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.

Writing About What’s In My Head 5/10


Sitting down and putting my thoughts out there can be daunting enough.  If I didn’t write about what was in my head, I would sit for hours staring at a blank screen.  I do some of that anyway trying to figure out how to express those ideas floating around up there.  10 Posts In 10 Days has helped me to be more conscious of my experiences because I know I’m going to have to write them down at some point.  Thanks Tina.  Sometimes during the day, a blog post opportunity will hit me and I’ve almost got it written before I sit down to actually write it.  I always write about what’s in my head.  I never try to “copy” what other’s are putting out there but some of what I read will stick with me and I’ll end up writing a “spin-off” blog post.  I guess that’s why we do this.

So what’s in my head? (Oops, left the door wide open.)  It has to come from somewhere.  It comes from my experiences that I share with many of you and also some that have never been here.  My thoughts, to a certain extent, come from your head.  I have often read other’s blog posts that I had a direct influence in because I can recognize the content from a conversation we had.  That’s awesome!  The ideas that are created here often have wings and then often come back around again.  This is how we see value in sharing.  But sometimes it’s kind of lonely out here and that’s expected when I’ve been inactive.  I have noticed that the hits to my site are increasing as of late.  I guess that’s what blogging more often and responding to others will do for your blog.

After my 10th post in this series, I’m not sure that I’m going to be posting everyday like I am now but I’m pretty sure I’ll be paying closer attention to what I’m learning day-to-day.  Self-reflection to so important for all of us and it’s just as important to model for our students and families.  My family has caught me writing often during the last few days and they ask what I’m doing.  I tell them and they go on their way.  I think that they think what I’m doing is kind of like keeping a diary which is something from the Brady Bunch or something only girls do.  At some point, they’re going to ask why I’m doing this.  So I’ll tell them, “I’m just writing about what’s in my head…if you read it, it will get into your head.”

The Symptom of a Greater Need 4/10


With a focus on improving numeracy in the province of Ontario, I can’t help but write about this topic for my contribution to 10 Posts In 10 Days.  We, like every board in Ontario, are trying to develop a cohesive plan that everyone teaching mathematics can understand and implement.  I have never been involved in such an undertaking but I’m getting the feeling that there isn’t a blueprint for this kind of process.  It’s a very complicated process because improving numeracy is a very expansive goal.  Ten people might have ten different suggestions on how to do this.  Just narrowing the focus on a particular area in numeracy is a difficult task.  All students aren’t the same and they don’t all have the same teacher.  How do we know that this area needs attention?  Do we have data that supports this?  Is the data reliable?  How will we know if improvement has happened?  Are provincial test scores going to be the measuring stick?

What complicates this even further is the traditional view that so many of us have about mathematics.  It’s the view that mathematics is only about right and wrong answers and getting them quickly.  So we tend to equate mathematics with basic facts and procedures.  Why are students coming to my classroom without basic fact skills?  So we drill those basic fact skills and they seem to get better for some students for the time being but next year their teacher says the same thing…Why are students coming to my classroom without basic fact skills?  That’s because they memorized them instead of “learning” them.  Without strategies for learning basic facts, drilling has a limited effect on improving basic skills for many students.  So… should basic facts be a focus for improving numeracy or is it just the symptom of a greater need?

Lukewarm Learning 3/10


I remember a time when I was teaching in a math classroom and teaching a course for the 10th or 11th time.  I remember how I could anticipate students’ struggles before they even happened.  Could that be because I taught it the way I always taught it?  Maybe.  But there were other times that I changed things up but observed those same struggles at the same time they always did.  Could it be that I needed to structure the learning so students could develop understanding on their own or with one or two other students?  Maybe.  Should I have got other teachers’ eyes on what I was doing and got some feedback before, during and after those lessons?  Maybe.  All of these things can definitely improve the way learning happens in a classroom but I think there was one really important ingredient that was missing in my teaching.

From time to time I question my purpose in education and what am I really supposed to be doing to improve learning.  What should I be seeing out there that tells me that all is good?  Ultimately, I want to see students engaged in learning because they value the process.  I want to see an urgent desire for learning not lukewarm learning.  But, as we know, modelling this process is so important.  How can we expect students to race after learning if the adults in their building don’t do the same?  Don’t get me wrong…this did happen in my class from time to time but not nearly often enough.  As the years past, I had a “been there done that” attitude and I’m sure the students picked up on it.  I know, for the most part, students liked me but I’m not sure they always liked the learning.  I’m not sure how often I “went through the motions” of differentiating the teaching and assigning rich tasks with multiple entry points and choice of how to complete them and show learning.  Maybe students needed to see me learning along side them with a contagious love of learning.  Does that mean some creative curriculum design?  Maybe.  How about time to explore multiple representations of the math and evaluate its merit?  If we did it this way, then there would be learning for me too and questions can arise from those connections which means more learning opportunities and more connections.  Students can make their own connections through this process and they can see value in this learning because they’re learning it their way not the way I want them to.

An Urgency To Share And Learn 2/10


For my second post in the Tina Zita 10 Posts In 10 Days Challenge, I want to highlight some learning and sharing in a 7-12 high school in the Algoma District School Board.  They are a smaller high school that has felt the effect of declining enrollment in the north.  Last year they embarked on a journey into project-based learning as a school focus and a significant part of that journey has been Webinar Wednesdays.  They gather as a staff at lunch every Wednesday and share and learn together.  Sometimes there’s just a few of them but often it’s standing room only.  I have had the pleasure of being invited to some of these meetings to learn with them and I usually leave there quite energized from the experience.  They have gathered to discuss their PBL philosophy which shapes everything that they will do moving forward and usually the meeting ends with next steps and action items.  Today, one of the teachers showcased a number of PBL projects that they have implemented and how they used technology to better collaborate as a group and connect teachers and students in multiple classes.  Technology is a focus for the month of January for them and this was born out of an inquiry that they conducted with teachers and students at the table trying to determine what their needs would be moving forward with PBL and the Windows books that all students in grades 7-9 would receive this year.  I’m hoping soon to invite them to expand their sharing and learning online with us.  It may take some time for them to see the value but I’m confident that once they do they will feel an urgency to share and learn in the same way they do now.

