Algebraic thinking is about recognizing, analyzing, and developing generalizations about patterns in numbers, number operations, and relationships among quantities and their representations. It doesn’t have to involve working with the x‘s and other stuff of algebra. In this post I propose a way of scaffolding learning of operations with integers and some properties of the set of integers by engaging students in algebraic thinking. I will focus on subtracting of integers because it difficult for students to learn and for teachers to teach conceptually. I hope you find this useful in your teaching.
In my earlier post on this topic, I discussed why teaching subtraction using the numberline is not helping most students to learn the concept. In this post I describe an alternative way to teaching operations with integers that would help students develop a conceptual understanding of the operation and engage their mind in algebraic thinking at the same time.
The table of operation is one of the most powerful tool for showing number patterns and relationships among numbers, two important components of algebraic thinking. It is a pity that most of the time it is only used for giving students drill on operation of numbers. Some teachers use it to teach operation of integers but more for mastery of skills and to show some beautiful patterns created by the numbers. Below are some ideas you can use to teach operation of integers conceptually as well as engage students in algebraic thinking. I promote teaching mathematics via problem solving in this blog so this post is no different from the rest. Use the task below to teach subtraction and not after they already know how to do it. Of course it is assumed that students can already do addition.
The question “Which part of the table will you fill-in first?” draws the student attention to consider the relationships among the numbers and to be conscious of the way they work with them. It tells the students that the task is not just about getting the correct answer. It is about being systematic and logical. Engage the students in discussion why they will fill-in particular parts of the table first.
Students will either subtract first the same number and this will fill the spaces of zeroes or they can subtract the positive integers. They will of course have to define beforehand which will be the first number (minuend) and which will be the second number (subtrahend).
Surely most students will get stuck when they get to the negatives except with the equal ones which results to zero. You may then ask them to investigate the correctly filled up parts of the table that could be of use to them to fill-in the rest of the table. Students will discover that the numbers are increasing/decreasing regularly and can continue filling-in the rest of the spaces. This is not a difficult task especially if the process for teaching addition was done in the same way. Encourage the class to justify why they think the patterns they discovered makes sense.
I have reasons to suspect that for a good percentage of students, the end of their mathematics career begin when they are introduced to subtracting integers. Well, for some, it’s when the x‘s start dropping from the sky without warning. In this post, let’s focus on the first culprit – subtracting integers.
One of the most popular tools for teaching addition and subtraction of integers is the number line. Does it really help the students? If so, why do they always look like they’ve seen a ghost when they see -5 – (-3)?
Teachers introduce the following interpretations to show how to subtract integers in the number line: The first number in the expression tells you the initial position, the second number tells the number of ‘jumps’ you need to make in the number line and, the minus sign tells the direction of the jump which is to the left of the first number. For example to subtract 3 from 2, (in symbol, 2 – 3), you will end at -1 after jumping 3 units to the left of 2.
taking away a positive integer
The problem arises when you will take away a negative number, e.g., 2- (-3). For the process to work, the negative sign is to be interpreted as “do the opposite” and this means jump to the right instead of to the left, by 3 units. This process is also symbolized by 2 + 3. This makes 2 – (-3) and 2 + 3 equivalent representations of the same number and are therefore equivalent processes.
But only very few students could making sense of the number line method that is why teachers still eventually end up just telling the students the rule for subtracting integers. Here’s why I think the number line doesn’t work:
taking away a negative integerour mind can only take so much at a time
The first problem has to do with overload of information to the working memory (click the link for a brief explanation of cognitive load theory). There are simply too many information to remember:
1. the interpretation of the operation sign (to the left for minus, to the right for plus);
2. the meaning of the numbers (the number your are subtracting as jumps, the number from which you are starting the jumps from as initial position);
3. the meaning of the negative sign as do the opposite of subtraction which is addition.
To simply memorize the rule would be a lot easier than remembering all the three rules above. That is why most teachers I know breeze through presenting the subtraction process using number line (to lessen their guilt of not trying to explain) and then eventually gives the rule followed by tons of exercises! A perfect recipe for rote learning.
The second problem has to do with the meaning attached to the symbols. They are not mathematical (#1 and #2). They are isolated pieces of information which could not be linked to other mathematical concepts, tools, or procedures and hence cannot contribute to students’ building schema for working with mathematics.
But don’t get me wrong, though. The number line is a great way for representing integers but not for teaching operations.
The tasks below are for deepening students’ understanding about the absolute value of a number and provide a context for creating a need for learning operations with integers. You may give the problems after you have introduced the students the idea of absolute value of an integer and before the lesson on operations with integers.
Tasks (Set 1)
1) Find pairs of integers whose absolute values add up to 12.
2) Find pairs of integers whose absolute values differ by 12.
3) Find pairs of integers whose absolute values gives a product of 12.
4) Find pairs of integers whose absolute values gives a quotient of 12.
These may look like simple problems to you but note that these questions involve equations with more than one pair of solutions. The problems are similar to solving algebraic equations involving absolute values. Problem 1) for example is the same as “Find the solutions to the equation /x/ + /y/ = 12. Of course we wouldn’t want to burden our pupils with x’s and y’s at this point so the we don’t give them the equation yet but we can already engage them in algebraic thinking while doing the problems. The aim is to make the pupils be comfortable and confident with the concept of absolute values as they would be using it to derive and articulate the algorithm for operations with integers later in the next few lessons.
Tasks (Set 2)
1) Find pairs of integers, the sum of absolute values of which is less than 12.
2) Find pairs of integers the difference of absolute values of which is greater than 12.
Encourage students to show their answers in the number line for both sets of task.
