Which kind of calculator promotes good algebraic thinking?

I teach high school math. Students bring scientific calculators to class, or they sometimes have to borrow one from me. I have two types available: immediate execution calculators and formula calculators. I’ve been wondering lately whether one type of calculator is better for learning algebra than the other.

Here’s how they work (see Wikipedia for a longer explanation: https://en.wikipedia.org/wiki/Calculator_input_methods).

Immediate Execution

The TI-36X Solar 2004 version immediate execution calculator.

These calculators work by performing calculations along the way as you type in values and operations. For example, you can evaluate the expression

\sin(3 \times 45)

by typing 3, multiply, 45, then the sine key. As you press operations and operands the calculator will evaluate what it can according to the rules of order of operations, or BEDMAS. For binary operands (those taking two values to produce a result, like multiplication), you put the values in order. For unary operations (those taking just one value, like squaring or taking a sine), the value must be present on the calculator screen when you press the operator key. These calculators usually have a bracketing feature to allow the user to work through complex expressions without using memory storage.

Formula

The Sharp EL-510R formula calculator.

These calculators work by waiting until the user has typed in a complete expression to evaluate, then evaluating the entire expression. The order of button-pushing is pretty much as the symbols are written in the expression, making them easier to use for a lot of folks. Once a value is calculated, it’s stored in an “answer” variable in case it’s needed for the next evaluation.

Algebraic Expressions and BEDMAS

When we write out algebraic expressions, we have a number of conventions to follow. The most important convention is order of operations, which people usually learn to remember with the mnemonic BEDMAS or PEDMAS:

  • Brackets (Parentheses)
  • Exponents
  • Division and Multiplication
  • Addition and Subtraction

When evaluating (simplifying) an expression, you first simplify the smaller expressions inside brackets. Then you evaluate exponents, then division and multiplication in the order they appear, and finally addition and subtraction in the order they appear. It’s useful to think of brackets as isolating sub-expressions, which then follow the same rules. It’s also useful to think of this order as the “strength” of the operation: multiplication is a stronger operation than addition, so it holds its operands more tightly together, and it gets evaluated first.

When a student is learning order of operations, it often feels like a set of arcane rules. There is no reason, from the student perspective, that it has to be this way. In fact, it didn’t really need to be this way, but the convention was established and now it’s important to abide by it (if you want to be understood, that is).

How a calculator helps (and hinders) learning arithmetic

People often lament that today’s youth can’t perform basic arithmetic in their head. It’s unfortunately true; I often see students reach for their calculator to evaluate 35 \div 5 or even 4 \times 6 . These are facts which prior generations had drilled relentlessly and now have available as “instant” knowledge. Younger people typically haven’t spent enough time practising these computations to develop facility with them. This is partly because the calculator is so readily available.

(Aside for parents: If you have kids, please do make them practise their age-appropriate facts. It’ll help them in the same way practising reading makes things easier)

This will draw a lot of heat, I’m sure, but I think calculators do have a strong place in even K-6 learning. They let students explore quickly without the burden of computation getting in the way of non-computational learning. It’s the same effect that web-based, dynamic geometry software can have on learning relationships between figures, lines, etc. (if you’re looking for awesome dynamic geometry software, try GeoGebra – free and wonderful).

But calculators are a hindrance when students are learning to compute fluently. They allow a student to bypass some of the thinking part of the exercise. Don’t let students (or your kids) use a calculator when they don’t have to. Only use them when students need the speed for the task they’re completing.

How a calculator helps (and hinders) learning algebra (?)

Here’s the part I don’t know about, but I’m speculating about.

I think immediate execution calculators require students to understand the algebraic expressions we write, where formula calculators bypass the thinking part of evaluating expressions.

As with arithmetic, if practising evaluating expressions is not part of the learning, and might be getting in the way of the goals for learning, then either type of calculator is fine.

But as students are developing their understanding of algebra and the order of operations, the immediate execution calculator displays the results of operations as they are evaluated, while the formula calculator obscures the evaluations in favour of a single result.

When a student types 5, add, 6, square, equals into an immediate execution calculator, they see the value 36 as soon as they press the square button. There is a reminder that the square operation is immediate. Similarly when a student wants to evaluate \sin(30+45) they must type 30, add, 45, equals*, then sine, emphasizing that the bracketed portion has to be evaluated first (i.e. before the sine function is applied).

*A student can use brackets, which is equivalent to pressing equals before sine. Also, I hope anyone using the sine function knows that 30+45 is 75 and doesn’t need a calculator’s help for the addition.

Is there research?

I perused the InterTubes to find research into this question, but either it’s not out there or I’m not skilled enough to find it.

I want to know whether one calculator is better than the other for a student who is learning to evaluate expressions.

Has no one looked into this? Help?

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2 thoughts on “Which kind of calculator promotes good algebraic thinking?

  1. This is a cool question. I’m inclined to agree with your reasoning.

    It seems to me like it would be difficult to isolate enough variables with regards to a student’s understanding of algebra to determine if the calculator type has a measurable effect. How would we know how much of their understanding is the result of other factors? Perhaps this is why we can’t find any studies.

    (I suppose we could take a sufficiently large sample of math classes and have half of each class use either type of calculator, then see if there is a significant difference on their results. I’m not sure how reliable such results would be unless we could also control for students using other calculators outside of class though.)

    I also share your belief that calculators are valuable tools for learning and investigation math, especially because they are all around us on our computers and in our pockets. It’s silly to expect students to go through school not using calculators, and it’s like making them learn with a hand tied behind their backs.

    That being said, even if one type of calculator has an edge over the other, is it enough of an edge to matter? And how much of an edge is it compared to compensatory advantages of the formula calculator, such as speed/error reduction?

    • Yup, it’s probably really difficult to study on a large scale (you’ll notice I didn’t volunteer), although perhaps small studies on isolated concepts would be feasible.

      As a computer programmer (and math guy) it’s important to understand *how accurate* calculators are. I like to show students examples of catastrophic cancellation when they argue that the calculator always gives the right answer. That discussion leads to the appropriate role of the calculator, and the reasonableness-checking we need to perform while using them.

      You ask a good question: is it enough of a difference to truly matter? And it’s clear that a formula calculator is faster and more accurate in the hands of a skilled user.

      I still miss my TI-36X Solar rev. 1993. :)

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