A Few Simple Beliefs

“One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world”
TIPS4RM: Developing Mathematical Literacy, 2005

A statements like this is easy to agree with.  Sounds great, doesn’t it?  The ideas in mathematics should connect!  The above quote speaks to what Richard Skemp calls Relational Understanding (an article you need to read!) which I believe is a major goal of learning mathematics.  However, I am not sure we would all agree on HOW we help our students achieve this relational understanding.
Dan Meyer has talked several times over the past few years about this quote from Daniel Willingham:
Willingham quote

I think Willingham is onto something here.  We have all become educators because we want our students to be successful… and we want to do our best to help them do well.  However, we are often so eager to get the results we want, that we don’t take enough time to allow our students to think… to explore… to make sense of the math… to realize WHY we are learning what we are learning.  In our eagerness to have our students get answers, we often miss the developmental pieces that our students need to be successful!

Dan Meyer gave us a nice little list of reminders that might help us allow our students more time to “develop the question” and student thinking here:


For me though, I think all of this can boil down to 3 simple beliefs:

Concept v Procedures

  1. We want our students to understand concepts (the why) first.  Thinking through concepts, making things make sense first helps build procedural knowledge.  Making sure my students have both conceptual understanding and procedural fluency is important to me, but I believe that the concepts need time to develop.  Jumping to procedures too quickly stunts concept development.
  2. We understand things at the visual level first.  All students can enter the conversation when we ask them to use visual clues first.  Starting with symbols, on the other hand (numbers, operation signs, formulas…), might disconnect the mathematics from what makes sense for many of our students.
  3. We want our students to be able to develop reasoning skills!  Spatial reasoning, algebraic reasoning, proportional reasoning… understanding mathematics is all about being able to THINK mathematically.  If we want our students to have the ability to reason, we need to start with tasks that help them develop their reasoning strategies.  “Answer Getting” strategies are helpful when we want 1 specific way to answer 1 specific type of question.  However, if we start here, our students will come to see mathematics as a series of unrelated things to memorize.  More often then not, students who are taught “answer getting” strategies first lack the ability to translate their original strategies to new situations.

The 3 items on the left are about understanding… thinking… making the math make sense… while the 3 on the right focus on being able to do the skills of math.  Personally, I believe that concepts, visuals, reasoning are more important for our students to develop, however, I understand that not everyone would agree with me here.

If you do think that procedures or symbols or “answer getting” strategies are more important, then I think the best way to help ALL of our students to get there, is to focus on concepts, visuals, reasoning.  Slow down and make sure OUR STUDENTS are making sense of things before WE jump into symbols, procedures and strategies.

Let’s take a look at a quick example:


If we are to start learning about the “mean” of a set of data, how do we introduce it?  Instead of starting with a procedure or “answer getting strategy” (add them up and divide by the number of items), we need to think about how our students can make sense of the math using what they already understand.  What visual could we introduce that would help our students use their reasoning skills to develop a conceptual understanding of “mean”?  How about this:


On average, how many books does Maria read per month?  1, 2, 3, 4, 5, 6…?

If our students have never thought about “average” before, what might they do here?  How might their visual / spatial reasoning help them make this make sense?  Hopefully, we might notice that January could be redistributed onto other months.  How might students see this happening?


Maybe we offer students opportunities to problem solve with manipulatives.  Offer everyone Snap Cubes and post this problem:

What would the price be if each of the games cost the same amount?

Van de Walle – Teaching Student Centered Mathematics


Understanding this would help us understand 1 interpretation of how we can conceptualize “mean”.  However, other students might think about mean more like a statistician might by thinking about the “balance point”.  Take a look:

Van de Walle – Teaching Student Centered Mathematics

In the end, we need all of our students to be able to make sense of problems like this one:



How would you solve this problem?  Is there more than 1 way to calculate the mean?  Do I need to calculate the mean first?  How did you determine the test scores for Todd?


If we were to focus on mean conceptually, we would likely have students who understood the procedures in ways that they are ready to use them in different ways…  If we focused on visuals we would likely have students who could mentally reason these numbers on a number line…  If we focused on reasoning, we would likely have students who were ready to adapt because they were used to making sense of things…


I leave you again with Willingham’s quote:

Willingham quote

How do you help your students make connections between visuals and concepts… between various representations and symbols… between and among different but related concepts… between concepts and procedures… between reasoning and answer getting strategies?

When we help orchestrate situations where our students make these connections, we are building mathematical thinkers… we are building mathematicians!

