Making More Sense

     This is my first time teaching algebra 2. I have two goals: make sure my students master and enhance their algebra 1 skills, and have the students see/understand the material in a new way. So far, its early in the semester and we haven't gotten into anything too deep, but yesterday we started a familiar (but often over-complicated) topic, arithmetic sequences.
     "Given a sequence 4, 10, 16, 22, 28, ..., a) Find the next three terms, b) Find the 87th term, c) Find the sum of the first 26 terms." If you see this problem in a textbook, there is a good chance it is preceded by explicit and recursive formulas with some notation that often confuses students. Seeing an and an-1 is not the easiest thing to explain, nor is it overly easy for students to retain. It is not often used, certainly not consistently throughout math topics, and is usually only associated with this topic of sequences and series, geometric and arithmetic. When there is a great chance that the above is going to happen for my students, is there a way to teach this so that they will understand it and connect it to what we've previously done, or am I doomed to teach this tough notation and risk it getting in the way of their learning? Luckily I work with someone who thinks outside the box on nearly everything and has an incredible gift for showing students how patterns exist within math.
     We started the course by having students find linear functions with only a table of values. They are able to find the change in y and change in x, and from their develop the linear relationship. So instead of teaching arithmetic sequences in terms of explicit and recursive formulas, we related it to a skill our students already have.
     Why not put the sequence in a table of values and have our students come up with the 'explicit formula' themselves? Is an arithmetic sequence not linear, always? Since they can see there is a common difference, a rate of change if you will, why not have them evaluate sequences in the same way they evaluated linear functions? Instead of an notation we can simply redefine x and y in the context of the problem. For our purposes, x will represent the term number and y the term itself. Done. Arithmetic sequences have just been taught in 10-15 minutes with a full understanding from my students. What about the 'recursive formula' you say? Do you see that common difference (aka slope) you just came up with kids? Continue adding that and you have your recursive formula. No confusing an-1 notation, just sense-making within the material.

     I've never taught this topic before this semester, but teaching it this way has made immediate sense with my students. They like it, they get it, and they are good at it. When I see other students in our building that are learning it in a traditional textbook sense, they are confused and constantly asking questions as to what is going on. The notation is making them lost and keeping them from seeing what's going on. When I present it to them like this, light bulbs go off and they wonder why they didn't notice this relationship before. This has quickly become a topic a enjoy teaching because it is now something that is easily understandable. We haven't gotten to geometric sequences yet, but you can bet I'll be teaching it in a very similar matter, in the context of finding an exponential equation from a table. I expect the same kind of success when I get there, assuming I'm able to teach them the table portion of exponentials effectively.
     But what about the sum of n terms? Well, so far I've taught it just by developing the formula; showing that the first and last terms, second and second last terms, etc. all add to the same value and you have n/2 of those sums (with a little more exploration and detail that that). But, I could also follow a similar path as above and have my students generate a table with y being the cumulative totals for the terms x and below. I could teach my students how to find a quadratic equation from a table by using the second differences and they could go from there. I intend on making this connection once I start quadratics, but we're still in linear land. We'll get there.
     I'm all about teaching math in a way that makes sense to students, but doesn't sacrifice the meaning and understanding of the material. I don't want my students to get bogged down by notation and formulas. I want them to see math for what it truly is, a study of patterns. I want them to make connections to previous material that they've learned and see how it all connects. Very few textbooks do this, and I'm finding that I'm learning more ways to do this through talking to other math teachers and exploring the math myself from scratch (which I need to do more of).
     By the way, it also helped a great deal that I started teaching this topic by using Dan Meyer's triangle toothpick problem. I imagine providing a great context for the lesson helped tremendously with the engagement process from the students. I also continued and developed the concept with the help of Fawn Nguyen's visual patterns. A many thanks goes out to these great educators for creating and sharing these resources with the MTBoS.

Quadrillions of Pennies?

