Unintended Messages

I read an interesting article by Yong Zhao the other day entitled What Works Can Hurt: Side Effects in Education where he discussed A simple reality that exists in schools and districts all over. Basically, he gives the analogy of education being like the field of medicine (Yes, I know this is an overused comparison, but let’s go with it for a minute).  Yong paints the picture of how careful drug and medicine companies have become in warning “customers” of both the benefits of using a specific drug and the potential side-effects that might result because of its use.

However, Yong continues to explain that the general public has not been given the same cautionary messages for any educational decision or program:

“This program helps improve your students’ reading scores, but it may make them hate reading forever.” No such information is given to teachers or school principals.
“This practice can help your children become a better student, but it may make her less creative.” No parent has been given information about effects and side effects of practices in schools.

Simply put, in education, we tend to discuss the benefits of any program or practice without thinking through how this might affect our students’ well-being in other areas.  The issue here might come as a direct result of teachers, schools and systems narrowing their focus to measure results without considering what is being measured and why, what is not being measured and why, and what the short and long term effects might be of this focus!


Let’s explore a few possible scenarios:

Practice:

In order to help students see the developmental nature of mathematical ideas, some teachers organize their discussions about their problems by starting to share the simplest ideas first then move toward more and more complicated samples.  The idea here is that students with simple or less efficient ideas can make connections with other ideas that will follow.

Unintended Side Effects:

Some students in this class might come to notice that their ideas or thinking is always called upon first, or always used as the model for others to learn from.  Either situation might cause this child to realize that they are or are not a “math person”.  Patterns in our decisions can lead students into the false belief that we value some students’ ideas over the rest.  We need to tailor our decisions and feedback based on what is important mathematically, and based on the students’ peronal needs.


Practice:

In order to meet the needs of a variety of students, teachers / schools / districts organize students by ability.  This can look like streaming (tracking), setting (regrouping of students for a specific subject), or within class ability grouping.

Unintended Side Effects:

A focus on sorting students by their potential moves the focus from helping our students learn, to determining if they are in the right group.  It can become easy as an educator to notice a student who is struggling and assume the issue is that they are not in the right group instead of focusing on a variety of learning opportunities that will help all students be successful.  If the focus remains on making sure students are grouped properly, it can become much more difficult for us to learn and develop new techniques!  To our students, being sorted can either help motivate, or dissuade students from believing they are capable!  Basically, sorting students leads both educators and students to develop fixed mindsets.  Instead of sorting students, understanding what differentiated instruction can look like in a mixed-ability class can help us move all of our students forward, while helping everyone develop a healthy relationship with mathematics.


Practice:

A common practice for some teachers involves working with small groups of students at a time with targeted needs.  Many see that this practice can help their students gain more confidence in specific areas of need.

Unintended Side Effects:

Sitting, working with students in small groups as a regular practice means that the teacher is not present during the learning that happens with the rest of the students.  Some students can become over reliant on the teacher in this scenario and tend to not work as diligently during times when not directly supervised.  If we want patient problem solvers, we need to provide our students with more opportunities for them to figure things out for themselves.


Practice:

Some teachers teach through direct instruction (standing in front of the class, or via slideshow notes, or videos) as their regular means of helping students learn new material.  Many realize it is quicker and easier for a teacher to just tell their students something.

Unintended Side Effects:

Students come to see mathematics as subject where memory and rules are what is valued and what is needed.  When confronted with novel problems, students are far less likely to find an entry point or to make sense of the problem because their teacher hadn’t told them how to do it yet.  These students are also far more likely to rely on memory instead of using mathematical reasoning or sense making strategies.  While direct instruction might be easier and quicker for students to learn things, it is also more likely these students will forget.  If we want our students to develop deep understanding of the material, we need them to help provide experiences where they will make sense of the material.  They need to construct their understanding through thinking and reasoning and by making mistakes followed by more thinking and reasoning.


Practice:

Many “diagnostic” assessments resources help us understand why students who are really struggling to access the mathematics are having issues.  They are designed to help us know specifically where a student is struggling and hopefully they offer next steps for teachers to use.  However, many teachers use these resources with their whole group – even with those who might not be struggling.  The belief here is that we should attempt  to find needs for everyone.

Unintended Side Effects:

When the intention of teachers is to find students’ weaknesses, we start to look at our students from a deficit model.  We start to see “Gaps” in understanding instead of partial understandings.  Teachers start to see themselves as the person helping to “fix” students, instead of providing experiences that will help build students’ understandings.  Students also come to see the subject as one where “mastering” a concept is a short-term goal, instead of the goal being mathematical reasoning and deep understanding of the concepts.  Instead of starting with what our students CAN’T do and DON’T know, we might want to start by providing our students with experiences where they can reason and think and learn through problem solving situations.  Here we can create situations where students learn WITH and FROM each other through rich tasks and problems.


Our Decisions:

Yong Zhao’s article – What Works Can Hurt: Side Effects in Education – is titled really well.  The problem is that some of the practices and programs that can prove to have great results in specific areas, might actually be harmful in other ways.  Because of this, I believe we need consider the benefits, limitations and unintended messages of any product and of any practice… especially if this is a school or system focus.

As a school or a system, this means that we need to be really thoughtful about what we are measuring and why.  Whatever we measure, we need to understand how much weight it has in telling us and our students what we are focused on, and what we value.  Like the saying goes, we measure what we value, and we value what we measure.  For instance:

  • If we measure fact retrieval, what are the unintended side effects?  What does this tell our students math is all about?  Who does this tell us math is for?
  • If we measure via multiple choice or fill-in-the-blank questions as a common practice, what are the unintended side effects?  What does this tell our students math is all about?  How reliable is this information?
  • If we measure items from last year’s standards (expectations), what are the unintended side effects?  Will we spend our classroom time giving experiences from prior grades, help build our students’ understanding of current topics?
  • If we only value standardized measurements, what are the unintended side effects?  Will we see classrooms where development of mathematics is the focus, or “answer getting” strategies?  What will our students think we value?

Some things to reflect on
  • Think about what it is like to be a student in your class for a moment.  What is it like to learn mathematics every day?  Would you want to learn mathematics in your class every day?  What would your students say you value?
  • Think about the students in front of you for a minute.  Who is good at math?  What makes you believe they are good at math?  How are we building up those that don’t see themselves as mathematicians?
  • Consider what your school and your district ask you to measure.  Which of the 5 strands of mathematics proficiency do these measurements focus on?  Which ones have been given less attention?  How can we help make sure we are not narrowing our focus and excluding some of the things that really matter?

baba

As always, I encourage you to leave a message here or on Twitter (@markchubb3)!

Which one has a bigger area?

Many grade 3 teachers in my district, after taking part in some professional development recently (provided by @teatherboard), have tried the same task relating to area.  I’d like to share the task with you and discuss some generalities we can consider for any topic in any grade.


The task:

As an introductory activity to area, students were provided with two images and asked which of the two shapes had the largest area.

