LESSON STUDY
With all the great stuff happening with number lines around the #MTBoS, I thought it might be an interesting idea to bring to our cross curricular lesson study group in first semester of this year (201516). The original lesson was designed for a 75 minute period. I just repeated this lesson again this semester with two classes of grade 10 applied's but carried it over two periods. This was much more manageable than the original lesson which was done in one period and that felt very rushed.Kudos goes to Robin McAteer (@robintg) for the original idea which I experienced at a summer math camp for teachers in Barrie Ontario a few years back. I specifically asked Robin to join in this lesson study (she is an instructional coach at our board  and I am a huge fan) as I thought she would have some great insights (as she always does). Also, as always, Robin documented our debrief and summarized the exit cards. You really can thank her for the depth of this post.
This particular lesson study was a bit different. Let me tell you why. Normally we would plan a lesson one week and then the following week deliver and debrief. This particular lesson was planned on the afternoon of January 7th, 2016 to be delivered on the morning of January 8th, 2016. The reason being that we had a guest coming from LA to join us for the two days. Judith Keeney (@JudithKeeney) whom I met at a session she presented on at #TMC14 in Oklahoma on lesson study (and hit it off with by the way) was interested in joining us to observe and participate in a lesson study at our school.
This presented a slight challenge as the lesson had to be done by the end of the first day, including any classroom set up. That being said I had an idea of what the lesson would look like before the meeting ( so..... unfortunately our group did not really feel like they had ownership of this lesson). None the less something cool did happen. Normally we would talk for a whole afternoon and then the teacher delivering the lesson the following week would iron out the details and set up the classroom for the lesson on their own. This time though we planned for about half the afternoon and then we all went to my classroom to set up the physical learning environment. OK something cool happened  not sure how to describe it because we were just chipping in on the set up we started bantering a little. Lot's of jokes, pokes and other things. There was something about setting up the lesson together not just planning it together. A real sense of team  hard to describe. Lot's of talk the next day as to how to incorporate the set up as a group as part of our lesson study model. Of course here we are 6 months later and we have not managed to do this (would cost more $$$), but it is on the back burner.
Anyhow can't even start to thank both Robin and Judy enough as well as all our team members from first semester  it was a unique experience.
THE LESSON
Here is the planning document for the lesson.
Number Lines: A Lesson for Connecting
Representations and Developing Number Sense
·
Provide all observers with:
Photocopy of names with pictures
Observation sheet
Lesson plan
Random Groupings
Timing

Teacher Moves

Observations /
Improvements

Before students arrive

·
Desks arranged in pods
·
Random Groups of 2/3 by cards
·
White Boards, colored markers, number line paper,
Number lines pre hung for groups, Clothespins


As students arrive

Inroduction to
number line activity (day 1)
·
Place 10 and 10 and 0 together
·
Give each group (random) of two/three students 3 numbers
and have them place it on the number line (from 10 to 10) Decimals, Fractions, and Whole numbers
·
Clothes
line all the way across the room.
·
Discussion about placements  scale


Group number lines (day 1)
·
Give groups expressions and colors associated
with each expression.
·
Quadratics and linear expressions.
·
Give groups Table Markers (8 colours) and one non
permanent marker.
·
Give groups a number from 
·
For each colored equation find/calculate the y
value on your whiteboard for your given x valueindicate the equation on the
whiteboard so that I can verify your work.
·
Once
you are finished and Mr. Overwijk has verified. Then place the values with
the appropriate color on your number line.
·
If beyond range of values it is not included
on the number line.
·
Need to do minimum three
·
Once the group has the number line done for
that x value they repeat for a different x value until time is up. (7 groups
 hopefully 3 per group)


Generating graphs
of expressions using the 21 number lines (day 2  second time through)
·
Students bring their number lines and place
them vertically on that x value this should create a graph of the y=the
expressions
·
Look at characteristics of the expressions
from the equations and the graph  connections