     The other day, a colleague of mine and myself were discussing exponential growth and how to open this to his students. I've never taught it before, so I felt that I wasn't the best person to ask but thought I could learn something as well. He suggested the classic problem of 'would you rather have $10,000 now or one penny today, and twice as many pennies each day for a month.' I thought this would be great, and I mentioned the similar problem of if you have a bean on the first square of a checkerboard and you double the number of beans for each square, how many will you have on the sixty fourth square. He decided to go with that instead, but use pennies instead of beans. Long story short, his students were amazed at the fact that there would be 92,233,720,368,547,800 pennies on the last square, not to mention the total amount of 184,467,440,737,095,000 pennies on the board.
     Needless to say, both of us were also shocked. In my geometry classes, I've been trying to take our calculated values and putting them into a context that students can relate to. Saying that a box has a volume of 150 cubic feet means nothing to a student until they can see what one cubic foot looks like. So having a value so large was just that to his students: a large, inconceivable number. We tried to put it into a context that was relatable but we kept coming up short. Nothing that we did, relating it to the dollar amount, distance, etc. made sense to us or our students. We came up with one good comparison, but it still wasn't the best.
     I decided to use this as an opportunity to experiment. As we were discussing the number I had students in my room taking a test. We were speaking just about the numbers but did not mention the context. One student looked at me afterward and laughed saying, "You guys are such nerds, talking about big numbers." I went on and on about how cool it was for the answer to be that big, and then I realized he had no idea what I was talking about. I described the original problem to him, and he became interested. I then put it into context, saying that's like if you stacked pennies on top of each other, that stack would reach Pluto from Earth 1,911 times. His response: "Wow," and then he pondered. I could see the look on his face, focused on what I just described to him.
     Subject #2 conversation went like this - Student: "Mr. B what's this large number on your board?" Me: "Go ask Mr. Miller, he'll tell you." Student comes back: "Yeah, he said something about pennies and checkerboard. I wasn't really paying attention." Me: *describes problem* "That's how many pennies would be on the last square." Student: "Ok." Me: "That's like if you stacked all of the pennies on top of each other, that stack would reach Pluto." Student: "Woah. That's awesome." Me: "955 times!!" Student: *mind blown*
     I type this because context matters. I've tried this discussion with many other students and the response was always the same: they start out not caring but then are super interested when they are able to relate to what is going on. Now, I understand that there are some math problems and topics that can be engaging in the pure form that they are. I'm not saying that everything has to be taught in a real-world context. However, students need to be engaged and they need to have a reason to be engaged. I do not like when entire courses are taught without any context whatsoever. How does this help students? In my opinion, this helps to create students' hatred of math. They don't have a reason to care about it so they don't try, which results in them not succeeding, which results in anxiety, and the cycle continues (well, perhaps that's an exaggeration, but it certainly does not help). I want everything I teach to be engaging and relatable, whether its connected to the real-world or previous topics we've discussed. Nothing should be so abstract that they can't comprehend what's really happening. I believe math can be interesting to all students, we just need to figure out how to get them hooked from the beginning and everything we can to keep them there.

With A 'Lil Bit O' Algebra

So, I've recently been asked to rewrite our districts Algebra 1 curriculum (and eventually Algebra 2 I believe) to fit with the Common Core State Standards, again (woot!). Teaching something because the standards tell me to bothers me as opposed to teaching something because its relevant, but that's a rant for another time.
A colleague of mine and I were looking over the CCSS for alg. 1 and realized its kind of a hodge-podge of topics thrown together. I mean, all of the linear equation/function stuff works together very well, but then there is some beginning stats/probability and also rational expressions, polynomials, exponents, etc. thrown in as well. We were trying to find a way to organize this course so that it makes sense and there are logical transitions. As of now, the topics are taught in the order in which they appear in our textbook (UCSMP), which is no good (both the organization and the book). We stared for a while and threw out some ideas, and then I realized something. Every alg 1 curriculum that I've ever seen has always ended with stats topics, and they are part of the 'if there is time' category. I wanted to give this course some flow and a context, so I thought 'Why not teach it from a statistics perspective?" After looking over the standards again, we figured out that this just might work. Here's the tentative plan:
      We'll start off with calculating different types of probability. This covers the different types of numbers, how to order them, represent them, compare them, etc. We will then move on to different ways to represent data, bar graphs, pie charts, stem and leaf, box and whisker, etc., further emphasizing the importance of number sense. This leads nicely into scatterplots and line of best fit, which opens a door to teach all of the linear equation/function topics that are essential to an algebra 1 course. This is obviously a very loose description since I don't have all the info in front me, but I think it will work. Every topic will have a context and we'll be able to teach everything in a real setting. My hope is that this allows students to see how these can be used and provide an easy method to be taught. I'm very excited for this to happen and can't wait for the results. It makes me wonder why I've never heard of anything like this before.
      The only hicup - the rational expressions, polynomials, GCF, LCM, exponents topics that are to be included. How do we incorporate them into a stats context that flows well with everything else in the course? As we talked, the best we could come up with is 'throw them in at the end.' No context, no transition, no meaning in regards to the rest of the course. This upsets me, but, I've got nothin'. Any ideas?
I feel that this is a new and exciting way to teach algebra 1 that could produce some amazing results. I'm a little nervous about showing this to those who are teaching alg. 1 next year because its so different than the way it used to be, and also because I'm not teaching it (so that will produce some interesting discussions as well).

