Monday, August 29, 2016

Math Don't Break

Integer operations are always an interesting endeavor with 7th grade students because they come pre-loaded with so many rules.  So. Many. Rules.

We've been talking about making our own rules, so we have this sequence of products and I ask students to discuss what patterns they notice.

-3 (3) = -9
-3 (2) =  -6
-3 (1) = -3
-3 (0) =  0
-3 (-1) = ??

Stuff we noticed:

"It starts with a -3 every time."
"It goes down by 1."
"It changes by 3."

I zero in to the apparent contradiction in going down by 1 and changing by 3 so we can clean up the language a bit.  This starts an nice little exchange about whether or not going from -9 to -6 is an increase or decrease.  We conclude it's actually an increase.  I have to remember to take my time here because this isn't an insignificant point:  Kids seem to think in absolute value.  

So what comes next? 

I wrote down everything I heard.  

"3".  "-3".  "4".  "-4".  

"Wow!"  I say.  "We've got a great argument about to happen.  This is awesome!  So many different opinions.  So which is it?"

Some minds change when groups start to discuss.  The students who thought 4 or -4 were thinking of sums and not products.  That leaves 3 or -3.  

"Ok, so which is it?"

If I had a dollar for every time a student said "A negative times a negative is a positive" followed by "because my teacher told me", I'd have all the dollars.  

But then Isaac offers a reason worth looking at. 

"I think it's -3, because positive 3 times positive 1 is positive 3, so negative 3 times negative 1 is negative 3."

So I write the following on the board:

(pos) (pos) = pos
(neg) (neg) = neg

We talk about this pattern Isaac. has noticed.  "Does this work for you all?"

Jordan speaks up, "I don't think so.  It has to be positive three so that it doesn't break the pattern."

"Which pattern is that?"

"The pattern goes from -9 to -6 to -3 to 0.  It's increasing by 3 each time so the next answer has to be 3."

"Why would that be so?" I ask. 

Then Vanessa chimes in.

 "Because math don't break."  

Thursday, August 25, 2016

Strategy vs. Procedure

I really want to focus on students being mindful of their process.  What they are doing is important, but they really need to know why they're doing it.  We've been doing daily exercises, How Many Squares?  that are based on Michael Fenton's activity, How Many Peaches?

We usually highlight different student strategies and have spent some time developing a continuum of strategies that looks something like:

counting --> grouping/adding --> skip counting --> multiplying --> writing/evaluating math expressions

This student's particular strategy generated a nice conversation.

I asked whether or not students thought this was a strong strategy.  Responses were less than enthusiastic so it was time to move a little.

Me:  Alright, if you think this is a strong strategy stand on this side of the room;  if you think it's not move to the other.

It was 31-2 in favor of the strong.  So I walk over to the "not strong" side and make my case.

Me: It can't be a strong strategy because the answer is 84 and this student said it was 76.

About half the class moves to my side.  I figure it was an even split on who was convinced by the "right answer" argument and who was convinced by the "I'm your teacher" argument.

Two students on the strong side raise their hands.

Student 1:  I think it's still a strong strategy because he probably just made a mistake.

Me:  Probably?  Where does that fall on our argument continuum, gut level, some reason or convincing reason?

Student 1:  Some reason.

Me:  Ok, great.  Can anyone take it to the next level?

Student 2:  I think it's still a strong strategy because he just counted 11 instead of 12 across the top.  He still multiplied right, but he just used the wrong numbers.  Everything else was good.

Yeah, that'll play.

Monday, August 22, 2016

From the Gut to the Head

Keeping in mind that we often get what we measure, I started from day 1 talking to students about an argument continuum.

Gut Level Answer

We're all pretty good at this one.  Offer an answer, but when asked why we .   This is often a student's default, especially if they're used to an answer getting culture.  

Answer With Some Reason

This is a step above the shrug, but isn't entirely satisfying.  I'm ok with students being in this area for a bit--"I think because " even if isn't completely convincing.  

Answer With Convincing Reason

I'm not really pleased with the wording on this one, but the gist is that we are looking for a student's thinking to be able to stand the test of peer review.  Does it convince others?  Can others use your process and arrive at the same conclusion?  If so, then we'll call this good.

I think this is something that I've had in my mind for as long as I've been teaching, but being more explicit about it with students has been beneficial.  I hear things like "show your work"  which has morphed into "show your thinking"  and I think they both are trying to get at the same thing.  Unfortunately, I think students usually interpret these in a quantitative way that amounts to something to check off the list.  Did I write a number of things down because teacher asked me to?  Yep, so let's move on.

As students begin to look at the quality of their work, we all win.

Wednesday, August 17, 2016

Don't Call it a Comeback...

...I haven't been here in (what seems like) years.

The past couple of years have been a whirlwind of change.  Full time math to full time elective to elective/part time math coach and now finally...

One section Math 7, three sections of electives and afternoon math coach.

Oh, and five of my 17 kids are now 17, 14, 11, 9 and 6 years old.  The older two are a senior and freshman (respectively) in high school while the younger three are still reaping the benefits of having an amazing mother who is willing to donate herself to homeschool.

It seems like a lot has changed since this blog was more active, but I hope to catch up with you all.