Education ∪ Math ∪ Technology

Year: 2018 (page 1 of 2)

The End Of National Conferences

I have been to perhaps a dozen national conferences and to two dozen or so regional conferences across the United States and Canada. With the exception of one or two of these conferences, I regret having ever attended.

My regret stems not from finding the conferences uninteresting or not enjoying meeting people face to face that I had only ever met previously online, my regret stems from the fact that I think these conferences are fundamentally immoral when our world is in crisis.


Conferences that draw people from all over a country or all over the world require participants to fly to a single destination. This results in thousands of people flying to destinations whom otherwise would not be flying. Unfortunately, flying in an airplane carries with it a huge carbon footprint. One flight across a continent or across the Atlantic has roughly the same carbon footprint as using a car for an entire year.


Here is an aggregated list of conferences across the United States. I counted 1374 conferences occurring next year, each of which may have many hundreds to thousands of participants. That’s potentially millions of flights each year for people to attend these conferences.

Maybe these conferences would be worth their carbon footprint if people learned something significant from them that changed their practice. But my experience is that this is not true. In most cases, I suspect that if a few people took two days off from work and read the same good book about teaching and then planned together based on that book, they’d get as much (or more!) out of those two days than they learn from attending 10 different scatter-shot presentations. Sessions are just not typically long enough to result in tangible learning and presenters often just don’t know the actual audience of teachers they end up with well enough to plan a session that meets those teachers’ varied needs.

It’s true that once in a while I’ve attended sessions that really made me think. A few years ago, I attended the NCSM and NCTM conferences back to back, focusing only on sessions on instructional routines, only attending sessions by a group of people who worked closely with Magdalene Lampert on these routines. That was a hugely valuable conference for me! But 1 hugely valuable conference out of the 18 or so I have attended does not justify the environmental cost of these conferences.

I think that we might be able to replace national conferences with the following and to some degree, this may produce similar learning for participants:

  • Virtual conferences: Sessions are run via web conferencing software. These are ideal replacements for non-interactive (or minimally interactive) presentations that dominate most conferences.
  • Book study groups: Grab 2+ friends and take two days off from work. Everyone reads the same book on day 1, on day 2, everyone convenes to first describe what things they learned and then make plans to implement some of the suggestions.
  • Run smaller regional conferences: I know everyone wants to see Fawn Nguyen, Jo Boaler, or Dan Meyer speak at conferences, but I believe there is lots of local expertise in most parts of the world that could be drawn upon instead.

This year I cancelled my presentations at the NCTM annual conference, the NCTM regional conference in Seattle, and CMC South in Palm Springs. I went to one conference that I could drive to up at Whistler, the Northwest Math conference (it was really good). I do not intend to submit proposals to conferences in the future that require me to fly to the conference.


Geometric Constructions as puzzles

Geometric constructions are amongst my favourite things to teach in Geometry. Why? I see each geometric construction as a puzzle to be solved and I love watching children solve puzzles and share their solutions to those puzzles.

Puzzle #1: Given a line segment, draw a circle with its radius as the line segment.

Many constructions build on earlier constructions so that as students figure out how to do earlier constructions, they build the pieces they need to figure out more complex constructions. Further, more complex constructions embed all sorts of opportunities for practicing earlier constructions.

Puzzle #2: Draw another line segment with the same length as the given line segment with an endpoint on either A or B.

The invention of dynamic geometry software, like Geogebra, means that students can learn these earlier constructions without their early challenges using a compass and straightedge interfering with their ability to learn the mathematical ideas behind the constructions.

Puzzle #5: Draw a line segment that is exactly three times the length of the given line segment.

It is super helpful for students to have their prior work with constructions visible for themselves as examples to work from, and so once students have figured out how to do a construction with the digital tool, I have them transfer their construction to paper (ideally in their notebooks for reference) so they can access it later.

Puzzle #10: Draw three overlapping circles on the same line such that two of the circles have their centers on the middle circle.

