How Do I Know They’re Learning?


The morning bell rings.  Students file into my mathematics classroom.  The national anthem and morning announcements occur.  I start teaching.  I’m asking questions.  Some students are answering – I wish I had a different question for each of them and time to answer them all.  I hand out the practise work.  I circulate and answer questions.  There will probably be a quiz tomorrow.  The next day – same thing.  A week later – a review day.  The day after that – a test.  I may be oversimplifying here and not everyday looks like this in all mathematics classrooms but one question always hovers like a dark cloud over me… how do I know they’re learning?

Is it grades that tells me that?  I think that tells me a little bit about learning.  In the conditions that I’ve described above, I think it tells me more about how well they’ve figured me out in that one week window or the procedures that I’ve demonstrated.  I think about the kinds of questions that I usually answer during practise time.  They are almost exclusively procedure questions.  What do I do now?  How do I do that?  Is it the act of assessing the learning that hampers learning itself?  Do students see a finish line when we assign a grade so they don’t pursue any learning thereafter?  And if we are only assessing procedures and content, that sounds more like a 100 metre dash, not an obstacle course.

Is it observational evidence that tells me if students are learning?  If students are sitting quietly at their desks, are they learning?  If students are sitting together collaborating, are they learning?  I think learning occurs in both situations but 2 things have to happen.  There has to be some structure and expectation around each activity and you have to allow for a little bit of both.  Productive group conversation is effective for gathering and consolidating various perspectives but individuals need to struggle a little bit on their own to mold their own ideas especially in the case of an introverted student.

I should be mixing up my teaching style from day-to-day as well as within the class period.  Students learn in different ways and can quickly disengage if I teach in the same style that feels most comfortable to me.  Too often, I assumed that if I taught, they learned.  So if they disengaged, that was a choice they made.  Yes, that was a choice they made, but some of the reason they made that choice was for the same reason I choose a different channel while I’m watching television.

Up For Another Challenge? 10/10


How appropriate that I use my last blog post contribution to Tina Zita’s 10 Posts In 10 Days to highlight Donna Fry’s Showing Our Love For Bloggers Challenge.  From one challenge to the next – isn’t that what we want our students to do?  Thank you Donna and Tina for inspiring me to keep this going.

Donna’s challenge is a little different though.  We are committing to commenting 5 times (or more) on Ontario educator blogs between now and February 14 and tweeting a link to those comments with an appropriate hashtag.  Click on the link above and all the details are there including suggestions and guidelines on how to leave good comments.  See all of you there!


20/20 IEP 9 of 10


“Give me six hours to chop down a tree and I will spend the first four sharpening the axe.”  This is one of the quotes that I remember seeing daily when I first started a system level job for the school board that I work for.  It reminds me of the work that we doing to prepare to teach in a classroom.  Part of that preparation should include reviewing the individual education plans (IEP’s) of the students that we will be teaching.  An IEP is a legally binding document which addresses each child’s unique learning issues and includes specific educational goals.  The school must provide everything it promises in the IEP.  Accommodations in an IEP are like prescription eyewear for people with vision issues.  Students receiving accommodations through an IEP are not cheating.  They’re receiving these accommodations because of a medical diagnosis that says that their intelligence does not match their achievement without those accommodations.  An accommodation for a student with processing issues could simply include more time to complete a task.

Special education is an area that’s been of interest to me for a little while now and it’s certainly a work in progress.  The board that I work for is doing some work in this area especially in mathematics.  What’s impressed me the most so far in this work is the connection between a student’s behaviour during instructional time and the area of need in his/her IEP.  If a teacher understands that a student is disengaging because of a struggle that they have that is medically diagnosed and can be remedied by his/her IEP, that’s a giant leap toward understanding that student.  And understanding our students is important in building relationships and trust – the prerequisite to an effective learning environment.

Second Chance At A First Impression 8/10


Elementary and secondary teachers can have a tremendous amount of control in how they want their students to experience learning in their classrooms.  The nice thing about teaching in secondary, particularly in a semestered school, is that you get twice as many opportunities to establish an environment for learning that students will thrive in.

In most areas in Ontario, semester one is wrapping up and semester two is just around the corner.  This should be a time when secondary teachers are beginning to prepare what those first few days will look like.  When I started a semester, the first thing I did was photocopy the course outline with its list of units that we would study and its assessment break down and the homework and missed assignment policies.  And these are all really important pieces of information that we need to share with students… but on the very first day?  Is that the first impression that I want students to have of my class?  Is that the first thing that I want my students to understand…compliance?

I should be starting on the first day with the things that I value most about learning.  So that should be what I’m thinking about right now.  How do I show my students what I value most about learning and hence, what I want them to value about learning?  And certainly compliance is part of what we all have to understand but it shouldn’t be the very first thing that I want them to understand.  I think “rules” should be a little ways down the to do list.  If I really value a collaborative learning environment then I should be arranging my classroom accordingly and establishing norms and structures around how they will collaborate for learning.  Sitting students together and giving them an assignment to work on together doesn’t mean that they will magically learn together.  It’s like putting your basketball team through some drills in the first practice.  You need to evaluate their skills first and then build them up so they can use them effectively.  I made a number of assumptions about what my students were capable of during the semester but I may have underestimated some of their capabilities because I didn’t do enough work to prepare them for learning in the first couple of weeks.

