## Focus on Relational Understanding

Forty years ago, Richard Skemp wrote one of the most important articles, in my opinion, about mathematics, and the teaching and learning of mathematics called Relational Understanding and Instrumental Understanding.  If you haven’t already read the article, I think you need to add this to your summer reading (It’s linked above).

Skemp quite nicely illustrates the fact that many of us have completely different, even contradictory definitions, of the term “understanding”.   Here are the 2 opposing definitions of the word “understanding”:

“Instrumental understanding” can be thought of as knowing the rules and procedures without understanding why those rules or procedures work. Students who have been taught instrumentally can perform calculations, apply procedures… but do not necessarily understand the mathematics behind the rules or procedures.

“Relational understanding”, on the other hand, can be thought of as understanding how and why the rules and procedures work.  Students who are taught relationally are more likely to remember the procedures because they have truly understood why they work, they are more likely to retain their understanding longer, more likely to connect new learning with previous learning, and they are less likely to make careless mistakes.

Think of the two types of understanding like this:

Students who are taught instrumentally come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill.  Each new skill requires a new set of procedures.  However, those who are taught relationally make connections between and within concepts and skills.  Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.

While it might seem obvious that relational understanding is best, it requires us to understand the mathematics in ways that we were never taught in order for us to provide the best experiences for our students. It also means that we need to start with our students’ current understandings instead of starting with the rules and procedures.

Skemp articulates how much of an issue this really is in our educational system when he explains the different types of mismatches that can occur between how students are taught, and how students learn.  Take a look:

Notice the top right quadrant for a second.  If a child wants to learn instrumentally (they only want to know the steps/rules to solve today’s problem) and the teacher instead offers tasks/problems that asks the child to think or reason mathematically, the student will likely be frustrated for the short term.  You might see students that lack perseverance, or are eager for assistance because they are not used to thinking for themselves.  However, as their learning progresses, they will come to make sense of their mathematics and their initial frustration will fade.

On the other hand, if a teacher teaches instrumentally but a child wants to learn relationally (they want/need to understand why procedures work) a more serious mismatch will exist.  Students who want to make sense of the concepts they are learning, but are not given the time and conditions to experience mathematics in this way will come to believe that they are not good at mathematics.  These students soon disassociate with mathematics and will stop taking math classes as soon as they can.  These students view themselves as “not a math person” because their experiences have not helped them make sense of the mathematics they were learning.

While the first mismatch might seem frustrating for us as teachers, the frustration is short lived. On the other hand, the second mismatch has life-long consequences!

I’ve been thinking about the various initiatives/ professional development opportunities… that I have been part of, or have been available online or through print that might help us think about how to move from an instrumental understanding to a relational understanding of mathematics.  Here are a few I want to share with you:

Phil Daro’s Against Answer Getting video highlights a few instrumental practices that might be common in some schools.  Below is the “Butterfly” method of adding fractions he shares as an “answer getting” strategy.  While following these simple steps might help our students get the answer to this question, Phil points out that these students will be unable to solve an addition problem with 3 fractions.  These students “understand” how to get the answer, but in no way understand how the answer relates to addends.

On the other hand, teachers who teach relationally provide their students with contexts, models (i.e, number lines, arrays…), manipulatives (i.e., cuisenaire rods, pattern blocks…) and visuals to help their students develop a relational understanding.  If you are interested in learning more about helping your students develop a relational understanding of fractions, take a look at a few resources that will help:

Tina Cardone and a group of math teachers across Twitter (part of the #MTBoS) created a document called Nix The Tricks that points out several instrumental “tricks” that do not lead to relational understanding.  For example, “turtle multiplication” is an instrumental strategy that will not help our students understand the mathematics that is happening.  Students can draw a collar and place an egg below, but in no way will this help with future concepts!

Teachers focused on relational understanding again use representations that allow their students to visualize what is happening.  Connections between representations, strategies and the big ideas behind multiplication are developed over time.

Take a look at some wonderful resources that promote a relational understanding of multiplication:

Each of the above are developmental in nature, they focus on representations and connections.

So how do we make these shifts?  Here are a few of my thoughts:

1. Notice instrumental teaching practices.
3. Align assessment practices to expect relational understanding.

#### Goal 1 – Notice instrumental teaching practices.

Many of these are easy to spot.  Here is a small sample from Pinterest:

The rules/procedures shared here ask students to DO without understanding.  The issue is that there are actually countless instrumental practices out there, so my goal is actually much harder than it seems.  Think about something you teach that involves rules or procedures.  How can you help your students develop a relational understanding of this concept?

I don’t think this is something we can do on our own.  We need the help of professional resources (Marian Small, Van de Walle, Fosnot, and countless others have helped produce resources that are classroom ready, yet help us to see mathematics in ways that we probably didn’t experience as students), mathematics coaches, and the insights of teachers across the world (there is a wonderful community on Twitter waiting to share and learn with you).

I strongly encourage you to look at chapter 1 of Van de Walle’s Teaching Student Centered Mathematics where it will give a clearer view of relational understanding and how to teach so our students can learn relationally.

However we are learning, we need to be able to make new connections, see the concepts in different ways in order for us to know how to provide relational teaching for our students.

Goal 3 – Align assessment practices to expect relational understanding.

This is something I hope to continue to blog about.  If we want our students to have a relational understanding, we need to be clear about what we expect our students to be able to do and understand.  Looking at developmental landscapes, continuums and trajectories will help here.  Below is Cathy Fosnot’s landscape of learning for multiplication and division.  While this might look complicated, there are many different representations, strategies and big ideas that our students need to experience to gain a relational understanding.

Asking questions or problems that expect relational understanding is key as well.  Take a look at one of Marian Small’s slideshows below.  Toward the end of each she shares the difference between questions that focus on knowledge and questions that focus on understanding.

I hope whatever your professional learning looks like this year (at school, on Twitter, professional reading…) there is a focus on helping build your relational understanding of the concepts you teach, and a better understanding of how to build a relational understanding for your students.  This will continue to be my priority this year!