Paying Attention to OUR Understanding!

I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.

In my 2nd year of teaching I was assigned to teach a grade 6/7 split class with 35 students.  Being a new teacher, I taught my students the way I was taught… through direct instruction… and in a very procedural way… with the belief that learning happens by me telling, and students following the rules/steps that I shared.

Now, I could look back at my teaching and cringe with disbelief at my actions, however, I was doing the best I knew how!  Instead of looking back and thinking about how poor of a job I was doing, or even how far I have come, I think it is far more valuable to recognize the moments that helped me grow as an educator.  Here is one such incident:

So, my second year of teaching was well underway when the topic of adding fractions came along.  I knew how to add fractions, and so I taught my students the only way I knew how.  At the end of the unit I gave a test and was surprised at how poorly many of my students did.

I decided to offer those that did poorly a second chance, but assumed that first I needed to offer some help.  I called a student over to my desk and showed him a few of the errors he had made on his test.

His first incorrect answer looked something like this:


Pretty common error right!  In fact, for each of the questions I had given where the denominators were different, this was his strategy.

So, slowly I “re-taught” him the steps of how to add fractions… I asked little mini-questions to walk him through each small step in finding the solution.  It looked like he understood until he said:

 “I get that you are telling me that 3/5 + 2/7 is 31/35, but look, on my test I got 3 of the 5 right on the front and 2 of the 7 right on the back.  Why did you put 5/12 on the top of my test?”

I, of course, wasn’t expecting this, and completely didn’t know what to do… so, I again showed him my method of adding fractions, completely dismissing his question.

While I think the example above illustrates an interesting moment where I recognized that I didn’t know enough about the math to understand how to react, it is equally interesting to point out what I did when confronted with something I didn’t understand.  Take a look at Phil Daro’s quote below:

Daro - Gaps

When we don’t know the math deeply, we jump to “answer getting” strategies… we tell the students procedures to remember… provide them with tricks… we focus on notations… we provide closed questions that are easily marked as right or wrong…..  and our attempts to help students that struggle includes doing the same things over and over again!

This brings us to where we started:

I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.

If we want to get better at being math teachers, we need to learn more about the concepts our students explore!  We need to learn from knowledgeable others about how the concepts develop over time, and the experiences our students need to make sense of the mathematics!

For me, this has included learning from wonderful influential math leaders like Cathy Fosnot, Cathy Bruce, Van de Walle, and Marian Small (check out the links).

When the Ontario Ministry published Paying Attention to Fractions K-12 I finally saw the connection and differences between my understanding of 3/5 + 2/7 = 31/35 and my student’s comment showing 3/5 + 2/7 = 5/12.  Take a look.  Can you see how my student was seeing fractions?

While a blog might not be the easiest place to deepen our content knowledge, it can be a platform to encourage you to consider how we are challenging our understanding of the math your students are learning.

Do you have a knowledgeable other learning with you?  Do you read resources that challenge your current understanding of the concepts you teach?  How are you making connections between different representations, or between concepts?  How are you learning more about what is developmentally appropriate?

I think we owe it to our students to be continually learning!  This learning is often referred to as “Math Knowledge for Teaching” or MK4T.  Take a look:

Math Knowledge for Teaching

In the beginning of this blog I explained a time when I recognized that while I understood the content myself, I did not know what to do when my students struggled with the math.  This is where we need to spend our time learning!  This is what our focus needs to be as professionals!

Is This “Real World”?

I hear a lot about “real world” math.  While I agree that we need to make sure our students are engaged with their math, I wonder if making things “real world” is the answer to the engagement issue?  Take a look at a few examples of problems involving circles:

Problem #1:  Find the diameter/radius, circumference and area of some dessert treats.

Cirlce problem1

Problem #2:  Which has a greater area?


Which of the two preceding problems would be more engaging to you?  For me, the first problem’s context seems quite tacked on.  The author has assumed that the math itself isn’t interesting, so they will make it look more interesting by adding dessert treats.  Yet, in no way does the context help the student understand anything more about circles or challenge them to think in any way.  Most students who would receive a worksheet like this, would ignore the context completely and mindlessly plug in values into their calculator to get the right answers.  Dan Meyer has posted many blog posts about Psuedocontexts like these.

The second problem is quite different though.  It encourages us to think first… to notice pieces of the problem… to reason informally before any calculations are done.  Look again at the picture of the two circular objects.  Which has the greater area?  What does your intuition tell you?  Which one looks like it has a larger area? Problems like this ask us to do much more than calculate, they ask us to start to think mathematically and be engaged in the mathematical processes.

