Education ∪ Math ∪ Technology

Tag: The Reflective Educator (page 4 of 43)

Black and White

Everything in education these days is black and white, with no possible shades of grey.

Standardized tests are evil. Except when we agree with their findings. Or when they are used in research about teaching methods. Or when they highlight inequity in our society.

Lecturing to students is evil. Except when it’s coupled with peer instruction. Or when it is used to talk about peer instruction. Or when the lectures are short and have animations added. Or when the lecture is about creativity.

Common Core is evil. Except when it raises standards for students. Or when it communicates a common definition of what it means for students to do mathematics. Or when it encourages multiple ways of solving mathematical problems.



Network Improvement Communities

I recently read an essay published by the Carnegie Foundation for the Advancement of Teaching called “Getting Ideas into Action: Building Networked Improvement Communities in Education.” The basic argument of the essay is that while traditional research can be effective for finding out what works in specific circumstances, it frequently is too far away from practice to be useful in an educational setting, but that many of the ideas of research can be applied in practical ways through the creation of networked improvement communities (NIC) which use the design cycle to study and solve shared problems of practice. This to me does not say that research is useless, as it can inform this process, only that research as a tool for improvement is insufficient.

After reading the article, I thought to myself, in what ways is our online community of math teachers like or unlike a network improvement community? A clear and extremely important difference is that as a community we have no central objective. We also do not have many obvious structures which lead to sustained improvement and accountability back to the community. However, we do have a community of people who are all obviously interested in improving our own individual practice; quite simply it would otherwise not be worth the effort we put into our online participation.

Here’s a proposal. Let’s form some small groups of math educators (or alternatively, educators of math educators) who carefully study our practice together and think about improvements, using a process I learned about here at New Visions called Inquiry. This process is detailed in a book called Strategic Inquiry by Panero and Talbert (reading this book could be a useful starting place). We would meet once every couple of weeks (using a Google+ hangout, I guess unless there is alternate technology out there that is better) and take turns presenting on our work, and asking for feedback from the group. This will allow these small groups to collaborate around our teaching, our students, and mathematics, which Ilana Horn describes as among the most effective learning experiences for math teachers.

Anyone interested in participating?



Pause and rewind this until you get it

Watch this video. Every time you feel you are confused, just pause and rewind the video and rewatch it. Do this until the video makes sense.


Can’t do it can you? It doesn’t matter how much you pause and rewind a video, if it doesn’t make any sense to you, watching it again isn’t going to suddenly change the video so it makes sense.

One critical job of the teacher is to find out where students are, and help them at that stage. Explanations that don’t make any sense aren’t particularly helpful. In fact, since our goal is to help students learn the mathematics, and not just the words that represent the mathematics, explanations may not be helpful at all.


What is teacher inquiry?

The Teaching Channel recently published four videos which describe much of the work I do with teams of teachers in a project known as Accessing Algebra Through Inquiry (or a2i for short). One of the primary objects of the a2i project is to build the capacity of school-based teams of teachers working collaboratively to improve their practice and make evidence based choices about their instruction. In other words, a primary objective of our project is to change how teachers collaborate and increase their opportunities to learn about their practice.

One thing I do is help facilitate a meeting at a school, with the objective of working through an inquiry cycle where we look at student work, identify a common problem of practice, unpack what the probable models the students are using to think about this particular area of mathematics, and then decide together as a team on an intervention for the teacher’s group of students. Next the teacher implements the strategy, and we study the resulting student work to see if the intervention worked.

These two videos describe the inquiry process in more detail.

Another portion of my work is individual coaching of teachers. I work with teachers where I typically observe their classes, usually focusing on a particular aspect of the lesson. During this time, I gather information about what strategies the teacher employed, and how these strategies played out with the teacher. Sometimes I model a particular strategy with a teacher, and occasionally I rehearse a strategy with them, particularly if they have never used it before.

After the lesson is over, or sometimes before the lesson, I meet with the teacher and we discuss instructional strategies for their students. Sometimes we discuss classroom management strategies, sometimes we discuss the core mathematical content of a unit, sometimes we look at different instructional strategies, and sometimes I reflect back to the teacher the questions they asked, and we discuss the impact of these questions.

Here is a short clip from an individual coaching session between a former colleague of mine, Xiomara Gonzalez and one of the teachers in our project, Anna Tabor.

One of the most important benefit of our project is that we are providing a structure through which teachers get more feedback on their work when they collaborate with other teachers, and at the same time, get feedback from an instructional coach.