## How to change everything and nothing at the same time!

I’ve been thinking a lot about trends in education… Some are trendy because they are flashy and look neat… Others because there is a lot of hype from influential people.  But where my interest lies is in seeing if these things are actually helping our students have richer learning experiences than before.  Or will we end up seeing all kinds of changes in our programming that lead to the same experiences?

So before you continue, I need to tell you that I’m writing this blog as a way for me to reflect, not as a way to criticize any particular practice. Please don’t assume my intent is to vilify certain practices, I’m actually trying to see if we’ve changed how we believe students learn & therefore the experiences we provide for them, or if we’ve kept things relatively the same.  I am trying to figure out where to put my energy, where to focus my attention.

#### 1. Teacher lectures, student passively listens

A common practice of my math teachers when I was a child was for my math teachers to drone on for most of the period.  Tracy Zager quipped in her TMC keynote speech this summer that the only way she would be able to recognize her high school math teachers was if they turned around and had a piece of chalk in their hands.  While the joke got some groans, there seems to be some truth here.

The idea that teachers need to impart their knowledge through direct instruction while students are expected to follow along has thankfully changed much over the years.  For many students passively listening isn’t how learning happens.  Students need to be actively thinking, actively involved in the doing, using their reasoning skills to make things make sense.  While some students are actively thinking in a lecture, many aren’t.

Now let’s take a look at a few newer teaching strategies:

Which of these allows our students to be more active in the learning process?  Which ones promote passive learning?

#### 2.  Teacher explains things, then students practice them.

Similarly, as a child most math classes were about copying what the teacher just showed us how to do. For instance, the teacher would show us how to add fractions, then we would get a worksheet or textbook page directly related to adding fractions. All of the “problems” would obviously be about adding fractions  – no thinking was required, just copying the same strategy.

In recent years we have learned more about teaching through problem solving, the importance of productive struggle, building an understanding of the math together…  All pointing to students starting the learning process through rich experiences and problem solving…where the learning starts with the students’ ideas, not the teacher’s.

So I wonder which pedagogical strategies and resources are helping us make these changes and which are keeping things similar to before:

I think this list might cause some more thought. At least for me there are a few items here that might promote productive struggle in some situations, and rob students of thinking in other situations. Which ones do you see helping us allow our students to be able start with thinking instead of copying someone else’s thinking?

#### 3.  Learning isolated facts, procedures and tricks

What we focus in math class on has changed quite a bit from when I was in school.  Memorizing random math facts, learning tricks  or procedures about how to solve very specific problems…was the norm in my school. From alligator inequalities, to invert and multiply rules, to placeholder zeros with an X through it when multiplying, to mad minutes…  All of these were about getting the answers, without ever helping us understand what was going on.

Thankfully today we recognize that conceptual understanding is equally important and we are learning to build understanding together. But which resources are helping us make these shifts, and which might be encouraging us to continue with previously help beliefs?

• Nix The Tricks
• Math manipulatives
• Common Core Standards
• Computer apps/games
• Pinterest

Again, this list might require some actual discussion. Which ones do you see as helping us provide better experiences?  Which ones depend?  And what might things depend on?

It seems to me that if we want to make changes, we need to consider WHY those changes might lead to different experiences!  We need to consider how learning via that new strategy or tool will help our students learn.  Seeing past what is an easy change and past all of the gimmicks requires us to be really reflective!
Hopefully you know which items you see as valuable learning opportunities for your students  and which ones are just more of the same (with a shiny new package).

## Who makes the biggest impact?

A few years ago I had the opportunity to listen to Damian Cooper (expert on assessment and evaluation here in Ontario). He shared with us an analogy talking to us about the Olympic athletes that had just competed in Sochi.  He asked us to think specifically about the Olympic Ice Skaters…

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

Think about this for a second…  An ice skater trying to become the best at their sport has many influences on their life…  But who makes the biggest difference?  The scoring judges along the way, or their coaches?  Or is it a mix of both???

Damian told us something like this:

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 the 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. Their goal is for you to become the very best you can be!  They want you to succeed!

In the everyday busyness of teaching, I think we often confuse the terms “assessment” with “evaluation”   Evaluating is about marking, levelling, grading… While the word assessment comes from the Latin “Assidere” which means “to sit beside”.  Assessment is kind of like learning about our students’ thinking processes, seeing how deeply they understand something…   These two things, while related, are very different processes!

I have shared this analogy with a number of teachers.   While most agree with the premise, many of us recognize that our job requires us to be the scoring judges… and 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:

https://player.vimeo.com/video/136761933?color=a185ac&title=0&byline=0&portrait=0

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 don’t care if they are achieving.

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 will all improve.

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

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.

So, 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 do you use these learning opportunities to provide feedback on your own teaching?

As  always, I try to ask a few questions to help us reflect on our own beliefs.  Hopefully we can continue the conversation here or on Twitter.

## What does “Assessment Drives Learning” mean to you?

There are so many “head nod” phrases in education.  You know, the kind of phrases we talk about and all of us easily agree upon that whatever the thing is we are talking about is a good thing.  For instance, someone says that “assessment should drive the learning” in our classroom, and we all easily accept that this is a good practice.  Yet, everyone is likely to have a completely different vision as to what is meant by the phrase.

In this post, I want to illustrate 3 very different ways our assessments can drive our instruction, and how these practices lead to very different learning opportunities for our students.

##### Unit Sized Assessments

Some teachers start their year or their unit with a test to find out the skills their students need or struggle with.  These little tests (sometimes not so little) typically consist of a number of short, closed questions.  The idea here is that if we can find out where our students struggle, we will be able to better determine how to spend our time.

But let’s take a look at exactly how we do this.  The type of questions, the format of the test and the content involved not only have an effect on how our students view the subject and themselves as learners of math, they also have a dramatic effect on the direction of learning in our classrooms.

For example, do the questions on the test refer to the types of questions you worked on last year, according to previous Standards, or are they based on the things you are about to learn this year (this year’s Standards)?  If you provide questions that are 1 grade below, your assessment data will tell you that your students struggle with last years’ topics… and your instruction for the next few days will likely be to try to fill in the gaps from last year.  On the other hand, if you ask questions that are based on this year’s content, most of your students will likely do very poorly, and your data will tell you to teach the stuff you would have anyway without giving the test at all.  Either way, the messages our students receive are about their deficits… and our instruction for the next few days will likely relate to the things we just told our students they aren’t good at.  I can’t help but wonder how our students who struggle feel when given these messages.  Day 1 and they already see themselves as behind.