The two problems also differ in what our goals are.  The first set of problems aims to make sure our students have a skill (calculating), while the second problem’s goal is more about mathematical reasoning.  This difference seems to me to be a really important distinction to make!

Let’s take one last problem involving circles and see how engagement and “real world” math fits in:

Activate (Minds On):

Teacher: Can anyone tell me what you notice here.

Various Students: its a circle, looks like tiles on the ground, I noticed the symbol for Pi, the stones are of various sizes and shapes, some parts are normal colors and others are black or white…

Teacher: What do you wonder?  Do any questions come to mind?

Various Students: Why is it there? Where is this from?  What is the ratio between white and black tiles?  Is the ratio 3.14 times more white? How many tiles are there to make the circle?

Teacher: Let’s explore how many tiles there are.  I want you to estimate how many tiles you think there are in your head.  First, I want you to think about the range of possible answers you think might work.  Write down the lowest number of your estimate (there must be at least ____ tiles) and the highest number (there is no way there are more than ____ tiles).  Finally, write down your actual estimate.

Teacher:  Who wants to share the lowest possible number there is (50), anyone think the lowest possible number could be a bit higher (100, 200…).  Who wants to share highest….  (1000, 900, 600).  Many of you think the answer is somewhere between 200 and 600.  Write down your exact estimate and we will check in at the end to see how close you are, and whether the answer fits in our range.

Action (Acquire -Apply)

Teacher: With your partner, I want you to calculate how many tiles you think there are (without counting every tile).  What information would you need?

Various Students: Diameter, radius, circumference, size of each block…

Teacher: Which of these will help you?

Various Students: Radius…Diameter…

Teacher:  With your partner, calculate how many you think there are.  Be able to defend your answer.

Students – working in pairs – all find an answer

Connect/Reflect (consolidate/Debrief)

Students share thinking (Pi x radius squared) – most have same or similar answer (16.5 squared x Pi is about 855)

Teacher – Is this answer reasonable?  Was it within our estimated range? (most estimated way less)

Teacher – We calculated how many there should be, but do you think the real number is less or more than this?  (here is where the real-life piece comes into effect – the math doesn’t give us the actual answer, but helps us to estimate and make sense of a real situation).

Various Students: “I think less because of the grout between the tiles.” “No, there is grout in the diameter too.”  “I think less because the pieces are smaller in the middle.”  “Wouldn’t that make it more then if they are smaller?”  “Some of the black tiles look a bit larger”…  This is where I smile and sit back because the students are driving the conversation!  The whole class is engaged in rich mathematical discussion!

Was this problem a “real world” problem?  Would it engage your students?  More importantly, would it engage your students in the process of thinking mathematically, reasoning mathematically…?

Engagement to me is more than tacking on contexts that we expect our students to ignore to get their answers.  Instead, it is about getting our students to be curious… to wonder… to want to find the answer because they have invested thinking before any calculations were attempted.

Great prompts can lead to great discussion and a much deeper understanding of math.  I find the real-life component is best utilized when we provide real things (strive for curiosity and perplexity and we will no longer hear “when will we ever need this”).  And while making connections to the real world isn’t a bad thing, what we really want is for our students to be making more decisions, thinking more, be in a state of not knowing for a period of time so they actively want to resolve that not knowing.  Engagement is more about this active process of thinking and reasoning than making math about dessert treats!


PS – The students calculated 855 tiles, but there are actually only 837 tiles – the real world isn’t as straight forward as school would like you to believe.  Math is a tool to help us figure it all out.  Start noticing the world through a mathematician’s perspective!

Aligning our Instructional Decisions and Educational Goals

Aligning our Instructional Decisions

& Educational Goals:

Analyzing the decisions of two very different mathematics teaching approaches

I have been thinking a lot about why we teach mathematics in school.  Every teacher wants the best for their students, but have we really thought about what “the best” means?  Do we have an agreed upon set of goals that we are all aiming toward?  Many teachers might believe that having students being able to do the math is our goal, but I think it might be worth taking a few minutes and thinking about the underlying goals of the curriculum, and even broader goals beyond the curriculum.

There seems to be 2 very distinct approaches to teaching mathematics.  I believe the differences are based on a number of factors that might lead teachers to choose one of these paths (past experiences, personal successes…), however, it might be worthwhile for all of us to question our own assumptions about what good math instruction is, and what our underlying goals are.  Below are 2 very different examples of classrooms.

