Algebra in the Early Grades
"Early algebra differs from algebra as commonly encountered in high school and beyond. It builds heavily on background contexts of problems. It only gradually introduces formal notation." (Carraher et al. 2008)
What do we mean algebra in the early grades? How does it differ from what you know as algebra?
What is algebraic thinking?
Carraher et al. (2008) lists two components of algebraic thinking:
(1) mathematical thinking tools (problem solving, representation, and reasoning skills)
(2) algebraic ideas (functions, patterns, variables, generalized arithmetic, and symbolic manipulation)
Kaput (1999) describes four different forms of algebraic reasoning:
(1) Generalizing from arithmetic and from patterns in all of mathematics
(2) Meaningful use of symbolism
(3) Study of structure in the number system
(4) Study of patterns and functions
(5) Process of mathematical modeling, which integrates the first four
“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 functions. Far from a topic with little real-world use, algebraic thinking pervades all of mathematics and is essential for making mathematics useful in daily life.” (VandeWalle, 2007, p. 259)
“Algebra is often referred to as “generalized arithmetic,” meaning in part that letters are used to denote expressions or statements that apply to any number; for example, “Not only is 4 + & equal to 7 + 4, but x + y is equal to y + x for any numbers x and y.” To see Algebra only in this guise, however, is to misrepresent the role of generalization in algebraic thinking…. To identify algebra as generalized arithmetic can be misleading, as well--- especially if it oversimplifies the challenge and induces the strategy of merely telling students they need only work by analogy; that is, what they have done with particular numbers they now will do with letters.” (Driscoll, 2007, p.64)
So what is this role of generalization that Driscoll is talking about?
Let's investigate this through appropriate age-level problems. An inquiry and/or induction teaching technique should be used with these problems.
Candy boxes problem (Given to eight year olds, Grade 3)
The box in my left hand is John's box, and all of John's candies are in that box.
The box in my right hand is Mary's, and Mary's candies include those in the box as well as three additional candies resting atop the box.
Each box has exactly the same number of candies inside.
What do you know about the number of candies John and Mary have?
Represent with pictures what you know about the number of candies John and Mary have.
Representations can take the form of words, pictures, tables, graphs.
63.4% of the students at this age focus on a single case
36.5% of the students refrained from assigning values to the amounts (an indeterminate amount)
A representation of an indeterminate amount on the part of a student is a placeholder for the eventual introduction of a variable.
What do students do to represent an indeterminate amount?
<insert picture here>
A table can lead them away from "single case" thinking toward an "indeterminate amount" thinking.
|Table of possible outcomes|
Wallet Problem (Given 18 months later, Grade 4)
Mike has $8 in his hand and the rest of his money is in his wallet;
Robin has exactly 3 times as much money as Mike has in his wallet.
What can you say about the amounts of money Mike and Robin have?
Represent with pictures what you know about the amounts of money that Mike and Robin have.
Representations can take the form of words, pictures, tables, graphs.
62.2% of the students used algebraic notation to capture the functional relationships among the variables in the Wallet Problem.
|In Mike's Wallet||Mike (in wallet and hand)||Robin|
How to help students make initial sense of variables and variation
Use tables to draw attention to multiple possibilities
Use graphs to help them visualize the covariation
Use algebraic symbolic notation to lead them to generalize
There are two facets of symbolic knowledge:
Production - writing the number sentences
Recognition - ability to determine whether the number sentence represents the context of the problem
Children's ability to grasp the meanings of a symbol is determined by three major factors:
The representation space - which includes the database of symbols, metaphors, and representation structures
The development of symbolic thinking -
The conceptual mathematical knowledge space -
Definition: The Purpose of Early Algebra
The purpose of early algebra is to provide students with the kinds of sense-making experiences that will enable them to engage appropriately in algebraic thinking (namely the role of generalization).
Two great problems from, Fostering algebraic thinking: a guide for teachers grade 6-10.
Activity 1: Skyscraper Windows (from Driscoll chapter 4)
You manage the Milwaukee Skyscraper building. This building is 12 stories high and is covered entirely by windows on all four sides. There are 38 windows per floor. Once a year, all the windows are washed. The cost for washing the windows is $2.00 for each first-floor window, $2.50 for each second-floor window, and $3.00 for each third-floor window, and so on. You have budgeted $2500 for window washing for the next year. Will this be enough to wash all of the windows? By how much are you off (+ or -)?
Some students will work this problem arithmetically by multiplying 38 times $2.00, adding that to the product of 38 and $2.50, and so on until the twelfth floor is reached and then compare to the budgeted amount of $2500.
In order to get students thinking algebraically, ask students questions like: “Suppose the building is 30 stories tall and we have $40,000. Is that enough money? Instead of figuring it out like the last one, what can we do to make it easier? Try making a chart, and what do you notice about how the numbers group, how might we make an easier way?” Students can then learn how to create a formula or “generalization”.
Activity 2: String of Pearls (from Driscoll Chapter 4)
On a string of pearls, the largest pearl is in the center and the smallest pearls are on the ends. Each of the small pearls on the two ends cost $1; each of the next larger pearls costs $2 each; the third pearl from each end costs $3 each; and so on. On the basis of this plan, how much would a string of 9 pearls cost? 12? 25? n?
A necklace with 7 pearls can be figured arithmetically like this:
1 + 2 + 3 + 4 + 3 + 2 + 1.
For a necklace with 8, it can be figured like this:
1 + 2 + 3 + 4 + 4 + 3 + 2 + 1.
But we want students to work towards thinking algebraically by creating a formula that works no matter what the size of the necklace. For the case of n pearls, a way of expressing the sum 1 through n is n(n+1)/2
So if n is even, we need to compute the sum of 1 through n/2 and multiply the sum by 2. Using the formula, the sum is (n/2)[(n/2) + 1]/2 which equals n(n+2)/8. Twice this is n(n+2)/4. So, the answer for n even is n(n+2)/4 dollars.
Important questions to ask in order to guide students towards generalization:
How does this expression behave like that one?
How is the calculating situation like/unlike that one?
When I do the same thing with different numbers, what still holds true? What changes? (Driscoll, 2007, p. 65-66)
Article Summaries and Questions:
Thinking Algebraically across the Elementary School Curriculum
This article discusses ways that elementary teachers can not only teach algebraic thinking throughout math curriculum, but how they can incorporate it into different areas of the classroom. It talks about how one teacher used literacy to enforce algebraic thinking in her third grade classroom, and then further into science and social studies. It addresses using arithmetic skills and how they become more meaningful when students’ minds are geared towards algebraic thinking.
Lannin talks about a progression which, "...helps students identify which factors vary and which remain the same" (p.343) What is that progression? And what does he mean by factors that vary and remain the same?
Lannin also gives six strategies students attempt to use as they generalize. Rank them from simple to complex. He then goes on to explain how student like to use a sort of proof by examples rather than a more formal proof by induction. How do we move students to a more formal type of proof for their generalizations?
Developing Elementary Teachers’ Algebra Eyes and Ears
This article talks about using activities to get students to think more algebraically. For example asking students questions such as, “How do you know this is true?” or “Does this always work?” in order to guide them towards generalization. There are many examples included of how a teacher has used activities and discussion with the students to foster thinking that helps develop the algebraic skills necessary in later math at an early grade.
Sores, Blanton, and Kaput
state that teachers can take a regular problem in the elementary curriculum and
algebrafy it. What does it mean to algebrafy a problem? What
components would you change? And how might your approach change?