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:
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:
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!
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:
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 remaine focused on the mathematics itself, and not expect all of your students to use YOUR strategies. That’s how you kill your students’ relationship with mathematics! We would never tell our students to pick C for every answer, that will only work 1/4 of the time. In the same way, many of the strategies we provide for our students will not work for all students and not for all questions… and rarely will these strategies actually help them reach our actual goals for our students:
(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;
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)!
I remember as a student being asked to estimate in mathematics class on a few occasions. It was either an afterthought from my teachers telling the group to estimate before we do our work to help us with the “reasonableness” of our answers, or during measurement activities where we had to estimate then measure items around the classroom. As a student though, when asked to actually estimate, I always did the calculations or measuring first, then wrote down a number that was near the actual answer as my “estimate”. Was I confusing rounding with estimating? Or was I avoiding thinking???
It might come as a surprise to consider just how much estimating we do outside of school. Within the same day we might determine how much milk to pour on our cereal so it is covered but won’t get soggy, think about how early we need to leave for work to make sure we aren’t late, figure out if we can safely squeeze our car into a parking space, consider how loud to speak to someone across the room, determine an appropriate amount to tip the waitress at dinner, think about if there is enough time during the commercial break to use the restroom so you don’t miss any of your favourite show… Whether we know it or not, nearly every minute of the day, we are estimating about physical spaces and numbers.
In school, however, the practice of estimating is often neglected. Many of our students are estimating all the time without realizing it, but others might not be aware of the mental actions others are doing and don’t engage in the same active thinking processes! Because of this, I believe we should be estimating more than we probably realize. The skill of estimation is directly related to our Number Sense and our Spatial Reasoning, so we need to make estimating a priority!
Kinds of estimating:
Situations in which we estimate involve: computational estimations, measurement estimations, numerosity estimates (how many) and number line estimates. Computational and numerosity estimations are directly related to students’ Number Sense (i.e., size of numbers, doubling, how much more or less…) and often involve students approximating numbers. While estimates involving measurement and number lines involve our students’ Spatial Reasoning (i.e., considering the size and space of objects). However, if we really delve deeply into any of the 4 kinds of estimating, they probably each deal with our Number Sense and each deal with our ability to think Spatially.
Questions that ask students to estimate distance or length:
Take a look at the following 5 questions. Which type of question do you think is most common in school? Which type of question is less common?
For many students, estimating a measurement is about “guessing”, then actually measuring. To a student, the act of estimating becomes useless in this scenario. If they are going to measure anyway, why did they estimate anything? Many of the questions above ask students to go beyond guessing and ask them to develop benchmarks then think about subdividing or iterating those benchmarks. While some of our students will naturally develop these benchmarks and strategies to subdivide/iterate, many others will not without rich tasks and discussions. For these reasons, we likely need to spend more time than we might realize estimating and discussing our strategies / thinking.
Questions that ask students to estimate area:
Take a look at the 5 questions below. Which ones might help your students better understand the concept of area? Which ones might help them consider the attributes of area you want them to notice?
To many students, concepts like area are easy… plug in the numbers to a formula and you get your answer. Estimating, however, requires far more thinking and understanding of the attributes than simple calculations. For this reason, it is probably best if we start by asking our students to estimate well before they are ever given any formulas, and continually as they learn more complicated shapes.
Each of the above problems ask our students to actually consider the size and shape of the things they are thinking about. Hopefully this happens ALL the time as our students learn measurement concepts!
Questions that ask students to estimate angles:
As a student I remember learning types of angles and how to read a protractor – very knowledge based and procedural in nature. However, as a teacher I regularly see students who can easily tell me if an angle is greater or less than 90 degrees, but make seemingly careless mistakes when actually measuring an angle. Personally, I believe that the issue isn’t about students being careless, it is more about a student’s experiences with angles. Specifically, too many students are asked to DO something instead of them being asked to THINK about something when it comes to topics like angles. Above are 5 problems / tasks that ask students to think first by estimating. When the task is about estimating, it adds motivation for students to actually measure to see how close they might be!
Questions that ask students to estimate number or computations:
Look at the 6 problems/tasks below. Which ones do you think are more common in classrooms? Why do you think they are more common? Which ones require your students to consider the space numbers take up? Which ones help our students develop and use their number sense?
When the problems / tasks we give are about estimating, our students think about what they already know and use this as a basis for learning new things. While the aim for many of us is to help our students determine reasonableness for our students’ answers, in real life, estimates are likely good enough most of the time!
