Jacob Disston asks his students to focus on a category of symbol strings (expression, equation, inequality), separate them, and discuss how they know how to categorize them. Students are asked to make statements like "I know this is an equation because..." or "I think ... what do you think?"
JACOB DISSTON: All right, so we talked about equations. Somebody else, I need you to tell me "how do we pull the inequalities all together?" How do we tell inequalities?
STUDENT: The difference is that it has the "less than greater than" symbol like that.
JACOB DISSTON: And what is that symbol? What are you looking for when you say that's an inequality?
STUDENT: The crocodile mouth looking...
JACOB DISSTON: The crocodile looking...
JACOB DISSTON: Yeah, so do all of these have that?
JACOB DISSTON: Yeah?
STUDENT: Inequality got the "less than or greater than" and then the "less than or equal to" or the "more than or equal to" or "greater than or equal to."
JACOB DISSTON: Ok, all right. How many symbols are there that an inequality could have that we can say "that's an inequality?"
JACOB DISSTON: 5?
STUDENT: No, no...4
JACOB DISSTON: What's the fifth?
STUDENT: The equal sign.
STUDENT: That's not an inequality.
JACOB DISSTON: Why not?
STUDENT: Because then it's an equation.
JACOB DISSTON: Then it would be called an equality, not an inequality? Ok, so we got inequalities. How do we tell...here's the last one before we move on. Can this group...can somebody tell me "how do we tell the expressions?" What would you guys say?
STUDENT: Expressions don't have an equal sign.
JACOB DISSTON: Expressions don't have an equal sign. People agree?
STUDENT: Yes, because they don't have equal signs or "less than or greater than."
JACOB DISSTON: So they don't have equals signs, they don't have inequalities? Can they have any signs?
JACOB DISSTON: They have operations. What are operations?
STUDENT: Plus, minus...
JACOB DISSTON: Plus, minus, division, multiplication, exponents, square roots.
STUDENT: Can they have percents and decimals?
JACOB DISSTON: Can they have percents and decimals? That's a good question.
STUDENT: And also for expressions, you can also combine "like terms" right, can you?
JACOB DISSTON: Did everybody hear what she just said?
STUDENT: Yeah, yeah you can.
JACOB DISSTON: Maria say it again.
STUDENT: That for expressions you can combine "like terms" I think.
JACOB DISSTON: What does she mean? Can you turn to your group and say what she means by you can combine "like terms."
STUDENT: Where it says that 4p plus 3p equals (inaudible) [4p+3p=?]. You can do 4p plus 3p equals 7p plus 2p [4p+3p=7p+2p] or you can do 4p plus 3p plus 2p which equals 9p [4p+3p+2p=9p] It's like you're combining...
JACOB DISSTON: So let me ask you a question. Maria brought up this idea...she said in expressions I can combine "like terms." Is that special to expressions? Can I combine "like terms" in equations?
JACOB DISSTON: Sometimes...tell about that.
STUDENT: It's like some that don't have any variables that are the same or anything like that because sometimes they are, so you can combine them.
JACOB DISSTON: So sometimes there are expressions and sometimes there aren't. So what about equations? Would I ever use that idea of combining "like terms?"
STUDENT: I think.
JACOB DISSTON: Ok, so these are good questions. I'm really glad that you are asking these questions.
COMMENTARY BY COACH LINDA FISHER: These clips bring up the importance of creating opportunities for students to use mathematical vocabulary for a purpose. While everyone is using the same language of equations, students seem to have a variety of definitions for what that means. Some students name it an equation because there’s multiplication. For others the variable signifies equations. Students seem to puzzle over words for describing signs: there’s addition, subtraction, and other signs like equal. They understand that the difference is significant, but don’t know how to classify it. How do you think about things you can’t name? As adults, we sometimes take meanings for granted. Equation seems like an obvious or given definition, but as I listen to students in the clips I hear many meanings or unintentional learning that they have attached to the word. Notice that even the writing promotes further discussion and negotiation of the definitions.