# Abstracting the Abstract – More Math GIFs and Function Talk

Creative individuals often take something that is concrete or complex and abstract it in some way that makes it more meaningful or provides a more useful perspective. When charged with the task of creating an abstract representation of my concept (functions) I was first at a loss. How do I create an abstract representation of something that is fundamentally abstract? The descriptions and media below describe the two methods I chose. For a summary of how these abstractions impact my teaching and understanding of functions scroll to the end of the media.

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Method 1

As I thought about this, I realized that when we write a function, we often only talk about a specific function. Yes, occasionally we will discuss a “family” of functions. But what does that mean and students really understand it? When we discuss a family of functions we are really talking about an infinite number of functions. What does that mean? How can we visualize such an abstract concept? Yes, sliders can help, but it can be visualized more effectively. Below are my ideas for abstracting various families of functions.

This also made me think…Could an abstraction be (at the surface level) more complex than what you’re abstracting? The GIFs show a more complex picture than just a curve with a slider (y=ax^2 with “a” as a slider). A family of functions is an infinite number of functions, right? How do we abstract an idea that is so complex? The GIFs attempt to abstract infinity (or an infinite number of curves) by suggesting what the escalation to an infinite number of curves would look like. (Click a GIF to look at a slideshow of each one individually.)

y=ax^2

y=a•e^x

y=a•sin(x)

y=(x-h)^2

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Method 2

The task set before me was to abstract my concept in two different ways. The second method I chose for abstracting the concept of functions was to pare downs functions down to their simplest…well…abstraction. Then, I sketched these abstractions and below are the images of my sketches. I decided that at their fundamental level functions have three things: an input, an operation (something that modifies the input), and an output. The images all contain these fundamental pieces.

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I think both abstractions can impact my teaching. The GIFs would be something that I could either show my students to give them a visual for the concept of a family of functions. A good project would be to have students create similar GIFs for different functions and their transformations. This solidifies the concept in a visual way and gives us something we can anchor back to throughout the rest of each unit.

The second abstraction might be useful in a different context. I like that the images provide visuals for the simplicity of functions. Often students don’t realize the vastness of functions and their stretch throughout the world. Most things with an input, operation, and output is a function. When you ask a student to give you a function they are likely to say something like y=2x+3 or f(x)=sin(x). They are unlikely to say “The number of cars in the school parking lot is a function of the time of day.” If students have the abstract concept of a function as their fundamental understanding then they might start to see function behavior in their worlds more frequently.

# Side-Side-Angle Ambiguity

For years (okay, it’s only been three…) I have struggled with how to teach the SSA ambiguity in the context of the Law of Sines. My first year, which is mostly a blur looking back, I’m not sure that I even understood it that well. It was in the context of geometry and I think we (geometry teachers) decided we would let Algebra II and precalc teachers handle that special case.

And then in my second year I taught Algebra II and precalc.

So I ‘ve been through this topic a few times and each time I feel like the kids get a glimpse of what’s going on but really have no deep understanding at all. After a botched geogebra demonstration in precalc earlier this year I decided to look to the Math Twitter Blogosphere for help.

From that tweet Matt Salerno sent me this activity/post from Dave Sladkey. I didn’t really want to mess with pipe cleaners so I modified it a fair bit, but the general concept is the same. Check out the activity.

In the activity we deal with a given acute angle first. Everything they learn from the acute angle exploration made the obtuse angle exploration fairly straightforward. One positive of this activity is that it is accessible to all students right away. “Draw lines using your ruler, then make observations. How many can you make?” Then I ask students to dig a little deeper into finding certain parameters that caused there to be one triangle, no triangles, or two triangles. By the end almost all of my students at least had a visual understanding of when each situation could occur. Once we did the activity (which took about 45 minutes) I showed them the slide below, and talked about the importance of the segment labeled “h”. Because they had just done the activity, the importance of this segment was fairly obvious! I just had to name it. We derived the formula h = b • sin A,  and then we finally solved a few triangles using the law of sines.

One misconception that I had to address was that many students said that one triangle could be made when side a was 5.5″, instead of greater than 5. Next time I need to build in a question or two to make them think about when side a is longer than the 5.5″ they drew in.

Any feedback you can provide or how you introduce and teach this topic would be greatly appreciated! Below are some of my students’ solutions.