Teacher A Teacher B
What does a typical lesson look like? Traditional teacher-led lesson.

Gradual Release of Responsibility model of teaching:
Lesson starts off with the teacher explaining a concept or skill, students interact with lesson by raising hands, teacher provides “enough” examples for students to understand the thinking strategies and content, students practice what they have been taught, finally the teacher assesses who has learned the material.

3-Part problem solving lesson
Lesson starts off with an activation.  Students are provided with a problem that they need to reason through to explain their thinking.  Students make decisions about what is important, where to start, how to communicate their thinking…  Lesson ends with a discussion about what was learned, different models, strategies… are explored.  Students learn from each other and connections are made between students’ thinking.  The lesson might be followed up with individual practice to consolidate learning.
Role of teacher while students are working Helper.
Students who have listened to the lesson and have learned how to “do” the math are ready to complete the follow-up assignment.  Other students will need more help so the teacher gives more guidance or examples to students (who have hands up, or form a line at the teacher’s desk) until the students can do the math independently.
Observer, Questioner, Assesser…
Teacher observes students working, attending to their thinking, strategies used, models for understanding, and other processes.  The teacher knows when and if they should intervene by posing questions that may illicit deeper thinking.  The teacher uses a variety of formative assessment strategies to know who understands what.  The teacher contemplates how to consolidate the learning after the lesson is complete.
Role of student during working Do the work
Students repeat thinking of the teacher by completing a set of related questions.
Think, plan, reflect, reason, justify, explain…
The students use this time to build their own understanding of the material.
What does a typical unit look like? The teacher ensures all curriculum content is “covered.”  Skills are taught in order.  Unit begins with mostly teacher-led instruction and may gradually move to some problem-solving experiences (now that the teacher believes that they are ready).  New skills are continually being introduced. Skills, concepts and thinking strategies are connected throughout the unit.  Progress in student thinking is what moves the unit along.  Thinking is continually deepened.  Practice and consolidation are an integral part of the formative assessment process.
Role of problems Teaching for problem solving
Lessons are taught by teacher so students will know what/how to think before they begin problem solving.
The teacher wants/expects students to get correct answers because they can evaluate their own teaching based on their students’ successes.
Teaching through problem solving
Problems take on the role of diagnostic assessments, formative assessments throughout the unit, and summative assessments.  Students learn by being challenged (productive struggle), through listening to others’ strategies, models and justifications, and by making connections between others’ ideas and their own.
Role of context Story problems
Real-life scenarios are used to show when the specific skills will be needed later in life.
The context does not actually help the student understand the math.  The teacher might actually show the student how to ignore the context by looking for key words…).
Problem solving
When contexts are used they are used to help students make sense of the mathematics.  The contexts actually lead students to think differently and learn about the mathematics more deeply.
Diagnostic assessments Diagnostic assessments typically track right/wrong responses.  They find out what students can do, and can’t do.
Purpose is for teacher to know what they learned last year, and who is “good” at math or might need extra help.
Diagnostic assessments aim to figure out how well students understand specific concepts.  They find out how students think, which concepts they have mastery over, and look at possible reasons for misconceptions.
Purpose is for teacher to know what to do next.
Formative assessments Students raising hands provide teacher with information about how to pace lessons (even though the majority of students do not raise hands).  Numbers of questions correct on assignments gives teacher feedback about who can “do” the math.    Formative assessments are used regularly to give feedback to the teacher about how to make instructional decisions.
Summative assessments Summative assessments again focus on right/wrong responses.  They find out what students can do, and can’t do. Summative assessments aim to figure out how well students understand specific concepts.  They find out how students think and which concepts they have mastery over.
Role of the teacher in this class Holds all the knowledge.  Makes all the decisions.  Primary source of examples, sharing ideas, providing thinking strategies and models.  Leads all conversations.   Finds the best opportunities for their students to grow and develop as mathematicians.  Asking effective questions.  Facilitates discussions after problem solving.  
Role of the student in this class Listener.  Passive.  Prove to the teacher that they can “do” the math. Thinker, justifier, reason maker, analyzer, communicator, listener…  Active.  Prove to themselves that they can think mathematically.
Demonstration of “Understanding” Questions tend to heavily favor knowledge with some application of math concepts later.  Procedures are checked for accuracy.  Students are assessed based on their products. A balance of open questions, problems and other strategies are used.  Students are assessed on their products, teacher observations, and conversations with the teacher.
Mindset Fixed
Some students are really good at getting the answers, others aren’t.Some students learn math quickly, some have a difficult time.Students rate themselves as good at math or not.