Another advantage of the digital geometry tools is that you can provide partial constructions for students. This way students can work on the part of the construction that is new. This doesn’t give students practice with the earlier part of the construction but it is a subtle way to give hints to students for particularly complex constructions.

Puzzle #11: Use the circle below to help you draw a regular six-sided shape (regular hexagon).

When a student shares their constructions with the class, I usually call up a volunteer that is not that student to come up to the front of the room and perform the construction, following the verbal instructions from the first student. This means that the pace of the construction is likely to better match the pace other students can follow or copy the construction which leads to more students understanding the construction. After the construction is complete, I can ask another student to restate the instructions while I annotate important features of the construction.

Puzzle #17: Construct an octagon (regular 8 sided shape).

The last few constructions, I provide the least amount of support for students since a goal of mine is to see if students can do these constructions independently. However, note that in the instructions for the constructions, I try to make sure that the language of the constructions isn’t a barrier for my students.

Constructions #1 through #17 are available in this Geogebra book and a paper copy of these instructions is available as Lesson 2 of this Core Resource.

Let me know if you have any questions and please share other ideas you have about introducing students to constructions.

Too Many Rich Tasks, Not Enough Rich Pedagogy

Great teaching is more than putting good tasks in front of students because a good task enacted with terrible pedagogy is still terrible teaching. While I think hardly any teachers are terrible, every teacher can be better than they are.

I see a lot of sharing of tasks, games, and activities via Twitter and blogs, but I see much less sharing of pedagogical strategies teachers would use with those tasks, games, and activities, which means a lot of people are losing potential opportunities to learn about pedagogy.

Often people share routines like Which One Doesn’t Belong or Connecting Representations which on the surface look like pedagogical strategies, but while the names themselves are somewhat descriptive, they aren’t sufficient to understand the routines they describe.

That’s part of the reason we created videos of the two main instructional routines embedded in our curriculum, Contemplate then Calculate and Connecting Representations.

Here is a (compressed) video of Kit Golan enacting Connecting Representations with his 6th grade students.

We also created slides, a pre-planner, a lesson plan, and a description of the routine to go along with these videos.

A new project we are working on is to share the instructional components that make up the routines. Here is a video showing different talk moves that can be used by teachers, either within the routines or whenever they are needed.

Here is another video showing Kit that focuses on the annotation he did while another student restated the strategy of another student, showing that these different instructional strategies can be used together and towards specific instructional goals.

It is important that explanations in the math classroom are clear and complete so that all students can follow the mathematical arguments presented. Here is one of our teachers describing how she supported students in creating clear mathematical arguments for each other to follow.

Are videos like these helpful? Would more videos sharing some of these strategies be helpful (if so, which)? And can we share more math pedagogy with each other?


A Story About Low Expectations

A friend of mine has been fostering a child that has been diagnosed with both autism and cerebral palsy. They have seen him grow from a child who could not talk and who had a great deal of difficulty using his body to a child who asks for help when he needs it, communicates his needs and interests with others, and who can climb up a climbing wall without difficulty.

My friend shared that they were quite shocked during recent parent-teacher interviews when they were shown her foster child’s “work” from the term. They were blissfully unaware during the first couple of months of school, during which their foster child enjoyed going to school and they believed he was also getting an education, that the he was in fact not being educated. Their biggest concern was that the work he did do, circling a few answers on 2 or 3 review worksheets each day, was not helping him progress. They wondered, what does he do all day?

Their position is that while it is true that this kid is behind, not giving him any productive work to do is not going to help him catch up. They are very worried that his needs are not being met, and I agree with them. I worry that his teacher is “meeting him where he is at” and that this means that he has little to no opportunities to grow. When this child has been supported and pushed to grow, he has responded by learning and growing immensely. The extremely low expectations that this child’s school has for him risk his future.

So what would you do if you were this foster parent?


Teach Better by Doing Less

There are two common activities teachers do that have either little to no impact on student learning but which do take teachers a tremendous amount of time, time that could be better spent on other activities.

