I’ve underestimated the importance of vocabulary

"Words" by Shelly on Flickr

I thought for a long time that I could get by teaching math while deemphasizing vocabulary. Obviously we would discuss the meaning of words, especially the ones that come up frequently. But I thought that if I was able to help students get a feel for the math, and show kids how to do math, without getting too caught up in what the new vocabulary meant, that would be success.

Part of this was time. Or rather to save time. Spending time helping students really understand vocabulary takes more time, especially if it’s something that is more easily shown/practiced. For example, I feel like one of my struggles with helping students understand domain and range is that I don’t do a good job at really helping them understand the words. In algebra II, if I present a new type of function to them and ask them to find the domain and range, they often struggle until they see a few examples. It’s as if they’re simply replicating the process for each type of function.

At risk of this turning into a domain and range post, let me explain a bit further. When we study quadratic functions I tell students the domain is always “all real numbers”. The student thinks, “Sweet. Whenever I see a question over domain on the quiz, I’ll just write ‘all real numbers’.” When we learn a new family of functions they have no understanding of how to find the domain, beyond “that’s something with the x values, right?”.

It’s not just that topic. In fact, the concept that propelled me to write on this topic was grading a quiz over factoring polynomials and finding zeros in polynomials. Way too many of my students don’t know the difference between factors and zeros and constantly get them confused. My most significant observation was that I find students are trying to get by with the least amount of vocabulary understanding, and I don’t think I’m helping things by demphasizing it.

Since I’m having this realization at this point in the school year, I think the fix going forward will be trying to find and develop small activities to help reinforce vocabulary. Simply emphasizing it more is a start. I’ve also done some activities, like concept maps and “functions back-to-back” which help with vocabulary understanding. Next school year I’d like to take a more systematic approach and deliberately build in vocabulary activities into each unit.


Drop your favorite vocabulary activities in the comments below or send them my way on Twitter. Thanks!

Image Credit: “Words” by Shelly on Flickr

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Three Arguments for a Mathematical “SSR”

I’m sure that at some point in your life you’ve either heard of or participated in sustained silent reading in school. The idea is that students simply spend a set amount of time reading anything they enjoy for an extended period of time. I remember doing it in middle and elementary school. Every English student in the high school in which I currently teach does it as well. In fact, since it’s implementation there has been a notable increase in our reading scores. This got me thinking….what would the mathematical equivalent of this look like and would it be valuable?

Choice

I think it might have a few components. One of the main premises of SSR is you get to choose what you read. In the realm of mathematics I don’t doubt that many students would need guidance in this area for a couple of reasons. First, many students see mathematics through the lens of their math books and previous math books that led to their current math book. This means that they are sheltered from a lot of math they might find interesting. Second many don’t know what doing math is like. For instance, have a look at this video by Vi Hart (who has one of my favorite Youtube Channels) in which by doodling she makes parabolas incredibly interesting. This is an exceptional example of where simply playing with mathematics can take you. Now, I understand that her mathematical background allowed her to draw and discuss parts of the video that would be over many student’s heads. The point is that there are many access points to mathematics that are both playful and creative. The teacher would have to front load some of the explanation for what constitutes mathematics, to broaden their horizons.

Being able to choose the mathematics students work on gives them some ownership of the content, even if it’s only for a small part of the week. Math catches a bad rep. Even certain students in my AP calculus would hesitate to brag about their love of math and a number of them don’t like math. I’m not contending that after implementing some sort of mathematical SSR that everyone will be running around jumping up and down about how great math is. I’m simply contending that if students view of mathematics broadens into something they think is enjoyable, the subject in general might be viewed in a better light. I would also hope that there would be a “spillover” effect in math class. This would stem from the notion that, although “what you’re telling me now isn’t particularly interesting, I can see that there are parts of this subject that are.” The goal would be that students would be (even slightly) more motivated to learn other mathematics.

