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 encourage 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.

How To Learn Math #5

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 fifth session is titled, Conceptual Learning, Part I: Number Sense.  There are 3 ideas that spoke to me here.

Number sense is built from the flexible use of numbers.  In my experience, teachers’ most urgent need for their students is number sense or basic skills.  Does that mean that students need to be able to recall multiplication facts quickly?  Does that mean that they need to remember to write the ones digit down and carry the tens digit when they add two digit numbers?  Does that mean they need more procedural fluency?  When students perform a question like 18+5, they may find it easiest to just start at 18 and count on to 23 or this might just be a known fact to them because they’ve seen it many times.  It’s impossible to know all possible addition facts and it becomes extremely difficult and requires significant cognitive function when the numbers get larger.  As a result, there is enormous room for error.  This is why students need to use numbers flexibly so they can decompose and recombine them to make friendlier numbers.  In the case of 18+5, they need to see this as 18+2+3.  So the 18+2 becomes 20 which is a friendlier number.  Adding 20+3 is easier to see than 18+5.  The beauty of seeing numbers flexibly, is that there are many ways to see the same question.  Some of us may see  18+5  as  5+5+5+3+5  and add up all the 5’s and then add-on the 3.  As the numbers get larger, there will be more varied opportunities to break up the numbers and then put them back together.

Studies have shown that the difference between high and low achieving students isn’t that the high achieving students learn more – it is that they learn to act upon numbers flexibly.  Flexible use of numbers and shapes is critical to all higher level math learning.  In fact, it is foundational for pretty much everything that comes beyond in algebra, calculus, and beyond.  The problem that low achieving students face is that they fail to engage with math in this way and rely solely on their counting method.  As a result, they come to believe that math is only about the “right” answer.  However, counting gets more complex as problems become more difficult.  These students are often identified early on and given more drill and practice but without the proper strategies, this is probably the worst thing for them.

Mathematics must be taught as a conceptual subject.  When we learn anything conceptually, a process called compression occurs in our brains.  New learning takes up a lot of “space” in our brains.  Eventually, if conceptualized, our learning is compressed and it becomes easier to access in the future.  If new learning never progresses past methods and procedures, it will not be compressed and will become more difficult to retrieve in the future.  This is why many students have difficulty with a mathematical process days or weeks later.  If teachers and students aren’t engaging in conceptual thinking during mathematics instruction, students won’t benefit from compression.

Number Talks are easy and effective for developing number sense.  Number talks is a process where teachers lead discussions with students on the flexible use of numbers as described earlier.  Generally, students work individually, then together with one or two others on a math calculation like 18+5.  As a group, various solutions are displayed and discussed.  Right answers, wrong answers… it doesn’t matter because the process of how an answer is arrived at is more important than the answer itself.  This process of taking numbers apart and then regrouping them as friendly numbers helps to conceptualize students’ understanding of operations with numbers and at the same time, teaches them number fluency and automaticity.  Students can really benefit from number talks because they show how math is a creative and flexible subject not just a series of rules and methods to memorize.  This flexible use of numbers is a very important building block in number sense and mathematics which is the foundational base from which all other mathematics builds.  Here are a couple of resources for number talks.

Number Talks, Grades K-5

Number Talks, Grades 4-10

How To Learn Math #4

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 fourth session is titled, Teaching for a Growth Mindset.  The focus of this module is on the various strategies that teachers and parents can use to promote a growth mindset in mathematics.  There are 3 ideas that spoke to me here.

Grouping strategies can result in lower achievement.  We call it tracking and the research that Jo Boaler points to concludes that tracking within and throughout classes sends fixed mindset messages to students especially the high achievers.  Furthermore, tracking tends to confine students to a lower track when tracked early on.  In countries like Finland and Japan, who track the least and latest, students tend to show more success.

When students are tracked there tends to be different teacher expectations applied to different ability groups, even when teachers try not to or don’t want to have different expectations.  Students in lower groups are generally given easier work which limits their achievement.  Teachers tend to teach to the middle of the tracked group assuming everybody is the same so everybody gets the same work at the same pace.  So there tends to be little differentiation in tasks assigned and in instruction.  Even in a tracked group, there tends to be a range of ability because often some are misplaced.

In untracked groups, teachers are more likely to recognize that there is a range of ability and that they need to provide differentiated opportunities.  These opportunities include giving more “open-work” that students can take to any level and differentiating the instruction for students.  When the environment is structured properly, untracked students are more likely to choose higher level work, they achieve more, students choose more advanced courses, and students are more likely to pass national tests.  These results occur across the achievement range.

The kinds of tasks we ask students to work on can negatively affect their mindset.  If they’re short, enclosed, with one right answer, students are either getting them right or wrong.  Under these conditions, it’s really hard to develop the idea that mathematics is about growth and learning.  If a student is constantly getting the wrong answer, it can be really hard to believe that they can do math.  So tasks need to give students the space inside them to learn and to see that math means learning, not just performing.  Teachers need to design tasks so that students can see them in different ways and can start it from where they’re at by choosing a strategy that suits them.  There should also be clear learning goals and adequate opportunities for feedback.  In many cases, teachers only need to modify tasks that they already use.

Summative assessment should probably be eliminated.  It tells students how well they’re doing compared to other students but it doesn’t tell students about their path to growth and improvement.  Effective assessment should tell students where they are, where they need to be, and how to close the gap between the two.  I know this as formative assessment.  Studies have shown that among students given only summative assessment, only formative assessment, and a mixture of both, the students given only formative, achieved significantly higher than either of the other two.