Desk with a huge pile of papers


Grading Student Work

There’s limited evidence that putting marks on students’ papers leads to students either being more motivated to work harder or that these marks leads to increased student learning. In fact, some of the literature on feedback suggests exactly the opposite is true:

When teachers pair grades with comments, common sense would tell us that this is a richer form of feedback. But our work in schools has shown us that most students focus entirely on the grade and fail to read or process teacher comments. Anyone who has been a teacher knows how many hours of work it takes to provide meaningful comments. That most students virtually ignore that painstaking correction, advice, and praise is one of public education’s best-kept secrets.

Source: Dylan Wiliam


But grading student work is extremely time-consuming and so if this effort doesn’t lead to student learning, why do we do it?

  • To communicate progress to children and their parents,
  • To evaluate students,
  • We are expected to grade student work.

In a recent parent-student-teacher interview, the teacher had samples of my son’s work in front of him. He shared directly what he liked about his work and where he thought my son could improve and we never talked about the numbers at the top of the paper at all. It is easier to communicate progress using artifacts of student learning.

Schools, in the interest of focusing on activities which have clear connections to student learning, should stop grading students and focus on time-efficient ways to solve the problems grading is intended to solve. Should teachers still look at student work? Definitely, but this should be as part of their process of planning future lessons and thinking about opportunities for feedback for students.


Creating Our Own Curriculum Resources

I spent years writing tasks for each of my classes, borrowing from other teachers when I could, but mostly making my own resources from scratch. These resources were not of much higher quality than what I could get from a textbook but I understood how they were designed and how I intended to use the resources. Writing curriculum took me SO much time.

Now I write curriculum about 50% of the time for my day-job and can see that the curriculum I currently create is far superior to the curriculum I used to create and far more complete and coherent. But it probably suffers to some degree from the same problem that led me to create my own curriculum as a teacher — it is extremely difficult to make sense of someone else’s lesson or task.

One feature of the curriculum I write that I think helps mitigate this problem of understanding curriculum is that it contains instructional routines. Once one knows a particular instructional routine, the task of understanding tasks to go along with that routine is far easier. Teachers who know the Connecting Representations instructional routine can look at tasks and are better able to make decisions about which tasks to use and why.

Evidence on use of curriculum suggests that all teachers benefit from access to high quality curriculum resources. In this experimental study some teachers were given access to Mathalicious  and others were not, and the teachers who had access saw better performance from their students.

What I think teachers should have more time to do is modifying and adapting curriculum for their particular context and their particular students. This is why our curriculum is licensed with an open license and why we share the curriculum in Google Document format — it makes it much easier for teachers to adapt and modify the curriculum rather than having to recreate a document from scratch just because the original is locked in PDF format.

We also wrote our curriculum to be largely sequenced but with lots of opportunities for teachers to make choices within the curriculum or to design their own tasks. Our tasks are aligned to the evidence of understanding we expect to see in students, which means the blueprints for constructing their own tasks are available to teachers as a support – leading to the best of both worlds for teachers: the autonomy to construct their own curriculum while not needing to reinvent the wheel each day.


Teaching Better

If teachers stopped grading all student work and writing all their own curriculum from scratch, then they would have more time for other tasks that contribute more to student learning such as:

  • Designing responses to evidence of student achievement (eg. formative assessment),
  • Collaborating with colleagues to investigate instructional strategies (eg. micro-teaching)
  • Work with individual or groups of students to support their learning.

What else could teachers do with their time to support student learning if all of a sudden they had more time available?

Handing out workbooks is not teaching

A number of times over the past few years, I have been confronted by a phenomena I do not understand: children being handed workbooks and, with very little to no instruction, expected to complete the workbooks and develop an understanding of the material contained within.

A variety of math workbooks


For some children this works out fine. The workbooks, mostly being designed for independence anyway, are sufficient for children who have some background in the material and see the patterns developed within the workbooks. For many children however, the experience is potentially one of frustration and humiliation: they have been given a task that their teacher clearly expects them to be able to do, and they cannot.