# “Yes, but did it work?” & “How are you going to test us on this?”

Yesterday I did this activity in which, using sticky notes, as a class we built a histogram of random data that created a normal distribution. Overall I think that activity went well. It took a bit longer than I anticipated, mainly because it simply takes time for students to find the average of each of  ten sets of ten numbers. Next time I might assign this piece for homework, so that they come to class prepared to creat the histogram.

So we created the histogram and talked about normal data and everything went well. At least, I think…

While reflecting two things occurred to me. First, I began to wonder, did this activity “work”? Did it accomplish what it was set out to do? At first blush it seems that it did. Everything went smooth, students appeared to be engaged, and I’ve put an anchor in their mind that I can draw back to for the rest of the unit. I then realized that there are some lessons/tasks/activities that have results that are not immediately seen or are quite subtle. It may the quiz next week where I see this pay off. It might be tomorrow in class discussions. It might be in four weeks when a kid is reading Time magazine and sees a graph similar to this and thinks “hey, this looks like normal data…” My sincere hope is that it helps them in the rest of the unit and in life, but if it doesn’t help some students then at least I gave them a visual they can relate back to. At least I didn’t begin the unit by saying “Today we are going to learn about z-scores…”. The benefit of some math tasks may not be immediately seen, but that doesn’t mean they shouldn’t be utilized.

The second thought I had was while I was reflecting on a comment a student made during this activity. She asked, “Mr. Cresswell, how would you test us on this?” My response was simply that I wouldn’t test her on this directly, but that the concept we were about to discuss would be valuable. This student seemed to believe that anything that happens in math class should be directly assessed on the test. If it wasn’t going to be or couldn’t be, then why is it happening in class? I suppose I realized many students believed this during my first year of teaching, but occasionally it comes roaring to the front of my brain. Many students believe school is about passing tests and getting good grades. Assessment is viewed as an end, not a means to an end in which they learn and become better thinkers. We need to continue to try to shift this paradigm because, I believe, students will buy into these types of tasks if they’re viewed as a piece of the learning process and not something that is irrelevant because it isn’t going to be tested. (Sorry the wording of that sentence wasn’t great…)

What do you do to to help shift that paradigm?

# I need help with an explantion

I need help explaining this problem to a student. The picture of the problem and his solution is shown below in the image.

“Calculate the probability of tossing a coin 8 times and getting 4 heads.”

His reasoning was that since there’s a .5 chance of getting a heads on each toss, then there should be a .5 chance of getting 4 of the 8 tosses to land on heads. (Note he used the same reasoning to get .25 for #11.)

I tried to explain to him that it was a binomial experiment but that didn’t convince him of a flaw in his reasoning.

I then tried to show him an example of only 4 tosses, in which the question was what’s the probability of getting 2 heads. I showed him the entire sample space, shown in the image below. I then found the probability by looking at all the possible ways to get two heads, and dividing it buy the number of outcomes in the sample space. I then showed him that the binomial experiment formula got the same answer as my reasoning.

He countered that I was double (or triple) counting some combinations. His claim was, for example, that HHTT was the same as HTTH since order didn’t matter, and therefore the sample space was much smaller then I was making it. This implies, in his mind, that there are only four options.

I need your help explaining the flaw in his reasoning, because I  not doing a good job of finding it or convincing him. Is there a better way for me to explain/teach this?  (By the way, I love when students do this. I wish I had more discussions like this one, where students didn’t just accept my first response but really probed my reasoning.)

# Making Review Less boring

We are closing out our probability unit and instead of giving my students 30 problems of review to complete in class I designed a “station” activity. The station format idea actually came from a couple colleagues of mine, and it really helps to get kids up and moving. Also, as the title implies, makes review a little less boring.

I wanted to put a creative twist on this so I came up with the following station activity. There entire activity is self contained, meaning that you don’t need our textbook to use it as I designed the problems. You should be able to run it as is, or modify/improve it as you like.

Also, Ted-Ed deserves it’s own paragraph for it’s awesomeness. Now that I’ve actually gone through and used it to flip a lesson (or part of an activity really), I’m really excited about using it next year in my flipped classroom (or even my non-flipped classes)!

