How To Help Elementary School Students Develop The Mathematics Language Skills They Need To Succeed In Math

Mathematics language is an aspect of math teaching we may not often consciously consider when planning and delivering lessons, because we’ve already been through the process of absorbing and becoming fluent in it. 

But explicit teaching of mathematical language is one of the most important and impactful aspects of developing our students into able mathematicians.

With that in mind, let’s take a detailed look into why mathematics language is important, how it might be presented, and how we can teach it effectively.

What Is Mathematical Language?

Before we can define mathematical language, we must first consider what we believe the function of language itself to be. Language is defined as a human form of communication, either spoken or written, consisting of the use of words in a structured way.

So, mathematical language is the means through which children can communicate meaning and mathematical ideas and ensure it is presented in a structured way.

Whether written or spoken, we want children to be able to present their mathematical thinking, reasoning and understanding of mathematical concepts through the use of math vocabulary.

After all, if you are to be fluent in a language, you must be able to both think and speak in it. Learning mathematics is therefore also language learning.

The Common Core Math State Standards explicitly states that we must teach children to “…give carefully formulated explanations to each other (in elementary school). By the time they reach high school they have learned to examine claims and make explicit use of definitions”.

Through doing this, children can construct their own understanding and add to their existing schemas of knowledge. If we teach vocabulary instruction explicitly and place an expectation on children to reciprocate, it will deepen their understanding of the content we teach.

It’s important to note that this explicit instruction is not only for English language learners or English as a second language students, it is necessary for all students.

How Does Mathematics Language Differ From ‘Normal’ Written Or Spoken Language Used in Everyday life?

If we are to view language as the communication of meaning, mathematical language differs significantly, because it extends to include the use of graphs, charts, mathematical symbols, diagrams and numbers.

This variation in the representation of language can make it more difficult for children to construct meaning. It is evident that math, as a language, has a rather unique phraseology and vocabulary.

Our goal is for children to understand the disciplinary literacy of mathematics, so that they can become truly mathematically literate. It is this understanding of mathematical language where children often falter during reasoning and problem solving.

8 Dimensions Of Mathematical Language Found In Reasoning Questions

This is, however, understandable, as mathematical language can be complex and confusing. Below are eight dimensions of math language I have identified that children encounter in reasoning questions:

1. The range of synonyms linked to one of the four calculation types (take instead of subtract, product instead of answer, altogether rather than add)
2. An understanding of superlatives (biggest, largest, tallest, smallest)
3. Words that can have different meanings outside of a mathematical context (round, product, factor, prime)
4. Terms other than superlatives that suggest comparison (between, more/less than, each, share, in order, sorting, put in the correct place)
5. Their understanding of the difference between the right answer and the wrong answer (best estimate, explain why Jack is not correct, write the correct symbol in each box, circle the improper fraction that is equivalent)
6. Verbs implying mathematical meaning (remaining, left, combine, collect, spend)
7. Compression of vocabulary through nominalisation and noun phrases – prime number, improper fraction, roman numeral, perpendicular and parallel lines, 3D shape
8. Abstract nouns – circumference, multiplication, area, perimeter

Why Is Mathematical Language Important?

Language is not learned incidentally, nor through limited and infrequent exposure. Teaching vocabulary in this way leads to a superficial level of understanding and prevents the mathematical proficiency we want our students to develop.

A child cannot pursue mastery of a mathematics topic without understanding the vocabulary associated with it. It is this vocabulary that truly enables children to engage with concepts and build a deep understanding of them.

By ignoring explicit vocabulary instruction and the understanding it affords, we are preventing children from achieving the depth of knowledge we want them to achieve. How can a child reason out word problems or justify their arguments, if they cannot think in the language that we are asking them to communicate in?

Realizing that even the most proficient mathematics students could struggle with vocabulary sparked my initial interest in vocabulary instruction. When completing a state test, one of my most knowledgeable mathematicians put up his hand. He looked at the question below and asked me what ‘combine’ meant.

Difficulties Caused By Overlooking Maths Language Teaching

Naturally, I was unable to tell him, but he had enough understanding to make an educated guess, because he could gather information from ‘how much higher’.

What about his peers who were less knowledgeable of mathematical language? I can recall that more than one chose to subtract rather than add.

This is only the fourth question of the test I may add, where cognitive load was significantly less intense than the questions at the other end of the test.