I have seen workbooks handed out with minimal to no instruction in two very different kinds of circumstances. In one case, the teacher sees a class with a diverse needs and although all of the students had the same overall goals for the class, students clearly have different levels of preparation and interest in the course. The other case is when teachers have a multi-age classroom (which also contains students with different levels of preparation and interest) and no time to actually teach anything at all. Both of these are compounded by the preparation of the teacher leading the classroom to deal with varied needs.

I have been in the first case, leading a group of students with varied needs but with the same goal, many times during my career. The best solution I have learned is the use of instructional routines with designs for interaction to support a varied set of needs, while working together towards the same mathematical goal.

As for the second case, when teachers have a multi-age classroom, I don’t know for sure what I would do. If I found myself teaching in such a classroom,  I think I would do two things; teach mini-lessons (using a variety of formats depending on the mathematical goals, but again instructional routines seem like a good fit) and assign problems from a somewhat independent workbook, and then while some students are working in the workbook, engage the other students in a mini-lesson on a different topic. I would have to organize the space so that students who were working on similar material sat together and I would need to develop classroom culture so that students knew to talk to each other after first trying the math themselves and to wait until I was available to assist. I would also look for places in the mathematical goals for the year where potential alignment between different grade levels occurs and use these opportunities to engage the entire classroom. For some students the activity might be review and for other students the activity might be somewhat new material, but both of these seem like helpful ways to use classroom time.

Imagine you are teaching a classroom with 20 students roughly evenly distributed from grade 2 through 7 (age 7 through 13). What would you do to meet the needs of these learners?


What is Conceptual Understanding?

According to Adding It Up  (H/T Dan Meyer):

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)


Essentially, conceptual understanding is knowing more than isolated facts, it is also knowing connections between those facts and having those facts well organized.

This image represents someone who understands some isolated facts.

Isolated facts as separated points

And this image represents what conceptual understanding looks like:

Points connected with line segments

I propose that those line segments between the points, representing the connections between isolated knowledge, are themselves a type of knowledge, consequently our second image is really more like the first image.

Connected dots transforming into disconnected dots


What is clearly missing in this analogy is that conceptual understanding is knowledge that is well-organized, but I claim that organization of knowledge is itself another type of knowledge. Also missing is knowing why the knowledge is important and the contexts in which an idea are useful, but again these also seem just like other types of knowledge.


Let’s take a look at a problem I had to solve recently and see if we can use it to unpack what conceptual understanding is.

Algebra II August 2016 Regents Exam, Question #1

According to the solution key, the writers of this question are assessing students’ ability to solve quadratic equations.  However, that likely requires applying the quadratic equation to all four equations given, which would be time-consuming and potentially error-prone.

When I solved this problem, I saw four quadratic functions, each transformed from y = x2, and based on those transformations, I was able to eliminate (1) and (3) as options since they have real solutions. I also know that if 1 – i is a solution, then so is 1 + i, and that the sum of the roots of a quadratic function are related to the coefficient of x in the equation. I could not remember if the sum of the roots is equal to the coefficient of x or to -1 times the coefficient of x, so I multiplied out (xa)(xb) to double-check and determined that if a and b are the roots of a quadratic equation, then -(a + b) is the coefficient of x in that equation (assuming a lead coefficient in the equation of 1). 1 – i  + 1 + = 2 so (4) must be the answer.

My actual solution strategy doesn’t matter all that much, what matters is that in order to produce it, I needed to know some smaller things and that those smaller things could be seen as connected together to create a larger idea. Instead of seeing this as the difference between an expert and a novice, I prefer to think of knowing things as being a gradient between knowing a little bit and knowing more than a little bit.


Conceptual understanding is both knowing ideas in an interconnected and organized way and knowing more ideas along the novice-expert spectrum.

The reason why conceptual understanding is an important goal is because otherwise we might be tempted to rely on teaching kids tricks instead of mathematics.

What mathematics does a child learn from learning this procedure for adding fractions?