I also can’t help but wonder if this is helpful even for their skills anyway?  As Daro points out below, when this is our main view of assessment guiding our instruction, we often end up providing experiences for our students that continue to keep those who struggle struggling.

##### Daily Assessments

On the other hand, many teachers view assessment guiding their practice through the use of daily assessment practices like math journals, exit cards or other ways of collecting information while the learning is still happening.  It is really important to note that these forms of assessment can look very different from teacher to teacher, or from lesson to lesson.  In my post titled Exit Cards: What do yours look like?  I shared 4 different types of information we often collect between lessons.  I really think the type of information we collect says a lot about our own beliefs and our reflections on this evidence will likely form the type of experiences we have the next day.

When we use assessments like these regularly, we are probably more likely to stay on track with our curriculum Standards, however, what we do with this the information the next day will completely depend on the type of information we collect.

##### In-the-Moment Assessments

A third way to think of “assessment driving instruction” is to think of the in-the-moment decisions we make.  For example, classrooms that teach THROUGH problem solving will likely use instructional practices that help us use in-the-moment assessment decisions.  Take for example The 5 Practices: for Orchestrating Productive Mathematics Discussions listed below, might be useful as part of the assessment of our students.

1. Anticipating
• Do the problem yourself.
• What are students likely to produce?
• Which problems will most likely be the most useful in addressing the mathematics?

The first practice helps us prepare for WHAT we will be noticing.  Being prepared for the problem ahead is a really important place to start.
2. Monitoring
• Listen, observe students as they work
• Keep track of students’ thinking
• Ask questions of students to get them back on track or to think more deeply (without rescuing or funneling information)

The second practice helps us notice how students are thinking, what representations they might be using.  The observations and conversations we make here can be very powerful pieces of assessment data for us!

3. Selecting
• What do you want to highlight?
• Purposefully select those that will advance mathematical ideas of the group.

The third practice asks us to assess each of the students’ work, and determine which samples will be beneficial for the class.  Using our observations and conversations from practice 2, we can now make informed decisions.
4. Sequencing
• In what order do you want to present the student work samples?  (Typically only a few share)
• Do you want the most common to start first? Would you present misconceptions first?  Or would you start with the simplest sample first?
• How will the learning from the first solution help us better understand the next solution?• Here we ask students specific questions, or ask the group to ask specific questions, we might ask students what they notice from their work…

The 4th practice asks us to sequence a few student samples in order to construct a conversation that will help all of our students understand the mathematics that can be learned from the problem.  This requires us to use our understanding of the mathematics our students are learning in relation to previous learning and where the concepts will eventually lead (a developmental continuum or landscape or trajectory is useful here)
5. Connecting
• Craft questions or allow for students to discuss the mathematics being learned to make the mathematics visible (this isn’t about sharing how you did the problem, but learning what math we can learn from the problem).
• Compare and contrast 2 or 3 students’ work – what are the mathematical relationships?  We often state how great it is that we are different, but it is really important to show how the math each student is doing connects!

In the 5th and final practice, we orchestrate the conversation to help our class make connections between concepts, representations, strategies, big ideas…  Our role here is to assess where the conversation should go based on the conversations, observations and products we have seen so far.

So, I’m left wondering which of these 3 views of “assessment driving learning” makes the most sense?  Which one is going to help me keep on track?  Which one will help my students see themselves as capable mathematicians?  Which one will help my students learn the mathematics we are learning?

Whether we look at data from a unit, or from the day, or throughout each step in a lesson, Daro has 2 quotes that have helped form my opinion on the topic:

I can’t help but think that when we look for gaps in our students’ learning, we are going to find them.  When our focus in on these gaps, our instruction is likely more skills oriented, more procedural…. Our view of our students becomes about what they CAN’T do.  And our students’ view of themselves and the subject diminishes.

“Need names a sled to low expectations”.  I believe when we boil down mathematics into the tiniest pieces then attempt to provide students with exactly the things they need, we lose out on the richness of the subject, we rob our students of the experiences that are empowering, we deny them the opportunity to think and engage in real discourse, or become interested and invested in what they are learning.  If our goal is to constantly find needs, then spend our time filling these needs, we are doing our students a huge disservice.

On the other hand, if we provide problems that offer every student access to the mathematics, and allow our students to answer in ways that makes sense to them, we open up the subject for everyone.  However, we still need to use our assessment data to drive our instruction.

##### As a little experiment, I wonder what it would look like if other subjects gave a skills test at the beginning of a unit to guide their instruction.  Humor me for a minute:

What if an English teacher used a spelling test as their assessment piece right before their unit on narratives?  Well, their assessment would likely tell them that the students’ deficits are in their spelling.  They couldn’t possibly start writing stories until their spelling improved!  What will their instruction look like for the next few days?  Lots of  memorization of spelling words… very little writing!

What if a Science teacher took a list of all of the vocabulary from a unit on Simple Machines and asked each student to match each term with its definition as their initial assessment?  What would this teacher figure our their students needed more of?  Obviously they would find that their students need more work with defining terms. What will their instruction look like for the next few days?  Lots of definitions and memorizing terms… very little experiments!

What if a physical education teacher gave a quiz on soccer positions, rules, terms to start a unit on playing soccer.  What would this teacher figure out?  Obviously they would find out that many of their students didn’t know as much about soccer as they expected.  What would their next few days look like?  Lots of reading of terms, rules, positions… very little physical activity!

##### A Few Things to Reflect on:
• How do you see “assessment guiding instruction”?
• Is there room for all 3 versions?
• Which pieces of data are collected in your school by others?  Why?  Do you see thes as helpful?
• Which one(s) do you use well?
• Do you see any negative consequences from your assessment practices?
• How do your students identify with mathematics?  Does this relate to your assessment practices?

Being reflective is so key in our job!  Hopefully I’ve given you something to think about here.

Please respond with a comment, especially if you disagree (respectfully).  I’d love to keep the conversation going.

