Using Language that Supports Mathematical Understanding
ARTICLE
INTRODUCTION
Research shows that American children enter kindergarten with weaker skills in counting and arithmetic than children in Asian countries (Mark & Dowker, 2015). This is due, in large part, to our English number names. The English language has more than two dozen unique words for digits, compared to only nine in Chinese. In addition, Chinese number names promote understanding of the base-ten number system. English-speaking children, however, often experience difficulty learning numbers 11-19 because there is no meaning behind the names of these numbers. For example, the Chinese word for 12 is “ten-two.” Yet 12 in English implies nothing about place value. The “teens,” often referred to by kindergarten teachers as the “tricky teens,” are also difficult for English-speaking children. In Japanese, 17 and 71 mean “ten-seven” and “seven-ten-one.” Yet in English, 17 and 71 are both pronounced with “seven” at the beginning, which makes learning these numbers “tricky” for English-speaking children.
Of course we cannot change the entire language of our number system so that American children don’t have to manage this complexity in their early years; however, we can be more deliberate in the language we use to teach key concepts within mathematics so as to ensure that language facilitates productive learning.
DISCUSSION
From the very beginning, the language used to engage young learners in discussions around mathematics should support conceptual understanding. With this in mind, we’ve thoughtfully analyzed our use of language. Our mathematical language is distinct from traditional math-classroom speak. We want learners to recognize language as important in conveying fundamental, transferable mathematical ideas. Having the knowledge and ability to use language very intentionally will allow learners to access various concepts of the discipline in ways that are quite powerful. We have made deliberate choices in the language we use to help learners make sense of math concepts and prevent misconceptions that often interfere with a child’s understanding of mathematics.
We have realized that language needs to create positive associations with math. Teachers can unintentionally create negative attitudes or anxiety toward math. In keeping with this idea, we suggest that the language teachers use can impact student attitudes toward mathematics. One small example is that when we work with children we use the term “expression” instead of “problem,” which has a negative connotation. After all, asking students to simplify an “expression,” such as 17 - 9, is mathematically accurate and perhaps more helpful than imploring them to “solve this problem.” The term "expression" (or “equation” if an equal sign is present) is mathematically correct and is a small change we can make in our language as teachers.
We have also realized that it is important to use language that supports the understanding of the concept and ensures that the concept is transferable across all number types. A study published in 2011 by Hadley and Dorward in the Journal of Curriculum and Instruction found that “Higher anxiety about teaching mathematics was related to lower student mathematics achievement.” This study suggests that professional development which includes "helping teachers become comfortable in teaching the mathematics curriculum by gaining a deep understanding of the concepts they teach” could produce larger gains in student achievement. In keeping with this research, it is our aim at Long-View to deepen teachers' understanding of mathematical concepts by focusing on the ideas that underlie each concept. Through our work observing many mathematics classrooms, as well as our work with mathematics educators, we have seen how many concepts are first learned one way in arithmetic and then must be relearned when students approach Algebra. And it has become clear that it is important to construct concepts in arithmetic such that they will transfer to Algebra, but also it is clear that we can use more deliberate language to communicate ideas that support these concepts.
Mathematics language needs to be conceptually correct from the very beginning of instruction. When we introduce addition, we read 4 + 3 as “4 and 3” rather than “4 plus 3.” In the English language, “plus” implies more, a surplus, or a gain. When considering combining positive integers, this makes sense. If we have 4 plus 3, we end up with more; we will have more ones. However, this idea of “more,” associated with the word “plus,” becomes confusing when working with negative integers. If we have -4 + -3 = -7, we actually have less: the magnitude of the sum is lower than either of the addends. “And,” in contrast to “plus,” indicates that values or terms of an expression are simply being combined. Thus, “and” is not only a word young mathematicians conceptually understand, but, again, it more accurately conveys the mathematical concept for all mathematical situations in which it is used and prevents misconceptions to which many young learners are prone when working with negative integers.
In subtraction, most English speakers read 10 – 4 as “10 minus 4.” In our assessment, the word “minus” is meaningless outside of subtraction. Instead, using a word that describes the action itself makes a significant difference to young children. When reading 10 – 4, we say, “ten losing four.” When paired with a model of a vertical number line, young children easily grasp that 10 – 4, or “ten losing four,” indicates we are moving down four ones on the number line from 10. Of course it isn’t that we never use the word minus; our learners hear this word from time to time, but we regularly substitute “losing” for “minus.”
