Why take the road less traveled?

I love being in the woods. It is nearly always an adventure. Even if it's a path I have been on before, the possibility of a surprise around the next bend is exciting to me. I never know for sure what I will see or hear. That's not to say that I am always comfortable with this state of uncertainty. A few weeks ago, I heard two coyotes howling up ahead as I was walking on a logging road in the Upper Peninsula of Michigan. Or there was the time I encountered an overwhelming smell that I can only describe as wet dog as I walked alone along the nature trail at Seney National Wildlife Refuge. In cases like this, I try to find a big stick to "support" me as I walk.

What keeps me coming back to these roads less travel are the sights and sounds that come from exploring something new. For example, while walking down one of the access roads at Seney, I came across this beautiful Trumpeter Swan that, true to its name, announced my presence to others in the area.

I have found that this metaphor of taking the roads less traveled also applies to my teaching practice. When I first began asking my middle school students how they would solve a problem before showing them the right way, I was quite uncomfortable with the idea that they might share something new - something for which I was unprepared. Fortunately, I had mentors that assured me it would be alright and encouraged me to explore.

The first time I recall letting my eighth-graders lead the way was when we began a section on dividing fractions. I put three-fourths divided by one-half on the board and asked for a volunteer to share how they thought we ought to calculate the result. Here's essentially what happened (requires ShowMe log in).

Because the student had not used "invert-and-multiply" to calculate the quotient, I considered the effort incorrect and tried to determine what went wrong. First, the student was finding common denominators, so there was a chance the student was confusing this procedure with the one for adding and subtracting fractions. Then the student divided across, which resembles the procedure for multiplying fractions. Clearly, the student was mixing up the various rules for fraction computation and I would need to be explicit about the differences. But something was bothering me. The answer the student came up with using this mish-mash of methods was correct. I attributed it to the numbers I had selected and decided to try the method with another pair. It worked again. I was at a loss.

I don't remember what I did with the students, but I do remember exploring this approach more and finding out that it always works and why. This experience hooked me. Now I am not suggesting that the student was doing anything other than trying to apply various rules and happened upon a solution.  But I might not have ever heard or seen this approach if I had not been open to following this unfamiliar path.

If you are still uncomfortable with exploring the "wilderness" with your students, then take solace in knowing that you do not need to go too far off the regular roads to experience something wild. Simply being alert to students' thinking while covering familiar ground can allow for new ideas to be uncovered. Just like when I was able to film this Peregrine Falcon on the bike path near our house, it only takes being aware of something new and prepared to see where it goes.

Which way is ... ?

A preservice teacher leading a review for a quiz on rational number computation invited me to watch the lesson and work with her to improve on it. The objectives addressed in the review activity were from the Michigan Grade Level Content Expectations (GLCE):

• N. FL. 07.08 (GLCE): Add, subtract, multiply and divide positive and negative rational numbers fluently 
• B. N. FL. 07.09 (GLCE): Estimate results of computations with rational numbers

But the preservice teacher was also interested in developing conceptual understanding - especially around the idea of how multiplying and dividing by numbers between 0 and 1 impact the result. A good activity from NCTM (pdf) was found and modified in an effort to achieve these goals.

Move down or sideways (never up) through the maze from Start to Finish. You may not retrace any steps. Begin with 10 and as you move along a segment do the indicated computation. Record your steps on the scorecard. Your goal is to find the path that results in the largest (or smallest) value when you reach the Finish.

After the lesson, we discussed ways of using the activity more effectively. The first idea was to model what a path looks like. Because this kind of activity was new to the students, it took them a while to understand what was expected of them. For example:

What if we just followed along the left-most edge?

1. 10 x 0.9 = 9; 
2. 9 x 1.75 = 15.75
3. 15.75 + 5 = 20.75

Next, without doing any computation, we would ask the students to predict the path that would result in the greatest result or the least result. Making predictions is a great way to develop interest in a task. Student would share their predicted path and the rationale for their choice. This would provide some insight into the students' number sense related to multiplication and division of rational numbers. 

Then the students would estimate the results of several paths. This would allow them to check out their predictions and refine our list of which paths might represent the largest (or smallest) value. Also, this would address objective B. N. FL. 07.09 from above.

