adding and subtracting integers Archives - Mathematics for Teaching

From whole numbers to integers – so many things to “unlearn”

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A lot of online resources on integers are about operations on integers especially addition and subtraction. Most of these resources show visual representations of integer operations. These representations are almost always in the form of jumping bunnies, kitties, frogs, … practically anything that can or cannot jump are made to jump on the number line. Sometimes I wonder where and when in their math life will the students ever encounter or use jumping on the number line again. If you want to know why I think number line might not work for teaching operations, click link – Subtracting integers using number line – why it doesn’t help the learning.

Of course there may be other culprits apart from rote learning and the numberline model. Maybe there are other things that blocks students’ understanding of integers especially doing operations with them.

Before integers, students’ life with numbers had been all about whole numbers and some friendly fractions and decimals. So it is not surprising that they would have made some generalizations related to whole numbers with or without teachers help. I pray of course that teachers will have no hand in arriving at these generalizations and that if indeed students will come to these conclusions, it should be by the natural course of things. Here are some dangerous generalizations.

generalization about whole numbers — over-generalizations about whole numbers

These generalizations are very difficult to unlearn (accommodate according to Piaget) because based on students experiences they all work and are all true. Now, here comes integers turning all of these upside down, creating cognitive conflict. In the set of integers,

when a number is added to another number it could get smaller (5 + -3 gives 2; 2 is smaller than 5)
the sum of any two numbers can be smaller than both of the addends (-3 + -2 gives -5; -5 is smaller than -3 and -2)
when a number is taken a way from another number, it could get bigger (3 – -2 = 5, 3 just got bigger by 2)
you can get an answer for taking away a bigger number from a smaller number (3 – 5 = -2)
when a number is multiplied by another number, it could get smaller (-3 x 2 = -5)
when a number is divided by another number, it could get bigger (-15/-3 = 5)

On top of these, mathematics is taught as something that gives absolute result. So how come things change?

You may be interested to read my article on Math War over Multiplication. It’s also about overgeneralization.

Feel free to share your thoughts about these.

Assessment, Curriculum Reform, Elementary School Math, High school mathematics, Number Sense

Assessing conceptual understanding of integers

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Assessing students’ understanding of operations involving integers should not just include assessing their skill in adding, subtracting, multiplying and dividing integers. Equally important is their conceptual understanding of the process itself and thus need assessing as well. Even more important is to make the assessment process a context where students are given opportunity to connect previously learned concepts (this is the essence of assessment for learning). Because the study of integers is a pre-algebra topic, the tasks should also give opportunity to engage students in reasoning, number sense and algebraic thinking. The tasks below meet these criteria. These tasks can also be used to teach mathematics through problem solving.

The purpose of Task 1 is to encourage students to reason in more general way. That is why the cells are not visible. Of course students can solve this problem by making a table first but that is not the most ideal solution.

adding integers — Task 1 – gridless addition table of integers

A standard way of assessing operations involving integers is to ask the students to perform the operation. Task 2 is different. it is more interested in engaging students in reasoning and in developing their number and operation sense.

subtracting integers — Task 2 – algebraic thinking and reasoning in numbers

Task 3 is an example of a task with many possible solutions. Asking students to find a relation between the values in Box A and Box B links operations with integers to the study of varying quantities or quantitative relationship which are fundamental concepts in algebra.

integers and variables — Task 3 – Integers and Variables

Algebraic thinking and subtracting integers – Part 2

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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.

The following subtraction table of operation can be generated by the students using the activity from my algebraic thinking and subtracting integers -part 1.

Now, what can you do with this? You can use the following questions and tasks to scaffold learning using the table as tool.

Q1. List down at least five observation you can make from this table.

Q2. Which of the generalizations you made with addition of table of operation of integers still hold true here?

Q3. Which of the statement that is true with whole numbers, still hold true in the set of integers under subtraction?

Examples:

1. You make a number smaller if you take away a number from it.

2. You cannot take away a bigger number from a smaller number.

3. The smaller the number you take away, the bigger the result.

Make sure you ask students similar questions when you facilitate the lessons about the addition of integers. See also: Assessment tasks for addition and subtraction of integers.

Elementary School Math, Number Sense

Subtracting integers using numberline – why it doesn’t help the learning

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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.

subtracting positive using number line — 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:

subtracting negative using number line — taking away a negative integer

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.

Click link for an easier and more conceptual way of teaching how to subtract integers without using the rules.

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