Ah, summer. A great time to kick back, relax, and have time to write reactions to things that bug me.

I read through the article on Slate titled ‘Why Johnny Can’t Add Without a Calculator’ and found it to be a rehashing of a whole slew of arguments that drive me nuts about technology in education. It also does a pretty good job of glossing over a number of issues relative to learning math.

The problem isn’t that Johnny can’t add without a calculator. It’s that we sometimes focus too much about turning our brain into one.

This was the sub-heading underneath the title of the article:

Technology is doing to math education what industrial agriculture did to food: making it efficient, monotonous, and low-quality.

The author then describes some ancedotes describing technology use and implementation:

- An experienced teacher forced to give up his preferred blackboard in favor of an interactive whiteboard, or IWB.
- A teacher unable to demonstrate the merits of an IWB beyond showing a video and completing a demo of an electric circuit.
- The author trying
**one**piece of software and finding it would not accept an answer without sufficient accuracy.

I agree with the author’s implication that blindly throwing technology into the classroom is a bad idea. I’ve said many times that technology is only really useful for teaching when it is used in ways that enhance the classroom experience. Simply using technology for its own sake is a waste.

These statements are true about many tools though. The mere presence of one tool or another doesn’t make the difference – it is all about how the tool is used. A skilled teacher can make the most of any textbook – whether recently published or decades old – for the purposes of helping a student learn. Conversely, just having an interactive whiteboard in the classroom does not make students learn more. It is all about the teacher and how he or she uses the tools in the room. The author acknowledges this fact briefly at the end in arguing that the “shortfall in math and science education can be solved not by software or gadgets but by better teachers.” He also makes the point that there is no “technological substitute for a teacher who cares.” I don’t disagree with this point at all.

The most damaging statements in the article surround how the author’s misunderstanding of good mathematical education and learning through technology.

Statement 1: “Educational researchers often present a false dichotomy between fluency and conceptual reasoning. But as in basketball, where shooting foul shots helps you learn how to take a fancier shot, computational fluency is the path to conceptual understanding. There is no way around it.”

This statement gets to the heart of what the author views as learning math. I’ve argued in previous posts on how my own view of the relationship between conceptual understanding and learning algorithms has evolved. I won’t delve too much here on this issue since there are bigger fish to fry, but the idea that math is nothing more than learning procedures that will *someday* be used and understood does the whole subject a disservice. This is a piece of the criticism of Khan Academy, but I’ll leave the bulk of that argument to the experts.

I will say that I’m really tired of the sports skills analogy for arguing why drilling in math is important. I’m not saying drills aren’t useful, just that they are never the point. You go through drills in basketball not just to be able to do a fancier shot (as he says) but to be able to play and succeed in a game. This analogy also falls short in other subjects, a fact not usually brought up by those using this argument. You spend time learning grammar and analysis in English classes (drills), but eventually students are also asked to write essays (the game). Musicians practice scales and fingering (drills), but also get opportunities to play pieces of music and perform in front of audiences (the game).

The general view of learning procedures as the end goal in math class is probably the most destructive reason why people view math as something acceptable not to be good at. Learning math this way can be low-quality *because* it is “monotonous [and] efficient”, which is not technology’s fault.

One hundred percent of class time can’t be spent on computational fluency with the expectation that one hundred percent of understanding can come later. The two are intimately entwined, particularly in the best math classrooms with the best teachers.

Statement 2: “Despite the lack of empirical evidence, the National Council of Teachers of Mathematics takes the beneficial effects of technology as dogma.”

If you visit the link the author includes in his article, you will see that what NCTM actually says is this:

“Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education.”

…and then this:

“The use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills.”

In short, the National Council for Teachers of Mathematics wants __both__ understanding and computational fluency. It really isn’t one or the other, as the author suggests.

The author’s view of what “technology” entails in the classroom seems to be the mere presence of an interactive whiteboard, new textbooks, calculators in the classroom, and software that teaches mathematical procedures. This is not what the NCTM intends the use of technology to be. Instead the use of technology allows exploration of concepts in ways that cannot be done using just a blackboard and chalk, or pencil and paper. The “*and other technological tools* next to calculators in the quote has become much more significant over the past five years, as Geometers Sketchpad, Geogebra, Wolfram Alpha, and Desmos have become available.

Teachers must know how to use these tools for the nature of math class to change to one that emphasizes mathematical thinking over rote procedure. If they don’t, then math continues as it has been for many years: a set of procedures that students may understand and use some day in the future. This might be just fine for students that are planning to study math, science, or engineering high school. What about the rest of them? (They are the majority, by the way.)

Statement 3: “…the new Common Core standards for math…fall short. They fetishize “data analysis” without giving students a sufficient grounding to meaningfully analyze data. Though not as wishy-washy as they might have been, they are of a piece with the runaway adaption of technology: The new is given preference over the rigorous.”

