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?
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?
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:
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.
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.
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.
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.
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.
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.
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.
Given very little direction and lots of time to think about how to solve this problem, we saw a wide range of student thinking. Take a look at a few:
Some students used circles to help them find area. What does this say about what they understand? What issues do you see with this approach though?
Some students used shapes to cover the outline of each shape (perimeter). Will they be able to find the shape with the greater area? Is this strategy always / sometimes / never going to work? What does this strategy say about what they understand?
Some students used identical shapes to cover the inside of each figure.
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!
Provide individualized instruction based on where students currently are
Let’s take a closer look at each of these beliefs:
Those that believe the answer is providing all students with same tasks and experiences often do so because of their focus on their curriculum standards. They believe the teacher’s role is to provide their students with tasks and experiences that will help all of their students learn the material. There are a few potential issues with this approach though (i.e., what to do with students who are struggling, timing the lesson when some students might take much more time than others…).
On the other hand, others believe that the best answer is individualized instruction. They believe that students are in different places in their understand and because of this, the teacher’s role is to continually evaluate students and provide them with opportunities to learn that are “just right” based on those evaluations. It is quite possible that students in these classrooms are doing very different tasks or possibly the same piece of learning, but completely different versions depending on each student’s ability. There are a few difficulties with this approach though (i.e., making sure all students are doing the right tasks, constantly figuring out various tasks each day, the teacher dividing their time between various different groups…).
There are two seemingly opposite educational ideals that some might see as competing when we consider the two approaches above: Differentiated Instruction and Complexity Science. However, I’m actually not sure they are that different at all!
For instance, the term “differentiated instruction” in relation to mathematics can look like different things in different rooms. Rooms that are more traditional or “teacher centered” (let’s call it a “Skills Approach” to teaching) will likely sort students by ability and give different things to different students.
If the focus is on mastery of basic skills, and memorization of facts/procedures… it only makes sense to do Differentiated Instruction this way. DI becomes more like “modifications” in these classrooms (giving different students different work). The problem is, that everyone in the class not on an IEP needs to be doing the current grade’s curriculum. Really though, this isn’t differentiated instruction at all… it is “individualized instruction”. Take a look again at the Monograph: Differentiating Mathematics Instruction.
Differentiated instruction is different than this. Instead of US giving different things to different students, a student-centered way of making this make sense is to provide our students with tasks that will allow ALL of our students have success. By understanding Trajectories/Continuum/Landscapes of learning (See Cathy Fosnot for a fractions Landscape), and by providing OPEN problems and Parallel Tasks, we can move to a more conceptual/Constructivist model of learning!
Think about Writing for a moment. We are really good at providing Differentiated Instruction in Writing. We start by giving a prompt that allows everyone to be interested in the topic, students then write, we then provide feedback, and students continue to improve! This is how math class can be when we start with problems and investigations that allow students to construct their own understanding with others!
The other theory at play here is Complexity Science. This theory suggest that the best way for us to manage the needs of individual students is to focus on the learning of the class.
The whole article is linked here if you are interested. But basically, it outlines a few principles to help us see how being less prescriptive in our teaching, and being more purposeful in our awareness of the learning that is actually happening in our classrooms will help us improve the learning in our classrooms. Complexity Science tells us to think about how to build SHARED UNDERSTANDING as a group through SHARED EXPERIENCES. Ideally we should start any new concept with problem solving opportunities so we can have the entire group learn WITH and FROM each other. Then we should continue to provide more experiences for the group that will build on these experiences.
Helping all of the students in a mixed ability classroom thrive isn’t about students having choice to do DIFFERENT THINGS all of the time, nor is it about US choosing the learning for them… it should often be about students all doing the SAME THING in DIFFERENT WAYS. When we share our differences, we learn FROM and WITH each other. Learning in the math classroom should be about providing rich learning experiences, where the students are doing the thinking/problem solving. Of course there are opportunities for students to consolidate and practice their learning independently, but that isn’t where we start. We need to start with the ideas from our students. We need to have SHARED EXPERIENCES (rich problems) for us to all learn from.
As always, I leave you with a few questions for you to consider:
How do you make sure all of your students are learning?
Who makes the decisions about the difficulty or complexity of the work students are doing?
Are your students learning from each other? How can you capitalize on various students’ strengths and ideas so your students can learn WITH and FROM each other?
How can we continue to help our students make choices about what they learn and how they demonstrate their understanding?
Do you see the relationship between Differentiated Instruction and the development of mathematical reasoning / creative thinking? How can we help our students see mathematics as a subject where reasoning is the primary goal?