We also use the word “debt” when teaching subtraction. From the very beginning of work with subtraction, our learners understand that 10 – 4 (ten losing four) is equivalent or conveys the same idea as 10 + - 4 (ten and a debt of four). This is how we define subtraction: to lose four from a number is to add a debt of four. Young mathematicians do not have to wait until they reach a certain level of math to be introduced to negative numbers, only to revise their understanding of addition and subtraction. Debt, like “losing,” has a conceptual understanding that is inherent in the word. If we have 10 – - 4 (ten losing a debt of four), learners understand that to “lose debt” is to come out of debt, so we are gaining or going up on the number line (they have a strong mental model of both a vertical and horizontal number line). Using the words "loss" and "debt" makes the transition to working with integers seamless as the conceptual understanding, constructed through purposeful lessons and thoughtful use of language, holds a deeper understanding for the learner.
We believe young learners should be exposed to integers early so that we do not cause misconceptions around the concept of subtraction. In many classrooms, when students move to subtracting two-digit numbers (ex. 42 – 17), they are often told that you “can’t subtract a larger number from a smaller number” when they are using the traditional algorithm and focused on first subtracting the ones digits (ex. 2 – 7). This then leads to a need to "borrow." In reality, you can lose 7 from 2, and you would have a debt of 5. Your loss, or debt, can exceed what is available to cover or absorb it, which means there is remaining debt. Our learners might solve the expression 42 – 17 by subtracting the tens and subtracting the ones: (40 – 10) + (2 – 7) = 30 + –5 = 25. Our learners work with integers from the very beginning, and we use words like “debt” and “losing” so that they have a strong and full conceptual foundation that will translate across all number forms, including algebraic expressions.
When considering multiplication, multiplication facts are traditionally written like this: 4 x 6. Interestingly, the “x” symbol for multiplication was introduced in 1631 and was chosen not for any mathematical significance but for religious reasons, as it was meant to represent the cross. In Long-View classrooms, we use the “x” symbol infrequently; of course our learners see it so that they know what it is, but most of the time, and always with younger learners, we instead use parentheses, e.g. 4(6), and use this language: “four sixes” or “four of six.”
When you read 4 x 6 as “four times six,” it is hard for young children to know exactly what this means: Does it mean four six times or six four times? With a shift in language, both written and verbal, to 4(6) and “four sixes” or “four groups of six” or “four of six,” learners are able to more easily conceptualize multiplication. Learners recognize that there is a specific unit being counted and a certain number of that unit being specified. In the case of 4(6), the unit of six ones is being counted, and there are four of that unit.
In fact, this use of parentheses also allows learners to transfer their understanding of multiplication from whole numbers to fractions. When learners read “½ x 32” using the traditional language of “one-half times 32,” it is much harder to access the idea than when seeing ½(32) and reading this as either “one-half of 32” or “half of 32.”
The language traditionally used in long division is also problematic. For example, in the expression 8⟌320 , students are often asked to start by determining whether or not 8 “goes into” 3. What does “goes into” mean? More problematic, there is not actually a value of 3 in this expression. The “digit” 3 has a value of 300 and should be read as such. Ignoring the actual value of the digits can cause confusion for young learners and actually impede understanding of the concept.
In Long-View classrooms we instead emphasize the relationship of division to fractions. In the case of 320 ÷ 8, learners are prompted to see its relationship to 320/8 or 320(⅛). When evaluating 320÷8, they read this as “320 divided into groups of 8” and recognize that, when groups of 8 are formed, each object becomes one-eighth of each group of eight. Thus, 320÷8 means there are 320 one-eighths (320(⅛)). This relationship reveals how we think about division: x÷y = x(1/y). Thinking about division in this way is a big idea in mathematics and is often a missed opportunity in developing a deep understanding of division and fractions in elementary math instruction.
CONCLUSION
It is important to take time to think about our use of language in mathematics. Language can be a powerful model and can help students more clearly conceptualize an idea. Replacing words like “plus” and “minus” can impact the underlying understanding of mathematical concepts. The importance of language as a model for mathematical learning cannot be underestimated. As Mark Twain said, “The difference between the right word and the almost-right word is like the difference between lightning and lightning bug.”
References
Mark, W., & Dowker, A. (2015). Linguistic influence on mathematical development is specific rather than pervasive: revisiting the Chinese Number Advantage in Chinese and English children. Frontiers in Psychology, 6, 203. http://doi.org/10.3389/fpsyg.2015.00203
Shellenbarger, Sue (2014, September 15). The Best Language for Math: Confusing English Number Words Are Linked to Weaker Skills. The Wall Street Journal, Retrieved from https://www.wsj.com/articles/the-best-language-for-math-1410304008
Hadley, K., & Dorward, J. (2011). The Relationship among Elementary Teachers’ Mathematics Anxiety, Mathematics Instructional Practices, and Student Mathematics Achievement. Journal of Curriculum and Instruction, 5, 2. http://www.joci.ecu.edu/index.php/JoCI/article/viewFile/100/pdf