## “I like math because it’s objective…”

I was told the other day from a fellow teacher that they enjoyed math a lot as a child because it was really objective. They recalled problems and questions that had exactly one right answer and enjoyed learning the rules to be able to find that answer.

My response went something like this:

I bet others liked History too as kids since the answers to questions were always right/wrong as well. You know, learn the names and dates, identify key events… There was no room for debate, the facts are the facts, there’s no room to refute the year Columbus landed or which country invaded which other country…

…and I bet many other people really liked writing essays as kids. As long as you can spell, remember punctuation, use the correct grammar and use the right hamburger style paragraph format there would be no red marks on your page. Spelling a word is either right or wrong, there’s no room to argue over if there is a topic sentence or not, or if we remembered a period at the end of the sentence. As long as you included each step, you did well.

…and I bet others really liked Geography too as kids. I mean all you had to know were the definitions of terms, and the capital cities… There was no way to argue a definition or a capital city. Remember these things and you’d do well.

…and I bet many people loved reading in school. All of the questions our teachers asked had answers right there on the page. If you didn’t remember what you read, you could reread if to find the exact answer. There was no debate about if you got the right answer or not, the answers were all there in black and white…

Come to think about it, most subjects were very objective… there was no ambiguity or reasoning or creativity or thinking…just get the right answers and get a good mark!

Now while my colleague stated she completely agreed with my point, she decided to argue back stating something like:
But in math, 2+2 will always equal 4. That’s math!

To which my response was:
…and to many History is about Columbus landing in 1492, not about arguing over the different perspectives or consequences of major events…
…and to many writing is all about the skills of spelling and grammar and structuring paragraphs, not about trying to convince, entertain, explain, or inform…
…and to many reading is about restating what happened or finding key information, instead of understanding character development/motivation or determining the writer’s bias or making inferences…
…and to many Geography is about naming capital cities, instead of looking for patterns of why our planet is the way it is…

At this time we were about to leave the conversation agreeing to disagree, but I thought I would add one last piece:

If we look at any good curriculum and any teacher passionate about their subject, their focus is probably on understanding why things are the way they are.  They likely help their students develop by allowing them time to be creative & think critically.  They likely focus on developing students to reason and make sense of things.  They likely focus on using skills and knowledge as part of the process/product, without isolating them all the time.

While I think most elementary teachers are comfortable with the subjectivity of open questions in reading or writing or History or Geography where there is more than one possible answer, or more than one possible way of explaining an answer, I think many of us have a long way to go in math!

After leaving the conversation, I am now wondering why I believe this to be true?  Really what I think this boils down to is our specific knowledge related to the subject.  For mathematics, we call this Math Knowledge for Teaching.  Take a look below.  Which of these areas would you say you are strong in?  Which ones do you think you need to continue to develop in (hopefully we can recognize we ALL need to continue to grow)?

The more we learn the mathematics, the deeper we understand the content, the more we understand how the mathematics develops over time, the more we understand which representations and models are the most appropriate at the right time……..  the more likely we will see math as more than a subject filled with distinct rules and procedures all meant to be seen as right or wrong…….. and instead come to see math as an interconnected rich subject filled with thinking and reasoning!

When we start to make these shifts, we come to allow our students to truly appreciate mathematics!

## Learning Goals… Success Criteria… and Creativity?

I think in the everyday life of being a teacher, we often talk about the word “grading” instead of more specific terms like assessment or evaluation  (these are very different things though).  I often hear conversations about assessment level 2 or level 4… and this makes me wonder about how often we confuse “assessment” with “evaluation”?

Assessment comes from the Latin “assidere” which literally translates to “sit with” or “sit beside”.  The process of assessment is about learning how our students think, how well they understand.  To do this, we need to observe students as they are thinking… listen as they are working collaboratively… ask them questions to both push their thinking and learn more about their thoughts.

Evaluation, on the other hand, is the process where we attach a value to our students’ understanding or thinking.  This can be done through levels, grades, or percents.

Personally, I believe we need to do far more assessing and far less evaluating if we want to make sure we are really helping our students learn mathematics, however, for this post I thought I would talk about evaluating and not assessing.

A group of teachers I work with were asked to create a rubric they would use if their students were making chocolate chip cookies as a little experiment.  Think about this task for a second.  If every student in your class were making chocolate chip cookies, and it was your responsibility to evaluate their cookies based on a rubric, what criteria would you use?  What would the rubric look like?

Some of the rubrics looked like this:

What do you notice here?  It becomes easy to judge a cookie when we make the diameters clear… or judge a cookie based on the number of chocolate chips… or set a specific thickness… or find an exact amount for its sugar content (this last one might be harder by looking at the final product).

While I am aware that setting clear standards are important, making sure we communicate our learning goals with students, co-creating success criteria… and that these have been shown to increase student achievement, I can’t help but wonder how often we take away our students’ thinking and decision making when we do this before students have had time to explore their own thoughts first.

What if we didn’t tell our students what a good chocolate chip cookie looked like before we began trying things out?  Some might make things like this:

or this?

or this?

But what if we have students that want to make things like this:

or this?

or this?

Or this?

I think sometimes we want to explain everything SO CLEARLY so that everyone can be successful, but this can have the opposite effect.  Being really clear can take away from the thinking of our students.  Our rubrics need to allow for differences, but still hold high standards!  Ambiguity is completely OK in a rubric as long as we have parameters (saying 1 chip per bite limits what I can do).

What about the rubric below?  Is it helpful?  While the first rubric above showed exact specs that the cookies might include, this one is very vague.  So is this better or worse?

As we dig deeper into what quality math education looks like, we need to think deeper about the evidence we will accept for the word “understanding”!

…and by the way, are we evaluating  the student’s ability to bake or their final product?  If we are assessing baking skills, shouldn’t we include the process of baking?  Is following a recipe indicative of a “level 4” or an “A”?  Or should the student be baking, using trial and error and developing their own skills?  Then co-creating success criteria from the samples made…

If we show students the exact thing our cookies should look like, then there really isn’t any thinking involved… students might be able to make a perfect batch of cookies, and then not make another batch until next year during the “cookie unit” and totally forget everything they did last year (I think this is currently what a lot of math classes looks like).

