The Reflective Educator

Education ∪ Math ∪ Technology

Menu Close

Author: David Wees (page 6 of 94)

Teaching Problems or Teaching Mathematics

Is this curriculum?

Is this curriculum?


On the day before I first started teaching, the district coordinator came to me and handed me a piece of paper with twenty questions on it. “Here’s what you have to teach, David. If your students can answer all twenty of these questions by the end of the year, you will be fine.

Needless to say that this was insufficient curriculum. But what kind of curriculum would have been useful for me at that stage in my career?

A place many math educators, especially in our accountability-driven system, seem to start when teaching mathematics is teaching children how to solve questions. “Here’s how you solve this kind of question.” The connections between different problems are left for students to discover on their own.


If the triangles are a different size, the transformation is a dilation.

Strategy: If the triangles are a different size, the transformation is a dilation.


The advantage to this system is that you can look at the state assessment and check off of all of the question types and feel like you have done your job. For the old NY state exam, this approach works in the sense that students were able to sit down at the exam and feel like they had been prepared for every question since the assessment was so predictable, even if they didn’t always know how to solve the problems .

The problem with this approach is that students now have to remember each question archetype and each solution to each type of problem separately, leading to a relatively unorganized and overwhelming set of problem-solving schema for students. This leads to students forgetting how to solve individual problems, forgetting which solution strategy they should use when, or misapplying strategies to solve problems for which the strategy is not appropriate. Even when students do master all of the problem types, knowing how to solve problem x, y, or z doesn’t help students make connections when they start studying further areas of mathematics.


A type list of units in a math curriculumum

A typical list of units in a math curriculum


To their credit, every textbook author I’ve ever read takes a different approach (to some degree). Textbooks start by dividing the year’s worth of mathematics to be learned into units of study and apportioning mathematical principles into those units. Within each unit of study, specific problems are used to illuminate mathematical ideas and ideally students at the end of a unit can articulate the mathematical ideas they have learned, rather than just the problems to which they apply. Where textbooks often fall down is in making the connections between units and ideas explicit.

When well done, problems become vehicles for teaching mathematical principles. Mathematical representations (like graphs, tables, etc…) are embedded across the units so that students get multiple exposures to these critical representations and can use them to make sense of similarities and differences between different mathematical ideas. Ideally students explicitly learn connections between different mathematical ideas so that they see, for example, how solving a linear equation is related to solving a quadratic equation and how the graph of an absolute function is related to the graph of a linear function.


Source: Sierra1223


One way to support students in making these connections is to ask them to answer reflection questions like: How is what we learned today related to what we learned yesterday? How is this problem we solved today like the problems we solved last week? What did we learn today that we can probably generalize and use to solve other problems? At the very least while planning, make sure you can answer these questions yourself.

A major disadvantage of the second approach to teaching mathematics is that it takes much more work to organize curricular resources into the general themes and to make the connections between ideas explicit. Rare is the classroom teacher who has time to do this all herself. This is one of the reasons why I think well-organized curriculum of some kind is always going to be a helpful resource.


There is a gradient between the two approaches to teaching mathematics.

There is a gradient between the two approaches to teaching mathematics.


For further reading on a related topic, check out this post on instrumental versus relational knowledge.


Planning Lessons

When I first started planning lessons, each lesson took ages to plan. I don’t really remember exactly what I wrote except that usually the lessons were based on choosing example problems to go over, producing a worksheet for students to work on, and assigning homework questions. Eventually I finally had some textbooks for students and so these lessons ended up being reduced to references to page numbers and questions in those textbooks.


A sample lesson plan

At some point I decided that being very economical with my planning was the mark of good teaching and so my lesson “plans” ended up being really short. “Teach Pythagoras” was an actual lesson plan I wrote. Of course, “Teach Pythagoras” is not a lesson plan. It’s not even really a topic. It’s a short-hand for pick some examples, tell kids how to solve those examples in an ad hoc fashion, followed by make up some examples for kids to try and solve themselves.


