SMP #1: Make sense of problems and persevere in solving them.
On a weekend walk with colleague, Amy Marquis, we got into a conversation about perseverance. She had been sharing conversations her 8th grade AVID students had regarding the topic of personal struggle. The following struck a cord with me:
- A core group of her students felt that teachers didn’t regularly let students struggle with questions
- When that would happen, they felt they were denied an opportunity to learn on their own.
Paraphrasing student comments: Teachers will ask students if they don’t understand. Then we raise our hands. Then teachers give us the answer or explain the process again instead of allowing us to struggle with it. This doesn’t encourage or teach us how to struggle with things.
Today, I’m sharing a specific situation of what happened when a teacher gave a student the time and space to clarify his own misconception.
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Featured Teacher: Allan Saladino
The Dilemma
One day, I was working in Mr. Saladino’s 8th grade classroom. Students were solving Pythagorean theorem questions. Although they were skilled at solving the equation, they struggled with translating the word problem into the formula: a2 + b2 = c2.
I wanted to follow up this lesson with a short activity – Posing a Pythagorean theorem word problem to the students and having them draw the picture to represent it. That’s all. Nothing else. Allan and I would walk around and select specific pictures to display under the document camera and the class would have to discuss what they see in each picture. The class would decide which picture was correct. I shared this idea with Allan and he agreed.
Clarifying a Misconception
The problem (taken from our text book): An airplane is 5 miles above a house and 42 miles from the airport. How far is the house from the airport?
The requirements: Students were to draw a picture representing the situation and label 6 pieces of information: The airplane, airport, house, 5 miles, 42 miles and x.
As I was walked around the room looking at student work, I stopped at a student’s desk. Let’s call him Derek.
His picture (a recreation)
JGV (pointing to Derek’s picture): Derek, why did you put the 42 miles there?
Derek: The distance from the airport and the house is 42 miles.
JGV: What words in the problem prove the 42 is the distance from the airport and the house?
Derek: The distance from the airport and the house is 42 miles.
JGV: What’s your evidence? Would you reread the problem and find the words that prove you’re correct?
Derek: Because… It goes there. The airport is 42 miles from the house.
At that moment, Allan jumped in.
Allan: Derek, will you come up and point to the words that prove your picture is correct?
Derek goes to the front of the room. Along with the written problem, a picture similar to his is projected from a document camera onto a screen. Derek grabs a meter stick and starts pointing to the word problem displayed on the screen. He first explains the placement of the 5 miles. Then he starts reading the problem from the beginning pointing to each word with the meter stick.
Derek: The 5 goes here because the plane is 5 miles above the house. The airplane is 5 miles above the house and 42 miles from the air… from the h… from the …
Derek stops. He first was about to say “airplane” which was correct, but stopped because it wasn’t what he had been saying earlier. Earlier he was adamant that the distance between the house and the airport was 42 miles. But now, he was questioning that thought. Allan and I made eye contact and smiled.
Derek repeated: The airplane is 5 miles above the house and 42 miles from the air… from the hou…
After a pause, he said: The airplane is 42 miles from the … AIRPORT! The 42 goes here. (He points to the line connecting the airplane and the airport).
Mr. Saladino and I celebrated. Derek celebrated. “Ah ha’s” rippled through the room, for Derek had just clarified an error made by many students.
Mr. Saladino’s Reflection
It is amazing to see when a student has an “Aha” moment and is able to figure out things on his own and realizes his own mistakes. A picture is worth a thousand words as one may say and in this case the student was able to connect the word problem with a sketch or drawing that made sense in this problem situation.
As teachers, we were the facilitators in helping the student arrive at the correct answer. Time is always the issue as we want to move things along, but giving the students “wait time” to actually think what was going on made the difference and making the student think and use evidence to prove that the answer was correct provided more examination and breaking down of the problem.
This use of justification made this student analyze his own thinking and helped him to make sense of the problem. Sometimes, maybe more so, we just need to slow down to allow students to think and ponder to make sense of what is going on instead of rushing through things just to get through it!
A Couple of Months Later
The described lesson happened before winter break. In preparation for this post, I stopped in to see Derek. During the class openers, I handed him the same problem and asked him to draw the picture. I sat behind him and waited. After a period of time, Derek turned around and handed me this picture. I smiled. He had drawn it correctly.
JGV(softly): You remembered.
Derek looked me square in the eye and made a small smile.
JGV: Do you remember the day we discussed this problem?
His smile grew and pointed to the screen where he corrected his mistake.
Conclusions
That was a powerful moment for me – and I think for Derek too. At this point in my career, Derek’s experience defines the type of teaching moments I crave.
Steve Reinhart wrote the article, Never Say Anything a Kid Could Say. As a coach, I find myself coming back to this belief over and over. The above scenario took more time than expected. I quickly could have pointed out Derek’s error. Mr. Saladino could have pointed out Derek’s error but the powerful learning experience resulted when Derek uncovered his own mistake. Excitement fill the room, the moment Derek understood his mistake. Excitement that wouldn’t have happened if either Mr. Saladino or I revealed the misconception.
Communicating Mathematically 