Saturday, November 18, 2017

5 things I was NOT doing as a Math Teacher... that I wish I did

Recently, I heard a Harvard researcher state,
Sometimes the problem is not in what you are doing, but instead in what you are NOT doing.
This made me reflect upon my first 8 years of teaching math. What was I NOT doing that may have increased learning? If I could go back in time, here are 5 things I wish I did more (or even at all), in no specific order:

  • I never gave pictures to students and asked them, "Where should the origin be placed to best understand, or work with, this image?". Instead I would always supply images to my students with a Cartesian coordinate already drawn on. If I could go back, I would make time for students to discuss and debate on where is the "best place" for the origin to be placed on an image to solve the problem given.

  • I never explained what "simplify" truly meant. I would give students loads of questions and I would write "simplify" as a directing word. In my classes "simplify" meant: Add, subtract, factor, expand, combine like terms, rationalize denominator, etc. I wish I showed students where, and ultimately why, each form may be simpler than other forms; however, change the question and then a different form may be actually simpler.

  • I never asked students to actually measure needed quantities to solve problems. I usually gave students problems with all the information needed, even in the order they needed it, and then asked the question. If I could go back, I would have started asking questions such as "To solve this problem, what would we need to measure and/or determine?". I think it is important that students know how to measure, but more importantly they know what is worth measuring.

  • I never allowed students to be individuals, not only in the instruction process, but also the assessment process. Most of my tests required students to learn the required material by the same day, and then even asked my students to demonstrate learning the same way. If I could go back, I would allow students to demonstrate learning when they have mastered the material, regardless of the speed and pace of the other students. In addition, I would also have asked students to relate their learning, when possible, to their passions and interests.

  • I never built my course to allow connections to be built between essential learning outcomes. Instead, I built my course in units where I would teach outcomes as disjointed ideas and rarely make connections between each unit; I created silos of learning throughout the year. If I could go back, I would actually remove all notions of "units" in my course and instead weave big ideas throughout my entire course. Instead of teaching a big idea in September, and then only discuss it again during our "final exam review", I would ensure big ideas spanned the entire length of the course.
What are things, thinking back, did you NOT do?

Friday, May 6, 2016

Number Talks

How do you foster numeracy in a math class?

Very simply; Once a day, for no more than 15 minutes, complete a Number Talk.

What is a Number Talk?

Simply put, a Number Talk is a "naked number question" where students must use mental math to arrive at the answer.  This tasks removes the myth of "there is only one way", or "there are better ways than others to do math" and instead ensures that all students are aware that each of them have some sort of mathematical insight to offer everyone else in the class.

Here is an example...

In Grade 2, Jennifer Smith put this up on the board and asked "How many dots are there?". 

 After most of the students said "7", she asked "How did you count them?", and this begins the Number Talk.  I will let her share her story:

 I was surprised that they had this many different ways of counting the dots and my class had no trouble explaining their thinking. I even had a girl; say 5. I was careful and said come and show us and she pointed to the middle 5 and then said oh ya and 2 is 7.

Now the cool part!!! 2 girls even asked to stay in during lunch and continue to count the number of ways to count to 7,  (This is Grade 12 Math outcome!!)

Number Talks are a great way to get students talking, explaining, reasoning and ultimatley arriving at a deep conceptual understanding of how numbers work.  If you are interested in knowing more I would suggest you read the following book:

Friday, March 4, 2016

Is Streaming an Intervention Technique?

Streaming, or tracking, students occurs quite regularly around the world.  This means that, at some point in their K-12 education, they are grouped by ability or intelligence.  Many educators and parents support this idea.  The philosophy, behind this practice, is that it allows for teachers to teach groups of similar intelligence levels.  Also many believe that the high end students, through streaming, can be enriched on topics beyond the course.

This streaming occurs at different ages, or grades, in different countries.  In Alberta, streaming usually occurs at grade 10, while in USA this practice starts in middle school.  Finland, an international leader in education, has outlawed streaming entirely.  This then begs the question, "What is best practice for streaming students?"

Some educators, who support streaming, say that this practice allows them to teach to "like-minded" individuals and the need for scaffolding diminishes as there is a homogeneous group in front of them.  The material, consequently, is presented in such a way for the "average" of the group to understand; using the logic that all students are more or less the same.

This, unfortunately, contradicts almost all research around the growth of individual students.  No matter how well you group students based on ability, there will always be some students who may find a certain topic easy, and other topics more difficult.  Teaching to the "middle" will actually cause some students to struggle while preventing others from being enriched.  I also fear, in an education system where streaming is prevalent, the practice might become, "If you are not understanding, you must simply be in the wrong stream".

In a mixed-ability class, the teacher is forced to create material that all levels can benefit from, where the top students are challenged while the weaker students are comforted.   This results with all students learning at the highest of their levels and a philosophy of teaching that "all can succeed".

Also, when students are streamed, certain stereotypes occur.  Early in my career, this was a regular occurrence for myself.  I remember thinking while teaching a dash-2 course (a second stream in Alberta) "Well this is the lower stream so they won't be able to handle this..." or "We won't have time to do that project, as my students will take longer to learn...".  This ideology is harmful to students.