Here is a video that explains the impact of the feedback Anna receives from her colleagues, and from the work with her instructional coach, Xiomara.

It has become clear to me that large, impersonal conferences, workshops that are disconnected from teacher practice, courses which overly focus on the theoretical, and spaghetti-style professional development (throw the PD at the teachers and see what sticks) are not accomplishing their objective; getting teachers to reflect on their teaching right now, with this group of students. This process of inquiry that we are working as a team of instructional coaches to implement in our schools is designed to give teachers peer feedback on their teaching, and to help them grow and improve their skills while still connecting this learning directly to the work teachers do with their students.


Note: If you can’t see the videos above (because this post is in your email or your RSS feed reader), you can view them here.


PARCC sample assessment items for high school math

PARCC recently released some sample computer-based test items for ELA and high school mathematics, so I thought I would check them out since NY state is still officially planning (eventually?) to use the PARCC assessments.

First, some kudoes to the team that created these assessment questions themselves. In general I found that the questions were looking for evidence of mathematical reasoning, and would be difficult to game with classroom test-preparation. What I think is missing is an opportunity for students to demonstrate the complete range of what it means to do mathematics, including asking questions themselves that they answer, but for an assessment with this function, this seems much better than the current generation of standardized tests.

If you want to stop reading this and preview the questions yourself, feel free to do so (if you are only interested in looking at the sample math questions, you’ll have to skip through the sample ELA questions).

Here’s my preliminary thoughts from attempting the first few problems myself.

  1. Use of space is critical. The first assessment question does not do this very well. Look at the video below that explains my reasoning on this.


  2. The second question has two issues, one of which is really very minor, the other of which is something PARCC should make an effort to fix.

    I’m okay with questions that use approximate models for mathematics, but it might be at least worth noting to students that these models are approximations.

    Taking a test on a computer is already extremely distracting as compared to taking a test on pencil and paper. Given that there is research that shows that people generally read slower and with less understanding on a computer, PARCC should make an effort to mitigate the platform issues as much as possible. Imagine if your work flickered in and out of existence while you were writing it down on paper?

  3. I put the wording for the third question through a reading level estimation calculator, and it estimated that the reading level was grade 10. While it is reasonable to expect a certain amount of competence from students in this respect, we have to be careful that our assessments of the mathematical thinking of students aren’t actually measuring whether or not they can read the prompts in our assessment.
  4. Question 5 assumes a certain amount of cultural knowledge, specifically knowledge of playing golf. Having worked with students who do not have the this sport in their cultural background, I found assessment items like this frustrating. Usually, the questions are doable without the cultural knowledge, but imagine you are a student who comes across a question that contains an idea with which you know nothing. Regardless of whether or not the knowledge is required to do the mathematics of the problem, it impacts student confidence and therefore their performance.
  5. The sixth question assumes that students have some minor technical knowledge, which I would classify in the same genre as my fourth point; students with a minimal technical background may struggle with the mechanics of this task. This may not affect a huge number of students, but the assessment instrument should be as neutral as possible to allow the greatest number of students to interact with the mathematics of the task, not the mechanics of the task.
  6. The seventh question has a video. It’s probably between 4 and 10 megabytes in size. Can you imagine what this will do to your school’s bandwidth if every student in a particular grade is accessing the resource at the same time?


There are some things which I think are obvious to me about the computer based assessment that PARCC is working on.

The first is that many of these questions are still going to require actual math teachers, with some experience looking at student work, to look at. Most of these questions are not just reformated multiple choice questions (although some of them are). While this increases the per-student cost of the assessment, I do not think that there are computer programs (yet) that exist that can accurately capture and understand the full range of possible mathematical reasonings of students.

Next, some of the more adaptive and social aspects of the work Dan Meyer and company have put together, are not present in this work. This assessment is intended to capture what students think now, rather than what students are able to do once given more information. This is still an assessment of the content students have learned, and does not appear to do an ideal job of making sense of how students make sense of problems and persevere in solving them, attend to precision, or any of the other standards for math practice (SMP). While it is clear to me that students will have to use these standards when doing this assessment, I do not see how anyone looking at the resulting student work is going to be able to say with any accuracy what is evidence of each of the SMP.

Unfortunately, unless the standards for math practice get captured somehow by an assessment (a goal of ours during next year with our project is to make an attempt to do this systematically), it is unlikely that teachers will use them.