Speed and accuracy are valued.

The teacher might even put students in ability grouping (under the premise of differentiation).

Every student is learning and growing.  Students and teachers view all students as capable and expect high standards.Mindset beliefs of teacher match the messages they send students.
Answer to, “when will we ever need to know this?” This question is asked by a few, but thought regularly by many students because students don’t connect with their work.  The teacher attempts to make the concepts meaningful by adding contexts from .
Answer: “You use math all the time…jobs…at home…”  
Finding the answers to questions using specific skills isn’t the goal.  Thinking mathematically is!  This question is rarely asked because students are more engaged.  Real world contexts are present, but there are also many highly engaging problems used that have no context… they are mathematically relevant.
Resources used Resources promote parrot learning, rote learning, kill-and-drill…

Thinking questions in text books are skipped (“you don’t need to estimate, just give me the answer”).

Resources promote contextual learning that is based on student development, deep conceptual understanding…
Role of technology Technology may be used by the teacher to teach students a concept (i.e., Smartboards, Khan Academy…), or for students to practice skills that have been taught in class (kill and drill websites). Technology may be used by students to interact with their mathematics.  Many programs are designed to meet students’ developmental needs (i.e., Dreambox…).
Role of Manipulatives Teacher shows students how a manipulative can help them get answers.  Teacher demonstrates, students try to show understanding by following the teacher’s thinking.
Goal is for students to use these tools as long as needed, until they master a skill or concept.
Students are encouraged to select and use concrete learning tools to make models of mathematical ideas.  Students learn that making their own models is a powerful means of building understanding and explaining their thinking to others.  
Manipulatives help students see patterns and relationships; make connections between the concrete and the abstract; test, revise and confirm their reasoning; remember how they solved a problem; and communicate their reasoning to others.
Students see manipulatives as meaningful, not just for those who need it.
What is valued in this classroom? Calculating, skills…

Correct answers.
Speed and accuracy.
If students get the right answer, the teacher knows they “understand” the material.


Deepening Understanding, thinking, reasoning…
Students learn to think mathematically at the same time as they are doing the math.
If students can show, explain, justify… their thinking the teacher knows they “understand” the material.  Partially correct understanding is valuable information for the teacher to make instructional decisions.

Who is doing the majority of the thinking in this classroom? The Teacher.
The teacher plans the lesson, the teacher thinks of the best way to explain the concept to their students, the teacher thinks of the context, models and shows students how to do it.  If students help in any of this process, it is typically only a few who are involved in the lesson, and the teacher is the one who approves or disapproves of the ways students should think.  Students then practice their teacher’s thinking.
The Students.
The teacher plans a rich opportunity for students to think through the math.  Students reason, communicate, select manipulatives, models and/or strategies on their own or in a small group.  Students are expected to think their way through the problem before any instruction is given.  Process expectations are stressed.
What does the word Understanding mean? Instrumental understanding:
Teacher wants students to be able to “do” the math.  If a student can follow the procedures required to get an answer, they “understand” their math (with or without knowing why the procedures work).
Relational understanding:

Teacher views understanding not as black-and-white, has-it or doesn’t-have-it.  Instead understanding is constantly deepening.  Understanding is thought of as developmental (i.e., Doug Clement’s Learning Trajectories, Cathy Fosnot’s Landscape of Learning, Mathematics Continuum)

What are the goals in this classroom? Whether the teacher is aware of this or not, their goals are very short-sighted (doing well on the test next week, passing the exam, achieving on high-stakes testing).  In attempting to have their students do well on these assessments, the teacher inadvertently cuts out deep understanding of mathematics.  The teacher may believe that their goal is to provide students with a long list of math skills that they will need in later life and that instrumental understanding is good enough, but the problem is that most students who will learn in this class won’t use these math skills after they take their last math class – all that hard work is lost. The goals for this teacher are to develop life-long mathematicians who have the knowledge, thinking skills, confidence and perseverance to solve problems in their current and future lives.

Students begin to see math around them because they can connect with it.  They question things and make sense of their world because they are thinking mathematically.

Developing relational understanding will help students continue to think mathematically well beyond the day they take their last math class.  


