Sunday, November 26, 2017

The flaws of (some) textbooks

Recently, I was asked "Why do you dislike Textbooks?" and upon reflection, here are the issues I see:

Problem 1: Textbooks assume you need to be taught and shown how to solve a problem before you are given the problem.

This is absurd to me.  When I, and probably most, encounter something I do not know how to do, the first thing I rarely do is look for an instructional video on how to complete the task.

Ironically, the first thing I do is actually play with the problem and see how far I can get without any assistance.  That is right; I play!   This play cannot happen if my hand is being held and shown how to complete the task.  Learners, specifically children, are not afraid to be wrong, take chances, and try to truly problem solve, yet a textbook is designed around the idea that a child loves to be told what to do.

This ruins the fun!!!  I say again, this ruins the fun!  It is comparable to turning on a movie and someone telling you "The main character dies at the end!"; Joy lost!

Problem 2:  Pseudo - context

In almost every textbook I have seen there is always some sort of situation that can only exist in "textbook land", a magical place where the following is true:


Textbooks usually tell students all the information, in the order the need it, and then call it "Problem Solving".  My favourite question and answer was when I read the following question from a textbook:

Jason weighed a fish, and found out that if you took the weight of the fish and added it to half the weight, the result is 20 lbs.  How much does the fish weigh?

The best answer was from a student who exclaimed:

Ask Jason he weighed the damn thing!

Brilliant response to a horrible question!

Problem 3:  They "unitize" learning.

Again, a common problem I see with textbooks, is they assume you need to learn A, then B, then C, to master idea D.  In my experience, creating these disjoint learning situations, or what I call "silos of learning" causes problems for students.  

A common practice in textbooks are "Chapter Tests", which means the pages after these tests have rarely little or nothing to do with the previous pages.  In essence, the learning that happened yesterday will have nothing to do with tomorrow.  

A great practice is to weave essential learning outcomes throughout your entire course.  This contradicts the textbook.  If you feel a certain outcome is important for all to master, I would hope that your students work with that idea throughout the entire course and not just for a finite time (week, or month) and then move on and never relate new learning to previous learning.  

Disclaimer:  Does this mean I don't think textbooks belong in schools?  NO!

This means educators have to be aware of the shortfalls of textbooks.  The biggest idea we always have to remember is that these resources were created, usually in an office, to be sold across an entire continent or country.  They are not designed for "your" kids; or really anyone's kids for that matter.  

Textbooks should be used similar to encyclopedias in the classroom.  Reference material.  If a student is struggling with a concept, give them a textbook, show them a certain page and advise him/her to complete some (not all) questions.  When completed, have a conversation, and then ask him/her to return the textbook to the classroom shelf.

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