Butterfly Method for Adding/Subtracting Fractions

Source: Pinterest


Very little! It is extremely difficult for a child to connect this procedure to other mathematics that they know, consequently using a trick like the butterfly method is likely to lead to students knowing an isolated idea and not much else. Note that I am not opposed to memory aids, I just think they should be based on solid mathematical reasoning and they definitely should not be taught instead of the mathematics.




Two Cultures of Mathematics Education

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.



There are many cultures of mathematics, a question is, which one dominates k-12 education and why?

The majority of those two cultures are likely aligned to one of these two possibly overlapping goals of teaching mathematics.

  1. To enable students to be able to solve a specific set of mathematical problems.
  2. To teach students a body of mathematics.

If you look at what is assessed in mathematics, the majority of assessment focuses on students being able to solve specific mathematical problems. Here are some examples of problems that New York State feels are so important for students to be able to solve that they gave these questions to the tens of thousands of student who sat for the June 2018 Algebra I Regents examination.

The first 4 problems from the June 2018 Algebra I Regents Exam

I do not know if it is possible to assess students on their understanding of mathematical principles without giving them some type of problem to solve, but I have never seen an assessment that did not involve solving mathematical problems, so it is either systemic in the way mathematics is assessed or it is impossible to assess mathematical principles apart from problems they can applied to solve.

I have occasionally seen teaching that focused on students building understanding of mathematical principles rather than attempting to solve specific problems, but even in this case the basis of the activity could be considered a mathematical puzzle or problem. However, almost every time I have observed a class the focus is on students being able to solve a fairly specific set of problems with little to no opportunity to generalize from the experience.

According to Tim Gowers, in his essay Two Cultures of Mathematics, these two same cultures apply in the world of mathematicians as well, except the culture that dominates is reversed; most mathematicians focus on learning more mathematical principles and eschew pure problem solving as a goal.

As Sir Michael Atiyah (Atiyah, 1984) puts it:

MINIO: How do you select a problem to study?

ATIYAH: I think that presupposes an answer. I don’t think that’s the way I work at all. Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.


So if mathematicians value mathematics as a set of organizing ideas and mostly prefer to study the organizing ideas rather than the problems they can be used to solve, then why does almost all of k-12 mathematics focus primarily on problems to be solved and much less so on general mathematical principles?

Maybe the problem lies with the standards that kindergarten to grade 12 teachers are expected to focus on? Let’s take a look at the Common Core clusters each of the questions listed above is supposed to correspond, which are A-REI.B, F-IF.A, A-APR.A, and A-SSE.B respectively.

The text of those clusters is as follows:

  • A-REI.B: Solve equations and inequalities in one variable.
  • F-IF.A: Understand the concept of a function and use function notation.
  • A-APR.A: Perform arithmetic operations on polynomials.
  • A-SSE.B: Write expressions in equivalent forms to solve problems.

It certainly seems like 3 out of 4 of those are focused on students being able to use what they know to solve specific kinds of problems with verbs like solve, perform, and write. So the fault could lie with how standards are written for courses.

Let’s contrast these standards with the objectives from one of the courses Tim Gowers teaches to see how they compare:

The goals of a Tim Gowers Math Course

One obvious difference between the standards and Gowers’ syllabus is the inclusion of verbs in the standards, which besides being the intended foci of a high school algebra course are also intended to be a map for people writing assessments, and you cannot write an assessment without specifying what students will do in order to demonstrate understanding. Gowers’ syllabus is just a list of topics, but I note that I have no idea based on his syllabus what kinds of problems these topics would be useful for and maybe the absence of verbs is why.

There is a way of teaching that breaks this dichotomy between solving problems and learning mathematical principles and uses problems to teach mathematical principles but this style of teaching seems incredibly rare and is somewhat difficult to learn and somewhat difficult to do. The best tool I have found for learning this kind of teaching is through the use of instructional routines, particularly an instructional routine like Connecting Representations that focuses on students making connections between two different representations of the same mathematica principle.

I’m not really sure why this difference between the work of mathematicians and the goals of k-12 mathematics teachers exists. One possibility is that it’s a lot easier to see how mathematics can be applied to solving problems than it is to see the mathematical principles being used and very few k-12 mathematics teachers are as mathematicians.