The Treachery of Words

This is not a pipe


Words are not ideas, anymore than the picture above is a pipe (it’s a picture of a pipe).

When we communicate about ideas we are forced to use words (or gestures or images, which are also not the ideas themselves), and so consequently we are never communicating ideas directly. We communicate about ideas through the medium of language.

It is possible for someone to learn the words that represent an idea without learning the idea themselves, even so much as to be able to mimic the output expected of someone who understands the idea.

This happened to me. I learned about linear functions in school. I learned about “constant rates of change” and “y = mx + b” and “find the rise over the run” and many other phrases which were connected to finding and subsequently graphing the equation of a line. I even later learned how to think of a specific line as a transformation through a translation and a rotation from any other line.

However, in a fourth year topology class, my professor used a completely different set of words. He said, “Let’s do an easy example. We have line segment AB and line segment CD, and our objective is to find the mapping function from AB to CD. Think about that problem and bring your work to next class.” Well, I spent a week thinking about that problem and could not do it. I spoke to my professor about it, and he gave me a look and told me that it was “not my fault, this is unfortunately due to how you were taught.” Note how I was earlier able to articulate “transform one line into another”, but not able to repurpose this information to create a function. It turns out, the solution is a linear function.

Here I am, many years later, with a much better understanding of the wide range of things in our world which are linear functions. In fact, in only a few minutes, I am now able to “prove” to myself (at least experimentally) a more general version of what my professor asked me to prove all those years ago. I have moved past a “just the words” understanding of linear functions into a more flexible and useful understanding.

This is the objective of good mathematics teaching, and the reason why I think that we need far fewer lists of things to achieve with our students, and far more time spent looking at very similar things but in a very wide range of contexts. We need to move past students learning about the words that represent mathematics and giving them enough time to actually learn about the corresponding ideas.



Keynote on Formative Assessment

I recorded some video and the audio from a keynote presentation I gave a couple of weeks ago. It turns out the video wasn’t all that useful, but I did a screencast of my presentation notes, and added the audio from my keynote to it.



Were I to do this again, I would definitely do a better job of summarizing my main points at the end, and I would probably explore more closely some of the different concrete methods through which one can do formative assessment.



How I used mathematics to choose my next apartment

The house my wife and I live in was recently sold, and so we have started looking for another apartment. Our current lease expires in a year and a half, and so we decided that, given how challenging the rental market is in NYC, that we should start looking right away. We also decided, somewhat arbitrarily, that we would attempt to find an apartment in the next six months, if only because we knew we would get sick of looking pretty quickly.

During the first 5 weeks of our apartment hunt, we found five apartments that we thought were worth looking at. Over six months, I extrapolated that we would get to see about 26 apartments that would satisfy our requirements. What we discovered, with each apartment, is that we basically got to see the apartment and then pretty much decide immediately if we wanted to apply to rent the apartment or not.

It turns out that there is a nifty mathematical algorithm that one can use to optimize one’s chances at picking the best apartment possible. We expect to have 26 apartments to look at, each of which we inspect and then either accept or reject immediately, and we want to optimize our chances of picking the best apartment possible, from the 26 that meet our minimum criteria. This exactly matches a solved problem in mathematics; the Secretary Problem.

In the Secretary Problem, where one has to decide on the best applicant between n randomly ordered applications, an optimal solution is to reject the first n/e applicants, and then accept the next applicant that is better than any applicant you have seen before. The proof of this particular solution is here. I couldn’t reproduce this proof if asked, and there are details in it which are fuzzy for me, but I am pretty sure I understand why it works. Informally, the first n/e applicants act as a sampling space, and this gives you information on how good applicants will be, and that n/e happens to be where you achieve an optimal amount of information on applicant strength, allowing you to make the best determination you can of which next applicant to choose, without raising the probability too high that you’ve already rejected the best applicant.

For our specific apartment hunting problem, with 26 total apartments to view, 26/e (e ≈ 2.71828) ≈ 9.6  10 apartments. So, my wife and I looked at 10 different apartments, and while we did this, I informally ranked the apartments based on the criteria my wife and I agreed were important to think about (space, apartment lay-out, cost, location, quality of neighbourhood school, commute time). The 11th apartment either she or I looked was superior in many ways on all of these criteria than any of the others we had looked at, so I whole-heartedly threw in my support for it, knowing that this specific apartment is most likely to be the best apartment we will see.

We’ve submitted an (overly lengthy) application for the apartment. Wish us luck.



