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.)

Mathematics is Beautiful

I believe that mathematics is beautiful beyond comprehension.

Really.  Mathematics.  

You may disagree.  It is likely you will disagree.  

But wait.

If you disagree, I can guarantee you aren't really qualified to argue this point with me.

Year after year of theory without ever listening to a song or making music would give you the same distasteful feeling about music that you have for math.  If enjoying math is beyond the scope of your imagination, you are a victim of learning meaningless mathematics.

Consider Mount Cheam.  Chilliwack's Eiffel Tower.  It seems reasonable that a child growing up in this area should learn about Mount Cheam.  Since learning is separated into disciplines in our current education system, this new learning outcome will need to be placed in an isolated category.
Where would it fit best?  How would it be likely be interpreted?

Art:

Students will observe how seasons and times of day effect the colours, lighting, and shadows on the mountain peak and then choose a medium to capture its essence. 

History: 

Students will research the impact that Mount Cheam has had on people through the 20th century.  Students will gain an understanding of why the Sto:lo peoples called it the source.

Geography:

Students will learn how the elevations effect the plant and animal life on Mount Cheam.

Students will climb to the peak in order to more fully understand the effects of the local mountain ranges and the Fraser River on the Fraser Valley.

Mathematics:

Students will learn that Mount Cheam is approximately pyramidical, and then calculate its height, volume, and average slope.  

This is just how it seems to work these days.  There is a lot of room for beauty in each of those disciplines.  But hello mathematics:  The beauty will be drained right out of it.  I can just picture the Math lesson.  Students would hardly even need to see the mountain.  Well, perhaps they might look at a picture or look at it through a window.  Before students have a chance to appreciate the gorgeous mountain, their teacher will have it decontextualized into a pyramid or cone on the whiteboard, giving them some basic measurements (perhaps its height and radius at the base), and asking them to make some random calculations.

Actually, probably not quite true.  Most math teachers will try somehow to spice it up a bit.  Make it more interesting.  Tell a joke.  A story about when they climbed Mount Cheam.

The learning outcome has been met.  Check.  Move on in the course.  There's so much more to cover.

What is wrong with this?



In my description above the TEACHER has done the abstract and mathematical thinking for the students by turning Mount Cheam into a pyramid and giving them basic measurements.  Measurements that are probably almost impossible to physically measure directly.  The STUDENTS use some meaningless formulas and crunch out meaningless numbers.  They will walk out of the class with no extra appreciation for Mount Cheam, or mathematics.  They will walk out of the class with no extra skills they can apply to their life-outside-math-class.

I can think of a perfectly useless question that would turn the lesson into a rich experience.

Could we completely cover Mount Cheam using all your school uniforms?

Really.

Some student will say, "Of course NOT, Mount Cheam is HUGE and we only have a small school!"
Check.  This makes me happy.  A student realizes that mountain is pretty large.  I could respond in several ways.  Well, how many students would you need?  How much of the peak would be covered?

To solve a question like this, students will not only need to do math, but also think mathematically.  They need to make choices, think abstractly, make estimations, measure, collaborate, ...

From measuring the area of their own clothes, to realizing that Grade 1 students wear smaller clothes, to deciding if extra clothes at home should be included, to collecting raw data and using other sources, to calculating a reasonable approximation of the surface area Mount Cheam... students will be doing math.  The question in itself is simple, but the solution is complex.  There is no answer at the back of the book.  The answer isn't even all that important.

The beauty in this silly question is that it demands that students do math and will give them an understanding of various degrees of 'size'.  A similar question could make this even more meaningful.  If we turned all the trees on Mount Cheam into paper, how long would it take our school to use it up?  


Ok, I went off on a bit of a tangent with that.  What I wanted to say is this.  If you believe there is no beauty in mathematics, then you what you know to be 'MATH' is not really math.

Math is not about memorizing formulas and crunching out numbers meaninglessly.  Math class is not about training humans to become calculators.

There is something spectacular in the process of studying a fern, placing it into the abstract world of fractals, pinning a function to it, adjusting the function, and realizing that the minor adjustment draws out a tree.  You don't even need to know how it exactly works to be wowed by this.  As I said, it is beautiful beyond comprehension.

Anne Burns is a mathematician that designs Mathscapes.  It is so unbelievable that these nature scenes are completely designed with functions and equations.  Check out her gallery here: Mathscapes

When functions, equations, patterns, numbers, and shapes are taken out of context... it becomes easy to crunch through math curriculum because these concepts become desaturated of their usefulness and beauty.  Math becomes meaningless to students and people become proud to admit that they are not in that group that gets math.

Where is math in the world around you?  Here is a little video clip that shows three perspectives of Math.  The 'real world view', the decontextualized view, and the 'functions and equations' view.  The three are pretty much equivalent.  This made me re-appreciate the mumble-jumble of mathematics.  It's always just lurking there behind-the-scenes.

Beauty of Mathematics

I will start sharing glimpses into my classroom as I push ahead with changing the way I allow students to learn and do math in my classroom.  One of my goals is that students will feel amazed by the things that pop up in mathematics without me having to jump up and down with an excited voice "Look at this!! Isn't it so amazing?!! Why don't you look amazed?"

Why take the risk?

This year I made a few changes in my math classroom.  Drastic changes.

Sometimes, I don't have enough chairs for all my students.

Sometimes, I don't have any chairs for my students.

Sometimes, I don't have any tables for my students.

Sometimes, I let students work individually

Sometimes, I let students make their own groups.