If “sufficient grounding” here means students doing calculations done by hand, I completely disagree. Ask a student to add 20 numbers by hand to calculate an average, and you’ll know what I mean. If calculation is the point of a lesson, I’ll have students calculate. The point of data analysis is not computation. Just because the tools take the rigor out of calculation does not diminish the mathematical thinking involved.

Statement 4: “Computer technology, while great for many things, is just not much good for teaching, yet. Paradoxically, using technology can inhibit understanding how it works. If you learn how to multiply 37 by 41 using a calculator, you only understand the black box. You’ll never learn how to build a better calculator that way.”

For my high school students, I am not focused on students understanding how to multiply 37 by 41 by hand. I do expect them to be able to do it. Usually when my students do get it wrong, it is because they feel compelled to do it by hand because they are taught (in my view incorrectly) that doing so is somehow better, even when a calculator sits in front of them. As with Statement 3, I am not usually interested in students focusing on the details of computation when we are learning difference quotients and derivatives. This is where technology comes in.

I tweeted a request to the author to check out Conrad Wolfram’s TED Talk on using computers to teach math, and asked for a response. I still haven’t heard back. I think it would be really revealing for him to listen to Wolfram’s points about computation, the traditional arguments against computation, and the reasons why computers offer students new opportunities to explore concepts in ways they could not with mere pencil and paper. His statement that math is much more than computation has really changed the way I think about teaching my students math in my classroom.

Statement 5: “Technology is bad at dealing with poorly structured concepts. One question leads to another leads to another, and the rigid structure of computer software has no way of dealing with this. Software is especially bad for smart kids, who are held back by its inflexibility.”

Looking at computers used purely as rote instruction tools, I completely agree. That is a fairly narrow view of what learning mathematics can be about.

In reality, technology tools are perfectly suited for exploring poorly structured concepts because they let a student explore the patterns of the big picture. The situation in which “one question leads to another” is exactly what we want students to feel comfortable exploring in our classroom! Finally, software that is designed for this type of exploration is good for the smart students (who might quickly make connections between different graphical, algebraic, and numerical representations of functions, for example) and for the weaker students that might need different approaches to a topic to engage with a concept.

The truly inflexible applications of technology are, sadly, the ones that are also associated with easily measured outcomes. If technology is only used to pass lectures and exercises to students so they can perform well on standardized tests, it **will** be “efficient, monotonous, and low quality” as the author states at the beginning.

The hope that throwing calculators or computers in the classroom will “fix” problems of engagement and achievement without the right people in the room to use those tools is a false one, as the author suggests. The move to portray mathematics as more than a set of repetitive, monotonous processes, however, is a really good thing. We want schools to produce students that can think independently and analytically, and there are many ways that true mathematical thinking contributes to this sort of development. Technology enables students to do mathematical thinking even when their computation skills are not up to par. It offers a different way for students to explore mathematical ideas when these ideas don’t make sense presented on a static blackboard. In the end, this gets more students into the game.

This should be our goal. We shouldn’t going back to the most basic textbooks and rote teaching methods because it has always worked for the strongest math students. There must have been a form of mathematical Darwinism at work there – the students that went on historically were the ones that could manage the methods. This is why we must be wary of the argument often made that since a pedagogical method “worked for one person” that that method should be continued for all students. We should instead be making the most of resources that are available to reach as many students as possible and give them a rich experience that exposes them to the depth and variety associated with true mathematical thinking.

I wish that everyone would read this excellent analysis. Every day, I meet adults who proudly claim their dislike of or ineptitude in mathematics. I am certain that some of these feelings come from the soul-killing, mind-numbing experience of calculation after calculation after calculation with no connection to anything meaningful. It’s been a very long time since I’ve been in the classroom (as a teacher), and I would be terrified to face an IWB. BUT that tool is merely a tool. And I would gladly use it, if it meant helping students understand the *concepts* and *meaning* of the mathematics behind the computations. Students don’t learn math from drills. They learn math from (duh) exploring mathematical concepts and applications.

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.

While there are parts of this response that I agree with, I’m still troubled by a few things:

1) The belief that there is a way to achieve understanding in mathematics without computational fluency. I’ve not yet seen a student in my 13 years of teaching who truly had one without the other. I don’t disagree with the statement that “One hundred percent of class time can’t be spent on computational fluency with the expectation that one hundred percent of understanding can come later. The two are intimately entwined, particularly in the best math classrooms with the best teachers.” But this gets to a bigger point in Kakaes’ article, which this post ignores – there aren’t a whole lot of these “best teachers” out there, especially in K-8.

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.

I appreciate your thoughtful comments. I’ll try to address them as best as I can.

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.