A sample project

A sample project

When I started using longer-term projects, that meant planning lessons got even easier as I could assign a time-line to the completion of the project and in each day I would support students either with some examples for the whole class to move their work in the project forward or by circulating through the class to help students out.

Up until this stage, any student mathematical discussions that occurred were ad hoc and almost always initiated by students. Not once in my classroom teaching experiences did I plan for student discussion.


About three years ago, I started a new job as a formative assessment specialist. It was then that I first read Peg Smith’s Thinking Through a Lesson protocol. As I read the article, I realized right then that I had spent most of my career planning poorly.

Next I read Dan Willingham’s book on Why Don’t Students Like School? and realized that I had spent most of my career planning tasks for students to do and not planning the thinking I wanted students to do. Tasks prompt thinking but what thinking? Here’s an example of a lesson that could have been one my early lessons. What are children being asked to think about?


A sample anticipation of student ideas

A sample anticipation of student ideas

Fortunately, I had an opportunity to test out these new ideas around planning for myself. I started teaching my son and between 6 and 8 other kids close to his age in a Saturday class. I decided to plan the student thinking, to try out the Thinking Through a Lesson Protocol, and most importantly, to ask other people to comment on my plans. Here’s my first lesson plan for this class. It’s by no means perfect but it has far more detail on what I will do in response to what I expect students to do, and the first example I can recall of a lesson where I explicitly planned a whole group discussion.

But who has time to plan lessons at this level of detail five days a week, potentially 4 or 5 times each day? The level of planning linked above is unsustainable for classroom teachers.



A sample task for Contemplate then Calculate

Last summer we started introducing instructional activities to teachers across our project. Instructional activities, as defined by Magdalene Lampert and Filippo Graziani, are “designs for interaction that organize classroom instruction”. Essentially they define a set of moves a teacher makes to position students to talk to each other about mathematical ideas, surface student thinking about those mathematical ideas, and then orchestrate a classroom discussion around the ideas in order to focus students on a mathematical goal.

These instructional activities have the advantage of bounding the scope of decisions a teacher can potentially make when planning a lesson while focusing the decisions that are made ahead of the lesson on planning for the thinking students will do, and then in the lesson enactment, allowing teachers and students the space to think about and respond to each others’ thinking. The routineness of an instructional activity, if the same structure is used many times, allows thinking about roles, what’s coming up next, to fade into the background so that more thinking can be focused on the mathematical ideas.

When planning one of these instructional activities, I find myself choosing an appropriate task based on some understanding of anticipated student thinking, then imagining how students might approach the task and what they will think about, then considering how to sequence the different strategies student might use toward a big mathematical idea, and then creating the resources to enable me to use the instructional activity in the classroom. This level of planning is sustainable.


As I reflect on my own development of planning over the course of my career, it seems to me that I would have benefited from knowing about planning routines that other people use. I would have benefited from learning at least a few instructional activities so that I didn’t need to plan every aspect of my teaching. I would have benefited from access to tasks where student thinking was anticipated for me.  I don’t think a highly scripted curriculum would have developed me as a professional (but maybe my students would have benefited). I would have benefited from seeing how other people sequenced mathematical ideas.

In short, I would have benefited from more explicit teaching of how to plan lessons.



What is Ambitious Teaching?

A recent analysis by David Blazar has stirred up some interest about Ambitious Teaching. But what exactly is Ambitious Teaching?


According to Elham Kazemi, Megan Franke, and Magdalene Lampert:

“Ambitious teaching requires that teachers teach in response to what students do as they engage in problem solving performances, all while holding students accountable to learning goals that include procedural fluency, strategic competence, adaptive reasoning, and productive dispositions.”

While this defines some aspects of Ambitious Teaching, this is not a sufficient definition such that any two observers would agree that an episode teaching observed was Ambitious.

