Sunday, February 19, 2017

Fraction Concepts Day 2! Conceptions, Misconceptions, and Mathematical Language

teaching fractions
Well, things got even more interesting as my fraction unit unfolded on day 2!  For those of you who read my first fraction post, I discussed how we used paper folding to get our fraction concepts unit "launched", and we ran out of time to finish the investigation.  I gave them another 25 minutes today to work on folding their different fractional amounts and then asked them to spend some time writing in their math notebook on any of the following:

What did you notice?

What was challenging?

What was easy?

What patterns did you discover?

fraction unit
First I gave the students "alone" time to write and then let them go compare notes with their partner from the investigation...



fractions
Not a great photo...not great handwriting...sorry!
What really stood out to me was the very NEBULOUS (vocabulary word!) math language I was hearing.  A phrase that kept coming up was "small fractions" as in "It was harder to fold the small fractions."

So, being the pesky and annoying teacher that I am, I presented this question to the class:
fractions
Tee hee...I love to ask questions like this!
I didn't let the students talk . . . I just said that I was hearing MANY students use this phrase, so I thought we had better made sure we all knew what it meant.  I asked each student to write the answer to that question on a post it note and come slap it on the easel.
mathematical thinking
And--seriously--I only wish I had taken some photos of the notes up close.  SERIOUSLY!  I had everything from . . .

64

to

1/100th

to

"When you make it smaller"

to

"Small fractions are like tenths"

At this point, any concerns I had about backing up this far and moving this slowly were GONE!  I read off the notes to the students and then asked what we had learned about what a "small" fraction is and one of my delightful yet very shy students called out, "ABsolutely nothing!" and we all laughed hysterically.

So where do we go from here?  I reminded the students about how we have been talking about the Common Core and how important it is to use precise mathematical language.  I asked if "small fraction" was precise enough that anyone else would know what we meant. . . and we had complete agreement that it was not.  I asked them if, on our 4, 3, 2 , 1 rubric we were evaluate how clearly we communicated, we agreed that all our post its were 1's with a few 2's sprinkled in . . . until "T" asked, "If there really isn't an answer, how could someone get a 4 on this?"  Have I mentioned how much I love fourth graders?  He honestly asked--and honestly wanted to know!  I told him I had to think about it. . . always a good strategy for the tricky questions!  Don't answer too soon!

So I set the students to work on a little activity to set them up for tomorrow's lesson...I asked them to work completely alone to answer the following question:

fraction misconceptions
If it is too small to read, the question asks "Is this shape divided into fourths?  Explain your thinking."

I let them work for a few minutes and then on their way out to recess made them commit to either a "yes" or a "no" response . . . and it will be our kickoff "debate" tomorrow!  Remember, one key element of the Standards for Mathematical Practice is the ability to explain one's own work and to critique the reasoning of others . . . so tune in tomorrow to see how it all shakes out!

After they worked for a few minutes we gathered back together to talk about T's question . . . is it possible to get a 4 on a question with no answer?   Maybe I should save that for another blog post . . . this one is already plenty long--but I told him that yes, I do believe you can get a 4 on a question with no answer.  I reminded him that we are stressing explaining our thinking, and we could certainly do that with this question.  I modeled something sort of like this (this was done aloud, not in writing . . . I'm just trying to give you the gist of it)

"I know that there are different ways to look at the idea of being small.  Some people might think the numbers should be small--like in 1/2.  Some people might look at the size of the pieces to determine what a "small" fraction is.  Still others might think a small fraction is one where you only look at one piece of a whole--like 1/3 is small where 2/3 is big.  So for the question "What is a small fraction?", it is very important to understand that there might be more than one way to answer it."

Would I expect a fourth grader to be able to answer like this?  Maybe not today . . . but now that it has been modeled--perhaps.  We must push their thinking, push their comfort level, and push them into the rigor that is required.  If we make it meaningful, they can do it!  Thanks for sticking with me again--sorry these are getting so long!  Stay tuned to see how our debate works out . . . I'll give you a hint--the vote was 9 "yes" to 13 "no"!


***UPDATE***
This blog post is now a part of my comprehensive fraction unit available by clicking the image below.  Hundreds of teachers have now used it to change the way they teach fractions!  



Saturday, February 18, 2017

Fraction Folding--discovery learning

hands on fraction activities
Today we kicked off our fraction unit, and I think I am going to really try to do a lot of blogging about it over the next few weeks--because I will be immersed in it AND because it is such a critical component of the Common Core for intermediate grades.  I think it is vital that we dialogue about ways to help students build their understanding of fractions, so I invite you to share along with me as I "trace" the path of our unit as it unfolds.  I'll try to be clear--but you know how I tend to get wordy!  I'll try to include lots of photos and work samples as I go, and I am hopeful that the rest of you will share great ideas and resources that have been successful for you.

