Mathematically Speaking 3

0 user(s) rated this page

by Roger Alcock (continues)

Describe and Justify

The first principle I would like to discuss is, “describe and justify.” It is very difficult for many children to explain how they came up with their answer to a math problem. With practice however, they become much more proficient in explaining how they arrived at their answers. Even the children who always have the correct answer may have trouble explaining how they arrived at their solution. When you initially ask students to explain their answer, a typical response might be: “I just knew it,” or, “it just popped into my head,” or my favorite (from a high school student), “it is intuitively obvious!”

It is important for students to be able to explain their answers for many reasons:

* They have to understand what they did in order to be able to verbally explain it.

* Students are more likely to use the mathematical vocabulary to explain their thinking because it is efficient for them to do so.

* The students gain confidence in their abilities.

* The students gain ownership in their answers.

* The students retain the information longer because they have ownership.

* If standardized tests ask students to explain how they would solve a problem, they will be comfortable with the directions.

It is important for teachers to hear students explain how they found their answers:

* The teacher gains an understanding of how the students are thinking.

* The teacher has a better idea of how students arrived at a wrong answer.

* The teacher may be surprised at the level of understanding that a student has about a particular concept.

* As a student explains their reasoning, other students are hearing it also. Other students may understand the concept by listening to a peer, when they didn’t understand it from the teacher.

* Listening to students is one form of “using on-going assessment.”

* Cooperative Learning situations may be more productive because of increased student involvement.

When I first heard the “describe and justify” principle, I though to myself: “That’s a principle I have been using for years! I’m glad to know that it is an effective technique. I’ll use it more often in the classroom.” As I went back to my classroom and started using the “describe and justify” technique more often, I found that I had only used it when the students had an incorrect answer. When I started asking students, “how did you get that answer?,” their response surprised me. The students would stop, re-look at their work, and then say, “that is the correct answer.” I would tell them that it was the correct answer, and ask again, “how did you get that answer?” In a relatively short time span, students began to expect me to ask them how they arrived at their answer, and some would volunteer their explanation without my asking. Success!! “Whoever does the work does the learning!” Sometimes teachers do all the work and most of the learning. Shift the work, and therefore the learning, to the students.

(Note: It is the teacher’s responsibility to ensure that the classroom atmosphere remains “safe” for all students. “Put-downs” can not be allowed.)

“I don’t know.” Did you ever hear that response from a student when you asked them for an answer? I did, many times. I received some great advice while discussing this response with an elementary teacher a number of years ago. She gave me a great question to ask. The question was, “If you did know, what would your answer be?” Hmmm…that doesn’t sound like a logical question because the child already said they didn’t know the answer. Here is what it might sound like in the classroom:

Teacher: “Johnny, how much is 8x7?”

Johnny: “I don’t know”

(If the teacher stops here, and asks another student, he hasn’t learned anything about Johnny’s understanding of the 8x7 fact.)

Teacher: “Well Johnny, if you did know, what would your answer be?”

(The pressure is off! Johnny has already said that he doesn’t know the answer….so his response at this point will give the teacher a great deal of information.)

Scenario 1: Johnny: “56”

(This response tells the teacher that Johnny knows the correct answer but is not confident of his answer.)

Teacher: Great! You did know the correct answer after all! How did you get 56?

Johnny: Johnny explains his thinking.

Scenario 2: Johnny: “42”

(This response tells the teacher that Johnny doesn’t know the answer)

Teacher: “You’re right, you don’t know the answer, but you aren’t far off. How did you get 42?”

Johnny: Johnny explains his thinking, and the teacher can help him arrive at the correct solution.

Scenario 3: Johnny: “But, but I said I don’t know the answer!”

(This response tells the teacher that Johnny doesn’t have a clue! But now, the teacher knows that Johnny really doesn’t know.)

The teacher has learned a great deal of information about Johnny’s mathematical knowledge. If the teacher had simply asked another student for the answer, she would not have gleaned any additional information about Johnny.

“Wait Time.” Educational research tells us that if we wait for a deafening few seconds after we ask a question to call on a responder, we will get better answers from our students. Don’t rush the answer, and don’t rush the explanation. It seems like a waste of time perhaps, but we are training our students to think before responding, and that “thinking ability” will save us time in the future. “Wait time” also comes into play after the question has been answered. If we immediately say that an answer is correct, our students can stop thinking. If we give them a few extra seconds, they will often give us more in depth information.