I’ve been thinking a lot lately about the role of vocabulary in learning, teaching, and research.
Until recently, I didn’t have a very well developed opinion on whether or not we need to worry about kids’ use of mathematical vocabulary, at least at the elementary grades. There were a few ill-articulated assumptions underlying my lesson writing style, though. Generally, I believed that:
- Helping kids to learn definitions should never be the main point of a lesson. An idea is bigger than its definition.
- Similarly, perfect use of the vocabulary shouldn’t be the main learning goal of a lesson. I don’t think imperfect expression should invalidate an idea, especially if it is coming from a young child.
- On the other hand, if an idea is key to a lesson, there’s no reason not to introduce a term for it. I don’t believe in withholding words from kids because they are long or “difficult”. Words can be powerful tools.
- Even though precise use of vocabulary isn’t an appropriate expectation for kids, I do think teachers should try to be precise in their use of the terms. And that means that curriculum materials should be precise in their mathematical language, too.
I was kind of a stickler about that last point in my curriculum writing days. I think it sometimes annoyed my coworkers, who believed that no teacher was going to notice or change her practice if we wrote The sides are equal in length instead of The sides are equal or said The angle measures 42° instead of The angle is 42°.
I had to admit at the time that they were probably right about that. Teachers have limited time to read curriculum materials and plan and I’m doubtful they spend that time paying close attention to subtle differences in language. Still, when I caught language that was imprecise, I was stubborn about changing it. I justified this mostly by arguing that if any mathematicians reviewed our materials, this would give them one less thing to pick at.
I read and article this semester, though, that made me wonder if there was a bigger reason for precise language than that. Gentner (2010) published a lengthy argument for the reciprocal relationship between language and analogical reasoning. The first half of the paper summarized research suggesting that making comparisons facilitates learning. The second half was a more specific argument about the role of language in learning and cognitive development. Gentner argued that:
- Having common labels invites comparison of examples and abstraction of shared elements, and
- Naming promotes relational encoding that helps us connect ideas that are learned in different contexts.
To illustrate her argument, Gentner cited research about learning to apply natural number labels to quantities. Studies of cultures whose languages do not contain specific number labels showed that people of those cultures were able to estimate magnitudes, but were not very accurate at assigning specific number names to quantities, especially as the quantities got larger (Gordon, 2004 and Frank et al., 2008, as cited in Gentner, 2010). Other studies showed that children who speak languages with number names (like English) first learn the count sequence by rote, but by comparing sets of objects (e.g. two trains and two dogs) that have a common number label attached, they gradually bind the number names to the quantities (Gentner, 2010).
This explanation makes perfect sense to me. This is why words are powerful — they are a means of connecting examples at their most fundamental, definitional level. They prompt looking for sameness in contexts where things feel very different.
This got me wondering whether my stubbornness was better justified than I originally thought. Abstract mathematical terms like equal (and its symbol) are known to be poorly understood (e.g., Knuth, Stephens, McNeil, & Alibali, 2006; Matthews, Rittle-Johnson, McEldoon, & Taylor, 2012; McNeil et al., 2006). At least one study has concluded that the contexts in which the equal sign is used impact students’ understanding of its meaning (McNeil et al., 2006). I would not be surprised if a similar study examining how the word equal is used in sentences showed that these uses impact understanding of the word. According to Gentner (2010), we both consciously and unconsciously use words as labels, which invite comparisons, which invite conclusions about meanings. It seems reasonable to suggest that if we use the word equal with counts and measures, but use the term congruent with geometric figures, teachers could abstract more sophisticated and precise meanings of those terms than if we use the term equal sides.
I don’t know for sure, of course. But I don’t think precision in language, especially in resources that teachers can continually reference, can’t hurt the educative power of the resources.
Gentner, D. (2010). Bootstrapping the mind: Analogical processes and symbol systems. Cognitive Science, 34, 752–775.
Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 297-312.
Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3), 316-350.
McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and instruction, 24(3), 367-385.