The example of Teacher A shows a traditional path to teaching mathematics.  Many students will learn math in this classroom, but not all.  High-stakes test scores (provincial testing) might show that many students are on track, but as we know, many of these students will start to dislike mathematics as it gets “harder” (i.e., highschool or sooner).  When students do not have a relational understanding of the mathematics they have learned, they will find mathematics increasingly more complicated and disconnected from them.  Many of the students in classroom A, even those who did well that year, will lose their learning because the learning was procedural and was not truly understood.  

Students in Teacher B’s classroom will be able to meet the same standards as classroom A on high-stakes test scores, but will be able to retain their learning later into highschool and beyond.  Teacher B has taught their students to think mathematically, and that all students can learn math at the highest levels.

At the beginning of this article I suggested that we all question our assumptions about what good mathematics instruction is, and what our goals are for teaching mathematics.  I hope that you have taken a deep look into how our instructional decisions and our goals are linked.  Most teachers would fit somewhere in the middle, having traits of both the “traditional” teacher and that of a “modern” (for lack of a better term) teacher, however, it is important to understand that the decisions we make on a daily basis are aimed at achieving one ultimate goal.  So it is of the utmost importance for us to know what that goal is and to align our instructional decisions toward reaching that goal.  If I believe that I need to teach the concepts first, before problem solving begins then I am not developing mathematical thinkers, I am teaching for instrumental understanding.  I will see progress throughout each unit and throughout the year, however, it will lead to students who stop taking mathematics courses in high school as soon as they can, and adults who simply don’t use math.  If our goals are short-sighted like wanting our students to do well on a test, quiz or even a high-stakes test, then we might take many shortcuts that undermine our students’ thinking and development.  Don’t get me wrong, we all want our students to do well on their tests, quizzes and high-stakes testing, but students achieving on these shouldn’t be our primary goal.  These should simply be markers to determine if our students are learning and understanding math.  Our goals need to meet the real purpose of why we are all here – to develop students who can think critically, question and make sense of their world independently, who understand the importance of thinking and learning!

As a final thought, I want you to ask yourself, how can we better align our instructional decisions with our educational goals?

Purposeful Practice: Happy Numbers

In my head there are two things at odds:

  1. Practice is important in the consolidation process
  2. Mathematics is about thinking and reasoning… making sense of things.

So finding ways to practice a skill, while continuing to develop our ability to reason mathematically is a goal of mine.  While there are lots of ways to do this (problem solving, games, open-middle …), I thought I would give an example of how we can do both problem solving that requires thinking and reasoning, and practice a new skill at the same time.  So here is a problem that I give students that have recently started learning about square numbers.

Happy Numbers

A happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1.(definition from Wikipedia)

Numbers that result in the number 1, remain at 1 and are therefore HAPPY.

Numbers that start to loop will not result in the number 1, and are therefore called SAD numbers.

Take a look at two possible numbers:


The number 18 will continually loop (notice the 37s, you can figure out what comes next), while the number 19 quickly results in a 1 making it happy.

Once the idea of a happy number is explained to a class, I ask students to try to find as many of the happy numbers as possible between 1-100.  I also ask them to try and predict numbers, not just going in order from 1 through 100.

Students typically start at a random number and try to work through the process described above.  However, it usually isn’t long before students start finding numbers that will work.  Again, I encourage them to try to predict other happy numbers.

Some students realize that if the number 19 is a happy number, then so will the number 91.  Makes sense, since 1² + 9² = 9² + 1²

Other students realize that if 18 is a sad number, then all of the numbers in its pattern will also be sad numbers (18, 65, 61, 37, 58, 89, 145, 42, 20, 4, 16, 37…)  which helps them narrow down their search quite nicely.

Other students use this information to be able to predict other happy numbers.  For example, if 19 is a happy number and it becomes an 82, then 68, then 64… then 82, 68 and 64 should be happy numbers too!

In a typical 50 minute period, students square numbers dozens or hundreds of times, all searching for the 20 happy numbers that exist between 1-100.  This is a low floor-high ceiling task in that ALL students have an easy entry point square, add repeat… while it offers the challenge of looking for patterns, noticing, making predictions…

During our sharing time, we share the patterns we noticed, the ways we were able to predict future happy numbers.  Every time I have shared this problem with students, I usually end by asking: Why do you think we did this problem today?  What did we learn?  Student answers typically sound like:

  • To look for patterns
  • To predict happy numbers
  • To persevere with a hard problem
  • To cooperate with our partners
  • Being strategic
  • To learn from others in the class how they solved the problem
  • Mathematicians like to play with numbers
  • We have been learning about square numbers, maybe to practice them

I like ending lessons with questions like these because it helps us realize that there are a variety of things we learned!  Yes, I can say that every student had plenty of practice with squaring numbers, but I think that is only a small piece of the learning here!

