Tuesday, December 30, 2014

A Ball Roll Race

Belonging to a group that studies lessons has it's advantages. At least once a semester we work on a lesson for my class. In our pre-planning meeting I brought a few ideas of lessons that I really wanted to lesson study to see where we could take them. I proposed Barbie bungee, cup rolls, factoring and expanding quadratics and ....

These all got shut down as everyone said I already do these lessons and they wanted something fresh.

We came up with a ball roll race. This lesson was designed for grade 10 applied math in Ontario. Here is how it went down.

Day 1
Students spent some time white boarding some questions that reviewed similar triangles, sum of squares (Pythagorean theorem) and right angled trigonometry. In the last forty minutes of the class I created visible random groups (5 groups of three students) and gave the groups a piece of wood (all 5 groups had different lengths ) that they were to physically create into a three degree ramp. At my school we have a hallway that is inclined at 3 degrees for well over 11 meters-this would be where the ball roll race would occur.

Once groups started to create their ramps they immediately said "I guess we need to know the length of our ramp." Groups asked for measuring tools which I provided. They also asked for books to build up one end of their ramp. Once they felt they had it I came and measured the angle of their ramp with the blackboard protractor. (Really wish I had taken some photos of this. Groups either went back to the drawing board or they had a three degree ramp) Groups then put their ramps and the books they would need for the next day to rebuild their ramps in the corner of the room. Here is some of the whiteboard work the groups did to figure out height needed for the ramp to be 3 degrees. This part went well-some groups needed to adjust the heights as they just went to the top of the books not the top of the piece of wood.


Day 2
To start I asked all students to guess too low, too high, and best guess for each of the balls to roll down the 11 meter ramp. Here is a sample.

Students collected data for time and distance for each of the five balls. The five balls were a basketball, a red dodge ball, a large marble, a bocci ball and a tennis ball . Here they are.

Students collected multiple data points of time for each distance up the ramp. I suggested at least four different distances. If they measured the time multiple times they could average the times for each distance. Groups collected data for the period.

I anticipated a problem here. I felt that if we were fixing the distances and asking the students to measure the times we would be looking for trouble. Since we are measuring time it would be dependent and distance would be independent and we wanted this reversed for a quadratic relation.
So....I decided to create the data collection sheet for them. Here is what I came up with.

 I was hoping that this would prevent any confusion. I was clearly worried about the correct answer versus student thinking. I should have let the students record their data anyway they wanted. Here are some pictures of students collecting data.
Day 3
This was the day that my lesson study peers observed. Groups were given their data back and were told that they were to model and calculate how long it would take each ball to roll down an 11 meter ramp. Groups worked on whiteboards (VNPS). The groups strategies were varied.

Here is the rundown and the work. Time was tight. If this had not been a lesson study I would have given groups more time to work on their solutions.

Group #1 Chose quadratic relations. Never got to a time for the 11 meter race.
Group #2 Chose a strategy that I am not sure about. They did not get to a solution for each ball for the 11 meter race.

Group #3 Chose quadratic relations. Never got to a time for the 11 meter race.

Group #4 Chose a linear relation. They used time as the dependent variable and distance as the independent variable.
Group #5 This group tried proportional reasoning first.

Then they realized that the speed of the ball was not constant so they went to quadratics. Did not quite get all the times - almost.


So at this point we had about 30 minutes left. I then assigned each person in each group the letter A, B or C. So what I had was A1, B1, C1     A2, B2, C2     A3, B3, C3      A4, B4, C4        A5, B5, C5.
I then ordered all the A's to one whiteboard solution, all the B's to another whiteboard solution and finally all the C's to another whiteboard solution (3 groups of 5). Groups of 5 would then rotate around the room to look at solutions - at each board there would be someone in their group who worked on it and would be able to explain the group's thinking.


Before they started I handed each student a Keep It or Trade It sheet. They would fill this in as they observed each solution. Here is what the handout looked like.