In 1960, Rosenthal and Jacobson conducted an experiment to look at the impact of teacher expectations.  Students were randomly placed in two groups, regardless of ability or talent, and these groups were labelled as "smart" and "weak" for 2 teachers.  After a certain period, they determined that actually the "smart" class had scored at higher levels on IQ tests, while the "weak" class struggled with many concepts taught.  Expectations from teachers was the difference.

This stereotype, once placed, becomes almost unbreakable.  In England, where they stream by age 4, it has been shown that 88% of children remain in the same groupings until they leave school.  This should be alarming!! A label, we give to a person who should be playing with blocks, dolls, and laughing, will determine his/her success for life!  This, again, contradicts almost all research around child development and learning.

This label is not limiting only the weak students but also the "smart" ones as well.  Carol Dweck, found that the moment students are streamed, by ability or intelligence, the students who were most negatively affected were those going into the top rank.  Their positive growth-mind-set thinking reduced almost instantly, and they became fearful of making mistakes and consequently avoided more challenging work.  This was especially prevalent in high achieving girls.

Students and parents usually support streaming due to the fact that this practice can allow schools to prepare students more appropriately for their future.  This argument is actually flawed.  Jo Boaler, followed students from two different school experiences.  The first group came from a school where they organized students heavily by ability, while the second school mixed all abilities together.  She found that the students who experienced mixed-ability grouping, despite growing up in one of the poorest areas in the country, were now in more professional jobs than those who had experienced streaming.

What is even more interesting was the attitude of the students who learned in mixed group settings.  At first some of the brightest students were aphrenrsive around the fact that they would be constantly explaining their ideas to the rest of the class.  However after one year this changed as they were quick to realize that this practice actually helped them understand the concepts being taught.

What is the solution?

Simply don't stream students based on prior knowledge or ability.

A great practice I have seen comes from a colleague of mine, Jonathan Mauro.  He is the department head of a high school physical education group and has started a creative way of streaming; he lets the students decide.  Instead of saying, "this group has high physical literacy, and this group has low physical literacy", he provides the students options of engaging in physical literacy.  3 full classes are scheduled during the same time slot and then the 3 teachers teach each unit using a different sport or activity to address the PE curriculum.  Meaning, one student can learn through volleyball, while a different student can learn through badminton.  Regardless of previous ability or experience, each student is free to choose which activity would be most engaging to them.

Tuesday, March 1, 2016

More to the PISA scores

Many people try and use PISA test data to infer that there is a problem with Math education in their area.  I have seen this argument used in many blogs, papers, and social media outlets.

Initially, some people will show fancy graphs and try to convince others that the drop of PISA results has actually been caused by a change in curriculum.  Later they will conclude that unless we dig up some old curriculum these scores will continue to drop in the area of mathematics.

First, any arguments, at best, truly show that there is only a correlation between the type of math curriculum their area has and a drop in PISA data.  All arguments (which I have seen anyways) always fails to show causation.  What is the difference?

Correlation is when two or more things or events occur near or around the same time.  These things might be associated with each other, but are not necessarily connected to each other by a cause/effect relationship.

An easy example is when people get a cold during the winter months they usually end up with a runny nose and a sore throat.  These two events are correlated but we cannot conclude that a runny nose will actually cause a sore throat to occur.

Most arguments around PISA, tend to show some sort of data analysis and link it to a change in curriculum, and again this would be correlation, at best.  However, we can take a closer look at the data ourselves...

If we look at the Canadian results of PISA by province, we see that every single province dropped from 2009 to 2012, other than Quebec and Saskatchewan.  On the international level, Netherlands, Belgium, Australia, Denmark, New Zealand, France, and many other countries fell at similar rates to Canada.

If there is a Math Crisis in your zone, then there must also be a Math Crisis around the entire planet? Does this sound likely?  I would hope not!

Some "back to the basics" folk will ignore the fact that they are implying here is an International Math Crisis and simply tell us that:
 PISA tells how well the math curriculum is in a certain area compared to other parts of the world.
 Well here are some stats that show how false that statement is:

Shanghai had the highest score on the 2012 PISA test with an average of 600.24, while Australia had a much smaller average of 515.01.

What can you conclude?  That Australia's math curriculum is much weaker than Shanghai's?

If we look at students who were born in China and moved to Australia before ever entering school, their average on the 2012 PISA test was 614.77... 14 POINTS HIGHER THAN SHANGHAI!!!

This must mean that Australia's math curriculum is superior to Shanghai's?  See the problem?

Using PISA scores, alone, to determine the quality of education in a province, state, region, or country is similar to judging someone's ability to drive simply by only watching them parallel park. While this single test can make many great observations around education on a global scale, we need to realize that it is exactly that: a single test.

There are much more variables at play in an Education system than simply the results on a test that some countries value more than others.  If you attend school in Scotland and are called to write PISA, you will be forced to watch champions winning Gold Medals for their country and informed that you have the ability to "bring home the Gold for Scotland".

