Yesterday my calculus students reaped some of the benefits of a flipped class situation - I made some videos on differentiation rules and asked that they watch the videos sometime between our last class and when we met yesterday. We spent nearly the entire period working with derivatives rules for the first time. The fact that the students were getting their first extended period of deliberate practice with peers and me around (rather than alone while doing homework later on) will hopefully result in the students developing a strong foundation what is really an important skill for the rest of calculus.

They were using Wolfram Alpha to check their work, something that I paid lip-service to doing last year but did not introduce explicitly on the first day of learning these rules last year. There was plenty of mistake-catching going on and good conversations about simplifying and equivalent answers. I needed to do very little in this process - good in that the students were teaching themselves and each other and being active in their learning.

It was also interesting doing this so soon after discussing the role of technology in helping students learn on the #mathchat Twitter discussion. There were many great points made regarding the content of technology's effective use across grades. It made me think quite a bit about my evolution regarding technology in the classroom. Many comments were made about calculator use, teaching pencil and paper algorithms, and the role of spreadsheets and programming in developing mathematical thinking. I found a lot of connections to my own thoughts and teaching experiences and it has me buzzing now to try to explain and define my thinking in these areas. Here goes:

### Developing computational and algorithmic fluency has its place.

In the context of my students learning to apply the derivative rules, I know what is coming up the road. If students can quickly use these rules to develop a derivative function, than the more interesting applications that use the derivative will involve less brain power and time in the actual mechanics of differentiation. More student energy can then be focused in figuring out how to use the derivative as a tool to describe the behavior of other functions, write equations for tangent and normal lines, and do optimization and minimization.

There was a lot of discussion during the chat about the use of calculators in place of or in addition to students knowing their arithmetic. I do think that good arithmetic ability can make a difference in how easily students can learn to solve new types of mathematical problems - in much the same way that skill in differentiation makes understanding and solving application problems easier. Giving the students the mental tools needed to do arithmetic with pencil and paper algorithms empowers them to do arithmetic in cases when a calculator is not available.

### Technology allows students to explore mathematical thinking, often in spite of having skill deficiencies.

One of the initiatives my colleagues took (and I signed on since it made a lot of sense) when I first started teaching was using calculators as part of instruction in teaching students to solve single variable linear equations. There was a lot of discussion and protest regarding how the students should be able to manage arithmetic of integers in their head. It wasn't that I disagreed with this statement - of course the students should have ideally developed these skills in middle school. The first part of the class involving evaluating algebraic expressions and doing operations on signed numbers were done without calculators in the same way it had been done before.

The truth, however, was that the incoming students were severely deficient in number sense and arithmetic ability. Spending a semester or two of remediation before moving forward to meet the benchmarks of high school did not seem to make sense, especially in the context of the fact that students __could__ use a calculator on the state test. So we went forward and used calculators to handle the arithmetic while students needed to reason their way through solving equations of various forms. They did learn how to use the technology to check the solutions they obtained through solving the equations step-by-step using properties. There were certainly downsides to doing things this way. Students did not necessarily know if the answers the calculators gave them made sense. They would figure it out in the end when checking, but it was certainly a handicap that existed. The fact that these students __were__ able to make progress as high school math students meant a lot to them and often gave them the confidence to push forward in their classes and, over time, develop their weaknesses in various ways.

I have seen the same thing at the higher levels of mathematics and science. I used Geogebra last year in both pre-Calculus and Calculus with students that had rather weak algebra skills to explore concepts that I was taught from an algebra standpoint when I learned them. Giving them tools that allow the computer to do what it does well (calculate) and leave student minds free to make observations, identify patterns, and test theories that describe what is happening made class visibly different for many of these students. If a computer is able to generate an infinite number of graphs for a calculus student to identify what it means for a function graph to have a zero derivative, then using that technology is worth the time and effort spent setting up those opportunities for students.

### Using skill level as a prerequisite for doing interesting or applied problems in mathematics is the wrong approach.

Saying you can't drive a car until you can demonstrate each of the involved skills separately makes no sense. Saying that students won't appreciate proportional reasoning until they have cross-multiplied until their pencils turn blue makes no sense. Saying that learning skills through some medium makes all the other projects and applications that some of us choose to explore in class possible does not make sense. It makes mathematics elitist, which it certainly should not be.

Yes, having limited math skills is a limit on the range of problem solving techniques that are available to students. A student that can't solve an equation using algebra is destined to solve it by guess and check. Never underestimate the power that a good problem has to entice kids to want to know more about the mathematics involved. Sometimes (and I am not saying all the time) we need to work on the demand side in education, on the why, on the context of how learning to think in different ways applies to the lives of our students.

### Emphasizing algorithms without providing students opportunity to develop context or some level of intuitive understanding (or both) has significant negative consequences.

I don't mean to suggest that teaching algorithms on their own can't result in students performing better on a type of problem. The human brain handles repetition extremely so well that learning to do one skill through repetition is not necessarily a bad way to learn to do that one thing.