Captureb.JPG

A variety of tools and manipulatives were handy, as always, for students to use to help them make sense of the problem.


Student ideas

Given very little direction and lots of time to think about how to solve this problem, we saw a wide range of student thinking.  Take a look at a few:

Some students used circles to help them find area.  What does this say about what they understand?  What issues do you see with this approach though?

Some students used shapes to cover the outline of each shape (perimeter).  Will they be able to find the shape with the greater area?  Is this strategy always / sometimes / never going to work?  What does this strategy say about what they understand?

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Example 3

Some students used identical shapes to cover the inside of each figure.

And some students used different shapes to cover the figures.

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Example 7
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Example 8
C-rods, difference
Example 9

Notice that example 9 here includes different units in both figures, but has reorganized them underneath to show the difference (can you tell which line represents which figure?).


Building Meaningful Conversations

Each of the samples above show the thinking, reasoning and understanding that the students brought to our math class.  They were given a very difficult task and were asked to use their reasoning skills to find an answer and prove it.  In the end, students were split between which figure had the greater area (some believing they were equal, many believing that one of the two was larger).  In the end, students had very different numerical answers as to how much larger or smaller the figures were from each other.  These discrepancies set the stage for a powerful learning opportunity!

For example, asking questions that get at the big ideas of measurement are now possible because of this problem:

“How is it possible some of us believe the left figure has a larger area and some of us believe that the right figure is larger?”

“Has example 8 (scroll up to take a closer look) proven that they both have the same area?”

“Why did example 9 use two pictures?  It looks like many of the cuisenaire rods are missing in the second picture?  What did you think they did here?”

In the end, the conversations should bring about important information for us to understand:

  • We need comparable units if we are to compare 2 or more figures together.  This could mean using same-sized units (like examples 1, 4, 5 & 6 above), or corresponding units (like example 8 above), or units that can be reorganized and appropriately compared (like example 9).
  • If we want to determine the area numerically, we need to use the same-sized piece exclusively.
  • The smaller the unit we use, the more of them we will need to use.
  • It is difficult to find the exact area of figures with rounded parts using the tools we have.  So, our measurements are not precise.

Some generalizations we can make here to help us with any topic in any grade

When our students are being introduced to a new topic, it is always beneficial to start with their ideas first.  This way we can see the ideas they come to us with and engage in rich discussions during the lesson close that helps our students build understanding together.  It is here in the discussions that we can bridge the thinking our students currently have with the thinking needed to understand the concepts you want them to leave with.  In the example above, the students entered this year with many experiences using non-standard measurements, and this year, most of their experiences will be using standard measurements.  However, instead of starting to teach this year’s standards, we need to help our students make some connections, and see the need to learn something new.  Considering what the first few days look like in any unit is essential to make sure our students are adequately prepared to learn something new!  (More on this here: What does day one look like?)

To me, this is what formative assessment should look like in mathematics!  Setting up experiences that will challenge our students, listening and observing our students as they work and think… all to build conversations that will help our students make sense of the “big ideas” or key understandings we will need to learn in the upcoming lessons.  When we view formative assessment as a way to learn more about our students’ thinking, and as a way to bridge their thinking with where we are going, we tend to see our students through an asset lens (what they DO understand) instead of their through the deficit lens (i.e., gaps in understanding… “they can’t”…, “didn’t they learn this last year…?).  When we see our students through an asset lens, we tend to believe they are capable, and our students see themselves and the subject in a much more positive light!

Let’s take a closer look at the features of this lesson:

  • Little to no instruction was given – we wanted to learn about our students’ thinking, not see if they can follow directions
  • The problem was open enough to have multiple possible strategies and offer multiple possible entry points (low floor – high ceiling)
  • Asking students to prove something opens up many possibilities for rich discussions
  • Students needed to begin by using their reasoning skills, not procedural knowledge…
  • Coming up with a response involved students doing and thinking… but the real learning happened afterward – during the consolidation phase

A belief I have is that the deeper we understand the big ideas behind the math our students are learning, the more likely we will know what experiences our students need first!


A few things to reflect on:

  • How often do you give tasks hoping students will solve it a specific way?  And how often you give tasks that allow your students to show you their current thinking?  Which of these approaches do you value?
  • What do your students expect math class to be like on the first few days of a new topic/concept?  Do they expect marks and quizzes?  Or explanations, notes and lessons?  Or problems where students think and share, and eventually come to understand the mathematics deeply through rich discussions?  Is there a disconnect between what you believe is best, and what your students expect?
  • I’ve painted the picture here of formative assessment as a way to help us learn about how our students think – and not about gathering marks, grouping students, filling gaps.  What does formative assessment look like in your classroom?  Are there expectations put on you from others as to what formative assessment should look like?  How might the ideas here agree with or challenge your beliefs or the expectations put upon you?
  • Time is always a concern.  Is there value in building/constructing the learning together as a class, or is covering the curriculum standards good enough?  How might these two differ?  How would you like your students to experience mathematics?

As always, I’d love to hear your thoughts.  Leave a reply here on Twitter (@MarkChubb3)

The Zone of Optimal Confusion

In the Ontario curriculum we have many expectations (standards) that tell us students are expected to, “Determine through investigation…” or at least contain the phrase “…through investigation…”.  In fact, in every grade there are many expectations with these phrases. While these expectations are weaved throughout our curriculum, and are particularly noticeable throughout concepts that are new for students, the reality is that many teachers might not be familiar with what it looks like for students to determine something on their own.  Probably in part because this was not how we experienced mathematics as students ourselves!

First of all, I believe the reason behind why investigating is included in our curriculum is an important conversation!   I’ve shared this before, but maybe it will help explain why we want our students to investigate:

Teaching Approaches - New

The chart shows 3 different teaching approaches and details for each (for more thoughts about the chart you might be interested in What does Day 1 Look Like). Hopefully you have made the connection between the Constructivist approach and the act of “determining through investigation.”  Having our students construct their understanding can’t be overstated. For those students you have in your classroom that typically aren’t engaged, or who give up easily, or who typically struggle… this process of determining through investigation is the missing ingredient in their development. Skip this step and start with you explaining procedures, and you lose several students!

Traditionally, however, many teachers’ goal was to scaffold the learning.  They believed that a gradual release of responsibilities would be the most helpful.  Cathy Seeley in her book Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver explains the issue clearly:Gradual Release

Cathy Seeley quote

In the two pieces above Cathy explains the “upside-down teaching” approach.  This is exactly the approach we believe our curriculum is suggesting when it says “determine through investigation,”  and exactly the approach suggested here:

Page 24 - paragraph 2
Page 24 of the Ontario Curriculum

At the heart of this is the idea of “productive struggle”, we want our students actively constructing their own thinking.  However, I wonder if we could ever explain what “productive struggle” looks / feels like without ever experiencing it ourselves?  How might the following graphic help us reflect on our own understanding of “productive struggle” and “engagement”?