Once each student was comfortable with their sheet I asked them to put their rank order of the balls in the race on a whiteboard. I also encouraged the teachers observing to commit. Here is what the board looked like.

Early in the day I surveyed my grade 12 class about the race. Here were their thoughts. 

And then it was time for the race. Here is what the situation looked like. 


Here are two links to the actual race.
Video One
Video Two

The next day I put all the data into a table.

Students did a linear and quadratic regression and we calculated the times for each ball based on these models. We then went back to the 11 meter ramp and measured the time for each ball. Here was the table and the final times.


Thought I would throw it out to twitter to see what kind of interest I would get - very little.
My only regret is that students were not given more time to flush out their ideas and come up with a calculated time for each ball. Covered lots of curriculum - proportional reasoning, trigonometry, linear relations, quadratic relations. Covered lots of mathematical processes - communication, representing, connecting, problem solving, reasoning, selecting tools. 
I would love to hear your thoughts.

Thursday, December 4, 2014

My Annual Christmas Beard

Let me make this clear-this is a bit weird- but there is a story behind it.

When I went to Carleton University in the 80's I was fortunate enough to play for the basketball team. By the way they have been the best team in Canada for over a decade.

When December first would roll around and basketball would take a break for exams I started a tradition of growing a beard till Christmas day. Anyone that has known me in my teaching career knows that I have kept this tradition since. Check out this kids comment on RateYourTeacher from "Apr 03, 2003 Go Overwijk. If you have this man, you'll like it best around Christmas. The reverse hair is hilarious. Nice tradition though." So that makes it a true story and a true tradition.

Now over the years I have started growing my beard much earlier than December first, much to the dismay of my wife, but tradition is tradition. Here is a photo of me today.

This year I decided to collect some data on beard growth. Here it is. Don't be too grossed out!


How about:
On what date did I stop shaving this year?
How long will my beard be on Christmas day?
How long would it take me to be Santa? (hint)

Any other wonderings? questions?
Also I am taking answers and solutions to the first two questions. Answers revealed on Christmas day!

UPDATE! December 28th, 2014
So before I show final work and reveal answers here is a couple pictures of the final product. Particularly proud. As Andrew Stadel @mr_stadel hashtagged #beardoweirdo !

Here is my solution to the original questions based on the data that I collected.

And now for the actual date I stopped shaving Tuesday October 22nd.
As for the length of my beard I took three samples and the longest was 4.45 cm. Here is a picture.

And what would be growing a beard without a little fun shaving it off.

And finally back to normal. #happywife. Hey is that a glass of red wine in the background? Gotta go!

Saturday, November 29, 2014

Serial position curve

This a nice one day activity on quadratics.

Students walk in and I spread them around the room and ask them to clear their desks other than a pencil and a blank piece of paper.

I tell them that we are going to do a memory task and that it is not a measure of their intelligence but a measure of their memory.

I have a list  of twenty words that I say pausing for a second or two between each word. The students are told that they are to remember as many as they can and then at the end they will write down as many as they can remember.

Once this is done I ask for a show of hands going through each word in order that I said it. I create a table of values of the position of the word versus the number of people that remembered it on the board. Time for students to get to work. I ask them to create a graph of position of the work versus percent of the people that remembered the word.

Here is what it would look like.

Here is some student work. This particular student did not use their model to answer all the questions.

Some questions:
a) What model is the best for this data? Why?
b) At what word position of the list did the least percent of people remember based on our model?
c) What percent of people will remember at this least percent position from part b)?
d) What position of the word would we expect 50% of our class to remember the word?
This is known as the serial position curve and leads to a nice discussion about short term and long term memory.
Ever go into a bar and ask what they have on tap? Tend to remember what is at the start of the list and the end of the list. If it was my bar I would list beers that I would want to sell at the start of the list or the end of the list.
Any other thoughts on where this might occur?
You can read more about it here and here.
I learned about this in a second year cognitive psychology course I took while I was teaching. You can also mess up the short term memory by delaying them in any way before they write down what they remember.
Fun activity.