Thinking

I constantly preach to my students that if you want to get better at something you have to work at it. No one wakes up one morning with the ability to shoot three pointers at 60%. Likewise, no one wakes up one morning with the ability to do and fully comprehend integral calculus. To this end, if we can get students thinking mathematically for a short period each week I believe that ultimately students would become better mathematical thinkers and problem solvers. Two of the critical components to the success of this is that a) students have enough time each week to make it worthwhile and b) students engage in activities that make them reason and use their logical thinking skills.

Focus

I don’t think I’m alone when I observe that many students in my class are trying to do math with a computer sitting next to them, lighting up every 15 seconds. This makes any kind of extended focus and concentration difficult. How are students supposed to “make sense of problems and persevere in solving them” if their phone is constantly distracting them from what their work? To this end a mathematical SSR would be phone/distraction free. I’m not sure if English classrooms implement it this way, but I imagine they do. One of the goals of this would be that students get better at concentrating on problems for longer than a minute or two. My hope is that students would begin to see value in distraction free work. They might even increase their ability to focus.

Nuts and Bolts

A few things remain to be worked out. For instance, what are the guidelines for something mathematical. Vi Hart spent a bunch of time drawing parabolas but the result was much more mathematical than if most of my students did the same. Here’s a list of activities that, I think, would be fit this time nicely.

  • Logic Puzzles
  • Creating Desmos Art
  • Sudoku, Kakuro, etc.
  • Reading and playing games on Math Munch
  • Something they find interesting from (gasp) the textbook
  • Watching Youtube videos from approved Youtube channels (I’m not sold on this one…)
  • Maker Stuff (Little Bits, Arduino, etc.)
  • Logic Games
  • Games (Chess, Guillotine, etc.)
  • Coding
  • Others (If you shoot me ideas then I’d love to add them to the list…)

This time would be explicitly not for remediation. I can think of no worse way for a student to spend this time than being forced to do math they don’t find interesting and are already struggling with. I can see the temptation for a teacher to fill this chunk of time with remediation but that completely misses the point.

Results

I have to believe that the end result would be better mathematical understanding in general. I also think that (another gasp) test scores would go up as a result. Many standardized test questions test reasoning more than given math skills anyway. I have no research to prove this, I just think that if students do more mathematical thinking, their math skills will improve. And to be quite honest, if the results are simply more students improving their reasoning ability and gaining a new appreciation for mathematics then I’d deem it a success.

On a final note, I think it’s important that the teacher does this with the students. This models what is expected and gives the teacher some time to explore the subject that they love. It would contribute to a culture of mathematics in the classroom and sends a message to the students that this time is valuable to the teacher as well.

This is just an idea that’s been pinging around my head for several months and I’m finally getting it out. I’d really love to hear feedback on this, including but not limited to “this idea sucks because…”.

%22The essence of mathematics resides in its freedom.%22

Is learning easy?

Something I’ve been thinking about for the last year is whether or not learning should be easy. I can think of times when I learned a great deal and it didn’t feel difficult at all. I can think of other times that learning was difficult and I didn’t feel like I learned very much. These are a few of the questions that bounce through my mind.

  • Is there some kind of payoff for learning something that is difficult to learn, beyond simply the thing you learned?
  • Is everything we learn ultimately worth learning, regardless of how difficult it was to learn?
  • If we teach things that are consistently difficult to learn then how do make sure those learning experiences end up being valuable?

I try to think about these questions from my students’ perspective. For instance, I dragged my extended (slow pace) algebra II students through a unit on quadratics. Realistically speaking my students were never going to use most of the mathematical concepts that we covered, at least not directly. So why do we teach them these things that are so difficult for many of them to learn (especially at a conceptual level)? Or maybe a better question is what do we tell them about learning when we teach them concepts they’ll never use and find difficult? What message are we sending about learning? A lot of algebra II (a requirement for every student in the state of Michigan) is an absolute struggle for many students, so how do we make this struggle meaningful?