If Nobody Shares, Nobody Learns 1/10


I am taking Tina Zita’s challenge of 10 Posts In 10 Days.  I’m hoping that it will e10-postsncourage me to focus more on putting my thinking out there and not to worry so much about whether it’s perfect or not.  My blog posts tend to be very serious in nature and almost too thought out and probably a little lengthy…sorry.  I’m just going to try and focus on sharing what’s in my head at the time and put it out there and maybe I’ll set a timer so I won’t get long winded.  And I think it’s appropriate that I use the above title for this series because ultimately, I think that’s the idea that led me to blogging.  I heard it first from Donna Fry but she has since told me that she heard it from Dean Shareski… how appropriate.  I may be a little late to this party but I just read Tina’s post today so I’m going to start from my start.  Thank you Tina for getting the ball rolling.

In my school board I’ve taken on the role of Secondary Numeracy Lead recently.  The past two years I worked as a special assignment teacher in data and technology.  I was a little apprehensive with the new role because I had started to become comfortable with the old one but that didn’t last long.  I find myself doing some of the same things that I had done for the past couple of years…sharing and learning with board staff.  It’s not all the same board staff but a lot of them are the same.  We’re just talking a lot more about mathematics now.  And I still do some work in my former role because of the relationships that I built.  I see this sharing differently now because it’s so easy to broadcast it.  It seems that I have so much more access now but that’s just because I’m more aware of where to find other’s thinking.  I choose what I want and how much I want without “keeping score”.  If my colleagues aren’t ready to put their learning out there, that’s okay but…if nobody shares, nobody learns.

How To Learn Math #6


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 sixth session is titled, Conceptual Learning, Part II: Connections, Representations, Questions.  The module focuses further on strategies to help students conceptualize mathematics learning.  There are 4 ideas that spoke to me here.

Visual representations of math thinking help all students.  When students or parents claim that they’re not visual thinkers, that’s really a misuse of Howard Gardner’s theories of multiple intelligences.  His theories were not intended to pigeon-hole people into categories of “all or nothing learning”.  In fact, brain research tells us that it helps everyone to engage in creating visual representations and those that indicate that they’re not visual learners, probably need visual representations more than anyone.  Visual thinking and solutions help us to see mathematics as conceptual and meaningful and that has become more evident to me as I make my way through this course.  Visual representation has never been my first choice when solving problems.  I usually approach them with a well-remembered process because it’s usually quick.  Visuals usually take more time which traditionally, has always been associated with less intelligence.  But what good is a quick answer if I have no strategies for checking its validity?  And what if I need to refine my answer?  How do I do that without a mathematical thinking process?

Mathematics students need to use a mathematical thinking process.  Many students, as I did, believe that  they should instantly be able to solve a math problem or they’re doing it wrong.  Part of the reason for this is that teachers start these problems with providing formulas to substitute numbers into.  Many students don’t understand them so they’re not useful to them and as a result they won’t remember them.  A mathematical thinking process is a necessary process for students and teachers to develop and participate in.  Some components of this process should include the following:

  • Process the problem without formal mathematics.
    • Read the question.
    • Think about it. (What’s it telling me?  Do I have a related experience?)
    • Create a visual. (drawn and/or manipulative)
  • Talk to others. Consult resources.
  • Estimate an answer or a range within it should fall.
  • Convert thinking to more formal mathematics.
  • Try something. Refine it. Revise it.
  • Verify the answer. (Does it look reasonable?)

This list does not imply a certain order that should be adhered to but I think it is certainly important to not rush to an answer immediately and I think a visual is important to create early in the process.  As well, students may want to verify reasonableness throughout the process.  Also, I’m not suggesting that all work in mathematics requires all steps in the process but certainly verbally dense algebraic problems require a mathematical thinking process and an integral component of the process is collaboration.

Mathematics classrooms should spend considerable time collaborating.  Collaboration is the act of sharing ideas with at least one other person.  It’s through this process that we’re invited to make connections and think conceptually about mathematical processes.  Active collaboration inherently shapes our thinking which is critical to conceptualizing our  mathematics thinking.  I’ve listened to many students question why they need to explain their thinking if they got the right answer.  When we explain ideas in mathematics, we are reasoning and reasoning is an important part of mathematics – it’s really what mathematics is.  A scientist proves scientific theory by finding cases for or against.  Mathematicians prove ideas by reasoning and making logical steps.  If students are not reasoning their way through problems, they’re not engaging in mathematics.

An inquiry relationship with mathematics helps students discover connections.  An inquiry relationship with mathematics is about being curious and using intuition to solve problems.  Students with an inquiry relationship with mathematics display courage and confidence in taking risks and are comfortable doing so.  We struggle with nurturing this relationship that students should have with mathematics and part of the reason why is our  educational policies focus on cramming knowledge into students’ heads.  Math knowledge is certainly important and we shouldn’t have one without the other but it seems that, in many cases, our educational focus has killed that inquiry relationship students entered school with or entered a class with.  Intuition is necessary for mathematics knowledge to grow.  This curiosity with math is hard for students to maintain when they’re being asked to remember lists of methods and to crunch numbers.  When students are curious, they use their own intuition to discover the connections in mathematics which in turn, sparks more curiosity.