Sunday, November 9, 2014

Edcampottawa Session 3/4 Restructuring the Math Classroom.

Bit of a ramble in this post as I wrote it live. Everything in this post is not thought out, everything in this post is not my thoughts, everything in this post is what I heard or said at #edcampottawa, everything is not linear (ideas are as they were heard), everything in this post is the collective of #edcampottawa. Have fun reading this. My apologies to everyone that was there as I was doing this (typing slowly) instead of truly engaging. I did this so I could share my experience so more people would see the value in #edcamp. Also I left this one after session three as I had to go coach my son basketball (Priorities)?

Struggle with how to change a math class.
How to pull it all together?

Good feedback, formative assessment, activities, warm ups, tasks.

Spiralling versus unit based.

Sounds like a nice continuum from elementary to intermediate to secondary in terms of talking about math / explaining your thinking /making it visible. 

Open number line. Start with a line with no ticks. They chose the ticks. Here is the number line for 100.  Where is 50? Where is 30?

Bruce's analogy of getting all students across the street at the same time. Kindergartens across the street holding a rope are slow. Let them go. They get there on their own time. Chaos. Scary for some.

Assessment- all the kids on the same day answer the same question. How about when they are ready?

Bringing math in their lives and showing how it relates there and maybe how it could relate to other things.

Homework meaningful? What is the purpose? Assessment for learning, assessment as learning-this should be the purpose of homework. Who is doing the homework? Is it just mimicking? Where is this strategy in their homework coming from?

How about the next year? If my class is so different do you get flack about it the next year?
Ok this is me ranting: I hate this. I am a teacher. I teach! Next year is next year. My job is not to prepare them for next year. My job is to engage them in learning this year.

Warm up activities . Estimation 180, visual patterns, headbanz, would you rather, fast fingers (adding by speed), number talks, Jessica Shumway stuff,  daily Desmos challenge, counting circles, bizz buzz, math mistakes.

Resiliency- mistakes are cool - that is how you learn. My best "no". Purposefully make a mistake in your work is OK. Non-permanent surfaces are better for mistakes-more willing to take risks.

Thanks to Mary Bourassa for leading this! Sorry I had to go :(

#edcampottawa Session One: Alternatives to Paper and Pencil Testing &Session Two:Turning Points in Assessment

Bit of a ramble in this post as I wrote it live. Everything in this post is not thought out, everything in this post is not my thoughts, everything in this post is what I heard or said at #edcampottawa, everything is not linear (ideas are as they were heard), everything in this post is the collective of #edcampottawa. Have fun reading this. My apologies to everyone that was there as I was doing this (typing slowly) instead of truly engaging. I did this so I could share my experience so more people would see the value in #edcamp.

My dear friend @BDMcLaurin picked me up at home and we immediately started talking about a lesson study we just did about a Ball Rolling Race. The premise is that the students will investigate ball rolls to create a model and then predict which ball will win and predict how long it will take each ball to go down an 11m ramp at an angle of inclination of 3 degrees. Anyhow.....there are some problems with the design of the lesson which I have to perform this Thursday. We were discussing this. On the way we also pick up @lkpacarynuk who immediately joined the conversation. This turned into how we as teachers ask kids to "investigate" but we value the solutions from students who get the connection or result that we want out of the investigation and not the process of "investigate". And then we assess/evaluate the connection or result that we wanted. We don't assess/evaluate the process of "investigate". In fact we don't even know the criteria for "investigate". Ok so I need to co-create the criteria for the verb "investigate" with my classes.

Went to the lobby and saw a bunch of people I know: @MaryBourassa, @hfxmark (from Virginia), @mmehmatte, @LiseGaluga, @Wheeler_Laura, @rswandel, @robintg all whom I have a great deal of respect for. I am sure I have missed some people.

Here is the edcamp board.

Bruce and I are heading to lead a session about getting away from paper and pencil testing. (or at least less of it)

Session #1
Introduction of what is happening in my class (collaboration and inquiry VNPS and VRG) and what is happening in my evaluations (traditional paper and pencil test).

