Posted by: degarcia | November 24, 2008

Algebraic Thinking

ALGEBRAIC THINKING

What is algebraic thinking?

“The development of algebraic thinking is a process, not an event” (Seeley, 2004). 

Kriegler believes that the term algebraic thinking “has become a catch-all phrase for the mathematics teaching and learning that will prepare students for successful experiences in algebra and beyond” (n.d ).

A Few Key Things:

Equality & the concept of equivalence: Many students have the misconception that the equal sign means “solve” or “the answer is.” In reality, the equal sign just means “the same as.” Teachers need to expose students to true/false equations about equality (i.e. 9+0=9 or 8=5+3). This is especially important when variables are introduced.

Inequality: Students need to reach a conceptual understanding of the greater than and less than symbols without memory tricks.

Positive & Negative Numbers: Early on, students need to be exposed to some negative numbers. If we always say “You can’t take 4 away from 2,” then pretty soon they will think that problem is impossible to solve.

Problem Solving & Critical Thinking: If students have these skills they can solve problems in new contexts and situations by applying conjectures and generalizations.

Making Generalizations/Conjectures: As students discover patterns and mathematical rules, it is helpful to make a growing conjecture chart for students to refer to.

Patterns: Patterns exist in math all of the time. Bahr insists, “We need to train…children to look for, and to expect to find, patterns in all math work that they do” (2008). Starting in kindergarten, students should frequently make and find patterns. As they do this, they will become more skilled with basic problems, which will prepare them to find patterns in their natural world, notice growing patterns, and make generalizations to harder problems. Likewise, when students find patterns in smaller problems, they are learning mathematical concepts that will equip them to solve more complex problems. (See http://www.learner.org/courses/learningmath/algebra/session1/part_b/index.html)

In the early grades, working with blocks and recognizing patterns in simple multiplication prepares students for later algebraic thinking, such as proportionality.

Variables: Variables are unknown and can change. They are represented by symbols. This may sound complicated, but children start solving for this “unknown” with simple problems as early as kindergarten. However, it is helpful to explicitly explain that we can abbreviate our thinking into one-letter symbols (Bahr, 2008). 

Relationships Between Numbers (Relational Thinking): Relational thinking focuses on the “why” behind the right answer. Yes, 8×9 is72 but why? It is because we are solving for 8 groups that are size 9. Bahr emphasizes that this kind of thinking can be practiced with number sentences such as: “58-27=59-r” where students solve for r. These kinds of math tasks require students to explain and justify their conjectures.

Symbolic Representation of Mathematical Ideas: Students need to learn to use equations to communicate relationships between numbers. However, it is crucial that they understand the meaning behind the symbols (i.e. variables, equal sign, etc.).

TOOLS vs. IDEAS

Many experts in the field of mathematics view algebra as tools and ideas.  Tools include problem solving skills, representation skills, and reasoning skills. Conversely, mathematical ideas include viewing algebra as generalized arithmetic, algebra as a language, and mathematical modeling.

In her article “Just What Is Algebraic Thinking,” Kriegler shared this figure about the components of algebraic thinking:

COMPONENTS OF ALGEBRAIC THINKING
Mathematical Thinking Tools Problem solving skills    

  • Using problem solving strategies
  • Exploring multiple approaches/multiple solutions

Representation skills

  • Displaying relationships visually, symbolically, numerically, verbally
  • Translating among different representations
  • Interpreting information within representations

Reasoning skills

  • Inductive reasoning
  • Deductive reasoning
Informal Algebraic Ideas Algebra as abstract arithmetic    

  • Conceptually based computational strategies
  • Ratio and proportion

Algebra as the language of mathematics

  • Meaning of variables and variable expressions
  • Meaning of solutions
  • Understanding and using properties of the number system
  • Reading, writing, manipulating numbers and symbols using algebraic conventions
  • Using equivalent symbolic representations to manipulate formulas, expressions, equations, inequalities

Algebra as a tool to study functions and mathematical modeling

  • Seeking, expressing, generalizing patterns and rules in real-world contexts
  • Representing mathematical ideas using equations, tables, graphs, or words
  • Working with input/output patterns
  • Developing coordinate graphing skills

But What Does This Mean?

Problem Solving: Having a toolkit of sorts that helps you solve unfamiliar problems. If you can see patterns and relationships, you can solve more complex problems (Kriegler, n.d.).