Sometimes, I let students write whatever they want on the board before writing a quiz.

Sometimes, I let students talk to each for a few minutes while they are writing a quiz.

Sometimes, I only teach 2 students a new concept and don't bother teaching the rest of the class.

This is extreme.  I realize that.  So, why do it?

I have taken this risk for the sake of my students.  As a teacher, it is my duty to provide students with a classroom environment that induces mathematical learning.  There is method to this madness.  

Believe it or not:

Math is not about memorizing multiplication tables.

Math is not about moving around letters in an algebraic equation.

When a Grade 12 Pre-Calculus student still pulls out their calculator to multiply 7 x 4, it doesn't really bother me.  Not everybody has the gift of memorizing the multiplication tables, so although it can be helpful, it does not make the student more or less of a mathematician.  I am more concerned if they don't know whether to 'add' the numbers or 'multiply' the numbers.

I cannot teach 'whatever I want'.  When I am assigned a course, I am obligated to teach a specific curriculum mandated by the government.  Check the curriculum document for mathematics, and you do not immediately find a checklist of things to teach.  THIS is what I find:

GOALS FOR STUDENTS

The main goals of mathematics education are to prepare students to: 

• solve problems 

• communicate and reason mathematically

• make connections between mathematics and its applications

• become mathematically literate 

• appreciate and value mathematics 

• make informed decisions as contributors to society.

In order to assist students in attaining these goals, teachers are encouraged to develop a classroom atmosphere that fosters conceptual understanding through:

• taking risks 

• thinking and reflecting independently 

• sharing and communicating mathematical understanding 

• solving problems in individual and group projects 

• pursuing greater understanding of mathematics 

• appreciating the value of mathematics throughout history.

Isn't this amazing!  These are skills that can benefit my students in life no matter what their personal or career goals are.  

My Grade 10 students need to learn how to graph a straight line.  I know the easiest way to graph a straight line.   I have done it hundreds of times.  You will agree that the most efficient way for my students to learn this concept would look like this:

1. Students sit quietly in straight rows.  Preferably assigned seating to minimize 'trouble spots' in the classroom.
2. I give a clear explanation of how to graph a line using some diagrams, hand-waving, stories, etc.
3. The students copy the diagrams/words into their notebook.  They are unable to transcribe my hand-waving and stories into their notebook.
4. The students try a practice problem and ask questions when they get stuck.
5. One student asks "Where would we ever need this?"
6. I talk about a practical application.  They look skeptical.
7. Students get started on homework.  Homework is a set of problems almost identical to the one I gave them.
8. Quiz next day will show me if the students completed the homework and understand the problem.

No surprise. The average student hates math.

This way of learning/teaching might work for some students/teachers.

But not for me.

Here are a few reasons why I have kicked 'traditional' mathematics out of my classroom.

I value collaboration and teamwork.  

So, I give it a prominent place in my classroom.  They need to learn to work well in a variety of situations.  In pairs. In large groups. 1 girl with 3 guys. 1 guy with 2 girls.  Creative thinkers with logical thinkers.  With friends. Without their friends.  This is why I change the groups every single day.  This is an important life skill.  

I value problem solving.  

So, I give it a prominent place in my classroom.  Students need to learn to solve a problems without 'parroting' my efficient solution.  They need to put effort into understanding the problem, choosing and carrying out a strategy to solve it, and checking if their solution is reasonable.  To practice this skill, students need to see problems they have not solved before.  Otherwise, it is not a problem.  It is a chore.  

I value persistance.

So, I give the students freedom to struggle.  They need to learn how to find another strategy when they get stuck instead of giving up.  When they fail several times before succeeding, their success is that much more meaningful.  They will also not forgot the strategy.  I think my students hate me a little bit when I purposely let them struggle on their own.  It pains me when they are confused and crying for help.  But if I step in and 'rescue' them the moment they start struggling, they will not learn the skills to rescue themselves when they are on their own.  

I value learning.

So, I give my students opportunity to learn.  Learning any new task, whether it be math or using a baseball bat, usually looks like this: 

Week 1: 

Try. Fail. 

Try. Fail. 

Try. Almost got it...   

Week 2: 

Try. Fail. 

Try. Almost got it... 

Try. Yes, I get it!

Week 3: 

Try. Almost got it. 

Try. Yes, I get it! 

Try. Yes, I get it!

In 3 weeks, this student has a 33% success rate.  Clearly, this student is a failure when it comes to hitting a baseball. Right? Perhaps talking to the student will help.  "You need to pull up your grades!  It is in your best interest to show immediate success when we start the next unit: volleyball."

Obviously there is a learning curve!  This student has shown drastic improvement.  This is why I no longer have my students write quizzes during this learning process that will affect their grades.   When I quiz them while they are still in the learning process, I may allow them to discuss the questions with others or look in their textbook.  I observe the students carefully.  Who is working independently?  What types of questions do they ask their peers?  What type of support do they need to master the content?  The quiz is now a powerful learning tool instead of burden that can sink a student into failure.

So, for the sake of the students, I have risked my 'safe' classroom environment in attempt to build a learning environment.  This is still a work in progress.  I hope to share some of my experiences on this blog as I shift my classroom from a teacher-centered environment to a learning-centered environment.


Oh, in case you are still wondering, when they don't have a place to sit, my students work through math problems on the whiteboards that hang on all four sides of my classroom or stand around the waist-high table that takes up space in my room.  Besides keeping them alert, the advantages of this are best left for a future blog posting.

Here is a video that has made some impact on my views of education: Changing Education Paradigms