This was only my second year of teaching. I had not even heard of explicit vocabulary instruction at this point in my career. This seems to be a common issue, as explicit vocabulary instruction is often overlooked in math classrooms (Adams, 2003; Riccomini and Witzel, 2010).

When we overlook language, we relinquish the opportunity to activate prior knowledge, to revise content already learned, for children to construct meaning and the chance to consolidate important language to long-term memory.

And while language may be ever-present in some math classrooms, it is often only a surface-level exposure. Some mathematics teachers may only display keywords on their working walls; others may misunderstand a mastery approach and rush children through content to get them to a problem-solving stage quicker.

Either way, vocabulary instruction is insufficient and the math classroom is being robbed of the vestiges of mathematical language. Too often we are tasking children with problem solving without having the language foundations to attempt questions independently and successfully; a realization I came too late to for my class taking their tests!

Teachers have tried to reduce the complexity of mathematical language by getting children to circle keywords, discuss with partners who have a lack of understanding or by using RUCSAC (Read, Underline, Choose, Solve, Answer and Check). These methods are trying to oversimplify a complex process. Mathematical language is not this straightforward.

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How Third Space Learning embeds math language teaching into 1-to-1 online tutoring

Helping students become mathematically fluent is a key part of Third Space’s philosophy, and effective teaching of mathematical language is woven into every part of the 7000 online, 1-to-1 interventions delivered to students every week.

Important vocabulary is highlighted by tutors in their session notes, which they can then use in lessons to help students identify and make sense of the topic(s) they are studying. Tutors are also given regular CPD sessions to ensure their own command of the ‘math language’ stays topped up.

The intervention lessons themselves reinforce the vocabulary being learned by repeating it through several layers of context so that students gain an understanding of both the words and the ways in which they may be used as part of a problem – which in turn makes it more likely that they will retain and fully understand what they have learned.

To find out more about how Third Space Learning’s online 1-to-1 interventions could help your students become more fluent mathematicians, book a demo today.

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Mathematical Language Must Be Explored To Be Absorbed

Simply exposing children to math language is insufficient. We must first understand the difficulties that language presents, before we can begin to counteract them and therefore teach children how to interpret more successfully. It is because of this that the three methods mentioned above are ineffectual and not fit for purpose.

Additionally, if meaning is only ever presented by the teacher and negotiated, then children will never consciously construct meaning for themselves. Meaning is derived from the context within which it manifests.

Therefore, it is the teacher’s duty to frame the dialogue for such meaning to occur. We must think carefully about the language we use during our inputs and to construct questioning, as well as the pictorial representations that accompany this language.

When we do not explore language through a range of pictorial and abstract representations, children do not form a depth of mental representation. A child’s visualization often ends up as just a representation of the concrete.

The Mastery Math approach is not achievable if visual representation is one dimensional – language must be embraced and explored throughout the thought process, whether through the concrete, representational, abstract approach or another similar method.

Try asking a large group of students to draw a rectangle. While the vast majority may draw it horizontally, very few, if any, will draw a square. Children tend to believe that a rectangle has two long sides, two short sides and is always represented as horizontal. Read more on math terminology here.

I grade state tests and I have noticed certain questions have caused children issues. In last year’s tests, this was particularly true of the question below.

In just over 1,000 instances of this question that I graded, I saw the word ‘composite’ in an answer just once. While this was not a necessity to achieve the mark, or points, I thought it to be indicative of an overarching issue – vocabulary is not taught effectively enough or in enough depth.

Additionally, a huge proportion of children answered this question incorrectly by stating that ’95 goes into 5 and 87 goes into 3’. These children have understood what a prime number is, that 89 is indeed the prime number out of the given choices and still they have received 0 points because of their disarticulation.

While we can all decipher what the children really meant, mathematically, their statement is far from correct. There is no leniency provided in state standardized testing, in general, to incorrect interpretations of language and nor should there be.

This was also true of children who discovered that 95 and 87 were not prime but claimed that 89 was prime, because it ‘can’t be divided by a whole number’ or ‘no numbers go into 89’. We can gather these children loosely understand the concept of a prime number, but they lack a depth of understanding that will gain them the point.

Mathematical Language In Tests

Alex Quigley from the Education Endowment Foundation posits that, “Vocabulary is a proxy for your background knowledge”. I agree with this statement and think it is particularly applicable to the reasoning questions. The more vocabulary a child knows from a topic, the more knowledge around that topic they have.