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Research Gate, Confusion can be Beneficial for Learning

I think it would be a wonderful opportunity for us to share problems and tasks that allow for productive struggle, that have student reasoning as its goal, problems / tasks that fit into this “zone of optimal confusion”.

In the end, we know that these tasks, facilitated well, have the potential for deep learning because the act of being confused, working through this confusion, then consolidating the learning effectively is how lasting learning happens!

Let’s commit to sharing a sample, send a link to a problem / task that offers students to be confused and work through that confusion to deepen understanding.  Let’s continue sharing so that we know what these ideals look like for ourselves, so we can experience them with our own students!

 

 

 

 

 

 

 

 

How to Answer Multiple Choice Questions

It’s that time of year when many start to panic about the inevitable tests that will be given to students all over. And with this panic comes many test-taking strategies that will be told to countless students.  I thought I’d share a few of the “secrets” many students are told about how to ace these tests, specifically how to answer any multiple choice sections:

Tip #1 – Cover up the 4 answers before you read the question.

Many teachers give advice similar to this asking students to cover up their answers with their hand or with a sticky note…  Take a look:

sticky note

As you can see above, the student covered over the potential answers to help them think through the problem before they start looking for answer.  I believe many might suggest this approach when they notice students guessing, or not taking the time and thought necessary to solve a problem.  However, I’m not sure this strategy is appropriate for all students, nor will it even work for many questions.  How would covering over the answers help here:

Cover with Sticky notes2

Take a look at each of the 6 questions.  Which ones would be helpful if your students covered up the answers?  Which ones would be impossible?  Is this the best strategy for all of your students?  Would you use this strategy yourself all the time?  Personally, I don’t think I’d ever use this strategy!

I wonder what would happen if we told a group of students to do this every time and they encountered questions like #1 or #2 above?  We would be setting them up for failure.  Hopefully you are seeing this method might not be possible for every question, nor will all students benefit from using it at all!


Tip #2 – Highlight keywords in the question so you know what is being asked of you to do.

Many teachers might ask their students to highlight pieces of a problem or question in order to help them focus their attention on information that might easily be missed.

highlighting1

For the above question, it is possible that some students might miss some of the final words and even though they understand the question, might get their answer wrong.  Obviously this isn’t ideal!  However, in my experience, many students, even when using a highlighter, miss out on all kinds of information.  In the following question, more than half of the students in one classroom answered “d”, even though all of the students were expected to highlight important information.  Did so many in this classroom get this question wrong because they were highlighting?highlighting2

In fact, many of the students in this class highlighted nearly every word of every question.  And some of the students who didn’t highlight as many questions as were expected – actually did the best on the test.  I’m not suggesting that highlighting is bad, just that it likely didn’t help any/many of the students in this class, and actually got some students to miss out on information and get the wrong answers.


Tip #3 – Eliminate the obviously wrong answer first.

Again, many teachers give advice like this because they know it works for them.  Take the following question as an example.  Would you first eliminate the wrong answers?  

Elimintating

Or would you place each of the 4 numbers in the box to see what the answer might be?  Or possibly work out the question without notice to the options, then see if your answer is there?  

When I watch students think, I typically don’t see students attempting to find the wrong responses. Would it be a feasible strategy for any/some/all of these questions:

eliminating nope.JPG


Some advice.

If you haven’t already read it, please read my post Quick Fixes and Silver Bullets…  There I discuss some of the many unproductive beliefs and strategies many schools employ as they attempt to improve testing score, followed by more productive suggestions. 

Thinking specifically again about multiple choice questions, there are many different tips we can give students to solve a multiple choice question, but because every question is unique, and every student will have their own thinking and strategies, we might be putting too much emphasis on trying to find the quick fixes and easy answers.  Instead of teaching these strategies, I wonder why we don’t just provide students with rich tasks / problems, then encourage more discourse?

If we want our students to do well in our classrooms, we need to make sure we are focusing our attention on providing rich learning opportunities, facilitating meaningful discussions, and consolidating the learning effectively!  However, in some classrooms I wonder how much valuable class time is spent preparing for high stakes testing by “practicing” questions that mimic ones found on the test?  I wonder how much time is spent seeing IF you understand something instead of time spent on actually learning the curriculum standards the way they were intended to be learned?

The problem here is that many of these test questions are evidence OF learning, but they are often not the type of experiences needed TO learn the material!

I think Linda Gojak said it nicely in her NCTM president’s message entitled Are We Obsessed with Assessment?:

Linda Gojak assessment


Final thoughts.

Personally, I’m not a fan of multiple choice questions (see the late Joe Bower’s post for details). In some classrooms there is far too much emphasis on getting right answers on simple questions and far too little emphasis on development of deep understanding of the mathematics!  Too many attempt to raise scores by replicating the form and format of the test instead of focusing on the mathematics itself.  Yet research shows us the more multiple choice tests we give, the worse our students actually perform on standardized tests!


If you have to help your students prepare for these kinds of tests though, please make sure that you remaine focused on the mathematics itself, and not expect all of your students to use YOUR strategies. That’s how you kill your students’ relationship with mathematics!  We would never tell our students to pick C for every answer, that will only work 1/4 of the time.  In the same way, many of the strategies we provide for our students will not work for all students and not for all questions… and rarely will these strategies actually help them reach our actual goals for our students:

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(unless your goals are misguided – like trying to get a certain percentage of students to pass the test – in which case I’m not sure how helpful I’ve been here!).

Professional Development: What should it look like?

A few weeks ago Michael Fenton asked on his blog this question:

Suppose a teacher gets to divide 100% between two categories: teaching ability and content knowledge. What’s the ideal breakdown?

The question sparked many different answers that showed a very wide range of thinking, from 85%/15% favouring content, to 100% favouring pedagogy.  In general though, it seems that more leaned toward the pedagogy side than the content side.  While each response articulated some of their own experiences or beliefs, I wonder if we are aware of just how much content and pedagogical knowledge we come into this conversation with, and whether or not we have really thought about what each entails?  Let’s consider for a moment what these two things are:


What is Content Knowledge?

To many, the idea of content knowledge is simple.  It involves understanding the concept or skill yourself.  However, I don’t believe it is that simple!  Liping Ma has attempted to define what content knowledge is in her book:  Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China & the United States.  In her book she describes teachers that have a Profound Understanding of Fundamental Mathematics (or PUFM) and describes those teachers as having the following characteristics:

Connectedness – the teacher feels that it is necessary to emphasize and make explicit connections among concepts and procedures that students are learning. To prevent a fragmented experience of isolated topics in mathematics, these teachers seek to present a “unified body of knowledge” (Ma, 1999, p.122).

Multiple Perspectives – PUFM teachers will stress the idea that multiple solutions are possible, but also stress the advantages and disadvantages of using certain methods in certain situations. The aim is to give the students a flexible understanding of the content.