Again, knowing that the CCSS places a great deal of importance on fractional understanding, our district math team made a decision to build in two fraction units into our year--this one and another one later which will work to tie together decimals, fractions, and a more sophisticated level of understanding.

Today I started by asking students to reflect on what they already know about fractions and to rate their overall confidence (using our 4  3  2  1 ) scale (Click here to revisit earlier blog post about this!) and the results were quite amazing!  In a ten minute writing time, I got to witness a number of misconceptions, poorly explained reasoning, and a bunch of "3's" and "4's" in confidence!  Good thing I had planned on starting slowly!  Today we started our new math notebooks, and I explained to the students that we are "raising the rigor" one level more (poor things have heard this all year) and we are going to work hard to use our new notebooks to both record our thinking, our practicing, and our new learning.  I am going to try a version of interactive notebooks, knowing that I cannot do it in true form as some teachers do . . . I'm just not quite there yet.  I'll blog more about this later as it unfolds!  For today, we started the section we called "Fraction Concepts" and even talked about what the word "concept" means.  Fascinating!  I told the students that our job through this unit would be to determine some things we could determine to be "true" about fractions and that we would be working our way through a number of these "truths" during the unit.

Today's "truth" involved ensuring that students understand that fractions represent equal parts of something (I didn't really want to use the term "whole" yet--I don't really like to treat fractions of "objects" and "sets" differently until I have to!) and that we would be spending some time creating equal parts.

I put the students into pairs (I love popsicle stick picking!) and gave them each 3 minutes to find a classroom object that was either a square, rectangle, or circle and was bigger than a deck of cards and smaller than a book.  Each team was then assigned a color of paper and the following task:

With your partner, trace and cut out your shape.  You will need many of these as the investigation unfolds.  Your job is to find ways to divide your shape into equal parts. . . first in two equal parts, then three, and so on.  


fraction unit
I made sure we had some circles, some rectangles, and some squares...

I then showed the students the following charts--each is labeled "halves", "thirds", "fourths", and so on.  As they discovered a way to fold their shape, they were asked to put it on the correct chart. . . and I was on the prowl for work that was not accurate and precise.
teaching fractions
Voila!  Posters are taped to the ground with deliberate "aisles" so they don't get trampled on!

fractions
Each poster is labeled with the word for the fractional part--I will use the words and different symbols interchangeably throughout the unit.  NOTE:  The chart does NOT say "1/3" because we were not identifying 1 out of 3 parts.

Students dug into their work and did need some frequent reminders about using straight edges, working carefully, and so on (I know--shocking!), but the were very engaged and thoughtful.  I heard some pretty nifty stuff like:

"Hey--as you try to get more pieces, the pieces get smaller!"

and 

"Man--it's a lot harder to fold the odd numbered pieces!"

and

"If you fold it in half and then in half again, each half gets cut in half!"

fraction unit
Tracing...
fraction activities
Using a straightedge for precision...
fraction lessons
Circles are the toughest as this team found!

As the period unfolded (and it became clear that this was going to be a TWO DAY investigation!), our charts began to fill up and things moved a little faster.
fraction lessons
The charts started to fill...first halves and thirds, then fourths and eighths...we'll see how it "unfolds" tomorrow!  Got to love a little math humor, right?
Tomorrow we will finish up our investigation and then ask ourselves if we have discovered any new "truths" to record in our notebooks.  Although I am pretty confident that my students would have been able to answer correctly if given a sheet of "shaded fractions" to identify, today's activity showed me that many students are missing some very critical understandings about equal parts and about patterns that arise when dividing shapes.  I'm pretty sure that the work we will do over the next few days is going to be very important to build a foundation for the more advanced skills coming.  If you aren't familiar with what the CCSS requires from students regarding fractions, I would encourage you to dig in and follow along with us as we try to construct meaning over the next few weeks!

***UPDATE***
This blog post is now a part of my comprehensive fraction unit available by clicking the image below.  Hundreds of teachers have now used it to change the way they teach fractions!  



Wednesday, February 15, 2017

Studying Dialogue to Improve Reading AND Writing

teaching dialogue
Today is my day to post over at Upper Elementary Snapshots, and I hope you'll head over to read about a lesson I did last week to help slow down my readers and get them thinking more deeply about the books they read.  I am excited to see if our dialogue studies transfer to their writing next week--so stay tuned!  Want to learn more about what we did?  Just click the image above and check it out!