Beginning Algebra
What do letters mean?

This is taken mostly from Algebra: Children's Strategies and Errors by Lesley R. Booth. Published by Nfer-Nelson, 1984. Children, ages 14, 15, and 16, in London were given simple algebra problems and interviewed about the meanings of the problems and possible answers. Though pages 22 and 38 nicely summarized the findings, the recorded interviews were most interesting. They also put together a 6-day teaching program, with results summarized on pages 60-70. It had moderate success.

What do letters mean? Kids are reading letters and computing with numbers. Some have experience with a variable as an unknown. Suddenly they're asked to compute with letters. How does one do that?

Often students are told, in a problem, that a letter represents a number of objects. So they might think y means pineapples, instead of a number. One child thought y might be yachts or yams, since someone took the trouble to choose the letter y.

This might lead to rules that work (for a while): 5q - 3w can't be simplified because you can't subtract 3 chairs from 5 pineapples. But treating variables as representing objects causes some fundamental problems. What does it mean to multiply or divide by an object? What does it mean to square an object? This kind of thinking can make math very hard.

One common consequence of thinking of a variable as a number of things, is that the student may "gather them up." Seeing a perimeter of a pentagon with 4 sides labelled m and one side labelled p, a student might gather them up and say the perimeter is mmmmp or 4mp, instead of "4m+p".

This may also lead to a firm experience that variables are always whole numbers. Though students can learn that there may be a fractional number of things, some things (like chairs) don't break up into fractions well. And it makes the concept of negative numbers even more unnatural than it already is.

Variables don't represent things, they represent quantities. A variable represents either a number, possibly a generalized, non-specific number, or any value (or all values, one at a time) of a set of numbers.

Imagine a pentagon with 4 sides labeled m and one labeled p. What is the length of the perimeter?

Students may try to deduce rules from what the teacher demonstrates- anything to try to get a handle on this weird activity:

• Add up all the numbers, then put down each letter that occurs (once only). If x is 3 apples and y is 5 oranges, that if you add them up, you get 8xy, 8 apples and oranges. So the perimeter is 5mp.
• Add up all the numbers, then list the letters as they occur in the expression. (4mp, if the implied '1' quantifying the p is ignored.)
• Add up all the numbers, then put down the "highest" letter, the one that occurs most often. 5m.

(Personally, I think kids listen poorly. Part of it is probably due to distractions, but part is due to not having language for what they're hearing and seeing. Instead of opening their minds, many students probably guess at interpretations and try to put it into their own words, messing up what they're hearing. Perhaps modern kids are also used to taking in information from the TV, where they don't have to make sense of it. Most of it makes sense and if they miss the rest, there's no harm. This kind of passive listening doesn't work in math.)

Question: Is (a + b + c) equal to (a + z + c) some of the time, always, never, or we don't know?

Lots of kids say never.

A common misunderstanding is for a student to think letters at the start of the alphabet have values that are less than letters at the end of the alphabet. One adds with numbers, not letters, so a, b, c, etc must have values. Maybe a=1, b=2? So b < z.

Many students know that different letters have different values.

• "You put a different letter for every different value."
• "You probably wouldn't have two letters for one number."
• "When the letters are different, I've always found the values are different."
• If z and b stand for different kinds of objects, of course they're not equal.

This is concluded from their experience that variables are used for different unknowns.

If operations haven't been fully learned, they really can't be abstracted. If they have memorized 2+5=7, but don't understand why, how can they know what a+b is? Or maybe they understand column addition- how does that apply to letters? To abstract operations to letters, they have to understand that operations operate on values, not just on digits and sequences of digits.

When pushed, they might measure or assume a particular value for a letter to get an answer. So adding a and b could result in 3, or the student could keep it as (a + b), seemingly getting the correct answer. Some students find it very difficult to answer (a + b). One child said, "That's a sum, not an answer."

Or a student might have an algorithm for an operation (such as adding in a column) that doesn't generalize. For instance, to add numbers, a student might continue counting. In the perimeter problem this doesn't work because there are no numbers.

And though a student might know how to compute a perimeter by adding m 4 times to p, they might not correctly abstract multiplication.

Notation

As above, sometimes this meant 4 + m. Adding means "gathering together" or "putting together." So adding 4 and m would naturally be written by putting them together. It's one challenge to learn new abstraction, but this notation actually goes against something the kids know.

Another natural meaning for 4m would be to think of m as substituting for digists. So if m is 10, then 4m is 410. Note that in "410", putting the 4 next to the 10 is actually an implicit addition (after multiplying times the place value.)

One student said if y is 4, then 5y is 54. "Or it could be a 5 to the power 4, making it 20" and he wrote "54, 5⁴". Confusion between products and powers was quite common.

It is a shame that the multiplication symbol '×' isn't on the standard keyboard, and looks so much like the letters x and X.

There can also be a confusion between multiplication and powers. So (m + m+ m) = m³

Or, even with numbers, students might not understand that adding a number to itself 3 times is the same as 3 times the number.

The expression 2x is a bit strange. Consider if x is 10. We can't just substitute the 10 for x and write 210. (Perhaps in teaching algebra we should always use parentheses, or always write a times symbol.)

Sometimes students are reluctant to answer an algebra question because they don't have a numeric answer, similar to the ones they've been asked to produce for many years. Sometimes they don't know if their answer is supposed to be numeric or algebraic. (For instance, can you write down a universal definition for "simplest form"?)