In my mind we have a couple of options. The first option is to try to reduce the curriculum to it’s simplest form. We give the students the tricks, shortcuts, calculator programs, and everything we can to get them to put the correct answer in the blank on the assessments. This way we can get as many kids through the curriculum as painlessly as possible. This method is fairly attractive and I know I’ve been guilty of it on several occasions. The glaring problem with it is that we are essentially wasting the students’ time. We are not creating opportunities for them to think critically or grow as learners (not to mention how this destroys the beauty of mathematics). Also, it’s been my experience that students don’t retain the concepts over the long term.

With this, I often consider another path. Maybe instead we take an approach that encourages critical and independent thinking. A model that allows students to construct the concepts within learning experiences that, although seemingly more difficult, allows them to grow as learners and mathematical thinkers. This route is more difficult for a number of reasons. First, developing these kinds of tasks is difficult. (Although, to be fair, it is getting easier. Consider the MTBOS search engine this list of Common Core aligned problem based curriculum maps or the power of online professional learning networks.) Second, students hate it. (Okay, maybe hate’s a strong word, maybe it’s not every student, and I think the culture of the classroom can make them hate it less, but I’ll have more on that in another post.) In addition, there is a concern that we won’t get through all the content. If you teach in trimesters, where a student might have a different teacher from trimester to trimester, this becomes especially important. From a teacher’s perspective this option can seem daunting. We are going to take kids that already (probably) don’t like a subject partially because they find it difficult and then we make it more difficult for them. For many educators this choice is simple. Go for option number one.

I would add one note about the second option. We wouldn’t be making the content more difficult because we are evil. We’d be trying to create valuable struggle. The idea would be that we help them build the concepts so students would be doing more thinking during class and the teacher would stop giving away the interesting stuff so frequently.

I think most educators, given infinite time and patience, would pick the second option to implement. So the big question becomes:

If the second option is more difficult for both the student and the teacher, does the payoff (if there is any) outweigh the difficulties in implementing it?

To be honest with you I don’t know the answer. My idealism pulls me hard towards the second option but my practicality pulls me in the other direction. Also, not having infinite time and patience is a big factor. I apologize if this post doesn’t feel like it has a resolution. It doesn’t, because I don’t. However I’d love if it started a conversation. I think this is something that all math teachers and departments should be having an open discussion about.

Getting A “Feel” for Trig Functions

My good friend Steve Kelly always talks about getting creative ideas in the shower and to be honest I’ve always been jealous. It’s not frequent that I get a good idea in the shower. However, that changed a couple weeks ago. (Having been entrenched in learning about creativity for the last several months I realized that maybe the problem was I was trying to find ideas in the shower as opposed to letting my subconscious do the work. So, I stopped doing that, not only during showers but also workouts, driving, etc. and the ideas have come more frequently.) Prior to the idea I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. The idea I had was to somehow make a massive unit circle and Cartesian plane and show students the connections in a life size context. Below is the outline of the activity and below that, my reflections. Continue reading

Dimensional Thinking – Functions

I’ve encountered assignments in this course that I’ve struggled to apply to my topic area (functions) but this assignment seems built for it! Dimensional thinking is thinking about a topic or concept in various dimensions and possibly coming up with models to represent that concept. One of the great things about modern graphing technology is that it allows us to push, pull, and explore functions. I want to explore functions both at the minuscule scale, the large scale, in three dimensions, and even explore higher dimensions.

We often look at functions in a standard viewing window. We generally like integers, especially numbers -10 to 10, when we first look at functions. Consider the image below that I’m sure most people remember from high school algebra, y=x^2 in the standard viewing window.

Dimensional Thinking Image 1

y=x^2, Standard Viewing Window

There is nothing special about this viewing window. In fact, as many math students find out, lot’s of other viewing windows are more useful. For instance, suppose this graph was modeling position (in meters) versus time (in seconds) and we wanted to know the velocity at exactly 2 seconds. This problem becomes difficult without calculus unless we change our window to make finding the exact velocity easier. See the image below.  Continue reading

Observe and Re-imagine

My current project in CEP 818 – “Creativity in Teaching and Learning” was to observe an object tied to my topic area (functions in mathematics) and to re-imagine it, appealing to a different sense than originally intended.