Learning isn’t about following rules though!  It’s about figuring things out and making sense of it in your own way, hearing others’ ideas after you have already had a try at it, learning after trying, being motivated to continue to perfect the thing you are trying to do.  We learn more from our failures, from constructing our understanding than we ever will from following directions!

Creativity happens in math when we give room for it.  Many don’t see math as being creative though… I wish they did!

## Accessing Our Spatial Reasoning

I spent some time with a wonderful kindergarten teacher a while back. The two of us were chatting when a little boy, probably 4 or 5 years old walked up to us with a inquisitive look and asked if it was home time yet. Since it was only about 10 in the morning I smiled at the question. But the teacher in a very calm and candid manner spread his arms out wider than shoulder width and said, “There’s still THIS much time left.

The child, very satisfied with the response, happily headed over to another activity and continued to play.  I, however, was quite stunned at what just transpired.  The little boy wasn’t given an answer… or was he?

What was it about the gesture given that helped this child understand that there was still lots of time left in the day?  I found myself quite interested in the above conversation for some time until I started thinking about the types of experiences our students have with time as young children.

Clocks like those below don’t have any meaning for most 4 or 5 year-olds.  There are far too many numbers or symbols… different ways to read… for many to understand here!

But they have likely had many experiences looking at things like these:

Let’s think about these items for a minute.  Children by age 4 or 5 have likely been exposed to a cell phone, a tablet or some type of electronic device that has a battery meter.  Without any understanding of percent or without even being able to recognize large numbers, children instinctively know when the device is charged or when it is nearly “dead”.  When they watch movies, they can determine if we are currently near the beginning or end based on the progression meter (when they hit pause of fast-forward)… or when downloading, they can tell by how much space is left before they can access the app or game…   Even when waiting in a line-up, children can tell if it will be a short time or not without having to count the people in front of them.

Each of the pictures above show our students’ natural ability to determine information based on the relative space provided.  This is a great example of how we use Spatial Reasoning to make sense of things.

There is a lot of research about Spatial Reasoning and it’s relationship with mathematics.  Much of this idea was first brought to my attention through the article: Paying Attention to Spatial Reasoning.  While you probably haven’t read this, I think it is really important to read through the whole article.  Really… it will explain a lot better than I can.

##### If our students come into our math classes with an intuitive understanding of the spatial relationships behind concepts, shouldn’t we provide experiences that allows our students to access this form of reasoning BEFORE we start with formal rules, procedures, notations…?

For instance, how might a simple graph like this bring about a conversation of subtraction?

How might a number line be used so our students can visually see how subtraction works?  And how subtraction is related to addition?

It is blurry, but the 132-94 was moved to show 138-100.  How might this visual help us with future problems?

Or, if we know Mr. Stadel’s height is 193cm.  How might that help us think about estimating Mrs. Stadel’s height?

How did you think about this?  What spatial cues did you use?  Can you see how others might see this as a subtraction or addition problem?

Spatial reasoning isn’t just for subtraction though.  If we wanted to know how many chocolates were here what might students do that are just starting to learn about multiplication?

How many different ways can you think of solving this question?

##### An example that really illustrates the need to focus on Spatial Reasoning:

At the end of the year, the grade 7 students throughout our school board were given a year-end test.  One of the questions our students did poorest on was this problem:

While the vast majority of students got this problem wrong because of issues with converting units, several students instead took a path that allowed them to make sense of the problem.  Below is a diagram we saw a few students draw that represented what was going on in the problem quite nicely.  What did they do here?  What do you notice?

I notice dimensions that make sense.  I notice a piece of cloth being cut up, much like what we would actually do if this were a real-life problem.  I notice the student thinking about the space required to cut the bandanas.

So I am left wondering why did only a few students attempt to make a drawing?  Why did so many make errors with converting between units?  Why did so many students make errors that were not even close to an acceptable answer?  What experiences did they have… and didn’t they have that led them to attempt this problem the way they did?

While many teachers might tell students that there are 100 x 100 cm in a square meter, and expect students to understand and remember… or show conversion charts… or offer a page of conversion questions… I think we might be missing a big piece of the learning process.  We haven’t tapped into our students’ spatial reasoning at all!

After seeing  so many questions like this attempted by students in such procedural ways, I think next year we will start to think more about the reasoning our students already come into our classes with.  And continue to think about how we can support the development of our students Spatial Reasoning is in helping them make sense of things!

## So you want your students to have a Growth Mindset?

There has been much talk about Growth Mindset the past few years.  Many teachers recognize that there are a lot of students who exhibit a fixed mindset.  I often wonder when we introduce another educational term into the mainstream how many different ways the term can be misinterpreted.  Ask someone what they think about the terms growth or fixed mindsets and really listen to what they say.  I bet you would be surprised!

When others talk to me about mindsets, I often hear about students who struggle, needing to change their mindset… Makes sense doesn’t it.  We see a group of students in our school who don’t apply themselves, and we wish that they would just realize they could do much better if they just put forth more effort.  But is this what the whole growth/fixed mindset conversation is all about?

Those who see growth and fixed mindsets being about effort often think that encouraging their students with phrases or posters is what they need to succeed.  When we believe this, we might start to see a neat looking poster or bulletin board like these:

However, a poster alone might send a message to students that THEY are doing something wrong, THEY need to do something different… and that if they only tried harder, changed their words, became better people… that they would do better in math.

Personally, I don’t think this is all what the issue is with our students.  And I don’t believe this is what the research behind mindsets is pointing to either!  If we are looking to change our students’ mindsets, we need to do two things:

1. Learn what it means to have a growth / fixed mindset.  We might be misinterpreting the whole idea here!
2. Change OUR actions to promote growth mindsets.  Our students will only improve when WE change!

#### 1.  Learning about Growth and Fixed Mindsets

While there are a lot of places you can go to learn more about a growth mindset, I think we need to make sure we are hearing from the experts. Here are a few I think you need to take a look at:

While I don’t want to spend much time restating what is already in these articles/videos/books, I do really hope you will check out at least one of the new ones you haven’t already seen.