Basic Ideas – PUFM teachers stress basic ideas and attitudes in mathematics. For example, these include the idea of an equation, and the attitude that single examples cannot be used as proof.

Longitudinal Coherence – PUFM teachers are fundamentally aware of the entire elementary curriculum (and not just the grades that they are teaching or have taught). These teachers know where their students are coming from and where they are headed in the mathematics curriculum. Thus, they will take opportunities to review what they feel are “key pieces” in knowledge packages, or lay appropriate foundation for something that will be learned in the future.

Taken from this Yorku wiki

As you can see, having content knowledge means far more than making sure that you understand the concept yourself.  To have rich content knowledge means that you have a deep understanding of the content.  It means that you understand which models and visuals might help our students bridge their understanding of concepts and procedures, and when we should use them.  Content knowledge involves a strong developmental and interconnected grasp of the content, and beliefs about what is important for students to understand.  Those that have strong content knowledge, have what is known as a “relational understanding” of the material and strong beliefs about making sure their students are developing a relational understanding as well!


What is Pedagogical Knowledge?

Pedagogical knowledge, on the other hand, involves the countless intentional decisions we make when we are in the act of teaching.  While many often point out essential components like classroom management, positive interactions with students… let’s consider for a moment the specific to mathematics pedagogical knowledge that is helpful.  Mathematical pedagogical knowledge includes:


Which is More Important:  Pedagogy or Content Knowledge?

Tracy Zager during last year’s Twitter Math Camp (Watch her TMC16 keynote address here) showed us Ben Orlin’s graphic explaining his interpretation of the importance of Pedagogy and Content Knowledge.  Take a look:

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In Ben’s post, he explains that teachers who teach younger students need very little content knowledge, and a strong understanding of pedagogy.  However, in the video, Tracy explains quite articulately how this is not at all the case (if you didn’t watch her video linked above, stop reading and watch her speak – If pressed for time, make sure you watch minutes 9-14).

In the end, we need to recognize just how important both pedagogy and content are no matter what grade we teach.  Kindergarten – grade 2 teachers need to continually deepen their content knowledge too!  Concepts that are easy for us to do like counting, simple operations, composing/decomposing shapes, equality… are not necessarily easy for us to teach.  That is, just because we can do them, doesn’t mean that we have a Profound Understanding of Foundational Mathematics.  Developing our Content Knowledge requires us to understand the concepts in a variety of ways, to deepen our understanding of how those concepts develop over time so we don’t leave experiential gaps with our students!  

However, I’m not quite sure that the original question from Michael or that presented by Ben accurately explain just how complicated it is to explain what we need to know and be able to do as math teachers.  Debra Ball explains this better than anyone I can think of.  Take a look:

Figure-1-Mathematical-Knowledge-for-Teaching-Ball-Thames-and-Phelps-2008
Research Gate: Figure 1.  Mathematical Knowledge for Teaching (Ball et., al, 2008)

Above is a visual that explains the various types of knowledge, according to Deb Ball, that teachers need in order to effectively teach mathematics.  Think about how your own knowledge fits in above for a minute.  Which sections would you say you are stronger in?  Which ones would you like to continue to develop?

Hopefully you are starting to see just how interrelated Content and Pedagogy are, and how ALL teachers need to be constantly improving in each of the areas above if we are to provide better learning opportunities for our students!


 What should Professional Development Look Like?

The purpose of this article is actually about professional development, but I felt it necessary to start by providing the necessary groundwork before tackling a difficult topic like professional development.

Considering just how important and interrelated Content Knowledge and Pedagogy are for teaching, I wonder what the balance is in the professional development currently in your area?  What would you like it to look like?  What would you like to learn?

While the theme here is about effective PD, I’m actually not sure I can answer my own question.  First of all, I don’t think PD should always look the same way for all teachers, in all districts, for all concepts.  And secondly, I’m not sure that even if we found the best form of PD it would be enough because we will constantly need new/different experiences to grow and learn.  I would like to offer, however, some of my current thoughts on PD and how we learn.


Some personal beliefs:

  • We don’t know what we don’t know.  That is, given any simple concept or skill, there is likely an abundance of research, best practices, connections to other concepts/skills, visual approaches… that you are unaware of.  Professional development can help us learn about what we weren’t even aware we didn’t know about.
  • Districts and schools tend to focus on pedagogy far more than content.  Topics like grading, how to assess, differentiated instruction… tend to be hot topics and are easier to monitor growth than the importance of helping our students develop a relational understanding.  However, each of those pedagogical decisions relies on our understanding of the content (i.e., if we are trying to get better at assessment, we need to have a better understanding of developmental trajectories/continuums so we know better what to look for).  The best way to understand any pedagogy is to learn about the content in new ways, then consider the pedagogy involved.
  • Quality resources are essential, but handing out a resource is not the same as professional development.  Telling others to use a resource is not the same as professional development, no matter how rich the resource is!  Using a resource as a platform to learn things is better than explaining how to use a resource.  The goal needs to be our learning and hoping that we will act on our learning, not us doing things and hoping we will learn from it.
  • The knowledge might not be in the room.  An old adage tells us that when we are confronted with a problem, that the knowledge is in the room.  However, I am not sure this is always the case.  If we are to continue to learn, we need experts helping us to learn!  Otherwise we will continually recycle old ideas and never learn anything new as a school/district.  If we want professional development, we need new ideas.  This can come in the form of an expert or more likely in the form of a resource that can help us consider content & pedagogy.
  • Learning complicated things can’t be transmitted.  Having someone tell you about something is very different than experiencing it yourself.  Learning happens best when WE are challenged to think of things in ways we hadn’t before.  Professional development needs to be experiential for it to be effective!
  • Experiencing learning in a new way is not enough.  Seeing a demonstration of a great teaching strategy, or being asked to do mental math and visually represent it, or experiencing manipulatives in ways you hadn’t previously experienced is a great start to learning, but it isn’t enough.  Once we have experienced something, we need to discuss it, think about the pedagogy and content involved, and come to a shared understanding of what was just learned.
  • Professional learning can happen in a lot of different places and look like a lot of different things.  While many might assume that professional development is something that we sign up for in a building far from your own school, hopefully we can recognize that we can learn from others in a variety of contexts.  This happens when we show another teacher in our school what we are doing or what our students did on an assignment that we hadn’t expected.  It happens when we disagree on twitter or see something we would never have considered before.  We are engaging in professional development when we read a professional book or blog post that helps us to consider new things and especially when those things make us reconsider previous thoughts.  And professional learning especially happens when we get to watch others’ teach and then discuss what we noticed in both the teaching and learning of the material.
  • Some of the best professional development happens when we are inquiring about teaching / learning together with others, and trying things out together!  When we learn WITH others, especially when one of the others is an expert, then take the time to debrief afterward, the learning can be transformational.
  • Beliefs about how students learn mathematics best is true for adults too.  This includes beliefs about constructivist approaches, learning through problem solving, use of visual/spatial reasoning, importance of consolidating the learning, role of discourse…
  • Not everyone gets the same things out of the same experiences.  Some people are reflecting much more than others during any professional learning experience.  Personally, I always try to make connections, think about the leaders’ motivations/beliefs, reflect on my own practices… even if the topic is something I feel like I already understand.  There is always room for learning when we make room for learning!