Bruce described a whiteboard lesson and his experience with a reflection piece that the students filled out where he prompted them with:
Reasoning and proving - What was your conjecture of the most useful function for your model? What evidence did you use?
Selecting - What tools and computational strategies did you select?
Representing - What representation(s) did you use for your model?
Reflecting - Comment on the reasonableness of your results.
Collaboration - Did your group's solution include contributions from all members? Explain.
(Other ideas for prompts would be welcome as would revisions.)
People inquired to him about what it looked like, how it went, how did you coach them to get ready for this type of assessment. A thought that came out was that after a few times of doing this you would have some student exemplars (student work) to show and prepare students for what you would expect. Eventually will have student samples of these type of reflections. 

Laura brought up the app show me. The thought was that this app might help with this type of assessment.

I talked about a learning journal. The thought was that students would do an activity / whiteboard lesson / solve a problem and then reflect on it in a journal on a regular basis based on the prompts above. This would happen in class. I would circulate and provide feedback on how the student might improve the journal entry. This would be practise for when they had to do it for the assessment as Bruce described above.

Lynn talked about a journal being like a paper and pencil test. That there seems to be an anti-test sentiment. I said I would still test but I was looking for something to evaluate the collaboration and inquiry that is happening with VNPS and VRG. Robin in her beautiful way prompted the elementary people to maybe give us some ideas of how they journal and prompt. Someone mentioned Marilyn Burns and her work. 

How to do all this? (this was a concern brought up by many through-out the day)

Voice recordings and a QR code might be a possibility. The students record their learning/reflection and are responsible for their learning <could be live>.

Helene offered a grade 7 example. Film one student answering a prompt to a question/problem in the hall. Then that student becomes the filmer. Then the next student becomes the filmer etc.
They found that students did not help the next student when they filmed. They also observed that the students filming were very patient waiting for responses as opposed to how we as teacher might be (read less patient). Would need to be carefully planned task/question/prompt to film.

Robin told a story of a student who failed a test. The student came in a lunch and did the question on board. She asked the student to "Tell me what you are thinking". Prompts from Robin made it very clear that the student did not know what they were doing. Turned out the mark she gave the student on the written part was actually very generous. 

It all counts. These recordings could be marked in evidence record as R1, R2 etc.

Student's marks are my professional judgement. That means it all counts. We threaten with this but does it all count?

Subtle ways where we convince students that everything counts. Spiralling allows all students multiple opportunities to demonstrate their learning.

Voice and choice.

Assessment turns to judgement. Judgement sometimes . Feedback has increased in a vertical class. Exit ticket. I want to hear your thinking so I can judge it. That is the trick? How to do this without influencing them. They have this idea about right answer. As opposed to process? This takes me back to my opening conversation this morning about the "process of investigating"? 

Evidence record problem-have to put a level on it. Want to just check it-instead of recording it on evidence record with a level. 

How do we do all this? Composition book or scribbler. It would matche what we see in the evidence record. (Their journal matches their evidence record)
Evidence of pictures / recordings for each student. How do you mark this. What is the mark?

VNPS pictures of student work. How does this translate to a mark.

When we say, "That is interesting." or "That is good." we stop the learning and the process. Students want to please us. Need to give non judgemental feedback-not influence or shut down their thinking. 

Maybe we ask the question and provide the answer and students justify the process to get at the answer.

Or we change the question so it has multiple solutions. i.e. we opened up the question??? (whatever that means)

Everything they do has multiple ways and it all counts and they need to explain their thinking.

Time limit -efficiency.

I spoke about this.

Ok here was my big take away. (Remember I want what is happening in my class (collaboration and inquiry VNPS and VRG) and what is happening in my evaluations (traditional paper and pencil test) to be more inline with each other.)
I want the students to journal regularly. It will be their personal space to reflect/comment/document on their learning and then they will do collaborative whiteboard tests with reflection piece (like their journal). It would be just like what the students do regularly in their journal.
I would have to build confidence in students to journal their thinking and reflections. Journal is feedback to teacher . No judging the journal.

session #2

This is definitely married to session #1. I came in late. This is where the group was at when I walked in.