How can we foster playful experience for our students to learn important mathematics and effectively help all of our students develop at the same time?
What is the same for your students? What’s different?
I’d love to continue the conversation. Write a response, or send me a message on Twitter (@markchubb3).
In my role, I have the advantage of seeing many great teachers honing and refining their craft, all to provide the best possible experiences for their students. The dedication and professionalism that the teachers I work with continue to demonstrate is what keeps me going in my role!
One particularly interesting benefit I have is when I can be part of the same lesson multiple times with different teachers. When I am part of the same lesson several times I have come to notice the differences in the small decisions we make. It is here in these small decisions that have the biggest impact on the learning in our classrooms. For instance, in any given lesson:
Who gets the most talk time? You, all students, same few students…?
There are so many little decisions we make (linked above are posts discussing several of the decisions). However, I want to discuss a topic today that isn’t often thought about: Scaffolding.
For the past few months, the teachers / instructional coaches taking my Primary/Junior Mathematics additional qualifications course have been leading lessons. Each of the lessons follow the 3-part lesson format, are designed to help us “spatialize” the curriculum (allow all of us to experience the content in our curriculum via visuals / representations / manipulatives), and have a specific focus on the consolidation phase of the lesson (closing). After each lesson is completed I often lead the group in a discussion either about the content that we experienced together, or the decisions that the leader choose. Below is a brief description of the discussion we had after one particular lesson.
First of all, however, let me share with you a brief overview of how the lesson progressed:
As a warm up we were asked to figure out how many unique ways you can arrange 4 cubes.
We did a quick gallery walk around the room to see others’ constructed figures.
We shared and discussed the possible unique ways and debated objects that might be rotations of other figures, and those that are reflections (take a look at the 8 figures below).
The 3 pages of problems were given to all (see below). Everyone had time to work independently, but sharing happened naturally at our tables.
The lesson close included discussions about how we tackled the problems. Strategies, frustrations, what we noticed about the images… were shared.
While the teacher leader made the decision to hand out all 9 problems (3 per sheet) at the same time, I think some teachers might make a different decision. Some might decide to take a more scaffolded approach. Think about it, which would you likely do:
Hand out all 9 problems, move around the room and observe, offer focusing questions as needed, end in a lesson close; or
Ask students to do problem 1, help those that need it, take up problem 1, ask students to do problem 2, help those that need it, take up problem 2…
This decision, while seemingly simple, tells our students a lot about your beliefs about how learning happens, and what you value.
So as a group of teachers we discussed the benefits and drawbacks of both approaches. Here are our thoughts:
The more scaffolded approach (option 2) is likely easier for us. We can control the class easier and make sure that all students are following along. Some felt like it might be easier for us to make sure that we didn’t miss any of our struggling students. However, many worried that this approach might inhibit those ready to move on, and frustrate those that can’t solve it quickly. Some felt like having everyone work at the same pace wasn’t respectful of the differences we have in our rooms.
On the other hand, some felt that handing out all 9 puzzles might be intimidating for a few students at first. However, others believed that observing and questioning students might be easier because there would be no time pressure. They felt like we could spend more time with students watching how they tackle the problems.
Personally, I think our discussion deals with some key pieces of our beliefs:
Do we value struggle? Are we comfortable letting students productively keep trying?
Are we considering what is best for us to manage things, or best for our students to learn (teacher-centered vs student-centered)?
What is most helpful for those that struggle with a task? Lots of scaffolding, telling and showing? Or lots of time to think, then offer assistance if needed?
In reality, neither of these ways will likely actually happen though. Those who start off doing one problem at a time, will likely see disengagement and more behaviour problems because so many will be waiting. When this happens, the teacher will likely let everyone go at their own pace anyway.
Similarly, if the teacher starts off letting everyone go ahead at their own pace, they might come across several of the same issues and feel like they need to stop the class to discuss something.
While both groups will likely converge, the initial decision still matters a lot. Assuming the amount and types of scaffolding seems like the wrong move because there is no way to know how much scaffolding might be needed. So many teachers default by making sure they provide as much scaffolding as possible however, when we over-scaffold, we purposely attempt to remove any sense of struggle from our students, and when we do this, we remove our students’ need to think! When we start by allowing our students to think and explore, we are telling our students that their thoughts matter, that we believe they can think, that mathematics is about making sense of things, not following along!
So I leave you with a few thoughts:
Do your students expect you to scaffold everything? Do they give up easily? How can we change this?
When given an assignment do you quickly see a number of hands raise looking for help? Why is this? How can we change this?