If you attend school in China and are called to write PISA, your name will be broadcasted and people will cheer you on as you walk into the testing room.  The test you will write will be similar to the test preparation you have received over the previous months.

If you attend school in Canada and you are called to write PISA, you will be quietly removed from a class, brought to a room to write a test you know nothing about and have had no formal preparation for.

So I ask again, does a drop in PISA scores really mean a math crisis?  I think not!

Lastly,  we need to understand that a lot of arguments against current math practices are actually attacking teachers and not curriculum.

Further Reading:

Sam Sellar- Globalizing Educational Accountabilities
PISA Key findings

Wednesday, February 10, 2016

Algebra Day 1

Some points after reading Connecting Math Ideas by Jo Boaler on creating algebra.  Also a great activity to have with students, or yourself.

Tell your students the following everyday
  • Addition, subtraction, multiplication, and division is part of math but math is so much more!  While algebra is a focus in many classrooms, don't forget about geometry, probability, or data analysis.
  • You can't make understanding an algorithmic process.  Students need to experience confusion, misunderstanding, and failure as part of the understanding process.
  • Being good at math is not something that is easily explainable.  Accuracy and speed in procedural mathematics is only one way.  Other ways include being open minded, logical, discussing conjectures, and trying to answer "why?".  Everyone can be good at math and we need people who can do math in those different ways.
  • While a problem may only have one correct answer, it may have many different correct solutions.  While some solutions are more efficient, or effective than others, it is more important that everyone understands at least one way of arriving at the correct answer.  Math must make sense.
  • We can get better at certain skills of math by practice, however talking and discussing with your fellow students will allow you to understand mathematical ideas deeper.
When developing algebraic thinking with students, we shouldn't force students do what is the most efficient (as sometimes this is objective anyways), but to do what makes sense.  When presented with a set of data, or a pattern, Thorton (2001) argues that it is less important that students be able to find the algebraic rule than they recognize a rule can be represented in equivalent algebraic expressions.  When "finding the rule becomes the focus" most mathematical thinking is lost.

Lets, for example, look at the following grid.  I want you to tell me how many blocks are coloured in this 10x10 grid without counting the blocks individually.  It is crucial you do this without writing, talking to a partner or counting, as it will force your mind to make generalizations which will be the basis of deeper learning to come.  

As a teacher we should first ask students now to share answers, not strategies, with an elbow partner.  This would then force some to reevaluate their strategy and possibly pick up any common errors he/she may have made.

Now how did you do it?  Here are some strategies in Arithmetic form:
As a reader, are you able to explain how each of the expressions above arrive at the same answer? An important task in furthering one's mind into algebraic thinking.

Imagine if this was done in a classroom.  Jason stands up and explains why he simply went 4x10-4.  There might be some smiles, nods, confused looks.  Some students will have seen the same answer provided a different way.  Does this happen on a worksheet?

Of course now, you have realized that the correct answer is 36.  Would then the expression "30+6" be appropriate?  This is where conceptual differs from procedural.  Yes 30+6 equals 36 but unless a child can create meaning to the 30 and to the 6, I would disagree that 30+6 would be an appropriate response in this context.  This illustrates the idea of solution(30+6) vs answer (36). 

What if the grid was 8x8, how would the above expressions change?  Would a number change in every expression or only some?  This idea of visualizing a problem and solving it, is crucial to the advancement of one's knowledge.   

Going back to your strategy and stretching the square to an unknown length, could you create a verbal description of what would you do?

Now the algebra begins..but first...

Noss, Healy, and Hoyles (1997) point out that somewhere we stop seeing algebra as a tool but instead of the end point of a problem. The confusion starts when we see problems as a way to practice algebraic skills instead of using algebra to explore problems.

When bringing in letters into these expressions we must remember some powerful pitfalls:
  • The largest misconception is that letters are labels or initials. (pg 25)
  • Changing the variable to a different letter changes the solution.  (Let students pick the letter, and encourage different students with different letters)
  • The equal sign simply means "gets the answer". (Try writing 5=2+3 as often as 2+3=5 especially in earlier grades)
Taking these points into account, create a variable for the side length of the square.  Could you create an expression using your variable to show the amount of squares which would be shaded?  Knowing what you have done so far, how could you test your expression?

In a class at this point it would be helpful to ask the class "What will be staying the same? What will be changing? As we bridge the gap from numbers to a variable".   For example if the strategy you are using is 10*10-8*8, then the multiplication and subtraction will remain the same while 10 and 8 will change to x and x-2 respectfully.  I would not tell them this, but instead question it.

If you followed all of these tasks then you have now done the problem in 3 different approaches: Arithmetic, Verbal, and Algebraic.

The crucial part of this example, or if this lesson was to be developed into a classroom, is to ask questions and not simply give answers.  Remind them we are not seeking the solution, but a solution that makes sense to them.  Embrace students' wrong answers since learners who are given competing ideas, engage in cognitive conflict and such conflict promotes learning more than the passive reception of ideas that are always correct and seem straightforward (Fredricks, Blumenfeld, and Paris 2004).