Things have changed because we think someone has the right answer so we follow what they did. Then it does not work for us. This is because no one has the right answer. Conversations help a great deal. (Read Collaboration)
Most Teachers are working on levels. People like this. I can group students into a level on the quality of their work.
Pegging marks to a level. Some teachers are using this to convert because they are uncomfortable to give a 3 which means 70-79. So 3- is 72, 3 is 75, 3+ is 78. Things are changing.

We test with level marking. Then we give a numerical grade. In high school we stream students. Class to class evaluations are different let alone school to school. Consistency not probable. Mark should not drive our practise. Easy to get overwhelmed with assessment so teachers give up. Assessment and evaluation should be different. Assessment is to inform their learning and my teaching. Evaluation should not be important to me as a teacher for 90% of the time. 

Students are used to collaborating-then in high school we beat the collaboration out of them.
This is changing. The same could be said of teaching.

We have to believe that the teacher next door is teaching the way they teach because they believe what they are doing is the best for their students. We need to be very careful if we don't believe this.

Collaborate-share, respect the sharing!
Zone of proximal development- we need to meet students where they are at. Right - and this translates to teachers - we need to meet teachers where they are at.

Maybe it is not about the techniques of teaching/pedagogy. Maybe something else is going on. Is it personality? Is it all about the relationships?



Saturday, October 4, 2014

Volume = Length * Width * Height

This activity allows students to see the relationship between cubic polynomial equations in expanded form (volume form - y intercept form) and factored form  (length width height form - x intercept form)

Students were put in groups of 4 using visible random groupings.

Each group was given one of these equations.
y = x^3+4x^2+5x+2
y = x^3+5x^2+8x+4
y = x^3+5x^2+7x+3
y = x^3+6x^2+11x+6
y = x^3+7x^2+16x+12

Students were then told to get four different colors of cube-a-link blocks, enough of each color to generate each term from x = 1 to x = 4. Here is one groups work to figure out how many blocks they needed.
Groups were then told to make 4 piles of the blocks when x = 1, 2, 3 and 4. Once they had the blocks in piles they were to create the 4 stacks. Constant term on the bottom, then the linear term, and then the x^2 term and then the x^3 term. They looked like this.
The first one is y = x^3+4x^2+5x+2. The second one is y = x^3+5x^2+7x+3. The third one is y = x^3+6x^2+11x+6. Since the stacks are ordered from x = 1 to x = 4, looking at the second one, you see the tops of the black cubes is the graph of y = 3, the tops of the brown cubes is the graph of y = 7x + 3, the tops of the red cubes is y = 5x^2 + 7x + 3 and lastly the tops of the green cubes is y = x^3+5x^2+7x+3. Of course we are only seeing the graphs for the four points when x = 1 to x = 4.
Very cool!
Now if you take the x^3 cubes at each stage (x = 1 to x = 4) you can create a cube.
The x^2 cubes will allow you to create the appropriate number of squares.
The x cubes will allow you to create the appropriate number of lines.
And the constant cubes will always be what they are (ones). (points maybe?)
For example looking at this chart from before. (which is the middle stack picture above)
I will do it for x = 3. The 27 green cubes makes one cube 3 by 3 by 3. The 45 red cubes makes five 3 by 3 squares. The 21 brown cubes make 7 lines of cubes 3 long. Finally the 3 black cubes make 3 ones. Beautiful.
Now a few rules to creating the three dimensional rectangular prisms.
1. The Cube goes in the corner and the Ones can only touch the cube at a point. So basically they are in opposite corners.
2. The Squares must be attached on the faces of the cubes.
This is what they create. The first one is the example I have been using. The wooden blocks show it in general. So we end up with:
Volume =Length * Width * Height
x^3+5x^2+7x+3 = (x + 1) (x + 1) (x + 3)

I posed these photos to twitter and got a few responses.