In addition, my good friend and colleague Eric Beckman, recorded the activity for me. Here is my reflection, and the activity resources:

Station Reflection

Stations 1 & 2: In these stations students were asked to watch two Ted-Ed lessons the night before (Station 1, station 2). I then used some of the provided questions, and created my own, for them to answer after watching the lessons. I loved that I get great data on their responses and that students can participate in discussions. I can review all student responses, both open ended and multiple choice, as you can see in the screen shot below. I can also give feedback to the open ended questions, and students will be notified when I give that feedback. You can also download all the responses as a CSV file. The discussions centered around the problems seemed really thoughtful, so I was happy with that. The videos also provide a different perspective on the concepts for the students, which I think was helpful for some.

This did however take them more time than I anticipated. I had one student from each group create an account, which took time, and some of the questions were tougher for them then I anticipated. Next time I will have them set up with Ted-Ed accounts when they come to class, and will also likely reduce the number of problems they have to answer. Because some groups took a while, some groups didn’t complete all the stations.

Stations 3-5: These were the basic probability problems. Students did well on these to varying degrees. I could’ve given these problems all as one station, but breaking it into multiple stations broke it up for that students. Sometimes simple things like that make math more approachable for students. It’s also important to have the key available for students so they can get instant feedback if I’m not available.

Station 6: This station asked students to solve two problems and then create two short video lessons using my iPad to explain their solution. Every time I do this I get mixed results, but the good results outweigh the bad. The downside is that students really don’t like doing it so some push back a bit. However, it forces students to take their understanding to the next level. They will learn it better if they are forced to teach it. It makes them take an extra step in understanding, as they don’t want to explain it incorrectly on record, or mess up and have to re record it. I need to do more of this, as the students that really try get a lot out of creating short lessons explaining concepts.

Stations 7 & 8: More practice problems, similar to stations 3-5.

Activity Resources

Station Packet (PDF)                    Station Packet (Pages)                Station Packet (Word) – I make no promises about formatting….

Ted-Ed Lesson for Station 1        Ted-Ed Lesson for Station 2        (These are editable, so feel free tow tweak to fit your needs)

# It’s the Soft Skills!

This trimester I don’t teach in a flipped classroom. Precalc is done for the year, AP calculus is not flipped, and Algebra II is taught by myself and another teacher and is typically done very traditionally. This being the case, my goal this trimester is to “inject” solid higher level thinking activities into my lessons. Last year when I taught it (the first time) was very traditional (Lecture, assignment, repeat). The catch 22 of this that I haven’t had these students all year. They are coming to me from traditional classrooms and are not familiar with the different format of my classes (mega whiteboards, relaxed deadlines, higher level thinking, collaboration, etc.). I wanted to reflect on a few things that I noticed in the last couple days that I may have taken for granted.

Collaboration

For some reason, in the back of my mind, I just assumed that juniors in high school knew how to collaborate. This is not true. At least it isn’t true for my students. As you can see in the image my class is set up in pods and each day they come to class with a mega whiteboard and a few markers at their pod. (This is to encourage collaboration on “normal” days, not just days in which we have special activities.) Today I gave them an activity to help them discover the connection between combinations, binomial expansion, and Pascal’s triangle. I prefaced the activity with an emphasis on the need to collaborate and to share ideas with each other. I noticed that even the “best” groups struggled with this.

There were several specific problems I noticed. First, students didn’t use the mega whiteboards very often. Even when I explicitly said, “hey, this would be a good problem to do on the board.” Second, there was minimal communication between group members that “got it” and those that were still struggling. Third, there was minimal critique of each others work. for instance if one person had the correct answer, in many case everyone else just copied down the answer.

Tomorrow I will be making a point to talk about the best ways to collaborate. I failed to recognize their lack of skills in this area and I need to do a better job of setting them up for success in the area of collaboration. I also need to continue to try to build a community in which wrong answers are not shunned but are view as just a step in the learning process.

Communication

This is intimately connected to collaboration, but my students ability to communicate mathematics needs improvement. I think the more I help my students with this the better collaboration will be. How can I expect students to collaborate if they can’t communicate the math to each other?

I hope that I’ll continue to see improvement in these areas as the trimester continues. Please give me any ideas that you have to increase students’ ability to communicate and collaborate. I’d love to hear them!

# “Opening up” Math Class

In an effort to write more I’m going to be posting shorter posts on things that are on mind regarding education and mathematics. Writing helps me process and refine my ideas and I believe it will make me a better educator.

I often think about “opening up” my math class. By “opening up” I mean developing my class in such a way that students have time to explore ideas (preferably ideas that are of interest to them, but also concepts that are in the standards).  In this setting students would be encouraged to do a number of things on a regular basis.