So coming back to the notion of practicing skills, I think I have shared a possible way of re-purposing  practice through the lens of a problem.  Where do you think this fits on the Task Analysis Guide below?math_task_analysis_guide

I encourage you to think of a skill that you typically ask students to practice.  What concept/skill do you know you want your students to practice, but struggle to find engaging ways to practice it?  Let’s continue the conversation to help us find ways to practice our math, but in ways that evoke thinking, curiosity, and engagement.

What Does Day 1 Look Like?


So I have been thinking a lot about this chart found in PRIME:

Teaching Approaches

There is a lot to take in here, but I want to point your attention to the “Goal” and “Roles” rows.  Take a look again at these two rows.  I think a lot of the differences we see between classrooms, between lessons, between the beliefs we hear online… comes back to what our GOALs are… and therefore, our ROLES.

For some, their goal is to fill gaps, make sure our students can do a skill or group of skills.  When this is our goal, likely we are explicitly teaching.  The role of the student is to listen, or as the chart says, “passively” listen.  The focus of the mathematics is typically procedural and symbolic.  Learning happens here by the teacher showing, students trying to imitate the teachers’ procedures.  With this approach, “learning” comes from the teacher, is passed on to the student, then time is given so a student can master what the teacher just showed them.  The belief system at play here is that math is about remembering and following steps/rules!


For others, their goal is to guide students carefully into a deep understanding of the math… to connect learning together… to build conceptual understanding.  The teacher’s role here is to guide not tell, and the students role is to make things make sense.  There is likely more discourse happening here because the students are taking on more of an active role as they are expected to develop meaning.  Learning happens through lessons, but in a different way than a skills approach.  Here learning isn’t transmitted from the teacher to the student, students take an active role in understanding.  The teacher might allow time for students to talk with their neighbor, or work independently for a moment, or explore visual representations, or share their thinking…  The goal of this lesson is conceptual understanding, but also the development of procedures that make sense.  The belief system here is that visual representations and contexts can help students make sense.  Students are capable of developing a conceptual understanding which leads to a bridge between concepts and procedures…


Still, for others, their goal is to develop mathematical thinkers.  Students being able to follow rules isn’t good enough for these teachers.  They want students who can make connections between concepts, and develop reasoning skills.  Relational understanding and mathematical reasoning are their goals!  Using the process expectations are how these students learn.  Learning starts with low floor/ high ceiling problems where students can answer in ways that make sense to them.  From there students share their thinking and learn WITH and FROM each other.  The teacher’s role is more complicated here, since it relies on the students’ thinking.  The 5 Practices are used to make sure the learning is deep and meaningful.  The belief system here is that students learn best when they have had enough time to grapple with the thinking themselves first… and that learning HOW to think mathematically will help you with future learning!


I don’t want to make it sound like there are 3 different types of teachers… or that one of these is a better way to teach or learn.  In fact, I am not sure any teacher is in any one of these columns every day.  Rather, I believe that we are likely moving in and out of these different teaching approaches regularly.

I do, however, believe that there are two different ways of thinking about these approaches though:

  1. Moving from Skills, to Concepts, to Problems
  2. Moving from Problems, to Concepts and Procedures

This might seem like a subtle difference, but I think it is something that we need to reflect on, and better understand others’ thinking.

Moving from skills to concepts and then finally to problem solving can be thought of as a Teaching FOR Problem Solving approach.  Taking this stance seems to align with the goals and beliefs of the Skills Approach in that learning happens FROM the teacher and only when students are ready can they solve problems.

On the other hand, moving from problems to concepts and procedures can be thought of as a Teaching THROUGH Problem Solving approach.  Taking this stance seems to align with the goals and beliefs of the Conceptual/Constructivist approach.

Gradual Release.png

This is a really powerful caption from Cathy Seeley’s book Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver.  But again it shows me what the goal of the author is, “developing mathematical proficiency that includes the development of thinking skills and the ability to tackle problems that may not fit a particular format.”  Let me say that last part again… “…tackle problems that may not fit a particular format.”  A skills approach does not have this as the goal… rather the goal is typically simple… master this one thing today.