Below is a huge list showing example math vocabulary or phrases that appear in reasoning questions on tests. The amount and variation of language used demonstrates the depth of knowledge children are required to access the test papers:

1,000 more, 1 million, 1-digit, 10 times, 2-digit, 3/4, 3D, 3D shape, 4 times, accurately, acute, acute angles, add, add together, added, addition, adds, after, all, altogether, am, amount, amount of money, angle, angle measure, angles, another, answer, area, areas, arrive by, arrives, axes, axis, before, below, best estimate, between, both, calculate, calculation, centimeter, centimeters, center, change, chart, circle, closest, cm, colder, collect/collected, combined, common factors, common multiples, complete, contains, coordinate, coordinate axes, coordinate grid, coordinates, correct, correct symbol, correct time, cost, costs, costs the same, costs the same as, cross, cube, cubes, cuboid, day, days, decimal, decimal numbers, decrease, degrees, delivered, design, diagonals, diagram, diameter, difference, difference between, different, different area, digit, digital clocks, digits, distance, distance between, divide, divided, divides, division, double, down, draw, each, each box, each box contains, each day, each month, each sentence, each shape, each take, each time, earliest, edges, equal, equal width, equally, equals, equilateral, equivalent, equivalent fraction, equivalent fractions, estimate, estimated, even numbers, every, every four years, exactly, expression, extra, faces, factors, figures, fill, five, for every, formula, four times as many, four-digit, fraction, fractions, furthest, grams, graph, greater, greater than, greatest, grid, half, half a kilogram, half a liter, halfway between, has, heaviest, height, hexagon, higher value, holds, hour, hours, hours and minutes, how many, how many days does it last, how many left, how many more, how much, how much each, how much higher, how much is left, how much larger, how much left, how much more, how much more than, hundred thousand, hundreds place, hundredths place, identical, improper fraction, in order, increase, isosceles, kg, kilogram, kilometers, km, large, larger, largest, latest time, leaves, left over, length, less than, level, lightest, liters, long multiplication, longer, lower than, lowest, marked, mass, masses, maximum, mean, measure, measuring, mental method, meters, meters long and wide, midday, midnight, miles, milliliters, million, minute, minutes, minutes fast, mirror line, missing, missing combinations, missing digits, missing length, missing number, missing numbers, missing values, missing vertex, ml, money, money left, months, more, moves, multiple, multiplication, multiplication grid, multiplications, multiplied, multiplies, multiply, multiplying, narrower, nearest, nearest hundred, nearest million, net, new position, next, not correct, not to scale, number, number line, number pairs, number pyramid, numbers from, numerals, obtuse, octagon, off, one, one decimal place, one gram, one-and-a-half, one-quarter, one-tenth, only, opposite, order of size, original, other digits, other number, parallel, pattern, pay, pentagon, percentage, perimeter, perpendicular, pictogram, pie chart, pieces, place, plots, plus, pm, position, possible numbers, pounds, price, prime number, prime numbers, product, protractor, quadrilateral, radius, ratio, rectangle, rectangular, reflect, reflection, regular, remaining area, remaining two digits, represents, result, results, right, right angles, right-angled, round, rounding, rounds, rule, sale, same, same amount, same area, same volume, scale model, scales, seconds, sections, sequence, shape, share, share equally, shortest, show your method, sides, similar, six, small, smaller than, smallest, sold, sorting, spend, spent, square, square meters, square number, square numbers, square-based pyramid, squares, stands for a number, start with, started with, statement, statements, straight, subtract, subtracts, symbols, symmetrical, table, take, takes, tale, tall, taller, target, temperature, ten thousands, tens column, three, three-quarters, times, timetable, to reach, together, total, total number, translated, translates, triangle, triangles, triangular prism, triangular-based pyramid, true, twice, two, up, value, vertex, vertices, volume, warmer, warmer than, weeks, whole circle, whole number, whole numbers, wide, wider, width, x, y, years

A child did not necessarily need to understand every single one of these words to be able to comprehend and answer all the questions, but the table is indicative of simply how many words children are expected to know.

What is clear from the table above is that we cannot, as teachers, rely so heavily on everyday language during our inputs and modeling. We must look to use math language – and math language that is specific to the topic being studied. This is what Beck et al (2013) refer to as ‘tier three’ vocabulary, more commonly known as subject-specific vocabulary.

“We take from here and put it there” is not as effective as “we transfer from the hundreds column into the tens column”. The former is not context specific and could be used across more than one place value column. The latter serves to cement understanding more clearly and adds to the child’s pre-existing schema of knowledge.