  1. T. Gowers, The Two Cultures of Mathematics, Mathematics: Frontiers and Perspectives (2000)
  2. F. Atiyah, An interview with Michael Atiyah, Math. Intelligencer 6 (1984), 9-19.



Magical Hopes: Technology and the Reform of Mathematics Education

In 1992, Deborah Loewenberg Ball wrote an article called Magical Hopes: Manipulatives and the Reform of Mathematics Education. This article is intended to draw some connections between our use of manipulatives and our use of technology in math education, and hopefully offer some suggestions for improving the use of technology.

There is a similar magical hope that technology can be used to reform mathematics education and I think that some of this reform is misguided, and this is in fact why I no longer work as an educational technologist. Without reforming work we as educators do to make links between the resources we use and the learning outcomes we hope to see borne out by children, there will be little effective change in the overall learning experiences for children.


Here is a paragraph from Dr. Ball’s article that stood out to me. For some context, Dr. Ball starts her article describing a student explaining how they understand odd versus even numbers. For further detail, I recommend reading her article.

Some teachers are convinced that manipulatives would have been the way to prevent the students’ “confusion” about odd and even numbers. This reaction makes sense in the current context of educational reform. In much of the talk about improving mathematics education, manipulatives have occupied a central place. Mathematics curricula are assessed by the extent to which manipulatives are used and how many “things” are provided to teachers who purchase the curriculum. Inservice workshops on manipulatives are offered, are usually popular, and well attended. Parents and teachers alike laud classrooms in which children use manipulatives, and Piaget is widely cited as having “shown” that young children need concrete experiences in order to learn…


Here is an updated paragraph with the current reform efforts focused on utilizing technology in mathematics education.

Some teachers are convinced that [technology] would have been the way to prevent the students’ “confusion” about odd and even numbers. This reaction makes sense in the current context of educational reform. In much of the talk about improving mathematics education, [the use of technology has] occupied a central place. Mathematics curricula are assessed by the extent to which [technology is] used and how many “things” are provided to teachers who purchase the curriculum. Inservice workshops on [technology] are offered, are usually popular, and well attended. Parents and teachers alike laud classrooms in which children use [technology], and [Papert] is widely cited as having “shown” that young children need [access to technology] in order to learn…


Here is an example of technology intended to be used in mathematics education. This is a virtual manipulative I created which is intended to draw students attention to how function rules can be represented as visual sequences. Try changing the sliders and keeping track of what you notice and what you wonder.

Now consider, what do you think children might notice when looking at this manipulative? What will they wonder? And most importantly, what will they learn as a result of using it?

If you have time, I recommend actually trying this activity out with a variety of children at different age levels. What I have learned is that children rarely see things the same things as adults (especially if those adults are mathematicians and/or mathematics teachers). In particular most children with whom I have used this particular manipulative do not attend to the differences between terms in the same way that adults do, and very few children even notice that the manipulative above includes an equation. They tend to focus almost entirely on the elements of the virtual manipulative over which they have control (the sliders) and the elements that change the most (the visual pattern) and in particular generally ignore features not immediately visible (the difference between terms) or recognizable (the equations).

Technology that relies on children independently making mathematical discoveries is likely to fail for some children. Given that one’s current mathematical knowledge is a large factor in how easily one makes mathematical discoveries, this use of technology may increase educational inequity rather than decrease it.


Recommendation: Utilize instructional strategies intended to make the sharing of mathematical ideas explicit for all students.


Here is another paragraph from Dr. Ball’s article:

Manipulatives –and the underlying notion that understanding comes through the fingertips– have become part of educational dogma: Using them helps students; not using them hinders students. There is little open, principled debate about the purposes of using manipulatives and their appropriate role in helping students learn…


Fortunately, there is tremendous debate about the use of technology in mathematics education, but most of this debate has centered on grain sizes of discussion that are far too broad to be helpful. Instead of questions like, “Is technology helpful in mathematics education?” we should be asking ourselves much more specific questions that have the same form as “is this use of this technology helpful for these students in this situation?” And then when we come up with a first draft answer for that question, we should make sure that our second draft includes some evidence from our students that the learning experience in question was actually improved by the use of the technology.