In this same paper, E. Kazemi et al. go on to talk about instructional activities they are teaching to pre-service teachers which:

  • make explicit the teaching moves that are implied in the kinds of cognitively demanding tasks that are found in curriculum materials available for use by novices;
  • structure teacher-student interaction using these moves in relation to teaching the mathematical content that students are expected to learn in elementary school;
  • enable novices to routinely enact the principles that under gird high quality mathematics teaching including:
  • engage each student in cognitively demanding mathematical activity
  • elicit and respect students’ efforts to make sense of important mathematical ideas
  • use mathematical knowledge for teaching to interpret student efforts and aim for well-specified goals
  • be generative of other activities by including the teaching and learning of essential teaching practices (high leverage practices) like explaining, leading a content-rich discussion, representing concepts with examples, and the like (Franke et al., 2001).

In his recent analysis, D. Blazar uses a research instrument which he separates into two aspects, one of which called “Ambitious Mathematics Instruction” positively correlates the following teaching practices with student performance on an assessment given to those students.

  • Linking and connections
  • Explanations
  • Multiple methods
  • Generalizations
  • Math language
  • Remediation of student difficulty
  • Use of student productions
  • Student explanations
  • Student mathematical questioning and reasoning
  • Enacted task cognitive activation

From this we can infer that these labels describe teaching practices which are part of Ambitious Teaching. We can generalize these practices to other content areas, although the practices will look different in those different content areas.

Dylan Kane unpacks one aspect of Ambitious Teaching, teaching to big ideas, which is about making connections between different ideas rather than treating each lesson as being in isolation from each other lesson.

Photo of me teaching

Photo of me leading a whole class discussion while annotating student strategies.


One key element that distinguishes Ambitious Teaching from other teaching is that ideas that emerge in class are built up and extended directly from student thinking rather than the converse. This one element leads to added layer of complexity to teaching as suddenly all of the ideas in the classroom have to be surfaced and some subset of those ideas needs to be selected to talk about in public. Authentically listening to children’s ideas while leading a class with a specific objective and trying to move all of the students within that class toward that objective leads to complexity that most teaching does not contain — and so we call the teaching that requires this at its core to be Ambitious. The name of this kind of teaching is not a rhetorical device, it is a reminder that learning to teach is challenging.

Another added layer of complexity is that when ideas are shared or presented to the whole group, not only does a teacher need to be conscious of how their decisions in this theater impact every child, they need to make sure that whatever idea that is being discussed is clear for every child. In a mathematics classroom this can be done through use of thoughtful representation, asking clarifying questions, asking students to listen to and restate each others’ ideas, adding on explanations to the student explanations, etc… but all of these individual practices have the same goal — ensure that everyone understands what is being discussed while treating students as sense-makers.


Based on this photo (which has a portion of the strategy cut-off on the right hand side), what mathematical idea is being discussed here? How do you know?


In order to make better choices about what problems and what representations of those problems to use, teachers need to develop specialized expertise about how students understand, and often misunderstand, their subject area in order to surface those important understandings and support students in transitioning to other understandings.


What mathematical ideas are being connected here? Why might that be helpful? What’s missing in these representations?


Now to be clear, I cannot easily describe Ambitious Teaching in a single blog post. Magdalene Lampert wrote an entire book about it and she only described this kind of teaching in the context of a single 5th grade mathematics classroom. All I can hope to do is to generate questions about teaching and to hopefully suggest that there is greater complexity to even typical teaching than the selection of the three worked examples for the day.

This entire short talk by Richard Feynman is worth watching, but if you look at his point at about 5 minutes into the video, you’ll hopefully understand my next point better.


Most of the time when we talk about teaching, we do in such vague terms that our conversations slide past each other. This includes educational research which claim to show that this method of teaching is better than this other method of teaching without carefully defining what the methods being compared actually are and what the goals of either particular style of teaching might be. One goal of my work this year is to make certain teaching practices explicit for teachers in our project so that when we talk about teaching we can be reasonably sure that everyone understands the decisions being made and why someone might make one decision or another based on the objectives of the teaching for the day.