In order for formative assessment to be effective, there needs to be a level of awareness on the students’ part especially for knowing where they need to be.  So starting with learning goals becomes so important and using those learning goals to self and peer assess helps to build that awareness.  In another study, two groups of physics students were given equal time to work together on the subject matter.  One group simply discussed the work while the other self and peer assessed.  The latter outperformed the former on three different assessments and the greatest gains were made by those who were previously low achieving.  The research suggests that low achievers are often low achieving, not because they lack ability or because they’re slow, but because they just don’t know what’s important or what they’re meant to be paying attention to.

• Tasks – youcubed.org
•  Modifying tasks – Dan Meyer
• Differentiating instruction – Mathematical Mindsets – Chapter 9

How To Learn Math #3

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 third session is titled, Mistakes and Persistence.  There are 3 ideas that spoke to me here.

Making mistakes is necessary in learning.  The most successful entrepreneurs are often the ones who made the most mistakes.  They often failed but learned from it and moved on.  If this is necessary in the “real” world then it is necessary in school.  Brain research tells us that when we make a mistake a synapse fires.  Then, when we’re aware of the mistake, another synapse fires.  Yes, a synapse fires whether we realize we’ve made a mistake or not.  We’re not sure why it happens, but it does.  We also know that this process is more significant in a growth mindset brain than in a fixed mindset brain.  Students need to struggle with content but they need to feel good about the struggle.  That means we need to change the way we present this material to them and the way we assess it.

The testing culture has inflicted a lot of damage.  When tracking traditional vs PBL (problem based learning) schools and students who achieve at the same level, PBL schools score higher on national exams 3 years later.  Those students also acquire higher skilled, better paying jobs 8 years later.  This is because in those schools students are rewarded for experimentation and not necessarily for correct answers.  They are rewarded for having good ideas, for trying them and making mistakes.  This creates a culture where student struggle and thinking is valued.  In these environments, students persevere.

In the traditional school, assessments are performed under test and time pressure.  These conditions promote an atmosphere of anxiety and fear which blocks working memory in the brain.  Students do not get better or faster under these conditions.  In fact, it’s been shown that students work more quickly when anxiety and fear is taken away.  Math anxiety across the achievement range has been linked to timed fact tests in students’ early school years.  Even answering the first hand up during questioning reinforces the idea that faster means smarter.  Schools, in countries that are successful, don’t test until the end of the year.  In these schools, students aren’t afraid to try something new, of being creative, or of thinking in a different way.  The tasks assigned in traditional schools don’t tend to require deep thought and students are often told what method to use.  We, as teachers, assume that this process is good for students because they won’t struggle as much but in fact, we’re working together with students to empty the interaction out of learning – the didactic contract.  In these classes, students will always ask the teacher for help instead of risk being wrong.

There are things we can do.  Teach math facts differently. – Fluency Without Fear.  Give students time to struggle. – Faster Isn’t Smarter.  Don’t just praise mistakes, talk about why they’re important. – Show this to your students.  Assign work that encourages mistakes.  If they’re not making mistakes, they’re probably not doing work that’s challenging enough.  Mark mistakes on their papers with stars or a picture of a synapse firing to highlight their importance.  Allow students time to work more deeply by clustering expectations or teaching to the big ideas instead of to specific expectations.

How To Learn Math #2

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 second session is titled, Maths and Mindset.  There are 4 ideas that spoke to me here.

Most of us are capable of doing math.  Brain research tells us that the brain can change and grow.  Less than 5% of the population experiences severe special needs that will significantly affect their ability to learn.  But even in those situations, brain growth still occurs.  This means that our teaching has to be responsive to students’ needs.  If it’s not, students will become frustrated with learning math and soon, the trust they may have had in their teacher will fade.  Once that happens, classroom management becomes an issue and conflict starts to be of a personal nature.  And that can be the point of no return in the relationship between a student and teacher.

The brain can change and rewire itself.  It’s called neuroplasticity.  When learning happens in the brain, synapses fire which creates connections between neurons.  Returning to an idea strengthens neural pathways.  When students believe they can’t do math, it’s not because their brains aren’t wired that way or that their brains are incapable of being wired that way.  It is because they don’t believe that they are capable of learning math.

Media portrayal of math is awful.  The media usually portrays math as boring and inaccessible.  Those who are proficient with math are usually “nerdy” individuals, white, male, or Asian.  So, it becomes difficult for us to change those ideas because students, parents, and even some of our colleagues start to believe them.  How many times have you heard, I don’t get this new math?

There are two kinds of mindsets.  If one has a fixed mindset, he/she assumes there’s a certain amount of unchangeable intelligence.  They also believe that math ability is a gift and failure says to him/her that he/she is not “smart”.  Because of this, he/she will avoid challenging tasks.  This mindset is prevalent in high achieving girls and it will likely affect their choice of future course or career.  A fixed mindset can be encouraged with praise of a fixed nature – “You’re so smart.”  Someone with a growth mindset assumes that math ability or “smartness” grows with experience.  He/she sees mistakes as opportunities for growth and this mindset can be strengthened with praise that addresses effort instead of result.  “You worked really hard on that.”

Mindsets are established early but can also be changed.  Research has shown that the kind of praise that a 3-year-old receives can predict his/her mindset 5 years later.  However, early mindset intervention can also change this and often immediately.  This means that mindsets can be changed for the better or for the worse.  In cases of positive mindset change intervention, it’s been shown that African-Americans and girls show the sharpest increase in grades and valuing school.  Positive mindset change intervention also eliminates gender differences in achievement.  In fact, after 8th grade, gender gap achievement exists mainly among fixed mindset students.  As well, among girls with growth mindsets taking STEM subjects, they were more likely to ignore gender stereotyping messages and continue in those STEM subjects.