Home Base
·
Exit Card

Here is some groups work calculating the values for all 8 equations using their given x value. We spent a period getting all the number lines together this most recent semester. (21 in total from 10 to 10)
At the start of the next day I asked if anyone could tell me what we did yesterday.
S "We placed dots on number lines"
Me "Where did the numbers come from?"
S "We had 8 equations and an x value and we placed the y values on the line based on the colour of the equation."
Me "Ok great. So if we were going to place these from 10 to 10 would they go vertical or horizontal?"
Long period of silence.
S "They would go vertical because the dots represent the y values for the x value you gave us. The y values are up down."
And there you have it. So with some student help we slowly placed (taped) the 21 number lines. Here is what it looked like.
So some dots were in the wrong spot despite me checking their values on the whiteboards. It appeared as though most of the mistakes were groups putting a positive value as a negative value and vice versa. I will say as we put the lines together there were lots of comments like, "Oh we are getting the graphs of those equations!" I think I even heard "Holy S&it."
Here is what it looked like once we connected the coloured dots for the two different classes.
I then asked students to do some characteristic finding for the 8 equations. Once they were done they could go look at the graph and verify their answers.
Here is some sample work from groups for one of the linear examples.
And sample work from a group on a quadratic one.
THE EXIT CARD
At the end of the activity we asked students to fill in an exit card that looked like this.
Exit Card Name:__________________
About your Learning

Red

Yellow

Green

I understand how to place values on a number line (positive, negative, decimals, fractions)
 
I chose an effective tool or strategy to calculate the y values given an x value
 
I understand the connection between the equation and the x and y values
 
I understood how to place my y values on the table of values for the different equations
 
I understand the math vocabulary used today.
 
I worked well with my team (asking questions, explaining, on task, encouraging etc.)

