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# Math Vocabulary Words: How to Teach Elementary School Students Math Terminology So They Can Use Them Accurately

The math vocabulary words we use in the classroom has wide implications for student’s learning in elementary school. Here we look at some of the uncertainty that can arise in mathematical terms and how you can ensure that you and your students share a common math language and vocabulary.

With such a vast number of teaching methods on show in classrooms, it has never been more important to ensure that as teachers our language is as accurate as possible when talking and teaching math words and phrases to our elementary students. Ultimate Math Vocabulary Lists Grades K to 5

Download this free math vocabulary list, perfect for building maths fluency from kindergarten through to 5th grade. Download Free Now!

### How we use mathematical vocabulary in the classroom

There are four basic language abilities, and each one is also widely used in math.

• Students listen to us talking, presenting and explaining
• They read their textbooks, worksheets, and displays
• They present their math by drawing and writing
• They ask questions and discuss their ideas to improve their understanding of mathematical concepts

Understanding mathematical terms and vocabulary can stump many students, yet using mathematics language is a core competency for them. This leads us to think; how many math words do students hear in lessons, but fail both to understand and use properly?

The math terminology we use in elementary classrooms, from addend to unit fraction, has meaning but it can easily be misinterpreted by the students we’re teaching. Here’s an example adapted and inspired by David Wood’s book ‘How Children Think and Learn’.

Teacher: Do you know what volume means?

Student: Yes.

Teacher: Could you explain to me what it means?
Student: Yes, it’s the button on the TV remote control.

Math is particularly vulnerable to misinterpretation because of the peculiarities and irregularities of many math words in the English language, and this is exactly why we should never say, “Let’s do some sums.”

### The dual terminology of math

Steve Chinn (2004) in ‘The Trouble with Math’ argues that the dual terminology of math is a real problem. The colloquial nature of some common math words and phrases means that students may well have to cope with inconsistencies, which, as we know, doesn’t always turn out well!

Clearly students have a lot to cope with when they are sifting and sorting words from the everyday non-mathematical language they encounter, then ‘translating’ it in a math context. Here are some elementary math examples that you might have come across in your own teaching:

• Acute angle – acute pain
• Borrow (in subtraction) – borrow (lend)
• Bracket ( ) – bracket (shelf)
• Cancel (fraction) – cancel (ticket)
• Capacity (volume) – capacity (potential)
• Carry (addition) – carry (a bag)
• Degree ( ° ) – degree (BSc)
• Differential
• Ellipse
• Expression (xy + 3) – expression (on a face)
• Factor (15 = 5 x 3) – factor 5 sunscreen, X-factor
• Irrational (number)
• Mass (in kg) – mass (in church)
• Mean (average) – mean (nasty)
• Negative (-7) – negative (critical)
• Net (flat) – net (fishing)
• Rational (number)
• Solution (solve) – solution (dissolve)
• Take away (subtract) – take away (food)

As teachers, we can directly monitor and help students in their interpretations of language by being aware of lexical ambiguity in the words we are using. We need to think twice about the words we use but also leverage this ambiguity as a learning opportunity, to ‘translate’ familiar words into math.

### ‘False friends’ can be found in math terminology

Although as teachers, we might be vigilant about the everyday mathematical words we use in the classroom, there are some ‘false friends’ that might confuse even us. This means that we’ll then allow students to use them incorrectly too.

I’m thinking of the common question you will no doubt have seen a number of times in relation to a chessboard investigation. It asks, ‘how many squares can you see?’

The answer of course is none, because there are no squares – the question just isn’t precise enough.

What it should say is ‘how many square rectangles can you see?’

Precision in Mathematics matters. Let me explain.

### Why you need precision in your own mathematical terms

A lack of precise language can impede students’ understanding and can easily lead to a lifelong development of misconceptions in math. Although in our everyday life we can cope with a fair amount of ambiguity, in mathematics ambiguity causes problems so it’s not welcome.

A few years ago, I became all evangelical about the way students were taught mathematical shapes because I observed too many teachers passing on faulty math to their classes which often went unchallenged.

### Math terminology up close: Square and Rectangle

My number one bugbear was the use of the word ‘square’ as a noun rather than an adjective.

When I started to look at definitions in math textbooks, math dictionaries and online materials, I could see that there was a major problem as their definitions were often ambiguous too. Take a look at a math dictionary yourself and look up ‘rectangle’ and ‘square’ and you will see they are isolated from each other and seen as separate entities.

The fact is, a ‘square’ should never be called a square, but rather a square rectangle. Classroom posters showing common 2D shapes were often guilty of this too and misled students’ understanding.

The classroom environment is a second teacher and it plays such a vital role in learning. If you cover your classroom in defective posters, then expect student’s learning to be damaged too.

Curriculum materials and teaching must be meticulous in the use of math vocabulary for elementary math.