One model for adding fractions

When I first started tutoring students, I often noticed that they struggled to add fractions. The addition of fractions just did not make sense to them. Part of this is caused by students having a weak understanding of fractions, and part of this is caused by them not understanding why the typical algorithms used to add fractions make sense.

Here is one model that I developed for myself, so that I could understand why addition algorithms for fractions make sense, and then use this model to help students make sense of adding fractions.


Step 1:

Draw a picture to represent each fraction you want to add. So if I want to add 2/3 and 3/4, I would draw the following, making sure to use rectangles which are the same dimensions, since these two fractions must represent a fraction of the same unit (in this case, the rectangle).

Adding fractions - step 1

Step 2:

One problem with the above drawing is that it is not the only way to represent fractions. Anticipating that I will want to be able to match units, I realized that it is convenient to draw my fractions with different orientations, as below.

Adding fractions - step 2

Step 3:

Now, I can’t add the fractions above because the individual pieces of each fraction have different units. I cannot easily see what the total area would be for the two shapes, so I need to divide the two areas again, so that I end up finding the area of each fraction with the same sub-unit of the rectangle (in this case, 1/12).

Adding fractions - step 3

Step 4:

Now that the units of each fraction are the same, I can simply count the total number of units in the first fraction (9) and add it to the total number of units in the second fraction (8), and I arrive at my answer of 17 units. I have to remember though that the size of this unit, relative to the original unit, is a twelth, hence my answer is 17/12 in terms of the original unit.

Adding fractions - step 4


What makes this model difficult to understand, and perhaps this lies at the heart of understanding addition of fractions, is that I have switched from counting in one unit (the whole rectangle), to counting in different units (quarters and thirds), to then counting in yet another unit (twelths), and then being able to see this final total in terms of the original unit (the whole rectangle).

This suggests to me that the prerequisites to understanding fraction addition involve an understanding of what a unit is, understanding how to compose and decompose units, understanding different ways of representing fractions, and how different fractions can be equivalent in size, but measured in different units.



What does effective mathematics teaching look like?

The definition of what effective mathematics teaching looks like very much depends on what purpose we assign to teaching mathematics. A classroom where the primary objective is to teach students a specific set of mathematical skills for them to use later will look much different than a classroom where the primary objective is to teach students how to think mathematically, although there is obviously overlap between those two classrooms. For a good description of the type of classroom which achieves the first goal but fails at the second goal, see When Good Teaching Leads to Bad Results by Alan Schoenfeld.

I will describe a classroom where the primary purpose of the classroom is to encourage mathematical reasoning, with a secondary benefit of students practicing mathematics skills they have developed.


What are the students doing?

  1. Students are engaged in the standards for math practice.

    The Common Core Standards for Mathematical Practice, which are similar in many ways to the NCTM Process Standards, are a useful tool for understanding the types of activities students should be engaged in within a mathematics classroom.

    In order to really do mathematics, students need opportunities to problem solving, to use mathematics they know to model processes, and to do all of this in the socia-cultural contexts of their classrooms. Both of these sets of process standards do an excellent job of defining what it means to do mathematics, but are flexible enough to allow for a variety of different activities to qualify.


  2. Spending significant time solving rich mathematics problems.

    Routine problems with limited opportunity for investigation might be acceptable for students to use to practice skills they have learned, but they do not have the breadth necessary to allow students to do the inquiry necessary to learn mathematical reasoning. A significant amount of time in the mathematics classroom investigating, postulating, formulating, deciding, and analyzing mathematical situations is necessary if the habits of mind required for mathematical reasoning are ever going to be adopted by students.

    For excellent examples of rich mathematical tasks (some of which are used for assessment of understanding, and others are used more to prompt student thinking) see some of the web sites linked here.

  3. Students talk to each other about math.

    While there is definitely value in students spending at least some of their time thinking independently, there is tremendous value in students having opportunities to discuss mathematical ideas and problems with each other. The first is that it is through the repeated access to different linguistic and representative variations on an idea that we come to more than a superficial understanding of that idea. If I say words, you hear the words, and you might even think you can assign some meaning to those words, but it is only when you hear other variations on the formulation of the ideas behind the words, and see other representations (often physical or pictoral) of the ideas represented by those words, that you can come to a full understanding of the concept. For more information on the dangers inherent in a “linguistic-only” understanding of concepts, see Richard Feynman on Education in Brazil.