Summary

1. Error may arise as the result of children's ways of viewing letters in algebra, and that of particular importance in this regard are the following:
   1. that there is a confusion between letter as representing number and 'abstract' examples of the type 5x +bsp;8y.
   2. that letter-as-number is frequently construed as latter-as-specific-number, so that the possiibility of a letter assuming a range of alues is not entertained, coupled with a view that 'same letter means same number', and 'different letter means different number'.
2. Error may arise as the result of Error may arise as the result of alternative conceptions of the appropriate method and the need and/or ability to formalize and symbolize this method, and that this may be compounded of several aspects:
   1. the use of correct but more 'primitive' methods which do not readily lend themselves to algebraic representation.
   2. non recognition of the formalized operational model in the arithmetic case,
   3. unwillingness to present an algebraic expression as an answer.
3. Error may arise as the result of Confusions over algebraic notation and convention, particularly with regard to:
   1. Conjoining (writing 2 symbols next to each other) in algebraic addition, and/or
   2. the need to use it brackets to specified order of operations.

Instruction

They came up with a 6-day lesson involving a diagram of a computer with an instruction pad for input, a processing unit with boxes to contain variable values, and an output area. This had a fair amount of success, but the learning proved to be somewhat context dependent. Well after the instruction, students didn't use brackets because there was no longer a machine that needed them. They didn't perceive the need for brackets in general.

They also discovered another bit of math jargon, "If we add any number, call it n, to 3..." In math, this means any number that comes up, or that the problem writer thinks of. In English, this means the student gets to choose a number, say, 100.

So they amended the course. They gave examples where fractions were used instead of just whole numbers, and they added a lesson on brackets, such as asking kids to evaluate 4 × 5 + 2 two ways, and to offer similar expressions with answers, asking the kids to add brackets so that the computation would be true.

They gave instructions for the general case of "any", and made it clear that two variables might have the same values.

And they added a rationale for writing 3a instead of 3 × a, that the latter is too easily confused with 3 + a.

Note that some students also had a lot of trouble decoding a word problem into algebra. The researchers did not have time to tackle this issue.

One result of the work with the computer was that kids saw that sometimes a method, such as a + b, was also an answer. Still, students often had a hard time letting "the answer" not be a single number. They complained of not knowing that this kind of answer was wanted, or would be okay.

The areas of difficulty substantiated by the present research may be detailed as follows:

1. The interpretation of letters:
   
   1. Children have difficulty in grasping the notion of letters as generalized numbers. Evidence from children who have not previously encountered algebra suggests that children may have a natural tendency to interpret letters as standing for specific numbers, with different letters necessarily representing different values.
   2. While children may readily accept that test letters represent numbers (albeit specific ones), they may nevertheless in some instances handle them as though they were entities rather than quantities. This is particularly apparent in more abstract examples such as the simplification of 2a+5b+a, where children do not interpret the letters at all, but merely 'symbol push', inventing such rules as 'add all the numbers and then write down the letters.'
   3. some children are confused over the distinction between letters as representing the values or numbers relating to a measure or object, and letters as representing the measure or object itself. In line with this, some children interpret the particular letter used as being the first letter of the object represented (eg, 'y is for yachts')
2. The formalization of method:
   1. Children have difficulty in representing formal mathematical methods even in the arithmetic case. This is partly due to the fact that children often do not make explicit the precise procedures by which they solve problems.
   2. The procedures which children use in solving arithmetic problems are often informal methods which are difficult to symbolize concisely.
   3. The procedures used are often context-dependent so that they do not readily generalize to other examples (such as algebraic cases), and are symbolized (if at all) in an informal manner which requires reference to the particular context for interpretation.
   4. Children consider mathematics to be an empirical subject which requires the production of numerical answers. Even when children can formalize the required procedure and symbolize it correctly, they may not appreciate that this is an appropriate thing to do.
3. The understanding of notation convention:
   1. Largely as a consequence of the desire to obtain an (single term) 'answer' to algebraic problems, children may attempt to perform an algebraic addition, producing the conjoined term as outcome. That this association of the conjoined term and the algebraic sum is not the result of confusion with the algebraic product is shown by the fact that the same error is made by children who have no previous background in algebra.
   2. Children ignore the use of brackets, mainly because they consider them unnecessary. This belief is largely founded upon the view that:
      1. the context of the problem determines the order of operations;
      2. in the absence of a specific context, operations are performed from left to right;
      3. the same value will in any case be obtained regardless of the order of calculation.
   3. Children show signs of other notational confusions. For example, 4y may be interpreted as '4y's' (not the same as 4 times y) or as 'forty y' or as 4+y (note 3a above). Hence for a given replacement value for y, such as y=3, '4y' may be interpreted as 12, 7 or 43.

The the results of the teaching trials indicated that a teaching program based specifically upon the difficulties indicated above and designed explicitly to address the misconceptions upon which they are founded can be successful (at least in part) in restructuring children's cognition with regard to the points at issue. Observations from the teaching and test results further indicated that:

1. Gains in performance may be observed in the period following the teaching program (as in the period between immediate and delayed post-tests), and these gains may be independent of further teaching or practice.
2. There may be maturation-linked factors contributing towards the child's likelyhood of assimilating a given concept in the course of the teaching.
3. Many of the errors in elementary algebra investigated in the present research are also made by a naive (in of the sense of lacking previous exposure to algebra) pupils.
4. Children may respond correctly to questions requiring the use of certain notations or conventions (such as the use of brackets or the writeing of an algebraic sum), and yet to be unable to discriminate between correct and incorrect representation, i.e. be able to select the correct equivalents from a list which includes both correct and incorrect alternatives.

On pages 92-94 is a discussion on the implications of the research, including teaching about the meaning of letters and formalizing of methods that children do use and should use.