Participation in math class

Nothing you can do can guarantee that every child actively participates in your math class but there are some things you can do to increase the odds.

Source: Wikipedia

Source: Wikipedia

In a typical classroom a teacher asks a question, a student responds, the teacher indicates whether the response is incorrect or correct, and this is repeated until the class discussion is over. If this is the only type of interaction a teacher has with their students, this can be problematic.

  1. It’s very difficult to keep track of who has participated and who has not which often leads to skewed participation in the conversation.
  2. It’s impossible to know how each child understands the mathematical ideas being discussed.Here’s an example Dylan Wiliam uses to illustrate this point: If a teacher asks four questions in a row of four different children and the responses to the first three questions from children are incorrect but the fourth is correct, the only thing that can be concluded is that the fourth child was able to answer that question but what is typically concluded is that the class understands and so the teacher moves on. Unfortunately I myself am guilty of this many, many times. All too often we stop questioning when we get a correct response.

Fisher and Frey (2007) say the following on this matter:

“I’ll ask the questions,
a few of you will answer
for the entire class,
and we’ll all pretend
this is the same thing as learning.”

But what can we do differently? How can we minimize this kind of interaction? Note that it is probably not possible to completely eliminate these types of interactions, nor would it necessarily be advisable to do so, but there are some ways to reduce their impact on a classroom.

Turn and talk:

Prompt students to talk to each other about a question while you circulate and listen for responses (and hold students accountable for talking about what you want them to talk about). Come back to the whole class and ask a few people to share responses. This has the benefit of increasing the chance every student shares an idea with at least one person and increases the amount of time students have to process and think about the question.

Wait time:

The average amount of time teachers wait for a response after asking a question is less than one second. Just delaying this to five seconds (or even more sometimes) increases the amount of students who have time to process the question, think about it, and then formulate a response.


Instead of posing a question and waiting for one response, pose a question and ask students to, either individually or with a small group, construct a response that they can share on their whiteboards. This way you can walk around and see what ideas students have and every student has an opportunity to take their time to think about the question.

Use student generated questions:

There is some evidence that students do better when they own their learning. Having students generate questions they have and allowing students to answer each other’s questions (perhaps initially mediated by you) or you to answer questions they have, gives students more control over their own learning. Of course, students are often not used to this kind of interaction and will likely need support of some kind in generating questions.

Start with the problem:

It is extremely common to start class with a somewhat interactive description of how to solve a problem, give students opportunity for guided practice with lots of feedback, followed by independent practice, followed by a summary by the teacher or a student of what was learned that day. In this case, any questions that are asked within the context of the entire class can only really happen either during the initial describing of how to solve the problem at the beginning of the class or during the summary at the end.

An alternative to this approach is to start with a problem where you can be reasonably sure all students understand what the problem is and what they are trying to find out. Now when you initiate any full group discussion, students are much more likely to have questions and the answers to these questions are much more likely to be understood by everyone.


Note that I’m not a fan of cold calling students. With some students, having a teacher who cold calls raises their anxiety, making it more difficult for them to think about the mathematics. It might be a useful technique once you know every student feels comfortable and has something to say.


What other strategies can you use to encourage participation (even non-verbal participation) in your classroom?


Why is it important for students to talk to each other in math class?

Why should students talk to each other in math class anyway? I was asked this question recently and I’m trying to avoid a tautological answer (eg. it’s important because it’s important).

In a classroom where students speak to each other about mathematics, the ideas of those students are valued instead of ignored or potentially marginalized. This gives students agency in their learning. It also allows new ideas the students learn to extend from the existing ways they understand the world.

Supporting students speaking to each other means that mathematics is much more likely to become a way of knowing and being rather than just a body of existing knowledge (although the value of mathematics as a set of tools that have been developed over time should not be marginalized). As students develop their understanding of what mathematics is and what it is useful for, they are more likely to insert themselves in the role of the mathematician rather than imagining this to be someone else, potentially from another culture. They can see themselves being part of a mathematical community.

