If there is one thing that has characterized my work as math teacher and curriculum developer over all these decades, it is the use of all sorts of learning tools. My personal interest in geometric puzzles in my twenties gradually led me to increasing work with manipulatives. My involvement with the Logo movement gave me a theoretical framework for the use of learning tools of all sorts. Along the way, I made some mistakes and encountered some resistance, but navigating those helped develop my understanding.
I have already written a fair amount on the topic of tool-rich pedagogy. On this page, I will link to existing material as appropriate, and expand on any new insights.
By tools, I mean electronic and manipulative tools, and a few related pencil-paper tools. A broader definition of learning tools is possible, including for example Cartesian graphing. This is the approach I took with Anita Wah in What Are Themes, Tools, and Concepts? the 1994 introduction to our algebra textbookAlgebra: Themes, Tools, Concepts. That piece is definitely worth rereading, but without graphing technology, graphing cannot play the role of tool for the majority of students.
Here are the central arguments for a tool-rich pedagogy:
- Electronic, manipulative, and pencil-paper tools have the potential to make the classroom a more student-centered, collaborative environment, where learning is more likely to take place.
- They can help generate the sort of discussion and reflection that is a prerequisite for deep understanding.
- They can make some challenging ideas accessible to more students, while at the same time offering an opportunity for strong students to make connections they may otherwise miss.
- In particular, they can help bridge the visual and the symbolic, a connection that is fundamental to all mathematics and science.
- They can increase student self-reliance by providing a circumscribed microworld which they can navigate and master. (I came across the word "microworld" in the Logo movement, on which more below, but it applies to many tool-based environments.)
- And last but not least, they can bring variety to math classes, which are alas often the most mind-numbingly boring classes in our secondary schools, seen by teacher and student alike as necessary drudgery.
I made these arguments, more or less, in A New Algebra, where I illustrated them by showing how many powerful lessons can be created by the interaction of a rich theme (area) with a variety of tools.
Of course, tools can be used poorly, and often are. Manipulatives can be "taught" in a "this is how you do it, now practice" style that works so poorly with so many students. (See A Proposal for Early Math.) Electronic tools are often used in a "do this, do that, what do you notice?" style which suffers from the misconception that students will notice the thing we want them to notice. (See "Make These Designs" teacher notes.)
Intelligent lesson design using learning tools is often based on a reversal of traditional teaching practice, and the creation of "no threshold, no ceiling" problems in the microworld. The problems need to be engaging and accessible, but at the same time they should be worthy of reflection, collaboration, and discussion. A tall order, yes, but surf this site for quite a few such lessons.
- Here are a few examples:
- Algebra: Make These Designs
- Geometry: Angles Around a Point
- Functions: Doctor Dimension
- Blog post about rich curricular activities
Manipulatives vs. Technology
Actually, it's all technology, with manipulatives being "low tech" and electronic environments being "high tech". Still, comparing and contrasting the two is quite useful. The main difference is that the computer is typically fairly hypnotic and isolating, while manipulatives tend to generate collaboration and discussion. This is why as a teacher, I prefer the latter. However, distance learning has introduced another sort of tool, that stands somewhere in between: virtual manipulatives. I wrote about those on my blog (Part 1 | Part 2). You can find links to my own contributions to this genre on my Manipulatives page.
Other differences lie mostly in the social acceptance of the tools by different groups.
Some parents and some mathematicians object to any use of technology, because they assume that the way they learned math is the way today's students must learn math. This is one of the battlefields in the so-called Math Wars. In the long run, the anti-tech activists are doomed. The question inevitably will become how to use technology in math education, not whether. When I was in high school, I learned to use a slide rule and trig tables on the one hand, and to calculate square roots with pencil and paper, on the other hand. With the passage of time, no one seems to regret the passing of those techniques. As technology continues to permeate the culture, the process will continue.
On the other side, some academics who are very interested in electronic tools have resisted manipulatives. This is perhaps due to a their lack of experience with those, their shallow understanding of how physical materials can be used effectively, or the faddishness and research fundability of electronic approaches. I responded to many of their arguments in A Proposal for Early Math, The Pythagorean Geoboard, and Reply to Critic.
I am unswayed by these professors, and stand by my career-long endorsement of hands-on materials to enhance pedagogy at all grade levels.
Logo and its Descendants
For two or three decades, we've been in the midst of a constant, ongoing technological revolution. At the beginning of this era, Seymour Papert, the MIT professor who was the founder and leader of the Logo Logo was a computer language that swept through the
elementary schools in the early days of educational computing. movement, was a prophet of educational transformation through technological change. His utopian vision has failed to materialize, to say the least, but for some of us it triggered a fundamental shift in perspective. His concept of "objects to think with" (in the book Mindstorms) readily expands to "objects to talk and write about", and is in a lot of ways foundational to my tool-rich educational vision.