Moreover, as we think about the children we teach transitioning to middle and high school, can we be sure that the everyday language we rely on will be heard in the math classrooms they will enter? Will our colloquialisms pervade? Not every teacher will say ‘borrow’ and nor should they!

Keywords and mathematical language, however, will surely be a mainstay of their future mathematics education. It is the one way we can guarantee consistency in how math is taught across elementary, middle and high school. While pedagogies and methods may differ, subject-specific terminology cannot.

What Can Reasoning Questions Tell Us About Math Language?

The language used in math reasoning questions gives children a lot of information that they can use to help them unpack and interpret questions successfully. For example, if they understand that the term ‘change’ (money), always means a subtraction must be performed (i.e. amount given – cost), they are more likely to interpret a question that includes that term.

By teaching children to recognize such a term, we can begin to free up their working memory by getting them to access information stored in their long-term memory instead. In doing so, this can perhaps allow them to interpret two-step questions more readily, as they understand the term ‘change’ implies a subtraction as the final step. This allows them to focus on the first step instead.

Perhaps the most important language information children can understand for the tests is the synonyms or terms used that indicate a specific operation is required.

Terms Used to Indicate The Four Operations

Operation	Synonyms and short phrases used in reasoning questions to indicate these operations
Addition	More than, twelve minutes faster, mean, altogether, add, together, total, increase, adds, combined, more, plus, add, and, sum, after, another
Subtraction	How many left, difference, how much more, subtracts, how much larger, difference between, takes, less than, lower than, change, how many more, how much higher, how many warmer, pours out, how many colder, decrease, different, before, off
Multiplication	Multiple, long multiplication, product, total, area, multiply, times, each, for every, ratio of, multiplies, factors, together, three-quarters, times larger, scale
Division	Halfway, share equally, mean, half, divides, each, divided into equal sections, factors, multiple, how much is one or how much does one cost, one-quarter, three-quarters, scale, times larger

NB: All the words in bold above have suggested different operations in different questions, or that one of two operations could be used to solve one specific question.

We can therefore teach children that certain terms have very specific connotations. For example, ‘difference’ has always indicated a subtraction and the term ‘altogether’ has always indicated an addition.

As shown by the words in bold above, some terms can be applied to more than one operation – e.g. ‘scale’. This is an important teaching moment, as this can often mean a link between inverse operations (but not always – see ‘total’ above).

Children should be made aware that terms like ‘scale’ may mean a division or multiplication based on the context in which the word finds itself. They should see such a term in a variety of contexts so that they do not solely rely on the definition of the term itself.

If children are aware of the definitions, as well as the implications, of mathematical vocabulary, then they can begin developing reasoning skills more effectively. Language, after all, is about an understanding of both breadth and precision.

7 Strategies For Teaching Math Language Effectively At Elementary

It is obvious that language plays a crucial role in children accessing reasoning questions and answering questions effectively, but what does that mean for our classroom practice?

Here are just 7 key components of explicit vocabulary instruction that are necessary for providing children with a rich and deep understanding of mathematical language:

1. Define mathematical vocabulary in every lesson alongside pictorial representations
2. Link language back to prior learning on topics in previous grades
3. Explain synonyms, comparative and superlative terms and use them interchangeably on a regular basis
4. Avoid use of everyday language (e.g. don’t say borrow or take, say transfer)
5. Explain the connotations specific terms have (e.g. change means subtraction is needed)
6. Provide opportunity for children to reason in every lesson, both verbally and in writing
7. Structure our questioning so that it provokes thought and use of key terms in response (e.g. Sam thinks…Is he correct? Why?)

It is paramount that we enable children by providing them with the tools they need to access mathematical language. This represents the underlying fabric of mathematics teaching, whereby we must develop a child’s ability to reason.

Consequently, they will be able to interpret mathematical language successfully and begin to communicate their understanding in a structured manner.

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Bibliography

• Adams, T. L. (2003) Reading mathematics: More than words can say. The Reading Teacher, 56, pp. 786–795.
• Beck, I.L., McKeown, M.G. and Kucan, L. (2013) Bringing words to life: Robust vocabulary instruction. Guilford Press.
• Riccomini, P. J. and Witzel, B. S. (2010) Response to intervention in mathematics. Thousand Oaks, CA: Corwin Press.

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The content in this article was originally written by primary maths lead, maths SATs marker and experienced year 6 teacher Eliot Morgan and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza.