Seems to me that many schools and districts are asking questions about assessment in mathematics. So, I thought I would share a few quotes that might get you to think and reflect on your views about what it means to assess, why there might be a focus on assessment, and what our goals and ideals might look like. I want you to take a look at the following quotes. Pick 1 or 2 that stands out to you:

A few things to reflect on as you think about the quotes above:

Which quotes caught your eye? Did you pick one(s) that confirm things you already believe or perhaps ones that you hadn’t spent much time thinking about before?

Some of the above quotes speak to “assessment” while others speak to evaluation practices. Do you know the difference?

Take a look again at the list of quotes and find one that challenges your thinking. I’ve probably written about the topic somewhere. Take a look in the Links to read more about that topic.

Why do you think so many discuss assessment as a focus in mathematics? Maybe Linda Gojak’s article Are We Obsessed with Assessment? might provide some ideas.

Instead of talking in generalities about topics like assessment, maybe we need to start thinking about better questions to ask, or thinking deeper about what is mathematically important, or understanding how mathematics develops!

Please pick a quote that stands out for you and share your thoughts about it.

I’ve been thinking a lot about how to meet the various needs of students in our classrooms lately. If we think about it, we are REALLY good at differentiated instruction in subjects like writing, yet, we struggle to do differentiated instruction well in subjects like math. Why is this???

In writing class, everyone seems to have an entry point. The teacher puts a prompt up on the board and everyone writes. Because the prompt is open, every student has something to write about, yet the writing of every student looks completely different. That is, the product, the process and/or the content differs for each student to some degree.

Teachers who are comfortable teaching students how to write know that they start with having students write something, then they provide feedback or other opportunities for them to improve upon. From noticing what students do in their writing, teachers can either ask students to fix or improve upon pieces of their work, or they can ask the class to work on specific skills, or ask students to write something new the next day because of what they have learned. Either way the teacher uses what they noticed from the writing sample and asks students to use what they learned and improve upon it!

In Math class, however, many teachers don’t take the same approach to learning. Some tell every student exactly what to do, how to do it, and share exactly what the finished product should look like. OR… in the name of differentiated instruction, some teachers split their class into different groups, those that are excelling, those that are on track, and those that need remediation. To them, differentiated instruction is about ability grouping – giving everyone different things. The two teachers’ thinking above are very different aren’t they!

Imagine these practices in writing class again. Teacher 1 (everyone does the exact same thing, the exact same way) would show students how to write a journal (let’s say), explain about the topic sentence, state the number of sentences needed per paragraph, walk every student through every step. The end products Teacher 1 would get, would be lifeless replications of the teacher’s thinking! While this might build some competence, it would not be supporting young creative writers.

Teacher 2 (giving different things to different groups) on the other hand would split the class into 3 or 4 groups and give everyone a different prompt. “Some of you aren’t ready for this journal writing topic!!!” Students in the high group would be allowed to be creative… students in the middle group wouldn’t be expected to be creative, but would have to do most of what is expected… and those in the “fix-up” group would be told exactly what to do and how to do it. While this strategy might seem like targeted instruction, sadly those who might need the most help would be missing out on many of the important pieces of developing writers – including allowing them to be engaged and interested in the creative processes.

Teacher 1 might be helpful for some in the class because they are telling specific things that might be helpful for some.

Teacher 2 might be helpful for some of the students in the class too… especially those that might feel like they are the top group.

But something tells me, that neither are allowing their students to reach their potential!!!

Think again to the writing teacher I described at the beginning. They weren’t overly prescriptive at first, but became more focused after they knew more about their students. They provided EVERYONE opportunity to be creative and do the SAME task!

In math, the most effective strategy for differentiating instruction, in my opinion, is using open problems. When a task is open, it allows all students to access the material, and allows all students to share what they currently understand. However, this isn’t enough. We then need to have some students share their thinking in a lesson close (this can include the timely and descriptive feedback everyone in the group needs). Building the knowledge together is how we learn. This also means that future problems / tasks should be built on what was just learned.

We know that to differentiate instruction is to allow for differences in the products, content and/or processes of learning… However, I think what might differ between teacher’s ability to use differentiated instruction strategies is if they are Teacher-Centered… or Student-Centered!

When we are teacher-centered we believe that it is our job to tell which students should be working on which things or aim to control which strategies each student will be learning. However, I’m not convinced that we would ever be able to accurately know which strategies students are ready for (and therefore which ones we wouldn’t want them to hear), nor am I convinced that giving students different things regularly is healthy for our students.

Determining how to place students in groups is an important decision. Avoid continually grouping by ability. This kind of grouping, although well-intentioned, perpetuates low levels of learning and actually increases the gap between more and less dependent students. Instead, consider using flexible grouping in which the size and makeup of small groups vary in a purposeful and strategic manner. When coupled with the use of differentiation strategies, flexible grouping gives all students the chance to work successfully in groups. Van de Walle – Teaching Student Centered Mathematics

If we were to have students work on a problem in pairs, we need to be aware that grouping by ability as a regular practice can actually lead students to develop fixed mindsets – that is they start to recognize who is and who isn’t a math student.

Obviously there are times when some students need remediation, however, I think we are too quick to jump to remediation of skills instead of attempting to find ways to allow students to make sense of things in their own way followed by bringing the learning / thinking together to learn WITH and FROM each other.

To make these changes, however, I think we need to spend more time thinking about what a good problem or rich task should look like! Maybe something for a future post?

As always, I want to leave you with a few reflective questions:

I chose to compare what differentiated instruction looks like in mathematics to what it looks like in writing class. However, I often hear more comparisons between reading and mathematics. Do you see learning mathematics as an expressive subject like writing or a receptive subject like reading?

Where do you find tasks / problems that offer all of your students both access and challenge (just like a good writing prompt)? How do these offer opportunities for your students to vary their process, product and/or content?

Once we provide open problems for our students, how do you leverage the reasoning and representations from some in the room to help others learn and grow?

Math is very different than Literacy. Reading and writing, for the most part, are skills, while mathematics is content heavy. So how do you balance the need to continually learn new things with the need to continually make connections and build on previous understanding?

What barriers are there to viewing differentiated instruction like this? How can we help as an online community?

Throughout this year I have been thinking a lot about perspective. It’s why I wrote a post about Spatial Reasoning here, and shared templates for skyscraper puzzles here and why I shared my presentation at this year’s OAME that included perspective here. It’s also why I shared videos like this or this or this.

So why is perspective so important? Mathematically, perspective taking involves us being able to mentally rotate objects in our mind, it includes us composing and decomposing shapes and figures into other shapes and figures… Really, perspective taking is one small piece of what Spatial Reasoning (Read this monograph to learn more) is all about:

Another look at perspective:

Have you ever been in a situation before where somebody is discussing with you their students – the ones you will be getting next year? Was it enlightening or awkward, helpful or fear-inducing? In my buildings, this is the time of year when lots of decisions start to happen for next year. Administrators moving to new schools and new ones coming, new teachers being hired, experienced teachers being given new assignments within the school or in a new school, and of course, students about to be promoted to the next grade. And with change comes uncertainty and anxiety.

In all of this uncertainty we often have opportunities to discuss with others some of our personal thoughts and feelings about our year, those who we work with, those who we teach…… So I thought I would challenge you to think about what your typical conversations look like.

A few things to think about when discussing others in your building:

Do you tend to see the good in others and describe things in a positive light or aim to help others see the potential issues and risks?

Do you tend to talk in generalities or specifics?

How will the other teacher/administrator perceive the messages you have given?

What information would you like to hear from others? How specific and detailed of a description would you like?

Most importantly, where is your line? You know, the line where it’s too much information and is either starting to cause you more stress or where you realize that the information is too negative.

A challenge for you:

What conversations have you had lately to describe to others about any of these changes? I think it can be powerful to hear both positive and negative experiences so we can think about how to navigate through changes.

If you are changing roles, or schools, or even getting new students, what conversations have you had? What might you have done differently?

I’ve been asked to share my OAME 2017 presentation on Mathematical Intuitions by a few of my participants. Instead of just sharing the slides, I thought I would add a bit of the conversations we had, and the purposes behind a few of my slides. Here is a brief explanation of the 75 minutes we shared together:

I started with an image of the OAME 2017 official graphic and asked everyone what mathematics they saw in the photo:

I was impressed that many of us noticed various things from numbers, to sizes of fonts, to shapes and other geometric features, to measurement concepts to patterns…

I decided to start with an image so I could listen to everyone’s ideas (the group could have simply noticed the numbers visible on the page, or the triangles, but thankfully the group noticed a lot more!).

I then shared a few stories where students have entered into a problem where they have attempted to do a bunch of procedures or calculations without ever doing any thinking, either before or after, to make sure they are making sense of things.

You can read the full stories on these 2 slides here and here.

The bandana problem above is a really interesting one for me because it shows just how likely our previous learning can actually get in the way of students who are attempting to make sense of things. Most students who learned about how to convert in previous years in a procedural way have difficulty realizing that 1 meter squared is actually 10,000 cm squared!

In an attempt to explain the kinds of mental actions we actually want our students to use when learning and doing mathematics I showed an image shared by Tracy Zager (from her new book Becoming the Math Teacher You Wish You’d Had). We discussed just how interrelated Logic and Intuition are. Students who are using their intuition start by making sense of things. They start by making choices or estimates, which are often based on their previous experiences, and use logic to continue to refine and think through what makes sense. This process, while often not even realized by those who are confident with their mathematics, is one I believe we need to foster and bring to the forefront of our discussions.

I then shared the puzzle above with the group and asked them to find the value of the question mark. Most did exactly what I assumed they would do… but none did what the following student did:

Most teachers aimed to find the value of each image (which isn’t as easy as it looks for many elementary teachers), but the student above didn’t. They instead realized that all of the shapes if you add them up in any direction would equal 94. This student had never been given a problem like this, so didn’t have any preconceived notions about how to solve it. They instead, thought about what makes sense.

So, how DO we help our students use their intuition? Here are a few ideas I shared:

Contemplate then Calculate routine (See David Wees for more about this here or here, or purchase Routines for Reasoning by Grace, Amy and Susan)

The two images above show visual representations (thank you Andrew Gael and Fawn Nguyen for your images) where I asked everyone to attempt to think before they did any calculations. I used Andrew’s picture of the dominoes and asked “will the two sides balance… don’t do any calculations though”. For Fawn’s Visual Pattern, I asked the group to explain what the 10th image would LOOK LIKE (before I wanted them to figure out how many of each shape would be there, and then find a rule for the nth term).

We shared a few estimation strategies:

and a few “Notice and Wonder” ideas:

However, while I love each of the strategies discussed here (Contemplate then Calculate, Estimation routines, Notice and Wonder) I’m not sure that doing a routine like these, then going about the actual learning of the day is going to be effective!

Instead, we need to make sure that noticing things, estimating, thinking happen all the time. These need to be a part of every new piece of learning, not just fun or neat warm-ups!

Building our students’ intuition means that we need to provide opportunities for them them to think and make sense of things, and have plenty of opportunities for them to discuss their thinking!

If our goal is for students to think mathematically, and use their logic and intuition regularly, we need to operate by a few simple beliefs:

I ended the presentation with a final thought:

Here is a copy of the presentation if you are interested:

I’d suggest you scroll down to slide 49 and play the quick video of one of my students doing a spatial reasoning puzzle. It’s one of my favourites because it illustrates visually the thinking processes used when a student is using both their intuition and logic.

To me, there seems to be so much more I need to learn about how to help my students who seem to struggle in math class use their intuition. Hopefully this conversation is just the beginning of us learning more about the topic!

A few questions I want to leave you with:

What routines do you have in place that help your students make sense of things, use their intuitions and develop mathematical reasoning?

Do your students use their intuition in other situations as well (or just during these routines)?

How can you start to build in opportunities for your students to use their intuition as a regular part of how your class is structured?

What does it look like when our students who are struggling attempt to use their intuition? How can we help all of our students develop and use these process regularly?

Special thanks to Tracy Zager’s new book for the inspiration for the presentation.

As always, I would love to continue the conversation here or on Twitter

For the last few months, a team of kindergarten teachers and myself have been working together to deepen our understanding of early years mathematics, spatial reasoning, and the importance of guided play as a vehicle to engage our students to think mathematically. Below is a copy of our slideshow presentation we shared at OAME 2017, and some of the documents we have created over the past few months.

A quick synopsis of our work first:

While our research led us toward Doug Clement’s work about trajectories, and research about spatial reasoning and early mathematics, much of the tasks we actually did with students directly came from the book shown above (Taking Shape) which I can’t recommend enough – if you can, get yourself a copy!

We discussed the quote above to help us realize what actually underpins mathematics success. More details about how the quote ends here.

We shared research showing just how important early mathematics is, and specifically what the kinds of instruction could / should look like to accomplish this learning. Duncan et. al., is a widely quoted piece of research that has led many to realize that early math learning needs to be a focus in schools – even more so than early reading!

We played a few games that helped us stretch our spatial reasoning abilities. The image above was part of our “See It, Build It, Check It” activity (found in Taking Shape). Everyone saw the image for a minute, then was asked to build it once the image was removed. What we noticed is just how difficult spatial tasks are for us!

After we had the opportunity to play for a bit, we dug back into the research about spatial reasoning and the jobs typically chosen based on (high school) spatial ability. Hopefully you noticed something interesting on the graph above!

So, we know how important spatial reasoning is, but the 3 pieces above (taken from Paying Attention to Spatial Reasoning document) might help us realize how important a focus on spatial reasoning is for both our students and us.

In our time together, we learned a lot about the importance of observing our students as they were engaged in the learning. From the initial choices they made, to how they overcame obstacles, to understanding the mental actions that were happening… Observing students in the moment is far more powerful than collecting correct answers!

See the link at the bottom of the page for our connections to Doug Clements’ work.

We also discussed the specific connections between the mathematics behaviours and the learning that happened beyond. In our Kindergarten program document, our students’ expectations fall under 4 frames (see above) so we linked the learning we saw to the program document in a way that helps us see the depth and breadth of the kindergarten program (see document linked below).

We then ended our presentation with a synopsis of what we learned throughout our work together. While the slideshow might be helpful here (I’d love for someone to comment on those slides at the end), the conversations that Sue, Kristi and Kristen had with others have shown me just how valuable it is to spend time learning together. I couldn’t be prouder to be able to work with such reflective and dedicated teachers!

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

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

“This program helps improve your students’ reading scores, but it may make them hate reading forever.” No such information is given to teachers or school principals.

“This practice can help your children become a better student, but it may make her less creative.” No parent has been given information about effects and side effects of practices in schools.

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

Let’s explore a few possible scenarios:

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Practice:

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

Unintended Side Effects:

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

Our Decisions:

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

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

If we measure fact retrieval, what are the unintended side effects? What does this tell our students math is all about? Who does this tell us math is for?

If we measure via multiple choice or fill-in-the-blank questions as a common practice, what are the unintended side effects? What does this tell our students math is all about? How reliable is this information?

If we measure items from last year’s standards (expectations), what are the unintended side effects? Will we spend our classroom time giving experiences from prior grades, help build our students’ understanding of current topics?

If we only value standardized measurements, what are the unintended side effects? Will we see classrooms where development of mathematics is the focus, or “answer getting” strategies? What will our students think we value?

Some things to reflect on

Think about what it is like to be a student in your class for a moment. What is it like to learn mathematics every day? Would you want to learn mathematics in your class every day? What would your students say you value?

Think about the students in front of you for a minute. Who is good at math? What makes you believe they are good at math? How are we building up those that don’t see themselves as mathematicians?

Consider what your school and your district ask you to measure. Which of the 5 strands of mathematics proficiency do these measurements focus on? Which ones have been given less attention? How can we help make sure we are not narrowing our focus and excluding some of the things that really matter?

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

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

The task:

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

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

Student ideas

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

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

Example 1

Example 2

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

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

Example 4

Example 5

Example 6

And some students used different shapes to cover the figures.

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

Building Meaningful Conversations

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

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

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

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

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

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

We need comparable units if we are to compare 2 or more figures together. This could mean using same-sized units (like examples 1, 4, 5 & 6 above), or corresponding units (like example 8 above), or units that can be reorganized and appropriately compared (like example 9).

If we want to determine the area numerically, we need to use the same-sized piece exclusively.

The smaller the unit we use, the more of them we will need to use.

It is difficult to find the exact area of figures with rounded parts using the tools we have. So, our measurements are not precise.

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

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

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

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

Little to no instruction was given – we wanted to learn about our students’ thinking, not see if they can follow directions

The problem was open enough to have multiple possible strategies and offer multiple possible entry points (low floor – high ceiling)

Asking students to prove something opens up many possibilities for rich discussions

Students needed to begin by using their reasoning skills, not procedural knowledge…

Coming up with a response involved students doing and thinking… but the real learning happened afterward – during the consolidation phase

How often do you give tasks hoping students will solve it a specific way? And how often you give tasks that allow your students to show you their current thinking? Which of these approaches do you value?

What do your students expect math class to be like on the first few days of a new topic/concept? Do they expect marks and quizzes? Or explanations, notes and lessons? Or problems where students think and share, and eventually come to understand the mathematics deeply through rich discussions? Is there a disconnect between what you believe is best, and what your students expect?

I’ve painted the picture here of formative assessment as a way to help us learn about how our students think – and not about gathering marks, grouping students, filling gaps. What does formative assessment look like in your classroom? Are there expectations put on you from others as to what formative assessment should look like? How might the ideas here agree with or challenge your beliefs or the expectations put upon you?

