Learning in Action

Instant feedback.  Constant Assessment.

Those are two of the major changes I see in my classroom.  When students are working around the classroom at tables or on whiteboards, with a quick glance, I can quickly assess learning and progress.  It is pretty obvious which students are engaged in the task, when students are engaged in the mathematics, and when they aren't engaged at all.  

Some people wonder what I really do in the classroom, given that I don't 'teach' anymore.  (Does teaching have to look like a lecture?)

Here is how I run a typical math lesson.  (Not always... just in general).

1. Students stream in and find a place to sit, lounge, stand... whatever.

• I'm assessing: 
• Are they energetic, frustrated/complaining about something, sleepy, etc
• Should I change my tentative plan for the class based on their mood.
• Sleepy/droopy students should be up and moving.
• Extreme energy needs to be zeroed in on a concentration activity.

2. I try to turn conversation into a math task.

• I'm assessing: 
• Do they have the background knowledge? 
• How much information do I need to give them?
• Should I adapt my task to make it more accessible?

3. Students start working on the task/problem in random groups on paper, portable whiteboards, or vertical whiteboard (depending on the problem).  I walk around, step into a few group discussions with a comment or question, observe group dynamics, differentiate tasks when necessary.

•     I'm assessing: 
• Which students need guidance to learn the mathematics?
• Which students need questions to push them to a higher level of mathematics?
• Which students can explain what they are doing and why?
• Which students are arguing about mathematics?
• Which students are off task? Should I step in or wait it out?
• Which students need the task changed to meet their learning needs?

                              

Usually the room is humming with discussions and debates.  I especially love it when students get into an 'argument'!  It means they are thinking and learning.  It means they are confident in their beliefs instead of just following.  It means that they are going to find ways to prove their point and that means coming at it in a few different ways.

Two students (A and B) were arguing about the shape of a graph yesterday in class.  Both were fully convinced they had a correct graph.  Other students started coming over to watch/participate.  Student A managed to convince everybody else that he was correct, and student B started to resign himself.  Student A sat down, looking satisfied.  

I stepped in at this point.  "Are you actually convinced?" I asked Student B. 

"Not really. But whatever." 

"You could still be right... try convince him that yours is correct."

At this point student A jumps in again and says, "No, I know that mine is absolutely correct.  Here, look student A, let me show you again..."  

"Well, I don't really want to argue any more" says student B.

"I'm not really arguing to prove you wrong" replies student A, "but to prove to myself that I'm right."

I love that quote!  I'm not arguing to prove you wrong, but to prove myself right.

Here are some videos/pictures* of what learning in my classroom sometimes looks like:

*students/parents signed consent forms

Video 1: Students are trying to sketch the paths made by a point on the vertex, edge or inside of a triangular wheel.  They are working any type of triangle (not necessarily equilateral).

Video 2: Students have not learned about sine/cosine graphs yet, but are debating discussing the shape of a graph that shows the height of a bug on the rim of a spinning tire over time.  

Video 3: Students try to determine the time it takes to fill a water tank (Dan Meyer's 3 Act Task).  They first guessed and gave a time that was sure to be too low, and too high.  They were not happy their results outside their range.  In the video, two groups compare their solutions, and all 3 groups watch Act 3 to see what ACTUALLY happens.  One group is thrilled... others, not so much.  This task gave me so much feedback about the students' mathematical background and problem solving skills!  Maybe 2 of the 3 groups did not have a reasonably correct final answer, but they worked on--and learned--some great mathematics in the problem solving process!  Looking for growth in their mathematical abilities, not perfection.

A few groups gather around a table to 'argue' about mathematics.

Group on left works at table with heads together.
Student at board wanted to work through problems on his own.
Students gathered in circle discuss problems and avoid using a surface altogether! 

One way to save whiteboard work to continue working on it the next day.

Working in pairs... group in middle merged to form a super group.

Don't write this down!

Note taking.

Note taking is a key part of math class, right?  

What better place than math class to learn that essential life skill: Note-taking

Well, my students will just have to learn this skill somewhere else, because they won't be learning it in my math class.  Why not?  They just don't have time, they are too busy DOING math.

I already know how to solve systems of linear equations, factor trinomials, and calculate surface areas.  So what good does it do my students to watch and copy me while I work through endless example problems when we all learn better by doing?  If I teach by working out example after example on the whiteboard, I simply get faster and faster at solving the problems, and most of my students get more confused.  Then they need try complete homework practice problems by mimicking what I did.  Do you remember the feeling?  Everything 'made sense' when the teacher was explaining, but when you were on your own, it was just numbers and letters jumbled together!

