Fearless – Introducing Mathematical Proof

“Ugh.   It is the worst class ever.   It is so hard.”  I heard those words expressed in almost identical fashion twice, roughly 5 years apart when I was a student.    The first time, I overheard them in the hall spoken by a student a year older than me.   He was taking about his experience learning how to do a proof in his geometry class.    The second time, I was in college and a couple of upperclassmen were talking about their experience with proofs in Linear Algebra.   Both times, I went into the class with a bit of a knot in my stomach, expecting the worst.    I don’t know if it was because my brain just works that way or if it was because I was lucky enough to have some really good teachers, but I thought proofs were kind of fun.    (Strange – I know.)

As I thought about introducing my 6th graders to proofs, I wanted to make sure they didn’t walk away with the same sentiment that some of my classmates expressed so many years ago.    I thought a lot about how to go about it.    It’s been so long since I learned it, that I don’t really remember what my teachers did.   My course doesn’t use a single textbook so I couldn’t fall back on textbook lesson plans.   So, I ended up thinking about it and just trying some things.    I decided to go forward with a gradual release model in mind.

Prequel

I introduced my students to most of the properties that we are using now in our proofs during a unit last fall.   They are familiar with them and can use them, but don’t always remember which property is which.

Modeling a Proof

I began by modeling a proof with the whole class.   I felt like they needed to see a proof before they could do one, but I didn’t want them to just sit there and watch me, so I modified things a little bit.  The goal was to find a simplified, equivalent expression and use properties to justify the process.   I set up the original expression and the first step in the proof and then asked the class what property justified the step I had just done.  I then called on volunteers to provide the justification (property) for the step.   I repeated that process with each successive step.    However, each volunteer had to be a new voice.   I would not return to any one until everyone in the room had offered a justification.    I wanted every voice heard, but I also wanted to respect the fact that some of the students were going to take a little longer to venture a toe into this particular pond.

Collaborative Proof

The next proof, I modified my approach a little bit.   I did the proof with the whole class again.   However, this time I had students suggest each successive step and provide the property justifying the step.    Just as I had done in the previous proof, I took volunteers but did not return to a particular volunteer for another contribution until everyone in the room had offered input.     I thought this would be a way to chunk the process of a proof into manageable pieces so that it would not seem so intimidating.    The idea of coming up with a single step in the process is a little less overwhelming than looking at that blank table that has to be completely filled.

 

Partner Proof

By the time that we got to the third proof, I felt like most of the students had at least a place to start.  Some of them were still on shaky ground, though.    I decided that the next logical step was to have them complete the proof with a partner.    (My seating assignment is such that I have a student who is stronger in this particular strand sitting next to a student who is not quite as strong.    This enables students  who have less mastery to have access to the thinking of the students with a stronger grasp of the material through the small group discussions and partner work that they do.)  In this scheme, Partner 1 provides a step and justification, Partner 2 provides the next step and justification, Partner 1 provides the following step and so on.

I think this worked really well.    Both students were engaged, both students felt responsible/accountable, both students had a support system if they got stuck.   Every pair was able to successfully complete the proof.    They also had some great discussion about the “next step” in the proof.   This brought out the idea that there is more than a single solution path.   This led to a nice discussion of which solution path is more efficient and the question of whether efficiency matters.

Practice Makes Perfect?

Over the course of a week, we worked through the process of generating equivalent expressions.   Each day, the initial expressions were a little more complex.   Each day, we spent a little bit of time working with proofs.   The proofs weren’t the whole focus of the lesson, just a piece of it.   This revisiting of proofs each day gave students practice with proofs.   I think equally important though is the time they had to just live with it, the time to mull over the process without even realizing they were doing it.

As I watched students progress over the course of the week, the students who continued to struggle a little bit were struggling with identifying the property rather than figuring out the next step.   To help them, I created a set of cards on the different properties that we were using.    Each property had three cards:   the name of the property, a verbal description of the property, an algebraic expression illustrating the property.   I used the cards in several different activities over the course of a few days.    Each review activity took five minutes or less.

Quiz-Quiz-Trade

IMG_1143

One day, I used the cards for a Quiz-Quiz-Trade entry activity.   I had more cards than students so I used  all the algebraic expression cards and some of the verbal description cards.   I left the cards with the names of the properties out of circulation for this activity.    The students each looked at their card and determined the property.    Next, students found a partner (they move around the room to do this which incorporates a little movement into the period, which  Marzano has shown to be an important engagement strategy).   In each pairing, Student 1 shows his/her card.   Student 2 identifies the property.   If he or she can’t, they use a Tip-Tip-Tell strategy.   That is, Student 1 gives a tip (clue) and Student 2 tries again.   If he or she still can’t identify the property, Student 1 gives a second tip (clue) and Student 2 tries again.   If he or she still can’t identify the property, Student 1 tells him/her what the property is.    The pairing then reverses rolls and repeats.   Finally, they trade cards and find new partners.   I only continue the activity for about two minutes and challenge students to see how many pairings they can complete in the allowed time.     The time limit keeps the engagement level high (Marzano) and allows me to do a quick review of important information without taking too much time.

 

Properties Card Sort

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Students worked collaboratively with their table group to sort the cards.    They had to match the property name, the verbal description, and the algebraic expression representing the property for each different property.   I had them lay the cards out in rows so that I could quickly walk around the room and see how they were doing.   I like using this kind of formative assessment because it is a quick, easy way to find the gaps that I need to address.

You can download a copy of the Property Card Sort by clicking on the text below the picture or by going to the Resources page.

IMG_1142                                                                       Properties Cards

 

Next Steps

My next step will be to have students complete a proof that is fill-in-the blank.   I will provide some of the steps and some of the justifications.   Students will have to fill in the missing steps or justifications.

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