The point about K-8 preparation is an important one. Schools are flocking to solutions like Khan Academy and instructional software because these options have the appearance of being well-developed in both content and pedagogy, thereby bypassing any deficiencies teachers have in these areas. The reality is that these options are sorely lacking in both, and an effort is currently underway to show precisely how Khan academy falls short. It is easy to sit students in front of these tools and see rows of students staring at screens as evidence that learning is going on, so it’s also easy to conclude that this is a productive way to apply technology funds. My response was not meant to address the reasons that technology is often thrown in classrooms in inappropriate ways. I bring this up every time I see anyone excited about getting access to technology in classrooms without also knowing how it will change their instruction. Preparation programs for mathematics teachers at universities are, more often than not, painfully inadequate in preparing teachers pedagogically and mathematically for teaching K-12 mathematics.

The problem I still see in starting sentences with “until students are coming to us with a stronger background” is the idea that students can’t do without the background knowledge and can’t possibly even explore the mathematics that depends on it. Looking at the big picture through technology can give students the intuition to then understand the background knowledge through observing patterns and testing theories – exactly what mathematics is supposed to be about. Conventional wisdom is that students have to memorize the unit circle before making a graph of f(x) = sin(x). Why not have them make observations of the function graph and its properties? If we are willing to put students through drilling a concept or skill over and over again in order to follow subsequent procedures that require this concept or skill, why not reverse the process? Technology makes this possible. It can be used to give the answer, and then have students figure out where those answers come from. This is a much richer process than merely following procedures. With only a pencil and paper, this isn’t possible.

As for using Geometer’s Sketchpad or Geogebra – you are absolutely right that students should be trying to do things by paper and pencil. I go back and forth myself between technology and pencil and paper activities. My students get the tactile experience of plotting by hand, bisecting angles and segments, and comparing triangles cut out of cardboard, not just on a screen. This keeps things interesting, as having the same tool the whole time gets boring.

I also agree that conceptual understanding and computational abilities can go hand in hand, but they don’t have to. They are definitely related. I have had students with 100% computational ability with no conceptual understanding. These students typically lose most of their knowledge from my class after the final exam. The more typical example though is a student that makes arithmetic mistakes but can explain mathematical concepts and ideas clearly and can figure out an appropriate solution method for a problem. If I’m testing students on algebra, I want to test their abilities in algebra, not arithmetic. I think they are distinct. You can understand the idea of using inverse properties to solve an equation while also getting -18 + 14 incorrect. To show that both are important, a teacher must devote appropriate time to both in the classroom.

Finally, on the point about data analysis and statistics: there is too much data out in the world to not have students learn some tools to understand what it means. Constructing models for the real world is the most powerful thing that math allows us to do. Evaluating those models and knowing how to navigate the use of statistics in the real world is something our students must understand to be literate in today’s society.

I know I’ve said a lot here – I figure it is better to get it all out there. Skim as needed and call me out on anything you disagree with me on.

Thanks for the comments!

I think the idea that technology is only a tool is frightening to those that are looking for easy ways to improve math achievement in a classroom. I found that teaching with it initially provided more engagement to students because it was new and different. I made sure, however, to change the way I presented things so that I made the most of its capabilities in my using it to teach. There are many things that it made much easier than just using a blackboard or whiteboard, the big one being the elimination of dead time spent erasing or drawing diagrams. It took a lot of effort though to use it to change how ideas were presented, and this took time. I still required my students to practice their skills given the additional insight (I hope) my presentation added using the technology.

For technology to be useful (and Kakaes definitely points this out in the article and one written since) the teacher must know how to use it. This requires serious investment on the part of administrators for training and time spent playing around with the technology if it is to make a difference.

2) To point #3, on the Common Core Standards and data analysis – A better question might be: what’s the purpose of teaching statistics (beyond mean, median, mode, and perhaps – perhaps – the normal curve and standard distribution) for most students? Is there one?

3) Point 4 – you expect your students to be able to multiply 37 by 41 by hand – and I agree with you. But by the time I see them, in high school, it has been such a long time since they’ve done it that many don’t remember the algorithm. And again, without well-prepared K-8 teachers, they won’t.

4) Point 5 – I disagree that technology allows students to do mathematical thinking even when their computational skills are not up to par. As far as I can tell, the only part of mathematics that can be suitably explored and understood via technology without a modicum of computational skills to back them up, in my experience, is geometry. And the technology for that subject is great – I agree completely with you. It’s just too bad that many of our students are spending time with Geometer’s Sketchpad or Geogebra before they’ve had a chance to measure some angles for real with a protractor or bisect some angles with a real (!) compass and straightedge. But I’m not convinced that allowing students to explore linear (or whatever type of) equations on Desmos.com or Wolfram|Alpha accomplishes anything meaningful unless they have the number sense to comprehend the result.

5) To me, though, the real point of Kakaes’ article, and one I’d love to hear your thoughts on, is the amazing lack of mathematical preparedness of many K-8 teachers, and the concomitant rush to technology that schools are now going to in order to remedy this deficit. It’s wonderful that high school teachers and teachers of other grades who are math experts can use this amazing new technology in our classrooms. But until our students are coming to us with a stronger background – more computationally fluent AND with deeper basic understanding – we will be in the woods, wondering why all of this new technology is not developing greater knowledge of and appreciation for mathematics in our kids.