    There are other benefits of students talking to each about mathematics. One benefit is the person who describes their solution is either likely to see flaws in their reasoning (or at least receive feedback on those flaws) or in the articulation process of their reasoning, come to a better understanding of the concept. Another benefit is that instead of just one person in the room able to give feedback to students, every student in the room becomes a resource for each other. Finally, someone who has just learned a concept, and more importantly recently moved past their own flawed models of that concept, is often more able to explain the concept as compared to someone who learned the concept long ago, and no longer remembers their struggles with it.

  4. Students have the opportunity to revisit and reflect on mathematics they have learned.

    Human memory is limited. Essentially, our mind trims information from it that is not used frequently (or possibly archives it so that it is difficult to access).

    In a highly effective mathematics classroom, concepts students have learned are revisited, often as embedded practice within the current unit of study. For example, when students create graphs of linear functions by plotting points, they are also practicing plotting points. The skill they practice is used within the context of current problem solving. Note that this practice is only really effective once students have mastered the concept as practice without understanding leads to student confusion.


What is the teacher doing?

  1. The teacher uses formative assessment practices on a daily basis within their classroom.

    Aside from engendering the opportunities for “what students do” as described above, an effective teacher gathers evidence of their student’s learning in a systematic way. Formative assessment is a process through which a teacher assesses their students, and then uses this information to inform their teaching. It acts as a feedback loop within the cycle of teaching and learning. If we consider what is to be learned as being like a vast wilderness, then the curriculum the teacher follows is a map through that wilderness, and formative assessment is the process they use when checking their compass so as not to get lost.

    There are three basic indicators teachers can use to collect formative assessment information; what their students write, what their students say, and the body language students use that indicates how they feel. All of these are important markers for teachers.

    Unpacking formative assessment is not the goal of this blog post. For more information, I recommend reading Dylan Wiliam’s Embedded Formative assessment as a good starting place.

  2. Teachers must build a classroom environment where students want to talk about mathematics and have a growth mindset.

    Developing a positive and productive classroom culture is a critical component of effective teaching. Students must feel that their contributions to the classroom matter, and that they feel safe to make mistakes. Making mistakes, and learning from those mistakes is an important part of learning. The goal of mathematics classrooms should not be to prevent students from making mistakes, but to treat mistakes as opportunities for everyone to learn and to grow.

  3. An effective teacher uses questioning technique carefully and thoughtfully.

    Teachers ask a lot of questions. For example, in one classroom observation I did this year, a teacher asked 170 questions in a 40 minute period, which averages out to about 1 question every 14 seconds. Given that many teachers ask a great number of questions each class, improvements in questioning technique are therefore likely to improve overall teacher effectiveness, perhaps even dramatically.

    Good questions prompt students to think. Teachers with effective questioning technique do two things well; they have a set of generic questions prepared they can ask students to prompt their thinking which they use frequently enough that students begin to ask these questions of themselves before even talking to the teacher, and they actively listen to, and clarify their understanding of, student reasoning before responding.

  4. An effective teacher learns about the linguistic and cultural backgrounds of their students and adjusts accordingly1.

    Mathematics is a cultural activity. Therefore, as mathematics teachers, we are not only teachers of mathematics, but also teachers of the socio-cultural norms of mathematics. In order to do this effectively, we need to understand our students at a more than superficial level. A cautionary note here: This is an area where it is easy to fall prey to cognitive bias and judgemental attitudes. Teachers need to make their best effort to objectively understand their students’ cultures and their linguistic understandings and then make sense of how their students’ backgrounds impact what is effective for their students.

  5. An effective teacher uses technology to focus students on mathematical reasoning.

    Classroom technology, in an effective mathematics classroom, is used to support student’s mathematical reasoning. Rote practice exercises, even if administered via technology, do little to help students develop their reasoning skills, and because they lack context, have limited ability to help students develop connections between different areas of mathematics.

    Imagine a classroom where students are looking for connections between different forms of a quadratic function. They could plot these functions using pencil and paper, and then look for connections, but during the time students would take to draw the functions, they would lose track of the goal of the graphing. Every time we ask students to do another task in preparation for mathematical study, they lose active cognitive resources to keep track of the overall purpose of the task. Instead, in an effective classroom, the teacher would give students access to a graphing calculator or graphing software, and students would be able to focus on seeing connections between graphs, instead of creating the graphs.


What else would you add to this description of an effective mathematics classroom?



1. Suggestion offered by Ilana Horn. See this tweet.