At what point do you offer any help? What does this “help” look like? Does it still allow your students opportunities to think and make sense of things?
When we scaffold everything, we might be helping them with today’s work, but we are robbing them of the opportunity of thinking. When we do this, we rob them of the enjoyment and beauty of mathematics itself!
There seems to be a lot of research telling us how important feedback is to student performance, however, there’s little discussion about how we give this feedback and what the feedback actually looks like in mathematics. To start with, here are a few important points research says about feedback:
The timing of feedback is really important
The recipient of the feedback needs to do more work than the person giving the feedback
Students need opportunities to do something with the feedback
I will talk about each of these toward the end of this post. First, I want to explain a piece about feedback that isn’t mentioned enough… Providing students with feedback positions us and our students as learners. Think about it for a second, when we “mark” things our attention starts with what students get right, but our attention moves quickly to trying to spot errors. Basically, when marking, we are looking for deficits. On the other hand, when we are giving feedback, we instead look for our students’ actual thinking. We notice things as almost right, we notice misconceptions or overgeneralization…then think about how to help our students move forward. When giving feedback, we are looking for our students strengths and readiness.Asset thinking is FAR more productive, FAR more healthy, FAR more meaningful than grades!
Feedback Doesn’t Just Happen at the End!
Let’s take an example of a lesson involving creating, identifying, and extending linear growing patterns. This is the 4th day in a series of lessons from a wonderful resource called From Patterns to Algebra. Today, the students here were asked to create their own design that follows the pattern given to them on their card.
Once students made their designs, they were instructed to place their card upside down on their desk, and to circulate around the room quietly looking at others’ patterns. Once they believed they knew the “pattern rule” they were allowed to check to see if they were correct by flipping over the card.
After several minutes of quiet thinking, and rotating around the room, the teacher stopped everyone and led the class in a lesson close that involved rich discussions about specific samples around the room. Here is a brief explanation of this close:
Teacher: Everyone think of a pattern that was really easy to tell what the pattern rule was. Everyone point to one. (Class walks over to the last picture above – picture 6). What makes this pattern easy for others to recognize the pattern rule? (Students respond and engage in dialogue about the shapes, colours, orientation, groupings…).
Teacher: Can anyone tell the class what the 10th position would look like? Turn to your partner and describe what you would see. (Students share with neighbor, then with the class)
Teacher: Think of one of the patterns around the room that might have been more difficult for you to figure out. Point to one you want to talk about with the class. (Students point to many different ones around the room. The class visits several and engages in discussions about each. Students notice some patterns are harder to count… some patterns follow the right number of tiles – but don’t follow a geometric pattern, some patterns don’t reflect the pattern listed on the card. Each of these noticings are given time to discuss, in an environment that is about learning… not producing. Everyone understands that mistakes are part of the learning process here and are eager to take their new knowledge and apply it.
The teacher then asks students to go back to their desks and gives each student a new card. The instructions are similar, except, now she asks students to make it in a way that will help others recognize the patterns easily.
The process of creating, walking around the room silently, then discussing happens a second time.
To end the class, the teacher hands out an exit card asking students to articulate why some patterns are easier than others to recognize. Examples were expected from students.
At the beginning of this post I shared 4 points from research about feedback. I want to briefly talk about each:
The timing of feedback is really important
Feedback is best when it happens during the learning. While I can see when it would be appropriate for us to collect items and write feedback for students, having the feedback happen in-the-moment is ideal! Dan Meyer reminds us that instant feedback isn’t ideal. Students need enough time to think about what they did right/wrong… what needs to be corrected. On the other hand, having students submit items, then us giving them back a week later isn’t ideal either! Having this time to think and receive feedback DURING the learning experience is ideal. In the example above, feedback happened several times:
As students walked around looking at patterns. After they thought they knew the pattern, they peeked at the card.
As students discuss several samples they are given time to give each other feedback about which patterns make sense… which ones visually represented the numeric value… which patterns could help us predict future visuals/values
Afterward once the teacher collected the exit cards.
The recipient of the feedback needs to do more work than the person giving the feedback
Often we as teachers spend too much time writing detailed notes offering pieces of wisdom. While this is often helpful, it isn’t a feasible thing to do on a daily basis. In fact, us doing all of the thinking doesn’t equate to students improving! In the example above, students were expected to notice patterns that made sense to them, they engaged in conversations about the patterns. Each student had to recognize how to make their pattern better because of the conversations. The work of the feedback belonged, for the most part, within each student.