There are a number of Logo descendants available on the Web, each with its own strengths. One could spend a happy few weeks exploring those -- each is interesting in a different way. There is also a commercially available Logo-Lego robotics environment, which echoes the earliest Logo, which was originally used by children to program a physical "turtle" that moved around the floor. The most accessible of the Logo descendants are Scratch (from MIT) and its high school sibling Snap (from UC Berkeley.)
For many years, I was very involvedRead about it in The Turtle in the Age of the Mouse (1997). with another Logo descendant, perhaps the least accessible. This is a now-defunct environment called Boxer, which was originally developed at UC Berkeley's School of Education. It is the brainchild of Andy diSessa, who describes it and the accompanying philosophy in Changing minds: computers, learning, and literacy. I find diSessa's ideas compelling and paradigm-shifting, and highly recommend this book.
Boxer had an old-fashioned look and feel, and was so underfunded that it never fulfilled its promise, but I love the vision it embodies, that of an all-purpose educational computer environment that reflects the power and flexibility of the underlying machine. In it, beginners could carry out Logo-like experiments, learn to program, and get a sense for the architecture underlying such software as Photoshop (at a very basic level, of course.) Meanwhile, in the same environment, educators could create microworlds to enhance learning in many parts of science and math. Alas, it may well be that making all this possible in one easy-to-use software package is another unrealistic utopia, at least in today's political economy.
Still, I used Boxer in my Infinity class for many years. For the purposes of this class, I created several Boxer microworlds: a game for complex number arithmetic; tools for iterating real and complex functions, and for exploring Julia and Mandelbrot sets; and lessons about recursion. My students explored and enjoyed all these microworlds, and went on to create their own fractals via programming. This is a powerful mix, which hints at how a new sort of computational literacy that incorporates programming could enrich learning.
The Educational Technology Mainstream
The reality of the last thirty years is that at one end of the educational computing spectrum, the managed-learning environments continue to be pathetic imitations"No, Johnny, this answer is incorrect. You need to review lesson 2!"
"Yes, Johnny, this answer is correct!
Here is a virtual ice cream cone to reward you!" of bad teaching. At the other end, the prophetic visions of Papert and diSessa, which predicted synergy between programming and subject matter, have stayed in the margins. What has taken off dramatically and wonderfully is somewhere in between: polished, well-designed electronic environments for math education: the graphing calculator, and its spectacular heir, Desmos; interactive geometry, especially the powerful and protean GeoGebra; dynamic statistics and simulation software; and countless applets and platforms on the Web. The power of these tools is actually game-changing in their respective domains, as they bring previously unthinkable depth within reach of most students.
However this power does not come without its share of new challenges. Even some educators who embrace technology are uncomfortable with its relentless rate of change and they question calculators and programs that do "too much". Thus they see the inability of a tool to do certain things as a strength, and the ability of another to do them as a threat. Where they draw that line gradually changes. These days, that line seems to be at the gate of computer algebra systems (CAS), which are deemed by many (and not just reflexively anti-tech Luddites) to be too powerful for students.
To be honest, I don't (yet) know how to best use CAS, but I take comfort from a historical parallel: when graphing technology was new, many if not most teachers thought it would cripple students, because they would not know how to graph points and functions by hand. In fact, by now the evidence is overwhelming that the impact of this technology on learning has been positive. The short-term answer to some of the risks was to have students continue to do some graphing by hand. The long-term and in the end more powerful answer involved the development of new curricula that take advantage of the technology to go deeper in the study of functions, to extend the range and difficulty of applied problems, and to bring some content to younger students. I expect the same evolution to happen with CAS, especially now that this technology can no longer be hidden from students, as it is available to anyone with a browser or free software such as GeoGebra.
A tool-based pedagogy is wonderful for students, but it is challenging for teachers. First of all, it is a lot of work: manipulatives require endless housekeeping; computers require endless learning. Second, not all school schedules accommodate this style of teaching as readily. Third, this sort of student-centered learning may fly in the face of a well-established culture of "listen then practice".
As I see it, the work is well worth it, even if the richer learning it can engender requires some breaks with tradition: schedule changesSee Teaching in the Long Period, shifts in classroom cultureSee Group Work, and especially teacher collaborationSee my Teaching page.. I hope that this Web site will help you move in that direction.
Article updated in September 2020