Multiple Representations: Explaining thinking in a variety of ways deepens our mathematical understanding.

Reasoning: Being able to reason is crucial to algebra and math. Inductive reasoning involves finding patterns in specific problems and applying them to other situations. Students use deductive reasoning when they reach particular conclusions from problem types. 

Abstract Arithmetic: Early on, teachers should turn student focus to the multiplicative nature of arithmetic. Students develop primitive algebraic thinking at an early age. This can be as simple as representing a quantity with a model (i.e. holding up three fingers to show their age).

Interesting Video: http://www.learner.org/courses/learningmath/algebra/session1/part_a/

Later, students see relationships between addition (10+20=30 and 20+10=30) and they extend these patterns to all addition problems. Project M3, a 5-year collaborative research group from several universities believes that finding generalizations like this is early algebraic thinking (Project M3, n.d.).

Language of Math: Some teachers purport that algebra is the language of math. “In short, being fluent in the language of algebra requires both understanding the meaning of its vocabulary (i.e. symbols and variables) and flexibility to use its grammar rules (i.e. mathematical properties and conventions)” (Kriegler, n.d.). However, similar to oral language, if students do not understand the “words,” they will not be able to effectively communicate. For students to be able to use algebra to communicate their thinking, they must understand the meaning behind symbols, the relationships between numbers, and the enduring patterns in numbers.

Tool to Study Functions & Mathematical Modeling:  Algebra can be extended to the real world. Further, students can use equations and graphs to illustrate mathematical patterns and relationships.

Additional Helpful Websites:

http://www2.edc.org/cope/projsupport/msp_reta/opd_demo/lc/sessions/session1.asp

http://www.learner.org/courses/learningmath/algebra/session1/index.html

National Standards–National Council of Teachers of Mathematics (NCTM, 2006)

Kindergarten: Children identify, duplicate, and extend simple number patterns and sequential and growing patterns (e.g., patterns made with shapes) as preparation for creating rules that describe relationships.

Grade One: Through identifying, describing, and applying number patterns and properties in developing strategies for basic facts, children learn about other properties of numbers and operations, such as odd and even (e.g., “Even numbers of objects can be paired, with none left over”), and 0 as the identity element for addition. 

Grade Two: Children use number patterns to extend their knowledge of properties of numbers and operations. For example, when skip counting, they build foundations for understanding multiples and factors.

Grade Three: Understanding properties of multiplication and the relationship between multiplication and division is a part of algebra readiness that develops at grade 3. The creation and analysis of patterns and relationships involving multiplication and division should occur at this grade level. Students build a foundation for later understanding of functional relationships by describing relationships in context with such statements as, “The number of legs is 4 times the number of chairs.”

Grade Four: Students continue identifying, describing, and extending numeric patterns involving all operations and nonnumeric growing or repeating patterns. Through these experiences, they develop an understanding of the use of a rule to describe a sequence of numbers or objects.

Grade Five: Students use patterns, models, and relationships as contexts for writing and solving simple equations and inequalities. They create graphs of simple equations. They explore prime and composite numbers and discover concepts related to the addition and subtraction of fractions as they use factors and multiples, including applications of common factors and common multiples. They develop an understanding of the order of operations and use it for all operations.

Grade Six: Students use the commutative, associative, and distributive properties to show that two expressions are equivalent. They also illustrate properties of operations by showing that two expressions are equivalent in a given context (e.g., determining the area in two different ways for a rectangle whose dimensions are x + 3 by 5). Sequences, including those that arise in the context of finding possible rules for patterns of figures or stacks of objects, provide opportunities for students to develop formulas.

Bibliography

Bahr, D., & de Garcia, L.A. (2008). Elementary mathematics is anything but elementary. Boston: Cengage Learning.

Kriegler, S. Just what is algebraic thinking? Retrieved December 16, 2008, from http://www.math.ucla.edu/~kriegler/pub/algebrat.html

National Council of Teachers of Mathematics. 2006. Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston,VA: National Council of Teachers of Mathematics.

Project M3. Awesome algebra: Looking for patterns and generalizations. Retrieved December 16, 2008, from http://www.gifted.uconn.edu/ProjectM3/teachers_curriculum_3_unit4.htm.

Seeley, C.L., (2004). A journey in algebraic thinking. NCTM News Bulletin, September 2004. Retrieved December 16, 2008 from http://www.nctm.org/about/content.aspx?id=936.

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