Teaching Algebraic Thinking Skills
Teacher Resource
Have you ever heard your students say, “Why do I have to learn algebra?” This is a great question that deserves an answer. Algebra is one of the cornerstones of today’s technology boom. Without algebra we wouldn’t have TV, radio, telephone, microwave ovens or other gadgets that make our modern world so comfortable and interesting.
Algebra is fundamental to understanding mathematical thought. It encompasses an understanding of patterns and functions, ways of representing and analyzing mathematical structures and situations, models that represent quantitative ideas and relationship and approaches that analyze change in a variety of situations. Most people recognize that algebra is needed by scientists or engineers, but algebraic thinking and reasoning are also used by many professions, including health care providers, graphic designers and home builders.
Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and function.
But this is not the algebra you most likely experienced in high school. The focus now is on the type of thinking and reasoning that prepares students to think mathematically across all areas of mathematics. Being expected to teach algebra – at any level – can arouse anxiety. However, research support that algebraic thinking is necessary for today’s workforce. Blanton and Kaput (2003) state that teachers must find ways to support algebraic thinking and create a classroom culture that values “students modeling, exploring, arguing, predicting, conjecturing, and testing their ideas, as well as practicing computational skills.” Think about how to ‘algebrafy’ current curriculum by transforming existing activities and word problems from a single numerical answer to opportunities to discover patterns and make generalizations about mathematical facts and relationships. Encourage students to discuss why they believe a statement or solution to a problem is correct. Use the following prompts as ways to extend student thinking:
Tell me what you were thinking.
Could you solve this in a different way?
How do you know that is true?
Does that always work?
This type of relational thinking is crucial for students in the GED classroom, as well as for adults in the workplace. Research supports that without instruction, students bypass problem representation and try to just solve the problem. But before beginning the process of teaching algebra, teachers should be sure that students understand the basics. The key prerequisites for students to be successful in the study of algebra are to first understand the technical language of algebra; concept of variables; and concept of relations and functions.
When teaching algebra, teachers need to use practical experiences that go beyond the mere computation required by equations. When developing practice activities in the algebra classroom, be sure to:

Develop processes/procedures for students to use when approaching algebraic tasks

Create authentic exercises that highlight the critical attributes related to the concept being taught

Provide opportunities for students to verbalize the task and predict what type of answer is expected

Offer opportunities for students to discuss and write responses to questions dealing with key concepts being learned

Select authentic exercises that anticipate future skills to be learned

Design authentic exercises that integrate a number of ideas to reinforce prior learning as well as current and future concepts
Here are a couple suggestions to start to get your students involved with algebraic thinking. Be sure to set up Tcharts to help organize your results. Seeing the data in chart form can help students recognize the pattern that is occurring. Be sure to help them generalize their pattern as an equation
The Handshake Problem
How many handshakes will there be if each person in your group shakes the hand of every person once? Start with a group of three people, then increase the group size by one person each time. Students can share and discuss different solution strategies. Generalize the total number of handshakes for an arbitrary sized group.
The Table Problem
How many people can be seated at a table constructed on 2 square card tables pushed together on one side. Only one person can sit on a side. And no one straddles the legs of the table. Now push 3 card tables together to make one long table. How many people can be seated at this table. How many people could sit at a long table constructed of 10 card tables? How many could sit at a table made with “n” card tables?
A Visit to the County Fair
How much will it cost to ride the amusement rides at the county fair? It costs $5 to enter the fairgrounds. Amusement rides are $1.25 per ride. What is the total cost if a visitor rides 1 ride? 5 rides? 10 rides? What cost changes in the problem? What cost stays the same in the problem? If you want to spend no more than $20, how many rides can you ride? Are there other costs that influence your cost for a day at the fair?
Blanton, M. L. & Kaput, J. J. (October 2003). Developing Elementary Teachers’ “Algebra Eyes and Ears”. Teaching Children Mathematics, (10)2, 7077.
Teaching Algebraic Thinking Skills Teacher Resource
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