Recommendation: The links between what we hope children will learn from a particular activity and the evidence that they have learned should be made clear and investigated.


Here is another insightful quote from Dr. Ball:

My main concern about the enormous faith in the power of manipulatives, in their almost magical ability to enlighten, is that we will be misled into thinking that mathematical knowledge will automatically arise from their use. Would that it were so! Unfortunately, creating effective vehicles for learning mathematics requires more than just a catalog of promising manipulatives. The context in which any vehicle–concrete or pictorial–is used is as important as the material itself. By context, I mean the ways in which students work with the material, toward what purposes, with what kinds of talk and interaction. The creation of a shared learning context is a joint enterprise between teacher and students and evolves during the course of instruction. Developing this broader context is a crucial part of working with any manipulative. The manipulative itself cannot on its own carry the intended meanings and uses.


I sometimes see the same kind of thinking from mathematics educators (myself included); that just because a particular Desmos activity or Geogebra applet contains some mathematical idea, that children will learn the mathematical principles that are potentially generative in the activity. Sometimes some children see the mathematical principle, we have them present their work, and then we either hope or assume that this sample of children is representative of the whole class. Unfortunately, selection of students to present is a challenging activity that tends towards sharing strategies deemed as correct rather than surfacing a mathematical idea (like Sean’s observation in Deborah Loewenberg Ball’s article) that is fruitful to present and discuss even though an outside observer would consider the idea incorrect.


Recommendation: Before students engage in a technological activity, anticipate how we think they may consider the mathematical features. While students engage in a particular activity, listen to students talk, watch them work, and ask them questions to probe at their actual mathematical thinking. Use this information to potentially revise the activity.


Dr. Ball continues with this argument:

If we pin our hopes for the improvement of mathematics education on manipulatives, I predict that we will be sadly let down. Manipulatives alone cannot and should not–be expected to carry the burden of the many problems we face in improving mathematics education in this country. The vision of reform in mathematics teaching and learning encompasses not just questions of the materials we use but of the very curriculum we choose to teach, in what ways, to whom, and in what kinds of classroom environments and discourse. It centers on new notions about what counts as worthwhile mathematical knowledge. These issues are numerous and complex. For instance, we need to shift from an emphasis on computational proficiency to an emphasis on meaning and estimation, from an emphasis on individual practice to an emphasis on discussion and on ideas, reasoning, and solution strategies. We need to alter the balance of the elementary curriculum from a dominant focus on numbers and operations to a broader range of mathematical topics, such as probability and geometry. We need to shift from a cut-and-dried, right-answer orientation to one that supports and encourages multiple modes of representation, exploration, and expression. We need to increase the participation, enthusiasm, and success of a much wider range of students. Manipulatives undoubtedly have a role to play in these aims, by enhancing the modes of learning and communication available to our students. But simply getting manipulatives into every elementary classroom cannot possibly suffice to fulfill these aims.


This leads to some questions that we educators can ask ourselves before we embark on using some new technology with students (in the same way that at the time of Deborah Loewenberg Ball’s article was published that manipulatives were a new technology for many teachers).

  • Does using this technology help my students learn mathematics that they can use without the use of this technology?
  • How will someone who does not yet know the mathematics embedded within this technological tool see the mathematics?
  • Does this technology focus solely on the acquisition of a limited set of mathematical knowledge or is it possible for students to use deliberate practice to identify patterns across different problems and acquire new mathematical ideas?
  • Does this technology make it harder for my students to interact with each other and with me?
  • How will I learn how my students understand the mathematical ideas that are the focus of this lesson?
  • Who is the audience of this technology?
  • Does this technology exacerbate existing inequities in mathematics education?


Some of these questions were originally in this post.