With this in mind, I have removed note-taking and lecturing from my classroom.  I try to get my students doing math the full 50 minutes they are in my room.  They are not always doing the math correctly.  That is okay... because they are still in the learning process.  If they did everything correctly... they are probably in the wrong class.  Through making mistakes, comparing solutions with others, debating about which strategies make more sense, adjusting their strategy, making corrections, and all around doing and talking mathematics... the students are learning mathematics!  On top of learning mathematics, they are getting a deeper understanding of it.  They start to realize there are multiple ways to solve a problem.  They also learn to interact with their resources... textbook, peers, teacher, and computer.  Learning is no longer a transfer of information from teacher to student... but a learning network.

BUT...

There are still times that I 'lecture' my students.  By this I mean, I have them all sitting and listening while I explain something to them.  This is pretty much always AFTER they have worked on a problem for a while in groups, and in order to complete the problem, they need some new mathematical strategies.    The reason I do this is because there are some algorithms or procedures for which they need to learn a specific method.  [Especially in the Grade 10 course, since it has a provincial exam].  I introduce algorithms afterward, so they already have some hands-on experience with the underlying concept, and they can make better connections between the algorithm and the mathematics.  

Here is how I 'lecture' in my non-traditional classroom:

1. I tell my students: "Please find a seat somewhere in the room so you can see me and get comfy because I need to talk to you for a while".

2. As I explain a concept or procedure, I either point out or ask them to make a connection to some prior knowledge they have or to a recent task they completed.

3. I write notes on a whiteboard... usually non-linearly.  

For example, starting from a diagram in the middle and working out in all directions.  I don't really have a reason for this, except that it is usually the way I personally think and problem solve.  When I am brainstorming or problem solving, I rarely start work left to right, top to bottom.  

4. I try to keep all notes to only 1 whiteboard... or less.  

If it is full, it probably means lecture time is up.  What?  Not boards full of examples and explanations?  No.  A high school student can focus on a lecture for only about 10 minutes.  Or, somewhere between 8 and 20 minutes according to a few random online sights I came across, here's one link:  Why Long Lectures are Ineffective.  If they can only focus for about 10 minutes... they are going to start phasing out on anything important I have to say after that.  

5. The students don't copy down my notes.  Just because.

Because... When students copy down notes, the notes are static.  They do not hear everything I say.  They do not see where I am pointing.  They do not see how the labels and arrows are in response to questions. They are usually 'behind'... copying down meaningless equations and diagrams... while I energetically try explain new diagrams to the few students that have their eyes facing the board.  When they look back at the notes later, it is usually a confusing mess.  

6. Repeat. The students DO NOT copy down my notes.  

Because... When I make diagrams and notes, they are dynamic.  I am talking. I am pointing.  I am drawing. I am labelling. The students are asking questions. I am responding.  The diagrams are part of a bigger conversation.

7. I usually do not erase my notes.

After a 10 minute 'lecture' (I prefer to refer to this as a mini lesson), I do not erase my notes.  I also take a picture with my iPad, just incase a student wishes to see them again for some reason.  Then, I have the students work in their groups on whiteboards to solve a problem.  This is another reason I keep notes to 1 whiteboard... the students need to use the other whiteboards.  Students can easily refer back to my notes as they work through their problem.

8. Tools to become comfortable with.

I like to think of the mini lessons as a way for me to introduce new tools into the classroom.  In any classroom, some students catch on to a new concept really quickly, while others need a little more time.  When I give a mini-lesson, I am hoping that some students will immediately understand HOW to use it, and that the rest of the students simply know it exists.  Then, as they are working on problem solving tasks, the students will become more comfortable learning the tool.  If they don't know how yet, hopefully someone in their group will have the ability to demonstrate the tool.  Having random groups every day means that the students comfortable with the tools will always be working with different peers.  It also means that students confused about the tool will eventually be grouped with someone who can share their knowledge of the tool.