First, a note on the importance of observation:  Robert and Michele Bernstein, authors of Sparks of Genius, point out that “All knowledge begins in observation. We must be able perceive our world accurately to be able to discern patterns of action, abstract their principles, make analogies between properties of things, create models of behavior, and innovate fruitfully .”

And now, the re-imagining of the graphing calculator.

Any math student that graduated from high school in the last two decades probably used one and I can guess (although I might be wrong in some cases) that it was used to get answers quickly and easily. In fact, as I was observing this device from every possibly angle and sense (okay, I didn’t lick it…) I noticed that it screamed simplicity. It is a no frills, unexciting, almost heartless object. Take 30 seconds and look at the image below. There is not much about that calculator that screams “Hey you, let’s do math and it’ll be awesome!” To be clear, it is incredibly powerful, but it is often underutilized as an exploration device and over used as a “shortcut machine”.

CalculatorThis may seem irrelevant to the learning of mathematics but it doesn’t help the plight of the math teacher that is trying to motivate students. I hadn’t considered this before, but I started thinking “Is this a device that I want to use?” I love mathematics but this device, although it is simple, is not elegant and not a device that I look forward to using. It is fraught with ugliness. From the black and green display to keys that seem randomly colored, there is much to be desired in terms of design. I believe that if we redesigned this device to be aesthetically pleasing and still powerful, functions and mathematics in general would be more accessible.

If you haven’t bought into this yet, consider your favorite technological device. My guess is it’s your phone (and if it isn’t, humor me for a minute). First, I bet the ratio of the sides is close to the Golden Ratio, which creates the most visually appealing rectangle. I would also guess that it fits nicely in your hand, that the interface is intuitive yet attractive, and that it balances power with beauty. The feel and design of this device make you want to use it. What if a calculator had that same appeal for math students? (The GIF below represents the potential of moving mathematics to the devices we use most often.)

In addition, this is not how I see mathematics.  I see math as a beautiful framework within which we can describe the natural world (and even if we couldn’t, it is still beautiful). Functions are like an engine that can be constantly modified to describe an infinite number of situations. If we can find ways to relay the beauty in mathematics and functions to students then I think (and admittedly I’m aiming high) we could ultimately shift the culture to one that views math as elegant and powerful, not just an obnoxious requirement to get a diploma.

Desmos gif

 

A note on the above GIF: Each one of these graphs (with the exception of the smiley face) is a function or two functions graphed together. I used desmos.com to create the graphs. I think that tools like Desmos, which runs on all smartphones (the devices we actually like to use) can help students to visualize functions as well as other mathematics. (Here are some of the functions I used.)

References

Bernstein, R., & Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people (p. 30). Boston, Mass.: Houghton Mifflin.

 

Lesson Plan: Version 2.0, TPACK Revision

Throughout the next couple weeks we will take the lesson we have chosen (see version 1.0 here) and analyze/revise it in the context of various “lenses”. The first is the TPACK framework, which I have outlined below.

My goal for this lesson is to take it from it’s current state, very dry and not based in constructivist philosophies, to a more engaging and inquiry based lesson. I will be viewing this lesson through the Technology Pedagogy and Content Knowledge (TPACK) framework. The first context this framework focuses on is technology. I will determine what kinds of technology can best help my students achieve the learning objective. Pedagogy is the the various methods I will employ to help my students learn my objective. Content knowledge is the well of knowledge that I have about my content area that I will draw from as I design and implement my lesson. The intersection of these three contexts is the focus of the TPACK framework (Mishra and Koehler, 2006).

Lesson Plan Version 1.0 (Through the TPACK Lens)

Technology

This lesson plan, in it’s current form, uses minimal technology. I use a whiteboard for the main instruction and to introduce the concept via a proof. Partway through the lesson students will utilize the collaborative whiteboards located at each pod to work on example problems. I think that I am currently underutilizing the technologies available to me. Even if I don’t necessarily add technology to the lesson, I think I can use the current technologies (the white boards) in a much more effective fashion.