I don’t have answers to all of these questions unfortunately but I think as a community of mathematics educators, we should be at least trying to answer them. I also have not unpacked some really large scale and potentially damaging initiatives around online learning and personalized learning. Stay tuned for a Part 2 to this post focused more on these uses of technology.


A Teacher Reflects on Their Teaching

I received this email from a teacher I know and with their permission, I am posting it and my response here. Identifying information has been redacted from these emails.


So I tutor this Junior from Stuyvesant in Algebra 2. But her text book is College Algebra & Trigonometry.

Her parents feel the materials I bring are entry level or at level at best, but fall short of the Sty expectation. I mentioned to you before she’s doing limits & pre-cal/cal topics as an 11th grader & not honors. She got a 60 on the last exam because she miscalculated a few limits & recurring series.

When I asked the parents what can I do if they feel my supplements are minimal skills at best, they told me the teacher said they are getting their juniors college ready, not only in practices but content. By the end of the Junior year, they should be proficient in material covered in the SATs, which includes some Cal & Discrete Math.

When I said but she’s rocking all the Regents material, her father replied, that’s one test, what about what comes after?

So are we short changing our kids at [school redacted]? Have we lowered the bar/expectation from being jaded by the system & have given up on the challenge of keep them learning? In my own practice, I have strong students & I felt like this semester I may have served them a disservice by not teaching basic limits & next level mathematics as an extension.

I don’t need a response, just venting on a reflection. But to have that face to face with a parent who has specifically asked me to raise my level of academic material sucked some life out of me. Are these conversations lost in our own population with parents who struggle to get their kids to 1) show up then 2) complete the minimum assignments & work.

I’m not sure where to go from here…..


For reference, Stuyvesant is one of the highly selective public schools of New York City which currently selects students based on academic achievement on an entrance exam. The school my friend works at is not one of the selective schools.

Here is my response:


Hi [Redacted],

This is a powerful and tough reflection on your practice.

One of the reasons we started the a2i project was to improve mathematics instruction in NYC and one element of mathematics we hoped to improve was the quality of the mathematics content to which students were exposed. That’s likely part of the reason we used the SVMI materials as a starting place and why we are in the middle of writing our own curriculum; because we aren’t super happy with what’s out there and available for our schools.

On the other hand, it seems clear that if you have a bunch of kids who are already struggling, it is counter-intuitive that what you want to do is raise the bar, so we introduced instructional routines with the goal of supporting classroom instruction, and in particular, teaching teachers instructional components they can include in their teaching which increase access to rigorous and challenging mathematics for all students, consequently allowing teachers to raise the bar.

In your situation, you’ve noticed the contrast between two Algebra II courses from two different schools. This contrast is part of the reason that parents and students work so hard to get into the selective schools. However, this contrast is a result of the system in which you work and not so much because of your own personal fault.

Kids select Sty because they want to be challenged. Kids choose a neighborhood school sometimes because they don’t want school to take up too much of their time. At Sty, kids are engaged in academic discourse and push and support each other to handle content that might otherwise be too challenging for them. At your school, much of this peer support may not be in place, and so as a community, your learners are not as able to handle more challenging materials.

Of course, this inequity of experience isn’t really fair. How can we help all schools develop the kind of academic community that Sty and the other selective schools have that push kids to be better than they are?

I wrote this post a while back and it may be relevant. Basically it can be summarized to, kids are more likely to have higher expectations for themselves and meet those expectations when the adults that work with them have high expectations. But expectations are part of a system-wide bias that exists that is hard to look outside of and even harder to change. Even if you turn around tomorrow and start trying to tackle more challenging material with your students, they’ll still be students who spent the previous 10 years not having to work or think as hard.



Here is their response to my email.


I appreciate the response. As I explained the situation & context to my [partner], I’m still torn & feel responsible for “creating” a finite course that terminates with a Regents rather than raise the bar & teach “mathematics” to where the regents is absorbed along the way. With 32 instructional days left, I’m curious to see how I can elevate my current practices to which I can start higher next September.


What other advice or support would you offer this teacher?