field of mathematics. To view other products created by Shari, you
can visit her TPT
Store by clicking on the link.

## 2 comments:

Great post! I actually love teaching algebra to young students. It's like a game, or puzzle with a missing piece.

Thanks for the comment Alicia!

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