We need to estimate more!
Estimating needs to be integrated into more of what we teach, instead of it being an isolated lesson/ concept. Whether teaching probability or time or Geometry or patterns… we need to ask our students to think more before they start doing any calculation!
Hopefully you have heard of Andrew Stadel’s Estimation 180. This is an easy to use routine that can help us with some of the types of estimation I have talked about here!
One of the best parts about these routines is that it helps students build benchmarks, use number sense, think spatially and consider the importance of a range of reasonable answers instead of just “guessing”.
Jamie Duncan added to the conversation when she shared snapshots of how she helped her students refine their ranges:
After noticing her students’ ranges were quite large, she started asking her students to indicate their “brave” too low and “brave” too high estimates. Ideally we want our students focused on their range of values, not their actual estimate. Focusing on the actual estimate promotes guessing, while focusing on student ranges helps us think more about reasonableness. Brilliant idea!
And of course, for me, two of the most powerful images on the topic have been shared by Tracy Zager (If you haven’t purchased her book: Becoming the Math Teacher You Wish You’d Had you need to!). The image on the left shows the process we go through when solving a problem. So much of what we want our students to do involves them making sense of things and considering their initial thoughts. The image on the right to me is even more powerful though. It shows just how important our intuition is, and how building our students’ intuition is key for them to build their logic! These two go hand-in-hand!
A few things to reflect on:
What does estimation look like in your class? Is this a routine you do?
Many of the ideas shared above might be more specific to the content you are learning. How do you help your students see the importance of estimation when you are learning new topics?
When your class is estimating, how do you promote the range? Are some of your students still “guessing”? If so, how can you use the ideas of others in the room to help? How else can we improve here?
Many of the ideas I shared above involved estimating without giving a number. These tasks often directly involve helping our students use their Spatial Reasoning. How are you helping your students develop their Spatial Reasoning?
The students in our classrooms that are estimating all the time (even without them realizing it) do well in mathematics. Those that struggle often aren’t using their intuition. Why do some use their intuition more than others? What do WE have to do to help everyone use their intuition more often?
Did you notice any relationships between the coloured images above? All the yellow ones involve… all the red ones are… I wonder which ones you gravitated toward?
As always, I encourage you to leave a message here or on Twitter (@markchubb3)!
A while ago I was introduced to Skyscraper Puzzles (I believe they were invented by BrainBashers). I’ll explain below about the specifics of how to play, but basically they are a great way to help our students think about perspective while thinking strategically through each puzzle. Plus, since they require us to consider a variety of vantage points of a small city block, the puzzles can be used to help our students develop their Spatial Reasoning!
While I loved the idea of doing these puzzles the first time I saw them, I was less enthusiastic about having these puzzles as a paper-and-pencil or computer generated activity because it is difficult to help develop perspective without actually building the skyscrapers. So, I created several templates that can easily be printed, where standard link-cubes can be placed on the grid structures.
Below are the instructions for playing and templates you are welcome to use. Enjoy!
How to play a 4 by 4 Skyscraper Puzzle:
Build towers in each of the squares provided sized 1 through 4 tall
Each row has skyscrapers of different heights (1 through 4), no duplicate sizes
Each column has skyscrapers of different heights (1 through 4), no duplicate sizes
The rules on the outside (in grey) tell you how many skyscrapers you can see from that direction
Taller skyscrapers block your view of shorter ones
Below is an overhead shot of a completed 4 by 4 city block. To help illustrate the different sizes, I’ve coloured each size of skyscraper a different colour. Notice that each row has exactly 1 of each size, and that each column has one of each size as well.
Below is the front view. You might notice that many of the skyscrapers are not visible from this vantage point. For instance, the left column has only 3 skyscrapers visible. We can see two in the second column, one in the third column, and two in the far right column.
Below is the view of the same city block if we looked at it from the left side. From left to right we can see 4, 1, 2, 2 skyscrapers.
Below is the view from the back of the block. From left to right you can see 1, 2, 3, 2 skyscrapers.
Below is the view from the right side of the block. From here we can see 2, 2, 4, 1 skyscrapers (taken from left to right).
When playing a beginner board you will be given the information around the outside of your city block. Each number represents the number of skyscrapers you could see if you were to look from that vantage point. For example, the one on the front view (at the bottom) would indicate that you could only see 1 skyscraper and so on… The white squares in the middle of the block have been sized so you can actually make the skyscrapers with standard link cubes.
A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity. Hopefully, this post has helped you consider your own decision making processes!
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).