I will give a few key points about the whole mindset movement though:

• Students with fixed mindsets believe that they either have or do not have a math brain.  They see their intelligence as static or hereditary (a gift or a curse)
• Students and adults across the achievement spectrum can exhibit a growth or fixed mindset (high achieving students are just as likely to have a fixed mindset).
• Students with fixed mindsets are likely to avoid challenges and likely give up easily.
• It is more common for a student to have a fixed mindset about mathematics than other subjects.
• Our actions and words will either promote a fixed or growth mindset in others.

Do these 5 points fit your current understanding of Mindsets?  If any of these do not fit your thinking, please make an effort to read/watch some of the articles, videos linked above.

#### 2.  Change OUR Actions to Promote Growth Mindsets

I think this is the part of the conversation that is completely missed!  We tend to focus on what everyone else needs to do instead of how we need to change.

Think again about the message above.  What are the common practices that happen in many math classrooms that help our students believe they are smart at math, or that they are not smart at math?

Practices that promote fixed mindsets:

• Providing students with a large number of closed questions
• Math fluency is often conducted via timed tests
• Evaluating each assignment, especially early in the learning process
• Grouping / Sorting students by ability
• Providing different activities for different students
• Asking struggling students to speak first, then more advanced students next
• Competitive mathematics tasks (often based on speed)

Think about how each of these practices helps our students identify with mathematics.  Their mathematical identity is based on how WE present their successes.  Each of the practices above treats mathematics as a performance subject.  Students in classrooms like these are far more likely to believe that they are either a math person or not because the tasks ask them to recognize if they are better or worse than others in the room.

For example, students who are routinely given a page of closed questions to complete come to believe their role is to get them right.  Students who get them all right are likely to see that they are naturally good at math, while those who struggle through the questions are likely to see their “ability” in more negatively.  There is little room for students to see that they are growing!

On the other hand, teachers who view mathematics as a learning subject are more likely to have practices that allow students learn and grow.

Practices that promote growth mindsets:

• Provide students with open questions (open-ended problems with more than 1 possible answer, or open-middle problems with more than 1 potential strategy to achieve the answer)
• Math fluency/flexibility is based on reasoning and development of strategies
• Assessment of students is non-evaluative and instead focuses on feedback
• Grouping of students is flexible in nature
• Open problems (low floor, high ceiling problems) provide natural differentiated instruction as they offer challenge for every student
• All students are expected to contribute.  Students sharing in an important part of the learning, and all students recognize that all members can contribute
• Cooperative tasks where students learn WITH and FROM each other

Providing experiences like the items above will more likely allow your students to see that they are capable of growing and learning since the focus is on allowing your students time to develop.  These experiences help move the focus away from whether or not students are already good at math, toward a focus on the the learning today.  Allowing our students room for growth promotes a growth mindset.

Think about the lists above.  There are possibly a few that might challenge your thinking! Many hear some of the messages about the types of practices that promote fixed mindsets and struggle to know what to do instead.  Moving toward classrooms that promote growth mindsets is not about lowering the bar, in fact, classrooms that are successfully moving in this direction are expecting much more of their students… and are especially helping our students who begin the year struggling.

If you or your school or your district are interested in taking a focus on growth mindset, I encourage you to not just consider the pedagogy involved, but to actually focus on the practices that will help us to understand the math deeper ourselves!

For example, to focus on growth mindset, we as teachers need to be able to understand the math deeper in order to be able to teach mathematics in way that allows our students to see mathematics as a learning subject.  Here are a few things we as teachers, as schools and/or districts that we can focus on to help us make these shifts:

1. Consider focusing on tasks that help us see that we can develop number sense through the use of strategies, visual representations and the big ideas behind the numbers.  Number Talks or Strings are wonderful practices that both allow our students to develop appropriate number sense, think flexibly about operations, yet allow for creativity and reasoning to develop.  Our learning here needs to include understanding the developmental models, strategies and big ideas appropriate for our students to develop number sense.
2. Consider focusing on opening up problems.  Accessing resources like Marian Small’s Open Questions for the 3-Part Lesson/Eyes on Math/Good Questions… or Cathy Fosnot’s Contexts for Learning units… or other sources that might help us see how we can use open problems as part of a sequence of learning.  These, along with structures like the 5 Practices for Orchestrating productive mathematics discussions, can help us recognize how we can learn THROUGH problem solving.
3. Consider focusing on mini-lessons that focus on student reasoning, conceptual understanding, the use of visuals and spatial reasoning.  This can include using tasks like Estimation 180, Which One Doesn’t Belong, Fraction Talks, Visual Patterns, 101Qs Problems, SolveMe Mobiles… in a way that promotes student thinking, students sharing, and the development of reasoning together.  This can also include the development of and use of instructional routines like Notice and Wonder or Contemplate then Calculate.  Whichever of these we explore, we need to learn how to give our students the opportunity to think and reason… share with each other… learn FROM and WITH each other.  And to do these well, WE need to continue to learn the important mathematics behind the tasks, not just use them because they are fun or neat.
4. Consider a focus on assessment.  If we want classrooms that promote a growth mindset, we need to reconsider what information we send our students about what is important.  Learning about developmental progressions, how to give feedback effectively, tasks that allow us to make observations, listen to students as they are thinking, have conversations in-the-moment as students are problem solving.  When we deepen our understanding of the mathematics our students are learning, we will become better at using assessment effectively.  While many believe they are focusing on growth mindset by looking at spreadsheets, this is exactly the opposite of what needs to be focused on.

Whatever your focus, keep in mind the unintended messages we send our students about who is a math student… and continue OUR learning about the mathematics itself so we know how to help our students as they struggle (developmentally appropriate tasks/problems).

##### Teaching for a growth mindset will require us to make changes in our beliefs, but also in our practices!  Let’s keep learning my friends!!!

If interested, here are a few more posts that might help us see how to make these ideals a reality:

## Is That Even A Problem???

Ask others what problem solving means with regard to mathematics.  Many will explain that a problem is when we put a real-world context to the mathematics being learned in class… others might explain a process of how we solve a problem (look at what you know and determine what you want to know, or some other set of strategies or a creative acronym that we have likely seen in school).  Sadly, much of what most others would point to as a problem is not really even a problem at all.