About the Lesson

Liked

Didn’t Like

No Opinion

Using the number line strips
 
Learning with my group / working together
 
Building the tables of values as a class
 
Other?
__________________________________

Something I learned today was _______________________
Something I’m wondering about is _______________________
We got these results:
THE LESSON STUDY DEBRIEF
Notes from the lesson study as they happened. (student names deleted  names included are teachers)
Al’s Class  Debrief
Complex detailed lesson
 couldn’t have done it all with one person  would have taken 2 classes
 needed 2 or 3 people walking around checking
 lots of details had to be worked through in the morning
ex. doing up answers for quick checking  checking was time consuming!
Al
 Reflecting on planning a lesson and leaving an individual to finish it up vs. the team effort of putting it together like we did this time
 Maybe we should be doing this more often instead of leaving it to one person
 it was cool for us all to do it together
France
 How  maybe staying after school  it worked in this case because it wasn’t all of our lesson
 Can’t sacrifice the time spent making up the lesson
Robin
 Some groups have a rule that they don’t leave someone with a pile of work  they do what needs to be done
Dana
 France’s idea  stay after school to help that person  get it done  it’s fresh
Paula
 The tweaking is sometimes more than the whole design  liked that we started with the lesson somewhat designed and had time for tweaking and perfecting
 Sometimes when we build from the ground with so much input you don’t get to the point of tweaking
France
 Maybe we’re at the point where we can build on an existing lesson because we understand that the kids are doing the work… we are starting with a fairly solid idea  we’ve been doing this for a while.
Robin
 agreeing that the details / intricacies are important
Al
 Reflecting on Ball Roll lesson  people were taken aback when I did a teacher move that I had thought of by myself  they weren’t expecting it
France
 Have seen some lessons where the plan get’s abandoned because the person left with the planning loses confidence / doesn’t fully understand / gets scared
Al asking Judy
 Do you find that there are times when you don’t get to all the details so you aren’t pushing the envelope as you hope?
Judy
 Yes  people work in much more isolation
 it took three years to get to a point similar to where you are now
 it was really bumpy at the beginning  because we were learning how to collaborate
 we got to the point where someone comes with a lesson and we focus on the details
 we do a lot of electronic collaboration
Discussing pros and cons of various timing options
 need to finish up the planning fairly quickly while it’s still in your mind, but leaving a bit of time for it to settle and ideas to come is good too
 Al: still likes it being a week between the planning and the delivery, but sometime during that week do the in between work
 Paula: Sometimes I start to forget when it’s later in the week
 Anneke: Plan on monday, revisit on Thursday after School, implement next Monday  to give it a bit of thought but not too long
Reflecting on the planning
 turning point was when nonMath people said that they weren’t getting it
 we all slowed down  realized orientation was a problem  worked it through a bit  Paula’s idea to pull the strips off the wall was generated
Flexibility
 sometimes you need to start from scratch, sometime a partially planned lesson can work
 depends where the participants are at  for newer people the lessons take more time to plan and everyone benefits from the whole team being involved
How important is it for everyone to understand the lesson?
 observation is improved with deeper understanding, but it isn’t reasonable to expect nonsubject teachers to be experts in the nuances of the curriculum
 important thing is for everyone to understand the big idea / learning goal
 ex. in this lesson, making connections to the characteristics
 if the whole purpose is to watch kids learn and move, then the more you know the better
Al’s reflections
 intricate lesson, super busy for me, needed to make sure all of the pieces were right
 breakthrough in the last 15 minutes  they were getting quicker
 second time was way faster
 bailed on the table of values  that wasn’t going to happen
 it was hectic  I didn’t have time to notice who was learning what
Robin
Student A and Student B
 two weak students together randomly
 took a while go get going, but they were engaged for almost the whole class
 payed attention to the number line  quite engaged / focused
 Student A has a knack for knowing when things don’t make sense
 Grabbed the graphing calculator, but they didn’t know how to use it
 They were excited and meticulous after that
 both learned about the calculator tool
 they had fun today
France
Student C, Student D, Student E
 Student C had absolutely no idea from the beginning to the end
 she did nothing  she filled out the exit card
 Dana: she looked busy  from afar
 Al: She can play the game
 Student D did the thinking
 None of them understood how to start
 they saw the boys
 Student D started doing the calculation
 Student E started copying from the boys board
 was on her phone  facetiming at points
 Might be interesting to have her followed around to collect evidence about the time she spends on the phone
 Student D asked Student E a question at one point  she said “we’ll do both” she was involved to some extent
Paula  observing same kids
 it never ceases to amaze me how Student C can hide
 Al: has warned her that she has to show me something  she’s a worry for me
 She bluffs it
Al / Dana  pros and cons of battles with kids over the phones
Al: There are 5 or 6 in the class with this problem  decision to battle or preserve the relationship
France: Would be interesting to follow her around and document phone use as evidence for a phone addiction
Dana: I don’t think you destroy relationships … they resent it…
Al: It’s a battle  I know where this ends up  those kids stop coming and you lose them  it’s tough  the energy it takes 