9. I expect everybody to listen, but I don't expect everybody to 'get it'.

When I teach a mini-lesson, I am keeping it short so they can use it as soon as possible.  They are going to fumble and falter.  This is good, because we only build our brain by experimenting, making mistakes, making connections, and correcting.  They are going to ask good questions.  They are going to problem solve.  They are going to work in teams to build on their knowledge.  They are going to learn how to learn.  They will learn how to use their resources.  And, when they finally do 'get it', they will probably never 'forget it'!!  And those who did 'get it' right away, will 'forget it' right away... except now they have the opportunity to teach their peers and therefore for more chance of gaining and retaining a deeper understanding of the topic.

These two charts are slightly different... but I'm sure you get the idea!

Preparing the Environment

Teaching this year has been interesting.  I've really changed things up.  My classroom looks different, my teaching looks different, learning looks different, and assessment looks different.  Although I am enjoying the ‘new atmosphere’ of my classroom, and the students seem to be enjoying it… I have made the changes for a reason.  6 reasons, actually. 

When you teach any course, at any level, there is a document that lists the concepts (Prescribed Learning Outcomes-PLOs) students need to learn in that course/grade.  An example of one of these concepts in Grade 10 is, ‘Solve problems that involve systems of linear equations in two variables, graphically and algebraically.’   For Math 10, there are 18 concepts on that list. 

It would be easy enough to cover the curriculum, by just starting at the beginning of the list and teaching each concept and then having students practice it over and over again.  This is the standard approach to teaching math.  It is how I learned math.  It is how the textbooks are set up. 

Yet, the reason I need to teach those concepts is not just so students ‘know’ of them.  The official purpose of teaching students mathematics is so they:

1.     Recognize and understand the incredible order and perfect design in God’s Creation.

2.     Improve their ability to think critically and increase their responsibility in decision-making.

3.     Can describe and model patterns to make deeper connections within and beyond mathematics.

4.     Engage in problem solving to develop their mathematical literacy, reasoning, abstract thinking and visual-spatial skills.

5.     Obtain problem-solving skills which will be indispensable for future education, career, and personal life.

6.     Gain the confidence to take intellectual risks, ask questions, and pose conjectures.

That list is reason enough that I needed to make these dramatic changes in my teaching.  If I just ‘cover curriculum’… I feel that I am not providing students an environment that incorporates these higher goals of mathematics education.  So those concepts now become the context through which I teach mathematics instead of the goals of my lessons.

So, what are the dramatic changes?  I will share some of those in future posts.  

For now, just a glimpse into my classroom set–up.

A comfortable and flexible environment.  The fold-up tables and chairs are easily collapsed and stored or set up as needed.

This is what I have: 

• 6 computer stations for students to use whenever they need to.
• Whiteboards around the perimeter of the room for students to solve problems on.
• Mini whiteboards to solve problems sitting around a table.
• Supply station (paper, whiteboard markers, etc).  The students do not take binders.
• A few comfy chairs that I found around the school.
• A stack of folding chairs/tables.
• A round table.
• A kitchen table (A great craigslist purchase!)
• An apple tv connection to my projector.  This allows me to wirelessly project anything from a computer screen, my ipad, or iphone onto the whiteboard.  Great for taking a snapshot of student work and projecting it.  I also use the iPad for ‘saving’ any whiteboard work.

Dinner table from craigslist and a few of the comfy chairs.

Whiteboards around the room are mostly for students to use for problem solving and discussion boards.

The round table is great for group work and discussions.

Lots of coloured whiteboard pens and erasers!

Portable whiteboards

Note/grid paper pads, blank paper, formula/conversion charts, scrap paper...

Textbooks stored in a corner of the classroom so students don't need to drag them home unnecessarily every day or forget them when they do need them in class.  These are just another resource in the classroom... not a map of the course.

One of 6 computer stations.  Each computer is near one of the vertical whiteboards.

And a few pictures of my students 'in action'.  

Using Technology

I would like to share an activity that I used in two math courses on the last day of classes this school year.  In spite of the end-of-the-year buzz in the school, the students were quite engaged in the task.  Here is a video I put together that discusses the impact of students using technology in comparison to a pencil/paper environment for the same task.

Waterline by Desmos

Math in Art. Art in Math.

When I first discovered that there were connections between Math and Art, I was fascinated.

So, I started integrating some art into math lessons.  Originally, my purpose was two-fold: 1. Make math class fun. 2. Show students this new world of Mathematics.



Fail.  This was definitely not a good teaching strategy.  Students mostly enjoyed the tasks, no doubt.  But the way I presented the tasks... it was more show.  The students did not think they were actually doing math, but were mostly humouring me since it was definitely more fun than math.