Take a look further into her thinking about how we can develop mathematically proficient students:

Cathy Seeley quote.jpg

“Upside-down teaching.”  Hmmm.  It sounds like she is telling us that a Gradual Release of Responsibility model doesn’t work in mathematics.  However, I think it has more to do with our goals.  If my goal is to develop skills, then we are likely going to start with direct instruction.  The problem with this is, I will be building students reliant on MY thinking.  Taking the “upside-down” approach listed here, or the teaching THROUGH problem solving approach I mentioned earlier should help us build mathematicians who can reason mathematically, make sense of concepts and ultimately have a relational understanding!  But this leaves us with the question, what order and for how long should I be using each approach?

I really want you to think about what day 1 looks like for any topic you teach… and what day 2 looks like…  and day 3… and day X.  Here is a sketch a friend shared with me (they found it on twitter).  What would your graphic look like?  What might be similar?  Or different?

Day 1, Day 2...

The reason I’m writing this blog post is for us to really consider if we are reaching our goals… and to reflect on our own beliefs.

Does the role you take match your goals?


Please leave a comment



“Worst Problem Ever!”

A few years ago I was teaching a 6/7 split class.  We had been exploring surface area and volume of prisms (rectangular so far), but were quite early on in the learning cycle.  At the beginning of my math class I drew a picture of a rectangular prism and wrote the dimensions on the diagram like below:

SA and V 4

I asked everyone to calculate the Surface Area.  Really this was just a check for me to see if we understood surface area… the problem for the day was coming up.

However, as I walked around, I realized something VERY unusual.  EVERYONE got the answer of 250… but not everyone did it correctly.  Let me show you:


SA and V 3

Take a look at the 2 answers above.  Some students calculated the Surface Area correctly at 250 units squared… and others calculated 250 units cubed (they found the volume).

I apologized to the class for giving them the worst problem ever… THE ONLY POSSIBLE rectangular prism that has the same surface area as volume.  Then one student commented, “Is that the only possible prism?”  

I didn’t know the answer, so we started seeing if it was the only possible prism.

Over the next 100 minutes, every student in the class, some in pairs, some on their own, started drawing prisms and calculating the surface area and volume.

Eventually, a student told me that one of the dimensions couldn’t possibly be a 1.  I asked him to prove it… After a few minutes he showed me several examples with numbers that were small or large.  He attempted to find the limits:

SA and V

He explained to the class that if there is a 1 as any of the dimensions, the Surface Area would always be larger.   We explored why that happens.

Minutes later, another student told me that it was impossible to have a 2 as one of the dimensions.  I asked her to prove it… She showed me the work that she had completed and shared it with the class.

SA and V2

She explained that the numbers were getting close if she chose 2 of the numbers identical, but the end pieces of the shape would always add up to more than the volume would.

After lots of trial and error… guessing and refining ideas, several students started finding possible prisms that also had the same Surface Area and Volume…

The class started noticing a pattern between the dimensions and found limits between the smallest and largest possible prisms…  

By the end of class, all students had calculated dozens of surface areas and volumes… all students were making conjectures or testing out the conjectures of others.

My original problem, which I intended as a quick warm-up was not a quality engaging problem.  However, I want you to think about what made this lesson better?

WHO posed the problem that day?  Did this have something to do with the shared responsibilities that happened later in the lesson?

What if I just moved on?  would the learning have been as rich?

Think about how the students picked the shapes they were testing?  Some students worked in teams to work strategically… others made their own conjectures and followed those patterns.

In this lesson, choice was key, but the choices didn’t come from me… Students were working together to reach an ultimate goal, not in competition with each other…  Conjectures were made and tested, not because I told everyone to, but because it served our purpose!

I think about Dan Meyer’s “Real World vs Real Work” a lot.  Why were my students so engaged here?  There was NO real world connection.  That wasn’t what was motivating my students at all!


I also think we need to reflect on the level of cognitive demand we ask our students to be engaged in:


“Doing Mathematics Tasks” seems like something that is hard to do, yet, every one of my students were engaged in this problem… everyone eagerly searched for patterns, many drew pictures, used snap cubes, visualized what was happening… all in the name of better understanding the relationships between the dimensions of a rectangular prism, and its surface area and volume!

By the end of the class, my students had found exactly 10 rectangular prisms that have the same value of its surface area as its volume (using only whole numbers), and could prove that these were the only 10 possible.

I’d love to hear your thoughts about our problem… or why the students were SO engaged… or about the conditions that must have been present in the class… or how problem solving can be used as a purposeful practice of procedures…..