In order to completely understand the language we know, we have to use it, either in writing or ideally in conversation, and hear other people using the same language. So from a practical perspective, students need to talk in order to develop their use of language (mathematical or otherwise), and rather than students talking in serial, one at a time mediated through a teacher, it is far more efficient for them to talk in parallel, to each other.

We remember what we think about. When students construct ideas and communicate them to each other, they necessarily have to think about these ideas, which means that they are building memories. While this occurs no matter what students do, the focus is more likely to be on the thinking with student discourse rather than the activity (eg. completing a task).

Finally, students talking and writing to each other also provides their teachers with more information about the ways they are thinking which makes it easier for the teacher to orchestrate productive whole group discussions and to plan activities that respond to the ways students are actually thinking. It is difficult to plan lessons that build off of student knowing if you don’t know how and why students think the way that they do. When students talk to each other, their teacher can gather formative assessment information about not only what they understand but ideally how they understand it.

This should not diminish the importance of students having independent time to work quietly on mathematical problems by themselves. Students are better positioned to work together when they have had time to think about ideas themselves first. Also, some students find working with other students really difficult for a variety of reasons, so in some cases the benefits of students working together may be outweighed by the challenges some students face with this activity.

What would you add as reasons students should talk to each other in a mathematics classroom?



Coherent conversations about teaching

Imagine four teachers each of whom teaches in different schools in a different context. Even if they all teach the same course, their individual teaching looks different.

Typical teaching #1 Typical teaching #2
Typical teaching #3 Typical teaching #4

If these four people come to talk together, they will find it challenging to have a conversation since the way they are teaching is so different from each other. Each person potentially has a valuable perspective but they may have so little in common that it is difficult for them to talk to each other.

Typical conversation about teaching

Now imagine instead these four teachers are all focused on working on the same instructional activity, perhaps even with the same mathematical tasks within the activity. Their teaching is still different as they still teach in different schools in different contexts but their conversation about teaching becomes much more coherent as they have far more in common to talk about. Instead of talking about their individual teaching they can talk about teaching practice.

With instructional activities

Over the last two weeks we launched the instructional activity, Contemplate then Calculate with 100 or so teachers for this exact purpose; to make our shared conversations about teaching focused and coherent. After two days of professional development, which included rehearsing the instructional activity together, virtually all of the teachers indicated that they were excited to try out this activity and then come back together in October to talk about its impact on students.

At one point during the two weeks, some participants and I ended up in a whole-group discussion about when exactly we should annotate our students’ strategy sharing. It was the most specific conversation I’ve ever had with a group of teachers and I feel fairly certain everyone understood the point that was being debated and why it might matter one way or another.

All of these teachers are going to go back to their individual schools and teach according to what they know with their individual contexts and their individual students but now, hopefully, at least one aspect of their teaching will be similar enough that they can come back and talk about the differences.



Apps for the math classroom

Here is an incomplete list of companies making apps for the math classroom. As far as I know, every application made by these people is fantastic.

  • The NY Hall of Science has recently published a series of science and math apps for the iPad.
  • Motion Math has some great low-cost math apps.
  • DragonBox has 4 really high quality math applications, all of which I have personally played and tested with my son. These are the most expensive apps on this list but also some of the best apps available for learning math.
  • BrainQuake’s math puzzles will present a challenge for people of all ages while being accessible to young children.
  • Geogebra is cross-platform and a must-have for people interested in constructing and exploring their own interactive math activities.
  • Desmos is a fantastic online graphing calculator which can be installed and used on many different types of devices.

As I learn about more companies publishing apps for the iPad (and as I have time!), I’ll update this list. I know there are lots of apps that I’m missing but this is what I had time to put together this morning.

