Vocabulary for Developing Math Reasoning

Teach­ers of math­e­mat­ics need to rec­og­nize that there is a strong link between lan­guage, writ­ing, and prob­lem solv­ing. In most of the assess­ments that states use to deter­mine stu­dent and school suc­cess, a stu­dent must demon­strate math rea­son­ing abil­i­ties through writ­ing. This skill is not auto­matic, though. It devel­ops through a recur­sive process:

Vocab­u­lary & Lan­guage <—> Rea­son­ing <—> Talk <—> Writing

Begin­ning with vocab­u­lary and lan­guage, a stu­dent learns to rea­son, then to com­mu­ni­cate those thoughts ver­bally, and finally to write. Each of the lev­els feeds back to the pre­vi­ous one, rein­forc­ing and fur­ther devel­op­ing it.

Thus if we’re going to teach rea­son­ing skills effec­tively, it fol­lows we need to care­fully con­sider the vocab­u­lary we use.

It isn’t uncom­mon, espe­cially in the pri­mary grades, for teach­ers to sim­plify the lan­guage we use with chil­dren to explain com­plex con­cepts. Although this is use­ful, it can also lead to sloppy lan­guage if we aren’t care­ful. It is par­tic­u­larly impor­tant that we don’t per­mit stu­dents to use pre­cise math terms improp­erly and that we teach the “real” terms as quickly as pos­si­ble. Even if stu­dents don’t use them right away, they should be hear­ing the cor­rect ter­mi­nol­ogy in con­text from the beginning.

Here are a few exam­ples of sloppy math lan­guage that I often hear from older stu­dents. If these go uncor­rected, stu­dents will have a very dif­fi­cult time com­mu­ni­cat­ing well when they need to explain their thought process–a skill that is essen­tial to upper level math.

Instead of these…	Use these…
take-​​away	minus
“plussed”	added
“minused”	sub­tracted
“timesed”	mul­ti­plied
answer	sum, dif­fer­ence, prod­uct, quotient
amount	length, height, vol­ume, num­ber, etc.
num­ber	digit, addend, fac­tor, div­i­dend, etc.
big­ger, smaller	greater than, less than

I believe it’s essen­tial to require stu­dents to be pre­cise when they com­mu­ni­cate. Often when stu­dents don’t use the cor­rect term, or use a valid term improp­erly, it is a sign they just don’t have the right words.

I’ve heard teach­ers argue that young chil­dren just aren’t capa­ble of such sophis­ti­cated lan­guage yet. My father, a retired pro­fes­sor of speech/​language pathol­ogy, has often said, how­ever, that if sec­ond graders can learn and cor­rectly use terms like “Tyran­nosaurus Rex” and “Diplodocus”, why on earth can’t we teach them to say “sub­tracted” instead of “minused”? Vocab­u­lary instruc­tion should be as much an inte­gral part of math­e­mat­ics as it is of read­ing, writ­ing, and other con­tent areas.

Tomor­row I will tackle a more chal­leng­ing vocabulary-​​related issue in math­e­mat­ics: ver­bal and writ­ten expla­na­tions of a student’s cog­ni­tive process.

(This arti­cle is based on mate­r­ial I orig­i­nally posted in Grandé With Room.)