Finally, the students would be asked to compute the path they believed would result in the largest (or smallest) value [N. FL. 07.08]. Calculators are not allowed in this classroom but we decided that we might allow students to use calculators on up to half of the calculations. That way they would be exposed to the idea of using calculators strategically instead of with an "all-or-nothing" mindset.

As an extension, we might ask the students to find the easiest or hardest path to follow without using a calculator and why. We thought this would offer the students an opportunity to be metacognitive. It would also provide us with information on areas where students could improve on their fluency.

The preservice teacher was able to apply some of these subtle shifts to her later class with success. She writes:

...they did much better!  They were excited to do something "more fun than boring problems."  I was really happy with the responses I got...

What are your thoughts? How would you improve on this activity? Why?

What if we gave them the answers?

My experiment of teaching a course where preservice and inservice teachers share two hours of class time has been going well. (I introduced the concept here.) In fact, one of the preservice teachers said recently, "I wish all of my education classes had classroom teachers in them." I believe the following example explains why he feels that way.

That same preservice teacher was part of a small group (along with an inservice middle school math teacher and a community college instructor) who were analyzing middle school students' work on an algebra assessment. They were talking about how difficult it is to get students to share their thinking especially once they assume they have arrived at an answer. I concurred and explained that this was one of the reasons I focused on using metacognitive memoirs, saying, "I know the answer but I don't know what you're thinking." This gave the inservice middle school teacher an idea.

He wondered what would happen on the next test if he gave the answers and asked the students to focus on their thinking. A few days later, I (along with the preservice teacher and the community college instructor) received the following email:

Hi, I gave a test yesterday in my 8th grade math class and I gave them all of the correct answers at the beginning of the test to see if it would improve the work that they showed and how well they explained their thinking.  They were shocked, but they actually caught onto the idea quickly, I didn't even have to tell them why I was giving them the answers, they came up with it themselves.  While the test responses weren't perfect, students did a MUCH better job sharing their thinking than they ever have before.  I am excited about how this turned out and I anticipate doing this more often in the future.

I asked the teacher if he would mind me sharing this experience and the test on my blog and he agreed. Not only that - he also provided how he implemented this new approach, a sample of students' work, and students feedback.

After handing out the test, the teacher began:

• Teacher: "Listen closely. This is a test. You know the rules as far as talking, etc."
• Teacher begins reading off answers.
• Students are following directions, no questioning until after the page flip.
• Student 1: “Why are you telling us all the answers?”
• Student 2: “I like this!”
• Student 3: “Don’t stop him.”
• Teacher keeps reading answers. There is no contesting of getting the answer and the kids keep filling in right answers for remainder of test.
• Student 4: “I don’t understand this...”
• Student 5: “Why did you just give us the answers?”
• Student 6: “Do we have to explain what we did for the answers you just gave us?”
• Teacher: “You’re not going to get any credit for having the right answers. You’re only going to get credit if you can explain how you get the right answers. So all of you are starting right now with all the answers and a 0%.”
• Student 4: “I like the other way better.”
• Teacher: “Let me just say one more time...You all have the right answers, so the explanations are where you can earn the points. With that in mind, go ahead.”

Here is what the test looked like after the teacher had read the answers.

And here are some examples of what students wrote:

After the test, the teacher asked for students' feedback on this approach to assessment. These represent some of their responses:

• "My head hurts because I actually had to think."
• "I realize now that I've never done a very good job explaining my answers."
• “This was like an English test!”
• “It took forever...like, I know what I want to say but I can’t explain it.”
• “Didn’t like volume of writing and repetition.” (Felt like there was too much writing and they were answering the same questions over and over.)
• “Didn’t see the point of giving out the answers because you have to do all that thinking to get the answer anyways.”
• “Liked it. I always spend time figuring out the problem so I don’t explain. This helped cut out the calculation step.”
• “Didn’t like because I don’t like explaining myself.”
• “Would have prefered to find answers instead of trying to explain because sometimes I can just get it (in my head).”
• “Liked having answers, otherwise I spend a lot of time trying to get the answer. This way I know the answer is right.”

I hope that I was able to adequately articulate this approach to assessing students' mathematical thinking. If you have questions or ideas, please leave them in the comments and I'll be sure to pass them along. We have 8 more weeks together in this course. I'm looking forward to whatever else they come up with in that time.