First, they’d be encouraged to explore wrong answers. If a student got an answer wrong they would take time to figure out why, and represent the correct solution in multiple ways (graphing, algebraically, numerically, verbally, etc.). We so often don’t have time for this and don’t value this type of exploration. I think that should change.

Second, they’d be encouraged to take ideas further on their own, in class. A good example is synthetic division vs. long division of polynomials. We always tell students that synthetic division only works in certain situations, but what about that student that wants to know why? How do we support that student? Because if that student is allowed to explore that idea he/she will likely come away with an understanding of polynomials that is far deeper than if I just told him/her the reason. (God forbid the student came up with a reason I hadn’t thought of!)

Third, students would be encouraged to work on meaningful tasks involving mathematics in small groups. These might be “real world” projects or, equally valuable, deep explorations in mathematics. The objective for the group would be not only to solve the problem(s) but to be able to communicate the solution in a meaningful (dare I say visually meaningful and appealing) way.

I do some of this on a small scale in my various classes, but I am quite often up against two major adversaries: the curriculum and time. Although I am up against this, I think that if I “opened up” my class my students would become better thinkers, communicators, and self-motivated learners. In general I think they’d become more mathematically minded and I think it is incredibly valuable to have a society of mathematically minded individuals (more on this in a future post!). I think this is why educators have to be creative, take risks, and embrace technology. That combination, for me, has been powerful in helping me to take what steps I have toward the “open” math class.

If I think of more ways in which math class could be opened up I will be sure to update. Please give me your feedback and ways in which you “open up” your class (math or otherwise)!

# Introducing Properties of Logarithms (Find the pattern!)

When I taught properties of logarithms last year this was the first slide I showed the students:

Yes, I know, boring. Not only is it boring but the notation can be confusing. On top of that, these properties aren’t exactly obvious. With all that on my mind, coming into this lesson today I wanted to start with something other than definitions. I came up with this activity.

This activity, as I mentioned above, will by no means change the world, but it gives students a “feel” for the properties before anything is formally defined. This is now how I start the lesson. This took my Extended Algebra II students 10-15 minutes to complete and I think it served it’s purpose. It also  allows them to see the properties play out numerous times and should help them when they apply the properties to more complex problems. I also made sure to use natural logs and common logs so that they could quickly find the values and could use brain power to look for patterns.

Reflection: This went pretty well and even got a couple “wow, that is cool”s immediately following the activity. What I had trouble with was making it “stick” and helping students to extrapolate the rules to more complex expressions. I’m thinking about an extension activity in a similar format with variables and symbols instead of just numbers. This would allow them to immediately apply their new found rules and I think would be more advantageous than going straight into the direct instruction as I did this year.

As usual, any feedback is greatly appreciated!

This is my second year of teaching precalculus and last year when I introduced radians I had students construct a radian “tape measure” and measure different distances around the unit circle. Some students left with a good understanding but many did not. This year I decided to take a different approach (admittedly less hands on) to introducing the topic. I’m making an effort to introduce topics in an inquiry fashion as much as possible so this is the activity that I used to introduce radians this year.

I handed out the activity to each student and told them that they shouldn’t talk or confer on the problems. I wanted their own individual observations. I then should the video to the entire class in its entirety. (The only thing I told them was that radians were another way to measure angles.) Once I played the video in its entirety, I played it again and paused it at each position as needed (For #1 I paused when r=1, etc.). Once we did #1-6 We went through and discussed their observations as a class. After a whole class discussion I asked them to answer #7 without collaborating or conferring. This gives me a good idea as to how well each student understands radians as a concept.

Although Algebra II touched on radians many students come to precalculus with a vague (if any) understanding of the concepts. I think it is valuable to understand why a radian is what it is and why it exists. Students had many “ah-ha” moments, especially in the whole class discussion.

Here are some student responses to #7. I tried to give a good sample of the class.

“One radian is equal to the radius along the edge of the circle. It also brings pi into the fold because when the radian count is equal to pi the point is at 180 degrees.”

“Imagine an x-axis. Draw a circle around the origin. A single radian is the distance around the circumference that is the same as the radius.”

“A radian is a way of describing degrees on a circle. It is closely related to pi in which half the circle is one pi radians.”

“It is the measure of an angle and its distance traveled along the unit circle in a counter clockwise direction.”