Pedagogy 

In the lesson’s current form the pedagogy is mainly direct instruction. The proof at the beginning of the lesson is important to understanding the concept, and as I mentioned in the first blog post, I believe it needs to stay in the lesson in some way. I wonder about the location of the proof however. I’m not sure that the best place for it is at the beginning. As Bransford, Brown, and Cocking (1999) point out in their book How People Learn: Brain, Mind, Experience, and School, the authors explain that when faced with tasks lacking apparent meaning or logic, it will be “difficult for them (students) to learn with understanding at the start; they may need to take time to explore underlying concepts and to generate connections” (p. 58). This lesson currently does a poor job of taking that fact under consideration. Some of the pedagogy is okay. There is a period during the lesson when students will be working in small groups on example problems. This allows students to work collaboratively and to construct some meaning from the concepts, but only after a lot of the meaning has been given to them directly. They are not given time (or proper methods) to construct it for themselves.

Content Knowledge 

This lesson is conceptually difficult, even for me. I understand it for myself, but struggle to do an effective job of helping my students truly understand it. Understanding the Fundamental Theorem requires a solid understanding of the meaning of the derivative. Students also need to have a solid understanding of the definite integral, beyond just being able to complete the basic definite integral problems. A basic understanding of limits is also helpful. One of the reasons this concept is so difficult for students to understand is that it relies on the strong understanding of so many other concepts in calculus. Beyond the calculus concepts that underly the Fundamental Theorem, a strong understanding of the meaning of a function is also important. Many students make it all the way to calculus without a strong understanding of the meaning of a function. A misconception at any one of these concepts can make the understanding of the proof and it’s extensions difficult.

The Context

Much of the context of this lesson was explained above but I can’t stress the importance of taking this into consideration enough. There has to be a solid understanding the previous concepts. In addition to prior concepts, providing students a view of the big picture is also really important, so I need to help students see where the concept leads also (Bransford et al, 1999, p. 42). This concept helps us find antiderivatives for numerous functions that we would not be able to find otherwise. Providing students with this information should help them to better contextualize the concept.

Intersections: Technology and Pedagogy

The value in the TPACK model is in understanding that all of these pieces are connected. The pedagogy I utilize is directly affected by the technology I have available and vice versa. In it’s current form my technology (mainly the large whiteboard at the front of class and the “mega” whiteboards on each pod are being underutilized as a pedagogical tool. My lesson plan is currently very teacher centered and not learner centered. I need to spend some more time digging into the concept to develop other ways to better utilize my technology. I’m not sure yet if “new” technologies (like Wolfram Alpha, or other powerful graphing tools) will be beneficial or not.

Intersections: Content and Pedagogy

The important thing to understand about the intersection of the content and the pedagogy is that this concept is incredibly dynamic. The pedagogy utilized depends on the students’ construction of the prior knowledge leading up to the lesson, more so than many concepts. In a sense the quality instruction in the weeks leading up to this concept are as important as the lesson itself. One of my goals in this lesson revision is to spend time really deconstructing the content for myself and from this deconstruction find a more inquiry based approach.

Intersections: Technology and Content

Often there is an assumption that mathematics is married to calculators. In this lesson the calculator is almost a hinderance. Anything the calculator can do will essentially be a shortcut and will cause the students to create misconceptions. I want the technology that we use to help students reason their way through the concepts and develop meaning as they go. I want to avoid technology that will provide shortcuts but result in misconceptions.

Striking a proper balance between these three intersections should result in a quality lesson. My aim is to take a very teacher centric lesson, and turn it into a more inquiry based lesson in which students can better construct the concept of the Fundamental Theorem.

References

Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind, experience, and school. Washington, D.C.: National Academy Press.

Mishra, P., & Moehler, M. (2004). Using the TPACK Framework: You Can Have Your Hot Tools and Teach with Them, Too. Learning & Leading with Technology, 14-18.