Yet, it was not total failure.  Surprising 'aha' moments popped out of integrating some art into math class.  Take a look at the polynomial function in the picture above represented in a pie.  Suddenly the 'zeroes' of the function mean something more than that curvy line crossing the x-axis.  Zero means don't draw any circles.

 

Students get a deeper understanding of fractals, similarity, patterns, infinity.... when they actually need to physical use a pencil or paintbrush to draw out designs.  Take a look at the series of pictures above. The painting on the left is a 2' x 3' painting, and each consecutive picture is just zoomed in to the next level of detail.

This hexagon of Pascal's Triangles is about 2 meters across.  Surprisingly large!  How does it work?  We first set up a colour code: 0-green, 1-blue, 2-orange, 3-yellow, 4 purple.  Then each student painted one of the triangles in either Mod 3, 4, or 5.  So, what skills are they using here?  Nothing too intense.  The numbers they were adding together were less than 5.  A bit of modular arithmetic.  Yet, the aha moments that DID come out were surprising.  Grade 12 Math students making statements like... 'Hey, these are all zeroes, so if we keep adding zeroes it will still be zero.' Very true.  Great observation.  Funny thing is, they never needed to add zeroes together before. Even though they knew that loads of nothing is still nothing, it felt more powerful when a big patches of green zero-ness appeared.

If students were given an assignment to graph that extensive list of equations above, they would scream with the dullness of it.  Lines.  Circles. Parabolas.  What's the purpose of it all.  Give them the opportunity to design a face.... and students almost beg me to show them (ie teach them) about inequalities, restricted domains, transformations, etc.  Why?  Because they want to fill the eyes in with color, keep the smile within the face, or move that nose just a tad up so it doesn't overlap with the mouth!

I attempt to put a little more thought into integrating art into my math classrooms now.  Students need to feel that what they are doing is truly mathematics to appreciate the deeper levels of mathematics involved.  Hopefully, I can take this to the next level in my classroom next year!

The three pictured below were actually an art class assignment from my current students... but I still think they are pretty awesome!

Freedom to Choose

Pushing my students to do math has always been a battle.  Try to make it interesting.  Keep them entertained.

This has gotten easier over the past few months.  Instead of painting up the math with exciting colours, I have stripped it bare.  I stopped forcing them to work on an assigned list of problems that is due the following day.

Originally, my students had less freedom.
Term 1: Traditional.  Assigned seating.  Students quietly face 'front', listening, taking notes.  Practice problems build in difficulty. Homework checks (when I remembered).
Term 2: Non-traditional.  Random groups every day.  Work standing at whiteboards.  Big problems.

In the last little while, I have been allowing them some more freedom.
1. Work however you want.  (Paper or Whiteboard. Standing or sitting.)
2. Work with whomever you want.
3. Work on whatever you want.

Strangely enough, they push themselves more!  Instead of resisting and complaining about a list of assigned practice problems that are too hard... they are choosing to work on the problems that are hard! Why?  Because those are the ones that they know they need to practice.

Here's an example what happened in a review lesson.

Students walked in to 4 questions on the board, from 4 completely different units.  On a scale of 1 to difficult, most of them pushed difficult.

On another board I wrote:
1 - No idea what to do ( ~ 40%)
----------------------------------------
2 - Understand what to do after checking textbook or talking to somebody. (~60%)
3 - Understand what to do, but made a mistake. (~80%)
4 - Understand what to do, completely correct. (~100%)

I told them to try each of these problems to see what level of understanding they have so they know what needs work.  This worked as a great 'quiz' replacement.  Students were 'willing' themselves to get a 3 or a 4.  They were sitting elbow to elbow.  Perfect environment for cheating on a quiz.  But if its not for mark, who are you cheating?  The class was quiet for about 10 minutes as they sweated over the problems individually.  Slowly a few students pulled out their textbook (their resource) for some ideas.  A few whispered to each other.  They studied their textbooks in pairs.

Suddenly, it was no longer about right or wrong.  How many points can the student earn.  It was about 'How well do I actually understand this?'  For me to assess my class, I did not need a quiz average.  I just needed to watch and listen.

At the end, students did not ask me what the right answer is.  They asked me good questions.  How do I multiply terms with exponents?  When might I need to use the tan inverse function?  Why is any number to the power of 0 equal to 1?  They frowned at me when I gave them a little exponent rule 'x^0 = 1' because they wanted to understand the rule.  Check. Check. Check!!