Some things to reflect on:

As always, I like to ask a few questions to help us reflect:

  • Think of a time you came to make a change in your beliefs about what is important in teaching mathematics.  What led to that change?
  • Think of a time you tried something new.  What helped you get started?
  • Where do you get your professional learning?  Is your board / school providing the kind of learning you want/need?  If so, how do you take advantage of this more often?  If not, how could this become a reality?
  • Think about your answers to any of the above questions.  Were you considering learning about pedagogy or content knowledge?  What does this say about your personal beliefs about professional development?
  • Take a look again at any of the points I made under “Some Personal Beliefs”.  Is there one you have issue with?  I’d love some push-back or questions… that’s how we learn:)

As always, I encourage you to leave a message here or on Twitter (@markchubb3)!

Estimating – Making sense of things

I remember as a student being asked to estimate in mathematics class on a few occasions.  It was either an afterthought from my teachers telling the group to estimate before we do our work to help us with the “reasonableness” of our answers, or during measurement activities where we had to estimate then measure items around the classroom.  As a student though, when asked to actually estimate, I always did the calculations or measuring first, then wrote down a number that was near the actual answer as my “estimate”.  Was I confusing rounding with estimating?  Or was I avoiding thinking???


It might come as a surprise to consider just how much estimating we do outside of school.  Within the same day we might determine how much milk to pour on our cereal so it is covered but won’t get soggy, think about how early we need to leave for work to make sure we aren’t late, figure out if we can safely squeeze our car into a parking space, consider how loud to speak to someone across the room, determine an appropriate amount to tip the waitress at dinner, think about if there is enough time during the commercial break to use the restroom so you don’t miss any of your favourite show…  Whether we know it or not, nearly every minute of the day, we are estimating about physical spaces and numbers.

In school, however, the practice of estimating is often neglected.  Many of our students are estimating all the time without realizing it, but others might not be aware of the mental actions others are doing and don’t engage in the same active thinking processes!  Because of this, I believe we should be estimating more than we probably realize.  The skill of estimation is directly related to our Number Sense and our Spatial Reasoning, so we need to make estimating a priority!


Kinds of estimating:

Situations in which we estimate involve: computational estimations, measurement estimations, numerosity estimates (how many) and number line estimates.  Computational and numerosity estimations are directly related to students’ Number Sense (i.e., size of numbers, doubling, how much more or less…) and often involve students approximating numbers.  While estimates involving measurement and number lines involve our students’ Spatial Reasoning (i.e., considering the size and space of objects).  However, if we really delve deeply into any of the 4 kinds of estimating, they probably each deal with our Number Sense and each deal with our ability to think Spatially.


Questions that ask students to estimate distance or length:

Take a look at the following 5 questions.  Which type of question do you think is most common in school?  Which type of question is less common?

Measuring-paths

For many students, estimating a measurement is about “guessing”, then actually measuring.  To a student, the act of estimating becomes useless in this scenario.  If they are going to measure anyway, why did they estimate anything?  Many of the questions above ask students to go beyond guessing and ask them to develop benchmarks then think about subdividing or iterating those benchmarks.  While some of our students will naturally develop these benchmarks and strategies to subdivide/iterate, many others will not without rich tasks and discussions.  For these reasons, we likely need to spend more time than we might realize estimating and discussing our strategies / thinking.


Questions that ask students to estimate area:

Take a look at the 5 questions below.  Which ones might help your students better understand the concept of area?  Which ones might help them consider the attributes of area you want them to notice?

To many students, concepts like area are easy… plug in the numbers to a formula and you get your answer.  Estimating, however, requires far more thinking and understanding of the attributes than simple calculations.  For this reason, it is probably best if we start by asking our students to estimate well before they are ever given any formulas, and continually as they learn more complicated shapes.

Each of the above problems ask our students to actually consider the size and shape of the things they are thinking about.  Hopefully this happens ALL the time as our students learn measurement concepts!


Questions that ask students to estimate angles:

As a student I remember learning types of angles and how to read a protractor – very knowledge based and procedural in nature.  However, as a teacher I regularly see students who can easily tell me if an angle is greater or less than 90 degrees, but make seemingly careless mistakes when actually measuring an angle.  Personally, I believe that the issue isn’t about students being careless, it is more about a student’s experiences with angles.  Specifically, too many students are asked to DO something instead of them being asked to THINK about something when it comes to topics like angles.  Above are 5 problems / tasks that ask students to think first by estimating.  When the task is about estimating, it adds motivation for students to actually measure to see how close they might be!


Questions that ask students to estimate number or computations:

Look at the 6 problems/tasks below.  Which ones do you think are more common in classrooms?  Why do you think they are more common?  Which ones require your students to consider the space numbers take up?  Which ones help our students develop and use their number sense?

When the problems / tasks we give are about estimating, our students think about what they already know and use this as a basis for learning new things.  While the aim for many of us is to help our students determine reasonableness for our students’ answers, in real life, estimates are likely good enough most of the time!


We need to estimate more!

Estimating needs to be integrated into more of what we teach, instead of it being an isolated lesson/ concept.  Whether teaching probability or time or Geometry or patterns… we need to ask our students to think more before they start doing any calculation!


Special Thanks

Hopefully you have heard of Andrew Stadel’s Estimation 180.  This is an easy to use routine that can help us with some of the types of estimation I have talked about here!

Estimating slides-24

One of the best parts about these routines is that it helps students build benchmarks, use number sense, think spatially and consider the importance of a range of reasonable answers instead of just “guessing”.

Jamie Duncan added to the conversation when she shared snapshots of how she helped her students refine their ranges:

Estimating slides-25

After noticing her students’ ranges were quite large, she started asking her students to indicate their “brave” too low and “brave” too high estimates.  Ideally we want our students focused on their range of values, not their actual estimate.  Focusing on the actual estimate promotes guessing, while focusing on student ranges helps us think more about reasonableness.  Brilliant idea!

And of course, for me, two of the most powerful images on the topic have been shared by Tracy Zager (If you haven’t purchased her book: Becoming the Math Teacher You Wish You’d Had you need to!).  The image on the left shows the process we go through when solving a problem.  So much of what we want our students to do involves them making sense of things and considering their initial thoughts.  The image on the right to me is even more powerful though.  It shows just how important our intuition is, and how building our students’ intuition is key for them to build their logic!  These two go hand-in-hand!