Naming a shape a ‘square’ might not have a high human cost in terms of someone losing their life (as a misplaced decimal point might do in a medicine dose), but it does have a high human cost educationally in that millions of students aren’t receiving the precision, accuracy, reliability and consistency they deserve.

How math terminology can affect cognition

I once asked 120 students in 2nd and 3rd grade to draw a rectangle. A huge majority drew this horizontally as a prototypical oblong with two sides longer than the other, and three drew ‘an oblong’ shape but vertically.

Rather interestingly, the culture of their classrooms supported their representations as they all had 2D posters of math shapes showing an oblong with its longest sides drawn horizontally and labeled ‘rectangle’. Next to this shape was a shape named ‘square’.

I recognized this shape as a rectangle, a square rectangle! Yet none of the 120 students drew a square rectangle and they saw the two shapes as two quite separate shapes.

For students in this sample, rectangles were always seen as ‘long’ and having two long sides and two short sides.

### Math terminology matters

I repeated the square and rectangle experiment once when I delivered an inset session at a ‘gifted and talented’ math day. Initially most teachers laughed and shook their heads as I asked them to draw a rectangle.

After all, they had come to learn all about helping able mathematicians and looked bemused that they were being asked to draw a rectangle! There were 87 teachers present, out of which 56 drew a prototypical oblong drawn horizontally, 28 drew an oblong vertically, two drew an oblong resting on one of its vertices.

Just one person drew a square rectangle.

I think these teachers soon realized that it was time to shape up and go back to the drawing board after I pointed out that a square is actually a rectangle…

In other schools where I have taught, I have presented students with a typical math worksheet that asks them to ‘circle the rectangles given in the shapes below’. In this worksheet I present various square rectangles and oblong rectangles. The oblong rectangles are the shapes that get all the attention.

##### A problem soon became evident

While the above research was highly informal and unscientific, it did teach me that we have a problem. And this is a problem that is quite widespread. To be absolutely clear, a square is a more specific classification of a rectangle, just as a rectangle is a more specific classification of a parallelogram, and a parallelogram is a specific classification of a quadrilateral.

What I suggest is this; wouldn’t it be better for younger learners to start learning about quadrilaterals first as a whole rather than focusing on a few specific examples? This would enable them to discover attributes of quadrilaterals and then the names of different quadrilaterals could be introduced with discussions based around similarities and differences.

Saying, ‘This is a square’, ‘This is a rectangle’ and so on is both mislabeling and misguided. This is only one example of the issues present when teaching math vocabulary to elementary students, but fortunately there are ways to prevent confusion from happening.

### How I teach common elementary math terminology

If ever I am teaching a known math misconception, and there are many, I put on a ‘Hi Vis’ vest as a signal to the class that they are about to be challenged. They find the ‘Hi Vis’ jacket a fun novelty, but they also know that wearing one means that they have to be on their guard because I am going to challenge their thinking and they are going to challenge each other. This is a great opportunity for them to question, and not to trust what I say at face value.

Here’s an example. I might hold up a shape and say, “This is a square.”  And of course there won’t be many students who disagree. I say “of course” because the images of a ‘square’ and a ‘rectangle’ have become so entrenched that to challenge them just seems plain crazy.  But it isn’t. There is no such thing as ‘a square’. There are two types of rectangle: square and non-square (oblong).

Rectangle refers to any quadrilateral whose corners are all right-angled, opposite sides are equal and parallel and its diagonals bisect each other. A square rectangle is all those things with a couple of extra bits: all four sides are equal and its diagonals cross at right-angles. So a square is definitely a rectangle but it is equilateral and equiangular too.

All the other rectangles are non-square rectangles because they have one pair of sides longer than the other. These are oblong rectangles. A rectangle can be tall and thin, short and fat or all the sides can have the same length. So, a square is a special kind of rectangle.

‘Square’ is an adjective to describe the type of rectangle so separating squares and rectangles and making them seem different is wobbly thinking. How is a rectangle different from a square? It isn’t. A square is a rectangle.

The real problem is that students in elementary classrooms do little more than learn the names of shapes rather than identifying properties, collecting shapes that fit these properties and naming sets or families.

### Mathematical terms lesson idea: Geometry Shape Hunt

A very common activity when teaching geometry is to go on a shape hunt. Students are split into groups and look around the school, searching for examples of circles, triangles, squares, rectangles and more.

After a while, everyone comes back and shares what they have found and where they have found them. Then there’s a discussion about the shapes found and their properties.

Lesson idea summary

1. Send your students on a shape hunt all around the school.
2. Gather the resulting shapes and ask your students what they have found.
3. Discuss the shapes and introduce specific definitions for each one.

But this type of lesson does students no favors if the teaching materials separate squares and rectangles without saying ‘look for square rectangles’ and ‘look for oblong rectangles’. A potentially rich math problem can quickly become muddled, confused and inaccurate.

My experience as a teacher across grade levels has taught me that the vast majority of students, and many teachers, fail to understand the inclusive nature of shape.