Following this 'quiz', I told students to flip to the cumulative review of the whole year and work on whichever problems they want, however they want, with whoever they want.  Here's a few pictures.

You will see students working on paper, portable whiteboards, and vertical whiteboards.
True, the classroom was fairly buzzing with noise and students regularly go into off-topic conversations.  I haven't figured out how to totally prevent the off-topic-ness, but I believe that quality working time trumps quantity working time.

A few things that really stand out lately in this approach are:
1. Hope.  Instead of writing themselves off as a failure when they make mistakes... they have a sense of hope that they can master difficult concepts.

2. Fun.  Instead of complaining about work, students enjoy the casual, stress-free atmosphere where they have freedom to choose.

3. Challenge.  Nobody truly likes working on tasks that are 'brainless' or 'useless' when they are working for themselves.  So with the freedom to choose what they want to work on, students challenge themselves to questions that look difficult.

As a teacher, this environment of students working independently and collaboratively gives me the time to informally assess the needs of individual students, coach where needed, and to teach small groups when they require it.

The Rainy Day Problem

As I drove to school this morning it was raining really hard... and I knew it was the perfect day to bring out the Rainy Day Problem.

It went something like this:

"Wow, its sure raining out there!"  The students immediately reacted to this with stories about how wet they got or how it was actually a perfect day to stay at home and nurse their cold (which many of them had today).  
"I wonder how much water is actually coming down..." I say as we watch the dreariness.  "I think you should calculate it.  Find a whiteboard and figure out in your groups if the rain that falls in 1 hour in 1 square kilometre will be enough to fill a milk tank!"

I mentioned the milk tank because in students in Math 10 need to know how to calculate the volume of a cylinder.  They also need to convert between different units of measurement, so at some point I 'happen' to mention to each group that 1 cubic cm is exactly 1mL.

Most students felt a little overwhelmed at the problem initially, because they had nothing to work with.  Besides the 'square km' I did not have any numbers for them.  They stare a little blankly at the rain drops gushing out of the sky thinking, How can we ever count those all?

How much DOES it rain in an hour?  If you put a cup outside, will it overflow?  Probably not. Ok.  So less than 4 inches.  Most groups chose a number between 5 mm and 1.5 cm.

It is astonishing.  10 mm of rain over a square kilometre gives about 10 Million Litres of Water! What happens to all this water?

Different groups came up with a variety of interesting points:
-It would take about 120 milk trucks to carry the water that falls in an hour.
-It would take about 16,000 hours to fill (or drain) a milk tank through a small hole at a rate of 3mL/hr.
-If it rained milk, 3 families in one group could be supplied enough milk in 1 hour for more than 3000 years.  This lead to the calculation that all of their families (from the 18 students present today) would have enough milk for more than 200 years.

My students usually work on whiteboards... be it standing at vertical whiteboards or sitting around small portable whiteboards.  This for several reasons.

• It is easier for students to 'think together' when they can all write and see the workspace.
• Students put 'pen-to-paper' much quicker when the surface they work on is non-permanent.
• Knowledge permeates through the class quickly when students are standing and can see each other's work.   (As I was going around and suggesting it might be useful to know that 1 mL = 1cubic cm, I was surprised to see that information made it around the classroom before I did.  The group shown below had that written on their board before I could feed them that 'tip'.)
• I can quickly assess students progress and understanding from anywhere in the room.

I often take a quick snapshot of their whiteboards so I can analyze them.  Students usually do not receive a formal mark for this work, but it gives me a good idea what concepts need to be covered and what still needs some work.  

Taking a quick look at the whiteboards shown I gather some interesting information.

• The group above is clearly able to work in both imperial and metric units and make necessary conversions from one to the other.  This is a course requirement, so I have evidence that they 'get' that learning target.  
• The group below understands volumes, rates, and conversions but their understanding of units of measurements in 1D vs 2D vs 3D needs some work.  They are the only group that condensed a large number into an exponent.

As I clean up the room at the end of the day, my sister walks into the room.  She looks at some of the remnants of calculations left on the boards.  "You know," she says, "there's this really great problem I love to do with my students.  Have them figure out how much water falls in 1 day in 1 square kilometre!  I did that in my math class today."  I grinned.  It simply was the perfect day for the rainy day problem.

(Note: Thanks to D.O. for sharing the Rainy Day Problem with us a few years ago in a Pro-D session!  I, for one, have used many variations of it since.)