Elementary School Math Archives - Page 5 of 6

Math War over Multiplication

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The post It Ain’t No Repeated Addition by Devlin launched a math war over the definition of multiplication. Here’s an excerpt from that post:

“Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not. Multiplication of natural numbers certainly gives the same result as repeated addition, but that does not make it the same. Riding my bicycle gets me to my office in about the same time as taking my car, but the two processes are very different. Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.”

Multiplication is repeated addition is definitely not correct. Counterexample: 1/2 x 1/4. Try also doing it with integers like -5 x -4 (not that you need two counterexamples to reject a statement). But is it correct to say that in the set of whole numbers multiplication is repeated addition? I think not. You can get the result by repeated addition for this set of number yes, but that does not make repeated addition a definition of multiplication. An operation is not defined by the strategy of getting its result.

But, should teachers in the grades stop telling pupils that multiplication is repeated addition? YES! In fact, they should refrain from telling pupils any rule at all. The pupils are perfectly capable of figuring things like these by themselves given the right task/activity and good facilitation by the teacher.

And let us suppose that students get this conception that multiplication is repeated addition, is there really a problem? Their world revolve around whole numbers so it’s only logical that this will be their understanding of it. Generalizing is a natural human tendency. Something must be wrong if they will not make this connection between multiplication and addition.

What is wrong with “undoing” later? Mathematics is man-made and there’s also a lot of trial and error part in its development. That is why “undoing” and rejection by counterexample are legitimate processes . And, isn’t ‘undoing’ part of teaching? Good teachers are those who can find out or know what they should be ‘undoing’ when they teach mathematics. ‘Multiplication is repeated addition’ is only one of many ‘over-generalizations’ pupils will make that teachers need to carefully undo later. There’s “when you multiply, you make it bigger”, or “the sum of two numbers is always bigger than any of the two you added”, etc. One way to prevent an over-generalization is to offer a counterexample. But where will you get that counterexample when their math still revolves around the world of whole numbers!

As teachers, don’t we all love that part of teaching where we challenge students’ assumptions? I’m not saying that we should deliberately lead pupils to over-generalizations so we have something to undo later. For example, we don’t lead them to “division is repeated subtraction”? Most of the time oversimplifying mathematics is not a good idea.

Click link to know what others say about multiplication is not repeated addition.

Fractal as multiplication model

Elementary School Math, Geometry, Math investigations

Math investigation lesson on polygons and algebraic expressions

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Understanding is about making connection. The extent to which a concept is understood is a function of the strength of its connection with other concepts. An isolated piece of knowledge is not powerful.

To understand mathematics is to make connections among concepts, procedures, contexts. A lesson that has a very good potential for learning a well-connected mathematical knowledge is one which is organized around a mathematical investigation. This is because of the divergence nature of this task which revolves around a single tool or context.

Here is a simple investigation activity about polygons. Change the x by x unit to 1 by 1 unit if you will give this to Grade 5-6 students.

Investigate polygons with area 5x^2 units on an x by x unit grid.

Some initial shapes students could come up with may look like the following:

different shapes, the same area — Figure 1. Polygons with the same area

Note: This is a mathematical investigation so the students are expected to pose the problems they want to pursue and on how they will solve it. It will cease to be a math investigation if the teachers will be the one to pose the problems for them. The following are sample problems that students can pose for themselves.

What is the same and what is different among these polygons? How can I classify these polygons?

Possible classifications would be

a. convex vs non-convex polygons

b. according to the number of sides

What shapes and how many are there if I only consider polygons made up of squares?

Students will discover that while they can have as many polygons with an area of 5, there are only 12 polygons made of squares. This is shown in Figure 2. These shapes are called pentominoes because it is made up of 5 squares. I have arranged it here for easy recall of shapes. It contains the last seven letters of the english alphabet (TUVWKXZ) and the word FILIPINO without the last 2 I’s and O in the spelling.

Is there a way of constructing different triangles or any of the polygons with same area?

Figure 3 shows this process for triangle.

Preserving area — Figure 3.Triangle with same area

Click this or the figure below to see this process in dynamic mode using Geogebra.

Fig 4 – Preserving area of triangle in Geogebra

Possible extension of this investigation is to consider polygons with areas other than $5x^2$ .

Click this link to see some ideas on how you can use this activity to teach combining algebraic expressions.

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.

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