10 Posts In 10+ Days 7/10


Well, I fell off the Tina Zita 10 Posts In 10 Days wagon but I’m back on.  To finish off this series, I’m now calling it 10 Posts in 10+ Days.  I’m going to the office to get a late slip… I’m going to get a note from my parents… I’m going to serve detention.  I really love this idea of keeping myself accountable because I have to reach that magic number 10 post.  I’m seeing the value in a short burst of intensive work to sharpen skills.  But I’m not going to rush to number 10 because obviously, I need a little more time.  Sometimes, we’ll see a better product when we allow a little more time.  Sometimes, a little more time is necessary to shape better understanding.  Sometimes, more time is necessary for learning.

To Victory Lap Or Not 6/10


I’m writing a series of posts as part of a post challenge put out by Tina Zita.  It’s called 10 Posts In 10 Days.

When students are in their grade 12 year in high school in Ontario, there’s a burning question that they wrestle with for the better part of the year.  Should I graduate and go back to high school for another year or should I move onto a post secondary pathway?  In my parts, we call a fifth year in high school a “victory lap”.  I’ve heard a lot of the pros and cons for both sides of this debate and I think students ultimately need to make their own decision on this.  It’s really only one year and if things don’t really work out like they planned, so what.  How many times have we heard the argument that a two or a three-year college degree is a better choice than a university degree because you can get out into the real world and start earning some money a year or two earlier?  So does that mean an extra year in university is wasted?

What is the best way to train for a career?  Should this process be on a timeline?  Does our system adequately prepare our youth for their futures?  I’m not sure I can answer all of this here but I think one thing is for sure.  Our students and children need to be encouraged to do something they enjoy so they can be happy doing it.  They have a lot of living in front of them and if they choose a career that’s going to make them lots of money or improve their status but make them miserable, they may have less living to do than they think.  So if it takes them some time to figure that out, so be it.  As long as the time they take is productive in terms of exploring or researching or experiencing different things that they might like to do, that’s not wasted time.

How To Learn Math #7


Recently, I began a MOOC (massive open online course) called How to Learn Math – For Teachers and Parents (2).  More than 40,000 people took the last class – mainly teachers, parents and school administrators. 95% of people completing the end of course survey said that they would change their teaching or ways of helping as a result of the course.  The course offers important new research ideas on learning, the brain, and math that can transform students’ experiences with math and is based on the work of and narrated by Jo Boaler.  It’s divided into 8 sessions and each one ends with a prompt to write a one paragraph summary reflection on the ideas.  My plan is to post those reflections here.

The seventh session is titled, Appreciating Algebra.  There are 3 ideas that spoke to me here.

Students struggles in algebra stem from learning structural algebra vs procedural algebra.  Structural algebra is thinking of a variable x, as any number, like in the following problem:

How many total tires are there on 10 tricycles?


Structural algebra is more about studying and describing patterns.  It’s also about the most important part of algebra – learning to generalize or describe the general form of a pattern.  Procedural algebra is thinking of a variable x, as one single number.  So x in the equation, 2x+5=9 would be thinking of x as one single number.  Procedural algebra is about solving for x.  The difference between procedural and structural algebra seems to follow a common theme throughout this course.  And it’s that we need to concentrate more on the creative aspects in mathematics not only because it promotes more engaging learning and conceptual understanding but also because it promotes a better mindset in learning mathematics.  Research tells us that students find it very difficult moving from procedural to structural algebra.  So if we spend a lot of time asking students to solve for x or think of one single number for x in their early school years and then move to describing patterns and generalizing them, that’s a big conceptual leap.  It’s interesting though, the research says that studying structural algebra first and then transitioning to procedural algebra does not introduce the same struggles for students.  This is not a significant conceptual leap.

Students struggle with the meaning of the equal sign.  In the early years, if teachers focus their attention on addition and subtraction facts, students often come to believe that the equal sign is an instruction to do something.  An equal sign signifies a symmetrical relationship between each side.  So if students don’t recognize this relationship they run-on their answers attaching the next operation onto the last answer and the calculations end up on one line across the page – equal signs abound and equality, nowhere to be found.  Knowing that the equal sign means equivalent is an important idea in helping students make sense of algebra.

Learning algebra should be about enabling learners to use algebra to make sense of the world.  This isn’t a new idea in mathematics or education in general.  When learning requires us to solve real problems that are meaningful to us, there’s more engagement with the learning.  In teaching algebra we need to identify those situations that the students are excited to investigate and need to explain, predict, and model.  Students should be encouraged to use varying representations of their patterns with visuals, graphs, tables, and descriptions.  Collaborative problem solving is important in making sense of rich modeling situations as well as using new technologies to engage in and explain them.  In an effort to help students make sense of a problem, we often use letters or variables with the same starting letter as the object being described.  Consider the value, in cents, of a number of nickels and dimes as in  V = 10d + 5n. (d represents the number of dimes, n represents the number of nickels)  Research tells us that using the first letter of objects leads to a classic misconception in algebra.  It leads students to think that we’re adding 10 dimes and 5 nickels.  I can see how that could happen because in most Ontario grade 9 textbooks, operations with algebraic expressions comes before linear relations.  So the procedural algebra comes before the structural algebra.  It’s important to remember to use other letters in this case so as not to confuse the value of the coins with the total number of coins.