Estimating slides-26


A few things to reflect on:

  • What does estimation look like in your class?  Is this a routine you do?
  • Many of the ideas shared above might be more specific to the content you are learning.  How do you help your students see the importance of estimation when you are learning new topics?
  • When your class is estimating, how do you promote the range?  Are some of your students still “guessing”?  If so, how can you use the ideas of others in the room to help?  How else can we improve here?
  • Many of the ideas I shared above involved estimating without giving a number.  These tasks often directly involve helping our students use their Spatial Reasoning.  How are you helping your students develop their Spatial Reasoning?
  • The students in our classrooms that are estimating all the time (even without them realizing it) do well in mathematics.  Those that struggle often aren’t using their intuition.  Why do some use their intuition more than others?  What do WE have to do to help everyone use their intuition more often?
  • Did you notice any relationships between the coloured images above?  All the yellow ones involve… all the red ones are…  I wonder which ones you gravitated toward?

As always, I encourage you to leave a message here or on Twitter (@markchubb3)!

Skyscraper Templates

A while ago I was introduced to Skyscraper Puzzles (I believe they were invented by BrainBashers).  I’ll explain below about the specifics of how to play, but basically they are a great way to help our students think about perspective while thinking strategically through each puzzle.  Plus, since they require us to consider a variety of vantage points of a small city block, the puzzles can be used to help our students develop their Spatial Reasoning!

I’ve written before about how to help your students persevere more in math class and I still think that one of the best ways to do this involves physically and visually thinking about tasks that involve Spatial Reasoning.

While I loved the idea of doing these puzzles the first time I saw them, I was less enthusiastic about having these puzzles as a paper-and-pencil or computer generated activity because it is difficult to help develop perspective without actually building the skyscrapers.  So, I created several templates that can easily be printed, where standard link-cubes can be placed on the grid structures.

Below are the instructions for playing and templates you are welcome to use.  Enjoy!


How to play a 4 by 4 Skyscraper Puzzle:

  • Build towers in each of the squares provided sized 1 through 4 tall
  • Each row has skyscrapers of different heights (1 through 4), no duplicate sizes
  • Each column has skyscrapers of different heights (1 through 4), no duplicate sizes
  • The rules on the outside (in grey) tell you how many skyscrapers you can see from that direction
  • Taller skyscrapers block your view of shorter ones

Below is an overhead shot of a completed 4 by 4 city block.  To help illustrate the different sizes, I’ve coloured each size of skyscraper a different colour.  Notice that each row has exactly 1 of each size, and that each column has one of each size as well.

skyscrapers 1
Top View

Below is the front view.  You might notice that many of the skyscrapers are not visible from this vantage point.  For instance, the left column has only 3 skyscrapers visible.  We can see two in the second column, one in the third column, and two in the far right column.

skyscrapers 5
Front View

Below is the view of the same city block if we looked at it from the left side. From left to right we can see 4, 1, 2, 2 skyscrapers.

skyscrapers 2
Left View

Below is the view from the back of the block.  From left to right you can see 1, 2, 3, 2 skyscrapers.

skyscrapers 3
Back View

Below is the view from the right side of the block.  From here we can see 2, 2, 4, 1 skyscrapers (taken from left to right).

skyscrapers 4
Right View

When playing a beginner board you will be given the information around the outside of your city block.  Each number represents the number of skyscrapers you could see if you were to look from that vantage point.  For example, the one on the front view (at the bottom) would indicate that you could only see 1 skyscraper and so on…  The white squares in the middle of the block have been sized so you can actually make the skyscrapers with standard link cubes.

blank 4 by 4


Templates for you to Download:

Beginner 4 by 4 Puzzles

Advanced 4 by 4 Puzzles

Beginner 5 by 5 Puzzles

Advanced 5 by 5 Puzzles


A few thoughts about how you might use these:

A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity.  Hopefully, this post has helped you consider your own decision making processes!


I’d love to hear how you and/or your students do!

The Same… or Different?

For many math teachers, the single most difficult issue they face on a daily basis is how to meet the needs of so many students that vary greatly in terms of what they currently know, what they can do, their motivation, their personalities…  While there are many strategies to help here, most of the strategies used seem to lean in one of two directions:

  1. Build knowledge together as a group; or
  2. Provide individualized instruction based on where students currently are

Let’s take a closer look at each of these beliefs:

Those that believe the answer is providing all students with same tasks and experiences often do so because of their focus on their curriculum standards.  They believe the teacher’s role is to provide their students with tasks and experiences that will help all of their students learn the material.  There are a few potential issues with this approach though (i.e., what to do with students who are struggling, timing the lesson when some students might take much more time than others…).

On the other hand, others believe that the best answer is individualized instruction.  They believe that students are in different places in their understand and because of this, the teacher’s role is to continually evaluate students and provide them with opportunities to learn that are “just right” based on those evaluations.  It is quite possible that students in these classrooms are doing very different tasks or possibly the same piece of learning, but completely different versions depending on each student’s ability.  There are a few difficulties with this approach though (i.e., making sure all students are doing the right tasks, constantly figuring out various tasks each day, the teacher dividing their time between various different groups…).

 


There are two seemingly opposite educational ideals that some might see as competing when we consider the two approaches above:  Differentiated Instruction and Complexity Science.  However, I’m actually not sure they are that different at all!

 

For instance, the term “differentiated instruction” in relation to mathematics can look like different things in different rooms.  Rooms that are more traditional or “teacher centered” (let’s call it a “Skills Approach” to teaching) will likely sort students by ability and give different things to different students.

teaching approaches touched up.png
From Prime Leadership kit
If the focus is on mastery of basic skills, and memorization of facts/procedures… it only makes sense to do Differentiated Instruction this way.  DI becomes more like “modifications” in these classrooms (giving different students different work).  The problem is, that everyone in the class not on an IEP needs to be doing the current grade’s curriculum.  Really though, this isn’t differentiated instruction at all… it is “individualized instruction”.  Take a look again at the Monograph: Differentiating Mathematics Instruction.

Differentiated instruction is different than this.  Instead of US giving different things to different students, a student-centered way of making this make sense is to provide our students with tasks that will allow ALL of our students have success.  By understanding Trajectories/Continuum/Landscapes of learning (See Cathy Fosnot for a fractions Landscape), and by providing OPEN problems and Parallel Tasks, we can move to a more conceptual/Constructivist model of learning!

Think about Writing for a moment.  We are really good at providing Differentiated Instruction in Writing.  We start by giving a prompt that allows everyone to be interested in the topic, students then write, we then provide feedback, and students continue to improve!  This is how math class can be when we start with problems and investigations that allow students to construct their own understanding with others!


The other theory at play here is Complexity Science.  This theory suggest that the best way for us to manage the needs of individual students is to focus on the learning of the class.

Complexity1.jpg
What Complexity Science Tells us about Teaching and Learning

The whole article is linked here if you are interested.  But basically, it outlines a few principles to help us see how being less prescriptive in our teaching, and being more purposeful in our awareness of the learning that is actually happening in our classrooms  will help us improve the learning in our classrooms.  Complexity Science tells us to think about how to build SHARED UNDERSTANDING as a group through SHARED EXPERIENCES.  Ideally we should start any new concept with problem solving opportunities so we can have the entire group learn WITH and FROM each other.  Then we should continue to provide more experiences for the group that will build on these experiences.