One problem I see with this has to do with transferring this skill to something new, especially when the depth of available skills is not great. Toss a weak student ten one-step equations of the form x + 3 = -8, and then give them something like 0.2 x = 25, and chances are that student won't solve it correctly without some level of intuition about the subtle differences between the two. Getting this right takes practice and feedback really good opportunity for students to be reflective of their process.

It is also far too easy when applying an algorithm to stop thinking critically about intermediate steps. I spoke to a colleague this week about his students learning long division and we both questioned the idea that the algorithm itself teaches place value. We looked at a student's paper that was sitting on the desk and instantly found an example of how the algorithm was incorrectly applied but through a second error resulted in a correct answer. If we teach algorithms too much without giving activities that allow students to show some sort of understanding of **some** aspect of how the algorithm fits into their existing mathematical knowledge, it's undercutting a real opportunity to get students to think rather than compute. I like the concepts pushed by the Computer-Based Math movement in using computers to compute as they do best, and leave the thinking (currently the strength of the human brain) to those possessing one.

As often as we can, it is important to get students to interact with the numbers they are manipulating. Teaching the algorithms for multiplying and adding large numbers does provide students with useful tools and does reinforce basic one digit arithmetic. I get worried sometimes when I hear about students going home and doing hundreds of these problems on their own for various reasons. If they enjoy doing it, that's great, though I think we could introduce them to some other activities that they might see as equally if not more stimulating.

I do believe to some extent that full understanding is not necessary to move forward in mathematics, or any subject for that matter. I took a differential equations course in college trying to really understand things, and my first exam score was in the seventies, not what I wanted. I ended up memorizing a lot after that point and did very well for the rest of the course. It wasn't until a systems design course I took the following year that I actually grasped many of the concepts that eluded me during the first exposure. This same thing worked for me in high school when I took my first honors track math class after being behind for a couple years. My teacher told me at one point to "memorize it if I didn't understand it" which worked that year as I was developing my skills. Over time, I did figure out how to make it make sense for myself, but that took work on my part.

### Uses of technology to apply/show/explore mathematical reasoning comprise the best public relations tool that mathematics has and desperately needs.

I really enjoyed reading Gary Rubenstein's recent post about the difference between "math" and mathematics. I read it and agreed and have been thinking a lot along the lines of his entry since then.

Too many people say "I'm not good at math." What they likely mean is that they aren't good at computing. Or algorithms. Or they aren't good at ________ where __________ is a set of steps that someone tried to teach them in school to solve a certain type of problem.

On the other end of the perceived "math" ability spectrum, parents are proud that their children come home and do hundreds of math problems during their free time. These students take the biggest numbers they can find and add them together or multiply them and then show their parents who are impressed that their normally distracted kids are able to focus on these tasks long enough to do them correctly.

It makes sense that most people, when asked to describe their experiences in math, describe pencil and paper algorithms and repetitive homework sets because that's what their teachers spent their time doing. This, unfortunately, is the repetitive skills development process that is part of mathematical learning, but should not be the main course of any class. We show what we value by how we spend our time - if we spend our time on algorithmic thinking, then this is what students will think that we as teachers and as thinkers value as being important in mathematics.

This fact is one of the main reasons I started thinking how to change my class structure. My students were talking about not being good at a certain type of problem ("I don't get this problem...I can't do problems that need you to...") rather than having difficulties with concepts ("I don't get why linear functions have constant slope...I don't get why x^2 + 9 is not factorable while x^2 - 9 is).

If we as teachers want students to value mathematics as more than learning a set of problems to be solved on a test, then we have to invest time into those activities that allow students to experience other types of mathematical thinking. This is where technology shines. The videos of Vi Hart, Wolfram Alpha, the antics of Dan Meyer, the Wolfram Demonstrations Project, the amazing capabilities of Geogebra - all of these offer different dimensions of what mathematical thinking really is all about.

We can share these with students and say "check these out tonight" at the end of a lesson and hope that students do so. Sometimes that works for a couple students. That isn't enough.

I think we need to invest in technology with our students with our time. We need to deliberately use valuable class time to take them through how to use it and why it makes us excited to use it with them. It's really the only way students will believe us. Show that it's important, don't just tell your students it is. That's right - that valuable class time that we often plan out too carefully and structure so that they reach the well-defined goals we have for them - that time. Plan to use a specific amount of class time, and enough time, to let students play around with a mathematical idea using any of the amazing technology tools out there. Show them how *you* play with the tools yourself, but don't make this the focus of this time - do so afterwards, perhaps.

To be clear - I am not saying do this all the time. Students need to learn algorithms, as I have already stated. Students also need to be looking at interesting problems. We should not wait to show them these problems until after students have demonstrated automaticity because it gives students the impression that the algorithms came before the thinking that went into them.

I am saying that balance is key.

The only way we are going to change the perception of what mathematical thinking really looks like is by living it and sharing it with our students.