Anneke: I take ____ phone all the time  I provide a free service to kids by taking their phones from them
Student F / Student G / Student H
 Student F was really involved
 Al: He’s a smart kid  has missed so much  when he’s there, he’s there
 Anneke / France  it’s anxiety
 Student H  why is she in P? Al talked to her about it early on
 one of her first questions  can I just make x^2 + x x^3… maybe that’s why she’s in applied...
Anneke
Student I, Student J, Student K
 Student K had the marker and did most of the work  was the boss / leader
 Student J stood on the side with the graphing calculator
 Student J was very engaged  trying to learn  the whole time she was on
 Lots of conversation in Spanish
 Exit cards showed positive reflections
Student I
 was trying to figure it out using tiles
 Al: That’s a skill that they have in my class  she was turning it into length/width form
 it would help them to sub into a factored (more familiar?) form
 Robin: Was she understanding why she was doing it?
 Al: They have to sub x values in… so they are used to doing it when they are doing the vertex… so she wanted to rewrite it so she could sub in.
 Exit card for Student I  is this going to be on the test?
Student L and Student M Student N
 Student L was engaged
 Student N came in late, hung out with student O
 Making comments about everyone else including Mr. O with his harem
 Student M was engaged right from the beginning  started measuring right a way to figure out where the zero was  knew that he had to put zero in first
Dana
Student O
 wants to wander around the room the whole period and socialize
 he’s a huge distraction
 other teachers have inquired via email about him
 was quite engaged until the end when Student N came in
Exit Card
 they liked it, they took time to do it
 some kids just checked it off (ex Student O )  have to question the validity
Graphing Calculator
 some were using it as a tool by putting in the equations and going to the table
 others were using it just to calculate
 most weren’t using it to get a table
Dana:
Student N, Student P, Student Q
 Student P was putting values on the white board while Student Q told him the numbers
Is the process of putting in the numbers helping him? He’s really just dictating.
 Logistically  might be an idea to change who uses the calculator
 Student N wrote explicitly on the exit card that he liked the explanation
 hard to say what he meant exactly
 Student Q was focused, asking questions, very much engaged
 exit card  yellow on math vocabulary
Judy
Student R and Student O
 Took a long time setting up the equations
 Student O was driving the calculator  putting in every single operation
 missed negatives and operation signs
 During pauses (ex. waiting for checks) Student R took the calculator and tried to figure it out
 diligence  she was watching what he was doing, then practicing
 The second time they worked through it together in about 5 minutes
 double checked and made adjustments themselves
 Took a long time with the number line  getting the dots coloured in
 Student O was engaged the second time around
 Student R said she liked working with the group  had a positive impact
 Student O didn’t like working with the group  put yellow for worked well with his team
Thach Thao
Student N, Student Q , Student P
 Student N was the first at the board, but Student Q corrected some of his answers, so he stepped back and Student Q took over
 Student P just stood there watching
 Wanted to go to Student P and just say “does that answer look reasonable” would be useful to get the EAs to do that prompt
 Student Q asked Thach Thao to check their answers
 Thach Thao asked them to reflect on the impact of repeated removal (negatives)
 Student P said “that doesn’t make sense”  he does have some understanding
 Student P  language barriers / processing
 when he understands what your asking, and when you sit and listen, he can tell you
 he can do a lot of the stuff  he needs time for processing and articulating
 Paula : he’s under investigation for an IEP  he’s ESL so it’s hard to assess
 Al: interesting about the language  sometimes if I wonder if he is just being a goof and playing people
 Paula  this question has been raised before
 Lot’s of wondering about him  He’s a conundrum
 France  it would be nice to have a bilingual tutor in there
 Al: I’d like to know what Student O saying about me  he’d be badmouthing me non stop
Student R
 was very late
 with a group who was struggling
 jumped right in and tried stuff
Student S wasn’t there today
 maybe didn’t want to be exposed
Student T
she’s anxious  she refuses support of a scribe
 during tests  she needs me to ask her the questions individually or she’s not going to write anything down
 Paula: if she could connect really nicely with an EA  who could force themselves on her  get in there and support  she has a lot of potential
Lesson Reflections
 great to have the repeat built in to the lesson
 immediate feedback / positive reinforcement
Robin: Wondering about the learning of people in groups where there is a large gap in readiness
 ex. Student H, Student F, Student G
 Student H was chatting with others
 Student F was holding his own  learned how to convert fractions to decimals
 Student G  not sure  said that something he learned today was nothing
MY THOUGHTS ON THIS LESSON
1. Loved this creation of the graph. It would be a cool way to introduce the graphs of functions that you will be exploring in a course. For example in MHF4U do a rational, polynomial, exponential, logarithmic, trigonometric. Tons of potential with vertical y value number lines.
2. What about later in courses where students then knew how to solve for x given y. We could give them the y value, they would solve each equation for the x value, put on the coloured dot, and then place the number lines horizontally at the given y value. Again we would be generating the graphs.
3. HMMM Desmos activity Builder anyone?
Huge thanks to all involved. Never would have generated all this on my own. So much learning for me.
Also nice to blog about my classroom again!