They fail to understand that squares can be included within rectangles because they have been misinformed by a lack of accurate terminology and clear instruction. Very often this comes from published schemes and teacher resources, and it’s a totally avoidable math misconception.

As ‘square’ and ‘rectangular’ thinking has shown, many words are poorly understood and can lead to misunderstandings that confuse and distort understanding. We have to watch out for the imprecise use of language not just amongst students, but in the math words we choose as teachers too.

### 10 ways you can improve students’ usage of correct mathematical terms in the classroom

So how can we help elementary students improve their understanding and usage of math terminology? Elementary students need regular access to activities, specifically designed to develop awareness and use of math as this is a great way to improve terminology. We can also extend students’ knowledge of subject-specific terminology by:

• Displaying a math word on the whiteboard each morning for them to copy and learn
• Collecting math words, writing out a definition or example to illustrate it and add it to a list every day
• Making a list of words that they encounter that also have another meaning in everyday life
• Using math ‘technical’ words that students will meet in tests and exams and be precise and consistent
• Emphasizing important and de-emphasizing unimportant features of mathematical text (including word problems) by using different voices. Students can mirror you by emphasizing particular words in a similar way
• Helping students by providing strategies for remembering mathematical terms words through rhymes, raps and songs
• Allowing students opportunities to express and represent the words they encounter in a variety of ways (talking, writing, drawing)
• Making learning new math terms fun, through activities such as making fold-able shapes, creating word walls and pictures and publishing student led videos about shapes
• Direct pupils to appropriate books and other reference materials
• Incorporate oral activities into your lessons

### Other useful resources for developing accurate math terminology

At Third Space Learning, tutors are trained very specifically to use math terms correctly and there are a number of resources on math vocabulary that form the basis of this:

• This math vocabulary list for elementary students looks at the principal mathematical words that students should know and provides a free downloadable vocabulary list for students to use and add to themselves
• Originally created for parents, but now used much more widely, this online math dictionary for kids has expanded definitions with examples of terms like place value, prime numbers, venn diagram, horizontal axis or x-axis and vertical axis or y-axis to name a few.
• And you might also like this article that suggests a few math vocabulary games you can use to develop fluency and reasoning at elementary school.

### Mathematical terms included in the Ultimate Math Vocabulary List

2D shape, 3D shape, Acute, Addition, Adjacent, Alternate, Angle, Area, Area model, Ascending order, Associative property, Attribute, Average, Axis of symmetry, Baker’s dozen, Base, Base angles, Bisect, Capacity, Cardinal number, Circumference, Commutative property, Composite number, Congruent, Consecutive, Coordinates, Decimal, Decompose, Denominator, Descending order, Diagonal, Difference, Digit, Dimensions, Distributive property, Dividend, Division, Dodecagon, Dozen, Edge, Equation, Equilateral triangle, Equivalent, Estimate, Even number, Expanded form, Exponent, Expression, Exterior, Face, Face value, Factor, Fraction, Greater than, Heptagon, Hexagon, Horizontal, Improper fraction, Integer, Interior, Intersection, Inverse, Irregular shapes, Isosceles triangle, Length, Kite, Less than, Line of symmetry, Mean, Median, Mixed number, Mode, Multiple, Multiplication, Numerator, Oblong, Obtuse angle, Octagon, Odd number, Order of operations, Ordinal number, Parallel lines, Parallelogram, Parentheses, Partial product, PEMDAS, Pentagon, Perimeter, Perpendicular line, Place value, Polygon, Polyhedron, Prime number, Product, Quadrant, Quadrilateral, Quotient, Rectangle, Rhombus, Right angle, Right triangle, Rounding, Scalene triangle, Square number, Squared, Standard algorithm, Standard form, Subtraction, Sum, Symmetrical, Trapezoid, Vertex, Vertical line, Whole number, Width, Word form.

### We all need to be careful with the mathematical terms we use

Mathematics is often referred to as a universal language, but it is anything but. I see it as heavily influenced by cultural misconceptions and colloquialisms that can seriously impede learning. A coordinated approach to the use of language is essential in improving the quality of teaching and learning in math.

Language is central to learning math and the better students are at using math terminology by KS2, the better they will be able to show their math knowledge. Being precise begins with stumbling, but with opportunities to use language accurately students will become better mathematicians in the long run.

Purposeful communication in math starts with knowing that volume isn’t the switch on a TV remote control and that a ‘square’ is an adjective to describe a type of rectangle.

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The content in this article was originally written by Ofsted inspector and teacher John Dabell and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza.

x #### Ultimate Math Vocabulary Lists Grades K to 5

Download this free math vocabulary list, perfect for building maths fluency from kindergarten through to 5th grade.

#### Ultimate Math Vocabulary Lists Grades K to 5 Download this free math vocabulary list, perfect for building maths fluency from kindergarten through to 5th grade.