Students need opportunities to do something with the feedback
Once students receive feedback, they need to use that feedback to continue to improve. In the above example, the students had an opportunity to create new patterns after the discussions. After viewing the 2nd creations and seeing the exit cards, verbal or written feedback could be given to those that would benefit from it.
This last piece is an interesting one. Feedback, by definition, is about seeing how well you have come to achieving your goal. It is about what you did, not about what you need to do next. “I noticed that you have switched the multiplicative and additive pieces in each of your patterns” is feedback. “I am not sure what the next position would look like because I don’t see a pattern here” is feedback. “The additive parts need to remain constant in each position” is not feedback… it is advice (or feedforward).
In the example above, the discussions allowed for ample time for feedback to happen. If students were still struggling, it is appropriate to give direct advice. But I’m not sure students would have understood any advice, or retained WHY they needed to take advice if we offered it too soon.
So I leave you with some final questions for you:
When do your students receive feedback? How often?
Who gives your students their feedback?
Is it written? Or verbal?
Which of these do you see as the most practical? Meaningful for your students? Productive?
How do you make time for feedback?
Who is doing the majority of the work… the person giving or the person receiving the feedback?
Do your students engage in tasks that allow for multiple opportunities for feedback to happen naturally?
PS. Did you notice which of the students’ examples above had made an error. What feedback would you give? How would they receive this feedback?
There are several great unsolved math problems that are perfect for elementary students to explore. One of my favourites is the palindrome sums problem.
In case you aren’t familiar, a palindrome is a word, phrase, sentence or number that reads the same forward and backward.
The problem itself comes out of this conjecture:
If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome.
Let’s take a look at the numbers 12, 46 and 95:
As you can see with the above examples, some numbers can become palindromes with 1 simple addition, like the number 12. 12 + 21 = 33. Can you think of others that should take 1 step? How did you know?
Other numbers when added to their inverse will not immediately become a palindrome, but by continuing the process, will eventually, like the numbers 46 and 95. Which numbers do you think will take more than 1 step?
After going through a few examples with students about the process of creating palindromes, ask your students to attempt to see if the conjecture is true (If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome). Have them find out if each number from 0-99 will eventually become a palindrome. Ask students if they need to find the answer for each number? Encourage them to make their own conjectures so they don’t need to do all of the calculations for each number.
Mathematically proficient students look closely to discern a pattern or structure. They notice if calculations are repeated, and look both for general methods and for shortcuts. (SMP 7)
Some students will notice:
Some numbers will already be a palindrome
If they have figured out 12 + 21… they will know 21 + 12. So they don’t have to do all of the calculations. Nearly half of the work will have already been completed.
A pattern emerging in their answers … 44, 55, 66, 77, 88, 99, 121… and see this pattern regularly (well, almost)
A pattern about when numbers will have 99 as a palindrome, or 121… 18+81 = 27+72 = 36+63…
Thinking about our decisions:
What is the goal of this lesson? (Practice with addition? Looking for patterns? Perseverance? Making conjectures? …)
Do we share some of the conjectures students are making during the time when students are working, or later in the closing of the lesson?
How will my students record their work? Keep track of their answers? Do I provide a 0-99 chart or ask them to keep track somehow?
Will students work independently / in pairs / in small groups? Why?
Do I allow calculators? Why or why not? (think back to your goal)
How will I share the conjectures or patterns noticed with the class?
Are my students gaining practice DOING (calculations) or THINKING (noticing patterns and making conjectures)? Which do you value?
The smallest of decisions can make the biggest of impacts for our students!
So, at the beginning of this post I shared with you that this problem is currently unsolved. While it is true that the vast majority of numbers have been proven to easily become palindromes, there are some numbers that require many steps (89 and 98 require 24 steps), and others that have never been proven to either work or not work (198 is the smallest number never proven either way).
Some final thoughts about this problem…
After using this problem with many different students I have noticed that many start to see that mathematics can be a much more intellectually interesting subject than they had previously experienced. This problem asks students to notice things, make conjectures, try to prove their conjectures and be able to communicate their conjectures with others… The problem provides students with the opportunity to both think and do. It offers students from various ability levels access to the problem (low floor), and many different avenues to challenge those ready for it (high ceiling). It tells students that math is still a living, growing subject… that all of the problems have not yet been figured out! And probably the most important for me, it sends students messages about what it means to really do mathematics!
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.
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:
Notice instrumental teaching practices.
Learn more about how to move from instrumental to relational teaching.
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?
Goal 2 – Learn more about how to move from instrumental to relational teaching.
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!