Working differently

This summer I’ve been doing a lot of task-based curriculum development on a series of fairly short activities. We are trying to develop resources for use with an instructional activity created by Grace Kelemanik and Amy Lucenta called Contemplate then Calculate. A key part of this instructional activity is surfacing the kinds of things people notice that allow them to make mathematical connections and solve problems quickly and efficiently.

It turns out that Twitter is a really good medium for testing these activities because most people don’t have access to pencil and paper (a key component of Contemplate then Calculate) and there are a lot of people from a wide variety of backgrounds willing to try out the tasks.

What I do is first share the task on Twitter, kind of like so:

Then, I wait. Usually, I get a few responses right away and a few more responses over time.

Screen Shot 2015-08-20 at 9.57.42 PM

Notice how I follow-up to those responses with questions to find out what people were thinking. If my objective with the task is to engineer opportunities for students to think about mathematics, it’s helpful to know in advance how they might think about any task I give them. A key benefit to Twitter is that a variety of different ways of thinking about the task emerge quickly and in many cases have led to me modifying the original tasks.

Some tasks get more responses than others. I try my best to respond to everyone who takes the time to try the question, but sometimes I miss people.

I also work to make sure that share back the work of the community back to the community.

All of the tasks I’ve been working on, and so far we are up to about 30 of them with many dozens more in the pipe-line, are going to be shared, with a Creative Commons license, back to the math education community. Stay tuned for the URL.


Four blog posts about using student ideas

Wees Assessment Cycle Art

I wrote four blog posts for NCTM’s Mathematics in the Middle School blog on using student work to understand and plan around student ideas. Each post is about using student work to make inferences about how they understand mathematical ideas and then using those inferences to help you plan. Note that these posts are actually relevant for mathematics educators at all levels.

  1. What Knowledge Do You Need to Plan a Unit?
  2. Categorizing Student Work
  3. Using Student Work Meaningfully
  4. The Mathematics of Students

While these posts focus on qualitative information about student learning, this doesn’t mean quantitative information isn’t useful. I have just found that quantitative information abounds while qualitative information seems to be rarely used in systematic ways.


Learning at Conferences

This year when I attended the NCSM and NCTM annual conferences, I had a much different experience than in previous years. I thought it would be worth sharing some ideas I have about how to make it more likely that you learn from a conference experience.

  1. If you want to walk away from the conference with something different about your practice, focus on one or maybe two ideas during the conference and only go to sessions that will support you in learning and revisiting those ideas. On the other hand, if you don’t know what you are hoping to learn at the conference, you might be better off going to many different sessions in the first couple of days at least before deciding however note that this may something you can learn from the conference planner rather than attending a scatter-shot of different sessions.IMG_3167
    No one or two hour session is likely to lead to any significant change in your practice. 20 hours thinking about one part of teaching, especially in the different ways different presenters think about the idea, on the other hand may.
  2. Skip a couple of sessions and take the time to reflect on your learning and make a plan for implementing the new idea(s) from the conference into your teaching. If you don’t eventually come up with a plan for implementing new ideas in your teaching, you will never try them out. The sooner you have a plan, the easier it will be to come up with that plan while the ideas are still fresh in your mind. You should also make a note of questions you still have.IMG_3168
  3. Find someone with whom to share and discuss your learning experiences at the conferences. In my case, I went to NCSM & NCTM with my colleagues, but if you end up going alone, try and arrange a buddy or two to spend the week with early in the week. This is one potential powerful use of Twitter; if you are active on Twitter, chances are good you will know other people at the conference.IMG_3215

    It’s worth noting here that not every presenter at NCSM and NCTM is equally good at communicating what they know and so having someone to talk after attending a session is useful to fill in whatever parts of the story or idea you missed but maybe someone else caught.
  4. Take the time to network with other people at the conference, even if this means you may miss a session or two.
  5. Be strategic in the exhibition hall if you visit it at all. I find it overwhelming and draining. I typically take 30 minutes and scan through the entire hall quickly to see if there are any types of products I don’t know about and then come back to follow up later if necessary.IMG_3219