I think we all need to consider the real notion of what it means to problem solve…

George Polya shared this:

…Thus, to have a problem means: to search consciously for some action appropriate to attain a clearly conceived, but not immediately attainable, aim. To solve a problem means to find such action. … Some degree of difficulty belongs to the very notion of a problem: where there is no difficulty, there is no problem.

What Polya is suggesting here is that if we show students how to do something, and then ask the students to practice that same thing in a context IT ISN’T A PROBLEM!!!  A problem in mathematics is like the DOING MATHEMATICS Tasks listed below.  Take a look at all 4 sections for a minute:

My thoughts are simple… If a student can relatively quickly determine a course of action about how to get an answer, it isn’t a problem!   Even if the calculations are difficult or take a while.  The vast majority of what we call problems are actually just contextual practice of things we already knew.  Doing word problems IS NOT the same as problem solving!

On the other hand, if a student has to use REASONING skills, they are thinking, actively trying to figure something out, then and only then are they problem solving!!!

Marian Small has written a short article on her thoughts about problem solving:  Marian Small – Problem Solving

What are her main messages here?  Does this or Polya’s quote change your definition of a problem?

I’ve already written about What does Day 1 Look Like where I shared the importance of starting with problems.  So, why should we start with problem solving?  If we don’t start there, we aren’t likely ever doing any problem solving at all!

I also think that many hearing this might assume that this means we just hand students problems that they wouldn’t be successful with…  Ask everyone to attempt something that they wouldn’t know how to do.

Let’s look at an example:

Take a look at these two grade 8 expectations from Ontario curriculum:

• determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies;
• solve problems involving right triangles geometrically

Many teachers look at these expectations, think to themselves, Pythagorean Theorem… I can teach that… followed by explicit teaching on a Smartboard (showing a video, modeling how the pythagorean theorem works, followed by some examples for the class to work on together…), then finally some problems that students have to answer in a textbook like this:

This progression neither shows how we know students learn, nor does it even get students to be able to do what is asked!

Let’s take a step back and start to notice what the curriculum is saying more clearly:

• determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies;
• solve problems involving right triangles geometrically, using the Pythagorean relationship;

When we pull apart the verbs and the tools/strategies from the content, we start to notice what the curriculum is telling us our students should actually be doing that day!  Remember… these expectations are what OUR STUDENTS should be doing… NOT US!!!

Above I have colored the verbs blue (These are the actions our students should be doing that day) and tools/strategies orange (specifically HOW our students should be accomplishing the verb).

Now let’s take a quick look at how this might actually play out in the classroom if we are starting with problems like our curriculum states:

Let’s start with the first expectation.  Our curriculum often includes the statement “determine through investigation,” yet it is overlooked far too often!  Students need to determine this themselves!  We need to assess students’ ability to “determine the Pythagorean relationship through investigation.”

This doesn’t mean we tell students what the theorem is, nor does it mean that we expect everyone to reinvent the theorem… so what does it mean???

Well, it could mean lots of things.  Here is one possible suggestion…

Show the figure on the left.  Ask students the area of the blue square in the middle.  Possibly give a geoboard or paper and scissors for this task.  How might students come up with the area?  How many different ways might students accomplish this?

Share different approaches as a group.  (By the way, some calculate the whole shape and subtract the 4 corner triangles.  Others calculate the 4 black rectangles and divide by 2 then add 1 for the middle… others rearrange the shapes to make it make sense).

Now explore how the two pictures are similar / different.  What do you notice between the two pictures?

If the curriculum tells us to “determine through investigation” that is exactly the experience our students need to conceptualize the concept.  It is also what we need to assess.  This can’t be put on a test easily though, it needs to be observed!

That second expectation is quite interesting to me too.  At the beginning of this post we started talking about what is and isn’t a problem.  If our students have now understood what the Pythagorean Theorem is, how can we now make things problematic?

Showing a bunch of diagrams with missing hypotenuses or legs isn’t really problematic!

However, something like Dan Meyer’s Taco Cart would be!  If you haven’t seen the lesson, take a look:

Oh… and by the way… problem solving isn’t always about answering a question… really, at it’s heart, problem solving is about making sense of things that we didn’t understand before… reasoning though things… noticing things we didn’t notice before… making conjectures and testing them out…  Problem solving is the process of LEARNING and DOING MATHEMATICS!

So I want to leave you with a problem for you to think about: what does this have to do with the Pythagorean Theorem?

## Exit Cards – What do yours look like?

Classrooms that have moved toward a problem-based model often wrestle with the idea of individual accountability.  If we offer tasks that are worked on collaboratively, we need to make sure we learn more about how each individual student is thinking.  To do this, one common practice is providing our students with exit cards at the end of the lesson.

However, I wonder when we talk about terms like “exit cards” are we envisioning the same thing?  Take a look at the following headings, and some sample exit card prompts.  Which of the 4 purposes do your exit cards typically fulfill?

#### Questions designed for meta-cognitive reflection / connection:

1. How does today’s problem remind you of a problem you have solved before?
2. Explain one of the strategies discussed in class that was different than yours.  How was it different than yours?  How was it similar?
3. What do you believe was the purpose of today’s problem?  What did you learn?
4. What area gave you the most difficulty today?
5. Something that really helped me in my learning today was ….
6. What connection did you make today that made you say, “AHA! I get it!”
7. Describe how you solved a problem today.
8. Something I still don’t understand is …
9. Write a question you’d like to ask or something you’d like to know more about.
10. Did working with a partner make your work easier or harder. Please explain.

#### Questions targeted towards concepts:

1. Use a diagram to help explain how to compare 2 fractions with different denominators.
2. Create 2 addition questions, one that is easy to solve mentally and one that is harder.  Use a number line to explain how to answer both.  What makes one of the questions harder?
3. When would you ever need to know the area of a rectangle?  Pose a math problem using this information with an appropriate context and find the answer.
4. 1/2 + 1/3 = 5/6  When would you need to know this?  Pose a math problem using this information with an appropriate context.
5. When 2 even numbers are multiplied together we get an answer (product) that is even.  Is this always true?  Give examples to explain your thinking.
6. Draw a picture to explain why 2/3rds divided by 4/5ths is 10/12ths.
7. Two rectangles have the same area but very different perimeters.  What could the dimensions of the rectangles be?  Explain why you picked these rectangles

#### Questions targeted towards procedures:

1. Two fractions are added together.  Their sum is between 1/2 and 3/4 . Show 2 fractions that could possibly work.
2. Explain to someone unfamiliar with multiplying how to solve a 2-digit by 2-digit question.  Give an example.  Solve it in 2 different ways.
3. How many ways can you solve 68 + 18?  Explain each way.  Which was the most efficient for you?