Order Matters at 5-Below

I was walking through 5-Below this afternoon looking for some fun stuff for my daughter and saw this hanging on the wall: a customizable iPod case with 'over 17,000 possible combinations!' Whenever I see advertisements like this I always wonder if they have a math guy that figured that out for them or if they just estimate it. I also wonder how many people will change the way their iPod looks 17,000 times.
If I showed this to my students while teaching permutations and combinations, which would they argue that it is? Is this an engaging enough picture to get them to think about it, or would they quickly move on?

Oh, the possibilities...

Any (mathematical) Questions?

Curiosity?

Now I'm curious. I wonder what this would like. Would you be able to predict the number of letters in any number (even though it would be easy to count)? What kind of relationship exists? I smell a project....

Betelgeuse, Betelgeuse, Betelgeuse!!

I was talking with one of my colleagues about how we can improve problem solving skills within our students to encourage higher-order thinking skills. In the past, I've shared with him Dan Meyer's 3 Acts and he was as amazed as I was. Together, we've been working on creating problems for algebra1, 2, and geometry that are set up for this process, and he shared with me a video similar to this one that he showed his students. It's a simple video; there's nothing fancy about the way its made and there is no narration, but the possibilities are endless in the way its interpreted. He used it to illustrate scientific notation. My first thought was to ask the question, 'How many Earth's fit inside Betelguese? How many suns fit inside VY Canis Majoris?' You could relate it to similarity, 3-dimensional space, fractions, or a hundred other topics. Regardless of how I end up using it, I plan on removing the diameters from the video so that my students can do some research to find the information that they think they need to answer their questions.
This video further illustrates to me how many questions can be generated from any picture or video. The phrase 'a picture is worth a thousand words' has become more real. I now try to look at my everyday surroundings from a mathematical perspective. I always tell my students that if math didn't exist, nothing else would either, but they always blow me off. Now I'm beginning to gather solid, concrete examples that they can see. I think the way I'm going to approach it from now on is show my students these examples so we can explore them as a class, and then use them to illustrate that math truly is everywhere. If I can get them to believe this, them I'm one step closer to getting them to appreciate everything that happens around them at any given moment.

So, what are your thoughts? What questions come to mind when you see this video?

Students in the Real World

I've had the luxury of teaching the same courses for the past five years. I'm the only teacher in the building who has taught one of them throughout that time, which means I've been able to adjust the curriculum to meet the needs of the individual students from year to year. Today, I thought of an idea that may make the course much more interesting for my current crowd: I think I'm going to let them create their own math problems.
Now, they're not going to create meaningless problems, with random numbers and variables that have no context. I want them to live their normal lives, but when they see or hear something that captures their interest I want them to turn it into a problem. Their job will be to take pictures, videos, recordings, etc. of something the interact with and come of with a question that they (or someone else) can solve. They will have to be able to describe the scenario in words and visually, come up with a question or multiple questions, and also gather the information that would be needed to come up with an answer. I don't care what math topics they choose; it could be any high school level math. If it involved something they haven't learned, then we'll go through it as a class. If it uses information they already have, then they'll come up with an answer. Either way, its math in the real world.
Since I've started my career as a math educator this is something that I've spent a lot of time thinking about. Every student always wants to know why the topics they are learning are relevant. They don't settle for an answer of, "You're learning it because its on the test," anymore. They want substance and meaning. If I want my students to see math in the world (which everyone should want this), why not let them explore it themselves? I think if I organize this project correctly, some of their eyes might be opened to how much math is used.
I'm at the point where I see situations or objects and my mind jumps to 'How did they do that?' or 'What if it looked like this...' I want my students to become that way too. This will lead to them being life-long learners and it will exercise the creativity of their minds. If I can get them to analyze any situation and expand upon it or just wonder about it in a new way, I think I've achieved something.
Let's hope this goes well!