What did you do at school?

Yesterday, I was at a gathering where the meditation provided below was shared. It is about a local school district, Muskegon Heights, where many at the gathering volunteer their time and talents as a part of the Coalition for Community Development [CCD]. About a year ago, members of the CCD helped to open on closed elementary school library in the district (read about it here). They continue to work to re-open libraries in the district. Anyone who can help is encouraged to contact Kathleen Kleaveland by email at kmdkk5@hotmail.com. Thank you in advance for your support.

I Went to School

by Cindy Anderson

I went to school.

It was Tuesday.

Just a few miles away.

It looked like every school I’ve ever seen except for the litter scattered on the ground across the street --mostly beer bottles, cigarette packs and fast food wrappers.

I opened the door of the school and smiled.  It smelled like every school I’ve ever smelled; hints of floor polish, wet mittens, washed hands and a slight undertone of bananas.

It looked like every school I have ever seen; low drinking fountains, names on lockers, teacher made bulletin boards of dancing letters, banners on the walls encouraging team work and good behavior-- except this school did not have

an art room,

a music room,

a gym

or a library. 

I checked in at the office. 

The secretary was busy but welcoming, just like every secretary of every school I’ve ever entered. 

She graciously directed me to the appropriate classroom and wished me a good day. 

With excitement, I went to the kindergarten classroom, and met my first student. 

Just like every classroom I’ve entered in my life, the children were busy working-- except they were not working in centers.

They were not moving about the room. 

They couldn’t move about the room. 

There were 36 students in the kindergarten classroom. 

They were sitting very still in their tiny desks pushed closely together. 

There was a palpable feeling of pent-up energy. 

The beleaguered teacher smiled and introduced me to my student.

My student and I walked to our work spot down the hall.

His eyes sparkled and his energy pulsed.

He told me about his family

He wrote his name

He listened intently to a story I read.

He held my hand as we walked back to his class. 

He extracted a promise that I would come again.

I was in love.

I met my second student. 

Her skin was glowing. 

She had 48 barrettes in her braids—butterflies. 

She skipped when she walked and she giggled when she talked.   

She drew a picture of her family. 

She sang her ABCs

She listened intently as I read her a story.

She asked me to come back.

I was in love.

I met my third student. 

Her dimples bounced as the teacher called her name.

She told me all about her family.

She recited her ABCs .

She read her favorite words.

She listened intently as I read her a story.

She asked what day I was coming back.

I was in love.

I met my fourth student. 

He had braids all over his head and his nose was running a little. 

He walked slowly to our spot.

He didn’t want to talk about his family

He didn’t say his ABC’s

He didn’t write his name.

He listened intently to the story about a hippopotamus.

He asked questions about it.

We looked up hippos on my Ipad.

He picked up a piece of paper and drew a perfectly proportioned hippopotamus.

He did not ask when I was coming back.

He stayed outside his room for a long time before reentered.

I was in love.

I went to school.

Every week for the next thirty weeks.

We celebrated Halloween, Thanksgiving, Christmas, Valentines Day and the onset of spring.

My students grew.

They told me more about their families.

They asked questions about my family

I knew when their dads had birthdays,

When their moms got new jobs,

When they visited their grandmas,

When their cousins came to play.

They began reading stories to me.

They beat me at games.

They told me Tuesday was their favorite day.

I was in love.

I was also sad.

The children never had gym.

They never had art.

They never had music.

I was sad and angry that we disadvantage this group of children because we do not give them the opportunities to develop as whole children. 

But they DID get a library.

An amazing group of volunteers created a library within the school..

The children had access to books and were thrilled.

The last day was hard.

I was sad.

I told them I would be back next year.

I went to school.

I went to school because children are my passion.

I went to school because I believe all kids are important.

I went to school because I believe kids do best when they feel loved.

I went to school because I wanted to make a difference.

I went to school because I wanted to teach.

I went to school and I learned.

I encourage you to

go to school

Or go wherever you have a passion

The prison

The shelter

The political office

The wetland

The nursing home.

Save the seals

Save a species

Build a library

Build a relationship

Build a more humane world.

You will make a difference

You will learn

You will be glad

You will be thankful.

I am

I am forever glad

I am forever thankful

I went to school.