#### Questions focused on clarifying misconceptions:

1. Susan and Jamie are arguing over which answer is correct to 12.1 + 25.25.  Susan believes the answer is 26.46.  Jamie thinks the answer is 37.35.  Explain how both got their answers.  Who is right?  What misconceptions led to the incorrect answer?
2. Phillip explained that 100cm2 is the same as 1m2.  Explain why he is correct/incorrect.
3. Explain one of the disagreements we had today during our congress.  What did we learn from our initial misconceptions?
4. Is 2/4 larger or smaller than 2/6?  How do you know?
5. Tracy says you can reduce 8/10 to 4/5.  Dan says they are the same, but Tracy says that 8/10 is larger.  Who is right?  How do you know?

When we are asking our students to provide us with information individually, I think the type of information we want to collect from them tells us a lot about our own beliefs about what is important!

Let’s take a quick look again at the 4 types…

Questions designed for meta-cognitive reflection / connection

Providing students with these types of prompts tells our students that we want them to be monitoring their own thinking, and that we value their thoughts and care about their identity with mathematics.

Questions targeted towards concepts

Providing students with these prompts tells our students we care about the big ideas in mathematics, and that we want our students to deeply understand the concepts we are learning about.  Asking questions like these asks our students to really pay attention to the thinking and reasoning of others in class discussions.

Questions targeted towards procedures

Providing questions like these asks our students to think flexibly and make decisions.  Although they ask for our students to show they understand procedures, we still keep the prompts open to help reach all of the students in our class.

Questions focused on clarifying misconceptions

Providing questions like these might take a bit more thought on our end, but it tells our students that mathematics is about constructing constructing and critiquing mathematical arguements.  See SMP 3 – Construct viable arguments and critique the reasoning of others.

I also want you to notice what kind of exit card I didn’t mention.  I didn’t provide room for closed tasks, ones where there is 1 right answer, where the answer doesn’t require thought or reasoning… Maybe this shows my bias!

Here is a printable version of my Exit Card Types document.  Share it, use it for ideas, use it to reflect on our own decision making…

## Concept vs Procedure: An anecdote about what it means to be good at math

A grade 5 teacher approached me asking if we could plan for a lesson toward the end of her unit on multiplication.  To start, we grabbed our curriculum to see what the specific expectations told us.  Specifically, she told me that she had been trying to help her students reach this expectation:

-Multiply two-digit whole numbers by two-digit whole numbers, using estimation, student-generated algorithms, and standard algorithms

I asked her what her students had done so far and was given a list of experiences including working with manipulatives, word problems and time to practice…  I asked her how her students were doing so far, and was told that her students were doing very well, with a few exceptions (4 boys and 1 girl were still struggling a bit).  I then asked her to look at the curriculum expectation with me again, pointing out a piece I wanted us to look at deeper:

-Multiply two-digit whole numbers by two-digit whole numbers, using estimation, student-generated algorithms, and standard algorithms

I asked which of the three (estimation, student generated algorithms, and standard algorithms) she had spent the most time on… to which I was met with a stunned look.  She shared with me that all of her time had been spent trying to help her students be able to understand the standard algorithm, but she felt that her students were doing really well anyway.

I asked if we could try something a little different, and see how well her students could estimate, to which she was eagerly agreeable.  Together we came up with this problem:

Before we started the lesson, I asked her how she would go about completing this task.  She pointed to the pieces in the picture that she could eliminate and gave clear explanations as to why they were definitely smaller than others (i.e., 42 x 65 must be smaller than 40 x 91 because there are a lot more 40s…).

As we walked into class she turned to me and pointed out the 5 students (4 boys and 1 girl) she believed would really struggle with the task.  Then, together we introduced the task and explained to her students that they had to do as few as possible to show us which one had the largest product, then be able to prove to the class how they knew it was the largest.  Students were then given time to think and work and show their reasoning.

The result of this problem couldn’t have been more revealing than it was!  Almost every student silently worked on and answered EVERY ONE of the given problems… all except for 6 students… 5 of whom were the students the teacher had just identified as struggling students!  These 6 had eliminated problems they realized they didn’t need to do, only working out those they weren’t sure about.  Only these 6 did any estimating at all!!!

We gave time for each of the 6 to explain how they started eliminating questions that they knew they didn’t have to do, and the rest of the class were quite receptive to their thinking/reasoning.  The teacher and I drew open arrays that were approximately proportional for us to think about the dimensions involved to represent what the 6 students had been explaining.

In the end, we gave a quick exit card with a few questions again asking which was the largest, to work out as few as possible, and asked for their reasoning… but still many struggled with this and just did them all.

After being part of this lesson I couldn’t help but reflect on a few really big questions:

• I wonder what it means to be good at multiplication?
• I wonder what it means to be good at math?
• I wonder if many other students in other classrooms struggle specifically with the procedures, yet are capable of thinking mathematically?
• I wonder which is better:
1. A student who can dutifully enact the steps of an algorithm quickly and accurately, yet can’t estimate, visualize, conceptualize what the algorithm is doing… or
2. A student who sometimes struggles with the algorithm, yet has a strong grasp of what multiplication means?
• I wonder WHY students who were SO successful with their learning multiplication so far struggled with this problem?
• I wonder what our students think being successful in mathematics means?
• I wonder what messages we send to make them believe this???

I’m also left with a thought about the curriculum and how we perceive what it is asking us to do.  I believe that when we read our curriculum, we read it through the lens of what we value, what we understand, and our own personal experiences!  And for many of us, what we end up valuing is the skill… or the procedure… or the knowledge…  Yet, for me, I wish we also valued understanding… and thinking… and reasoning…