Helping all of the students in a mixed ability classroom thrive isn’t about students having choice to do DIFFERENT THINGS all of the time, nor is it about US choosing the learning for them… it should often be about students all doing the SAME THING in DIFFERENT WAYS.  When we share our differences, we learn FROM and WITH each other.  Learning in the math classroom should be about providing rich learning experiences, where the students are doing the thinking/problem solving.  Of course there are opportunities for students to consolidate and practice their learning independently, but that isn’t where we start.  We need to start with the ideas from our students.  We need to have SHARED EXPERIENCES (rich problems) for us to all learn from.


 

As always, I leave you with a few questions for you to consider:

  • How do you make sure all of your students are learning?
  • Who makes the decisions about the difficulty or complexity of the work students are doing?
  • Are your students learning from each other?  How can you capitalize on various students’ strengths and ideas so your students can learn WITH and FROM each other?
  • How can we continue to help our students make choices about what they learn and how they demonstrate their understanding?
  • Do you see the relationship between Differentiated Instruction and the development of mathematical reasoning / creative thinking?  How can we help our students see mathematics as a subject where reasoning is the primary goal?
  • How can we foster playful experience for our students to learn important mathematics and effectively help all of our students develop at the same time?
  • What is the same for your students?  What’s different?

I’d love to continue the conversation.  Write a response, or send me a message on Twitter (@markchubb3).

Who Makes the Biggest Difference?

A few years ago I had the opportunity to listen to Damian Cooper (expert on assessment and evaluation here in Ontario) give a talk at OAME.  He shared an analogy with us that I found particularly interesting.  He talked about the Olympic athletes that had just competed in Sochi, specifically ice skaters.

He asked us, who we thought made the biggest difference in the skaters’ careers:  The scoring judges or their coaches they had throughout their careers?

Think about this for a second, an ice skater trying to become the best at their sport has many influences in their life.  But who makes the biggest difference?  Who helps them become the top athletes in their field?  The scoring judges along the way, or their coaches?  Or is it a mix of both???Damian gave us some time to think and then told us something like this (I’m paraphrasing here):


The scoring judge tells the skater how well they did, however, the skater already knows if they did well or not.  The scoring judge just CONFIRMS if they did well or not.  In fact, many skaters might be turned off of skating because of low scores!  The scoring judge is about COMPETITION.  Being accurate about the right score is their goal.

On the other hand, the coach’s role is only to help their skater improve.  They watch, give feedback, ask them to repeat necessary steps… The coach knows exactly what you are good at, and where you need help. They know what to say when you do well, and how to get you to pick yourself up when you fall. Their goal is for you to become the very best you can be!  They want you to succeed!


Thinking about this analogy, I can’t help but wonder how we might reflect on our own practices.  In the everyday busyness of teaching, I think we often confuse the terms “assessment” with “evaluation”.   Evaluating is about marking, levelling, grading.  Any time we put a check on a question, a mark on the top of a page, or a grade or level on an assignment we are evaluating our students’ work.  

On the other hand, assessing is actually something quite different.  The term “assess” comes from the Latin “assidere” which translates as “to sit beside”.  Assessment is kind of like learning about our students’ thinking processes, seeing how deeply they understand something.  It is a process where we observe our students in action to understand their thought process, consider what a student was thinking when looking at an assignment, listen carefully to students’ reasoning as they share with us and others.  Assessment, while related to evaluation, are very different processes!

assidere


I have shared this analogy with a lot of teachers in a number of settings because I think it is helpful for us to consider our role in the classroom.  Are we handing out assignments and marking them (like a scoring judge), or are we providing opportunities for our students to learn, then observing them and really considering what we need to do next?  While most agree with the premise of the coach having a bigger impact, many of us recognize that our job requires us to be the scoring judges too.  While I understand the reality of our roles and responsibilities as teachers, I believe that if we want to make a difference, we need to be focusing on the right things.  Take a look at Marian Small’s explanation of this below.  I wonder if the focus in our schools is on the “big” stuff, or the “little” stuff?  Take a look:

Marian Small – It’s About Learning

marian-small-its-about-learning


Thinking again to Damian’s analogy of the ice skaters, I can’t help but think about one issue that wasn’t discussed.  We talked about what made the best skaters, even better, but I often spend much of my thoughts with those who struggle.  Most of our classrooms have a mix of students who are motivated to do well, and those who either don’t believe they can be successful, or aren’t as motivated to achieve.

If we focus our attention on scoring, rating, judging – basically providing tasks and then marking them – I believe we will likely be sending our struggling students messages that math isn’t for them.  On the other hand, if we focus on providing experiences where our students can learn, and we can observe them as they learn, then use our assessments to provide feedback or know which experiences we need to do next, we will send messages to our students that we can all improve.


Hopefully this sounds a lot like the Growth Mindset messages you have been hearing about!

Jo Boaler – Assessments for Learning Encourage a Growth Mindset

jo-boaler-assessment-for-learning

Take a quick look at the video above where Jo Boaler shows us the results of a study comparing marks vs feedback vs marks & feedback.  


As usual, I leave you with a few questions to reflect on:

  • Are you assessing or evaluating most of the time?  Can you see the difference?
  • Do your students see you as their coach or as a scoring judge?  
  • How do you provide your students with the feedback they need to learn and grow?
  • How do you provide opportunities for your students to try things, to explore, make sense of things in an environment that is about learning, not performing?
  • What does it mean for you to provide feedback?  Is it only written?
  • How can we capitalize on using various students’ thinking to help each other improve?
  • How do you use these learning opportunities to provide feedback on your own teaching?

Let’s continue the conversation here or on Twitter!

Quick fixes and silver bullets…

I find myself reflecting on what I believe is best for my students and best for my students’ beliefs about what mathematics is often.  When I get the opportunity to take a look at my students’ work and time to determine next steps, I can’t help but reflect on how my beliefs inform what next steps I would take.   However, I wonder, given the same students and the same results, if we would all give the same next steps?  Let’s take a look at a few common beliefs about what our students need to be successful and discuss each.


My kids need to know their facts:

Often we see students who make careless mistakes and wonder why they could have gone wrong with something so simple. To some, the belief here is that if we could just memorize more facts, they would be able to transfer those facts to the problems in the assessment. While I agree that we want our students being comfortable with the numbers they are working with, I’m not convinced that memorizing is the answer here. Our Provincial test includes a few computation questions (for grade 3 only) and none of these are timed. Most of our questions involve students making sense of things (across 5 strands), some with contexts and some without.