Time is always a concern. Is there value in building/constructing the learning together as a class, or is covering the curriculum standards good enough? How might these two differ? How would you like your students to experience mathematics?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Tip #3 – Eliminate the obviously wrong answer first.

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

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

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

Some advice.

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

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

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

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

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

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

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

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

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

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

What is Content Knowledge?

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

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

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

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

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

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

What is Pedagogical Knowledge?

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

Understanding how to help students construct knowledge and a belief that constructivist principles are necessary to help all students be successful;

Strategies to help promote thinking and discourse in all students (i.e., practices for orchestrating productive conversations, importance of wait time, cooperative learning opportunities…);

Which is More Important: Pedagogy or Content Knowledge?

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

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

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

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

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

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

What should Professional Development Look Like?

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

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

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

Some personal beliefs:

We don’t know what we don’t know. That is, given any simple concept or skill, there is likely an abundance of research, best practices, connections to other concepts/skills, visual approaches… that you are unaware of. Professional development can help us learn about what we weren’t even aware we didn’t know about.

Districts and schools tend to focus on pedagogy far more than content. Topics like grading, how to assess, differentiated instruction… tend to be hot topics and are easier to monitor growth than the importance of helping our students develop a relational understanding. However, each of those pedagogical decisions relies on our understanding of the content (i.e., if we are trying to get better at assessment, we need to have a better understanding of developmental trajectories/continuums so we know better what to look for). The best way to understand any pedagogy is to learn about the content in new ways, then consider the pedagogy involved.

Quality resources are essential, but handing out a resource is not the same as professional development. Telling others to use a resource is not the same as professional development, no matter how rich the resource is! Using a resource as a platform to learn things is better than explaining how to use a resource. The goal needs to be our learning and hoping that we will act on our learning, not us doing things and hoping we will learn from it.

The knowledge might not be in the room. An old adage tells us that when we are confronted with a problem, that the knowledge is in the room. However, I am not sure this is always the case. If we are to continue to learn, we need experts helping us to learn! Otherwise we will continually recycle old ideas and never learn anything new as a school/district. If we want professional development, we need new ideas. This can come in the form of an expert or more likely in the form of a resource that can help us consider content & pedagogy.

Learning complicated things can’t be transmitted. Having someone tell you about something is very different than experiencing it yourself. Learning happens best when WE are challenged to think of things in ways we hadn’t before. Professional development needs to be experiential for it to be effective!

Experiencing learning in a new way is not enough. Seeing a demonstration of a great teaching strategy, or being asked to do mental math and visually represent it, or experiencing manipulatives in ways you hadn’t previously experienced is a great start to learning, but it isn’t enough. Once we have experienced something, we need to discuss it, think about the pedagogy and content involved, and come to a shared understanding of what was just learned.

Professional learning can happen in a lot of different places and look like a lot of different things. While many might assume that professional development is something that we sign up for in a building far from your own school, hopefully we can recognize that we can learn from others in a variety of contexts. This happens when we show another teacher in our school what we are doing or what our students did on an assignment that we hadn’t expected. It happens when we disagree on twitter or see something we would never have considered before. We are engaging in professional development when we read a professional book or blog post that helps us to consider new things and especially when those things make us reconsider previous thoughts. And professional learning especially happens when we get to watch others’ teach and then discuss what we noticed in both the teaching and learning of the material.

Some of the best professional development happens when we are inquiring about teaching / learning together with others, and trying things out together! When we learn WITH others, especially when one of the others is an expert, then take the time to debrief afterward, the learning can be transformational.

Beliefs about how students learn mathematics best is true for adults too. This includes beliefs about constructivist approaches, learning through problem solving, use of visual/spatial reasoning, importance of consolidating the learning, role of discourse…

Not everyone gets the same things out of the same experiences. Some people are reflecting much more than others during any professional learning experience. Personally, I always try to make connections, think about the leaders’ motivations/beliefs, reflect on my own practices… even if the topic is something I feel like I already understand. There is always room for learning when we make room for learning!

Some things to reflect on:

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

Think of a time you came to make a change in your beliefs about what is important in teaching mathematics. What led to that change?

Think of a time you tried something new. What helped you get started?

Where do you get your professional learning? Is your board / school providing the kind of learning you want/need? If so, how do you take advantage of this more often? If not, how could this become a reality?

Think about your answers to any of the above questions. Were you considering learning about pedagogy or content knowledge? What does this say about your personal beliefs about professional development?

Take a look again at any of the points I made under “Some Personal Beliefs”. Is there one you have issue with? I’d love some push-back or questions… that’s how we learn:)

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