Posts Tagged ‘teaching and learning’

by Scott Adamson – January 4, 2017

Think about the teaching and learning of K-12 mathematics in a classroom. What images come to your mind? What are students doing? What is the teacher doing? What does the classroom look like? To what extent are the images in your mind dependent on the date? That is, would your images be different in terms of what students or teachers are doing if it was 1957 instead of 2017? Should your images be different?

These questions are of great interest to me right now and I don’t know if I have answers, but I have thoughts. Let me begin by explaining the origin of the question “is it 1957 or 2017?”

Allow me to describe, from a big picture viewpoint, an amalgamation of a mathematics classroom that could be at a school near you. Students are sitting in desks arranged in rows. The class period begins with a set of warm-up exercises that students are to complete in their notebooks. The teacher takes care of administrative details (attendance, paperwork). After a brief amount of time, the answers are displayed and students have the opportunity to have any issues clarified. Next, the class “goes over” last night’s homework. If a student has a question, they can ask and the teacher will show a solution. After several questions are answered, the students pass their homework papers to the teacher. A new topic is introduced and developed and students “take notes”. This means that they watch and copy what the teacher shows them to do. The new homework assignment is given and, with 4 minutes remaining in the period, students pack up their notebooks and prepare for the bell to ring announcing their transition to the next class.

So, what year is it? 1957 or 2017? Everything that happened in this class could have happened in 1957 nearly identically. Are the needs of learners of mathematics the same in 2017 as they were in 1957?

Let’s take a closer look at what happened in this classroom beginning with the content students were expected to learn.

Generally, students are expected to learn to do something. For example, students factor trinomials, solve equations, simplify expressions, compute derivatives and integrals, identify characteristics of a graph of a function, plot the graph of a function by hand, or memorize mathematical facts. Perhaps this was important in 1957 but given the technological advancements in recent years, every student in the classroom has access to a small device that will do, quickly and accurately, most of the things that they are learning to do with paper and pencil. Most students own (or have acccess to) a smartphone or tablet that, when connected to Wi-Fi, can access free and powerful tools such as the Desmos calculator (, Geogebra (, or WolframAlpha ( Sometimes when I visit classrooms, I challenge myself to see if it is possible to complete any tasks asked of students during that lesson using my cell phone and an appropriate tool like Desmos or WolfrAmalpha. Most of the time I can complete the given tasks and more! For example, in one case students were asked to find the equation of a line that is tangent to an implicitly defined curve. Using Desmos, not only could I get the equation of the line as requested, I could additionally create a graph of both the linear function and the implicitly defined equation! This provided a wonderful, visual confirmation that my thinking was correct.

I am fully aware of the tension between only pushing buttons on a device to make it do something and having a deep, well-connected understanding of the mathematical ideas. Am I able to get my cell phone apps to do the work needed because I learned the mathematics without these tools and understand it well?  How does the teaching and learning of mathematics change if we try to accomplish both the development of the conceptual understanding of the mathematical ideas and the effective and efficient use of technology to solve problem?

Part of my answer to these questions involves considering what mathematics we should teach. In 1957, the technological tools did not exist so it was more important for people to become effective and efficient at computations – perhaps appropriately using a slide rule when necessary. To be effective and useful in 1957 culture, certain skills and knowledge were needed to be mastered by students. Have the skills and knowledge needed changed to prepare students for 2017 culture? If part of the school’s responsibility is to prepare students for careers, have we kept up the pace to do so in 2017 by teaching students to be masterful at antiquated techniques like using synthetic division to divide polynomials?

I want to be a part of the discussion in the mathematics education community related to what mathematics we teach and then how we teach it given the tools we have access to and the cultural needs in the year 2017. Then, I want to see appropriate changes implemented.

How we use current pedagogical tools and the purpose they serve in teaching and learning needs to be addressed as well. I have two examples.

In many classrooms, document cameras, connected to a projector, are used to project images for all in the room to see. We can place anything under the document camera and its image is projected on the screen. This can be a very useful tool allowing students or the instructor to share ideas, methods, strategies, solutions, and procedures. In 1957, this same thing was accomplished (although less efficiently and with less potential in my opinion) using the overhead projector. Is that the difference between 1957 and 2017? We now have expensive* document cameras and projectors rather than overhead projectors but we use them in the same way? That’s it? Rather than writing the solutions to the homework exercises or warm-up assignment on an overhead transparency (which needs to be cleaned at the end of the day), teachers can now write the same thing in a notebook and project the image on the screen. Using a white piece of paper, teachers can cover up anything they don’t want the students to see yet and uncover solutions a little bit at a time – just like 1957 with an overhead projector!

The Smartboard is another example. Many classrooms have been equipped with Smartboards that look like a traditional whiteboard but are connected to a computer and projector. They can be used like a tablet device in that the board can be touched, written on (electronically), images moved, notes recorded, and more. However, in many cases, I observe teachers using the Smartboard in just the same ways they would have used a whiteboard – or a chalkboard in 1957! The Smartboard simply becomes a very expensive (but much smaller) chalkboard and its full potential is not realized.

Is it 1957 or 2017 in terms of the teaching and learning of mathematics? In many respects, it is 1957 when we think about the focus on the teaching of procedures, algorithms, and computations that are more efficiently and effectively completed by technology. It is 2017 when we think about some of the powerful tools that today’s classrooms have access to such as Smartboards and many internet resources. But it is 1957 when we think about how these tools are being used!

My goal for this series of posts is to help us to think about how the teaching and learning of mathematics might be and should be different in 2017 and beyond. Specifically, I hope to share thought provoking comments about these questions:

  • How should the mathematical content we teach be different in 2017 compared to 1957? Should we be teaching new things? Should we be teaching old things in new ways?
  • How can we best use 2017 pedagogical tools such as Smartboards, document cameras, handheld technology, or internet resources to mathematically prepare students for the world?
  • Is there a place for “old school” mathematical ideas in today’s mathematics classroom? For example, should students learn to perform computational skills like factoring, rationalizing denominators, or even computing a square or cube root by hand? If so, for what purpose?
  • How do we assess student learning? Suppose a decision is made to move toward teaching students to solve realistic problems using technological tools appropriately. It has been the practice to show students how to do something (e.g. divide fractions using the “keep-change-flip” algorithm), have them practice doing this skill so that they can get right answers frequently, then assess their ability by asking them to do a routine computation. If they do it – successful teaching and learning! If not, re-teaching and additional practice is needed. How will assessment look in 2017 and beyond?
  • What are the common objections to changing mathematics education and how are these objections refuted? For example, we have all heard teachers say, “what are you going to do if your calculator (now, cell phone, tablet) loses battery power? Then what will you do?” Or, “what will you do if you are stranded on a deserted island?”
  • How do we prepare our teaching force to make any proposed changes?
  • How do we prepare parents to support their children if any changes are made?


* I found a 3M 9100 Overhead Projector (the last I remember using) for $170 on Amazon and the Aver 5MP Document Camera (that I currently use) for $630.