Instead of spending more time worrying about memorizing facts, I wonder if other strategies have been thought of to help our students as well?  For example, other than the 4 questions in grade 3, all other questions allow students to use manipulatives or calculators, and all questions have space for students to write in the margins any rough work or visual models might be using.  The question below is one of the few computational questions a grade 3 student is expected to do.  Many who might use the traditional algorithm might accidently pick 41.  How might a number line help our students visualize the space between the two numbers here?

c5x1eyewyaaqx45


My kid aren’t reading the questions:

Often we notice that our students understand a concept, but the question itself requires several steps and students don’t end up answering what is asked. For many the solution is having students do some kind of strategy (whether they need to or not) like highlighting key words.

I wonder what answer students might get to the above question?  Will they get the right answer here?  What would you have liked them to do instead?

Highlighting specific words or filling out a standard graphic organizer isn’t the answer for all kids, nor for all questions!

Personally, I think the issue isn’t that our students can’t read the questions, it is that they are jumping to a solution strategy too quickly.  Instead of believing the solution is to have our students highlight or fill out graphic organizers, what might be more appropriate is to help our students slow down and think deeper about the questions they are being asked.  I wonder, however, about how our students’ prior experiences might be a big part of why they jump to solution strategies too quickly?  If students typically receive questions that are simple and closed, and typically follow a lesson directly telling us how to answer those questions, then I wonder if the issue is that our students can’t read or if their experiences have actually been counter-productive? If students don’t experience mathematics in ways that help them make sense of a situation, and instead see math as answering a bunch of questions, then it is no wonder why they aren’t reading the whole question!  They have been trained to believe math is about getting answers quickly, and that we get rewarded (less homework, better grades…) when we are fast.

Instead of more time practicing reading and highlighting questions, or filling out graphic organizers, we might want to spend more time building questions together, asking students to pose their own problems, or providing experiences for students to notice and wonder. What if we started by showing this:

1.jpg

What do you notice here?  What do you wonder?

What might our students see?  They might notice things we didn’t realize they were not even aware of (e.g., each row has 7 boxes, some rows are missing numbers, the numbers go in order, there are 2 Ss and 2 Ts at the top, the letters at the top probably mean days of the week…..).

What might our students be curious about?  They might wonder why April 1st is on a Monday and not a Sunday.  Or wonder about the “Chapter 1” part.

Then we could continue to show more of the question and again ask what students notice and wonder.

3.jpg

No matter the grade level or content, my students need to realize that mathematics is about the development of mathematical reasoning, not just quickly jumping to a solution strategy (especially not one that my teacher told me to use all the time).  Taking the time to think deeply about our mathematics is what I want from my students!  Numberless word problems, notice and wonder strategy, contemplate and calculate… any strategy that helps my students slow down, pay attention to visuals, and start to think about the situation more will probably help many of our students given enough opportunities.


They need more practice with questions like these:

Many believe that if students are doing poorly on something, that the best course of action is to continue practicing that thing.  For example, if our students are doing poorly on Provincial testing questions, then giving students more questions like these will answer all of the issues.

To some, the answer to the problem is to give sample questions every week in a package or even more frequently.  While there are times when practice is helpful, if our students are struggling with the content, giving more questions will not be helpful!  Take a look at Daro’s quote:

More often than not, the quick fix solutions like this (noticing our students struggle with something, then providing the same instruction or same types of practice again) will not be successful.  Developing our math knowledge for teaching is probably the most difficult aspect of teaching mathematics, and is definitely NOT a quick fix, but it is probably the answer here!  If our students are struggling with concepts we believe they should be able to do, it is likely that they haven’t had the right experiences to help them learn!  Have we provided experiences for our students to deeply explore a variety of representations?  Have we provided experiences where our students are able to develop reasoning skills?  Have we provided ample opportunities for our students to consolidate their learning?

Remember, questions that are designed to show evidence OF student learning is not necessarily the WAY students learn!  Handing out these questions toward the end of the learning is far more reasonable.


My students need more stamina:

Often when giving young students extended time to sit and focus independently on an assessment task we have many that struggle to remain focussed.  For some, the solution is to help students build stamina through quiet seat work regularly.  While I do agree that we should have our students work independently at times, I’m not sure this is the answer to the stamina problem.

To me, I think the issue has more to do with how our students experience mathematics.  Do they get lots of short closed questions where the right answer is apparent quickly?  Or do they experience rich problems where they reason through and figure out their own way of making the question make sense?  Do they learn math through independent think time and cooperative problem solving, or are they told material then asked to remember all of the steps and terms.  Is their mathematics class structured in a way where students come to rely on themselves (individually or within their group) to make sense of challenging problems, or do they feel the need to access their teacher every time they don’t know what to do?  When our students are working, are we monitoring all of our students’ thinking, or are we spending a lot of time guiding our students’ thinking?

If stamina is the issue, I wonder if we are allowing our students to productively struggle enough?  If we see a bunch of hands raised around the classroom all wanting us to help, this is a huge red flag moment.  Our students are asking US to think for them!  If we find ourselves sitting beside our students helping out a small group all of the time, this might be another red flag.  Our students are learning that they always have access to us right beside them when they learn, but the unintended problem is that we aren’t allowing our students to struggle enough!

Providing our students with a variety of manipulatives to learn and puzzle through their mathematics on a daily basis might be a big step in the direction of allowing our students to gain the confidence and stamina they need to do well every day.  Notice how these students are using manipulatives to help them make sense of their work:

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When our students learn their mathematics using manipulatives and have access to any manipulative at any time to solve new problems, we start to notice that our students come to realize their role is to slow down and make sense of things.  When our students have had various experiences with manipulatives and can see their role as “thinking tools”, we start to notice fewer hands asking for help, less need to have to sit down with a group, and more time for us to really notice our students’ thinking going on in our classroom.  When this starts to happen, we no longer see stamina as a big issue.


I want you to consider for a moment the differences between the beliefs I’ve mentioned. What messages are we sending to our students about what is important in mathematics?  Strategies that get us to do better on the test, or strategies that help us slow down and think more?  Is math about memorizing or figuring things out?  Is math about removing the context to mathematize a situation, or about using the context to make sense of things?

Sure I want my students to do well on any assessment they are given, but I want them to do well every day!  Quick fixes and silver bullets often don’t help our students in the long run though!!!


So I leave you with a few things to reflect on:

  • What are some of the quick fixes you’ve heard about?  Did you try any of them?  Did they work?
  • Is there a strategy that you see working for all of your students?  Was it actually helpful for everyone, or just some?  Do you expect everyone to use this strategy?
  • Have you been asked or told to use specific strategies?  Do you see it being successful for everyone?  Do you have the autonomy to choose here based on the students you have in front of you?  Do your students have any autonomy over the strategies they use?
  • When looking at your student work, are you determining next steps for your students, or for yourself?

It is far easier to determine what your students can and can’t do well, than it is to figure out what to do next.  While we absolutely need to help our students notice the things they could do to improve, we also need to do the hard work of reflecting on our own practices.  Our beliefs about what is important and how we learn mathematics have a direct effect on how our students will do in our classrooms!