Problems / Exercises

I wrote about this earlier today and I want to spend a few minutes trying to organize my thoughts.

A conversation on twitter today with David Wees (@davidwees) reminded me of a conversation with a former colleague. It also reminded me of a class I took in my master’s program. The course had the vague title Problem Solving and my professor (who was my advisor) had a long background in studying problem-solving. I remember (not clearly enough) that we had a working definition for what qualified as a problem. The definition revolved around the idea that there are certain questions we encounter in math where we immediately know what we are supposed to do – what formula to use, what definition or theorem to call upon – while there are other questions where we do not immediately know what we need to do. The first group we classified as exercises while the second group were called problems. It is not necessarily that problems are harder. I have certainly dealt with many challenging math questions where I knew exactly what I needed to do, it was just really hard to do it. I have a real fondness for problems in mathematics and I have developed the habit of writing homework assignments for my classes that should probably be called problem sets. For years, I was writing these for Honors Calculus, AP Calculus AB, AP Calculus BC, Honors Precalculus, and AP Stats. A few summers ago I wrote a Geometry text for our school and I wrote all the HW assignments as well. These students are fundamentally different in many ways from the students I was working with in those other classes. I do not necessarily mean that they are inherently less talented or anything like that. What I do believe is that they are younger, less experienced, and less patient in their problem solving. So, they are more likely to simply shrug off a problem and figure that we’ll talk about it the next day. Over the course of the year most of them have become more patient and they are aware that we will discuss these questions and that they will not be graded on their HW. After my twitter exchange this afternoon, I am (once again) rethinking this strategy and I am nowhere near a conclusion. I did share the tweets with my class after we struggled through the question I wrote about earlier today. I asked them to honestly share their opinion about whether it is a valuable exercise to struggle with questions like this one. A few were upbeat and said that they liked thinking about these questions and that it is helpful to try challenging problems. One student said something really striking. She said that it is really frustrating to work through these problems alone and that she wishes she could get the opinions/insights of others when she is struggling with these questions. This is certainly in line with what David suggested in our exchange and with what my former colleague (who I wrote about earlier today) mentioned as well. I have some thinking to do here. I do believe that it is powerful for students to wrestle with challenging questions. I do believe that by not grading HW I am helping to create a safer environment to struggle. I know that a number of students work together on assignments either here in our dorms or libraries at night or during study halls during the day. I also believe that conversations in class are richer when they have some ideas already thought out to toss around. However, I also recognize that this is frustrating for some students and may simply push them further away. I recognize that if I am going to say that I value collaboration that I need to commit to making the time for that when we are together. I also recognize that what works to motivate seniors in college level math classes might not work as well with 9th and 10th grade students in a required math class.

 

Lots of thinking to do, luckily the summer will afford me some valuable time.

Thinking Out Loud

Super brief post here – class starts in 15 minutes. One of my Geometry colleagues asked me about a HW question I had written. I asked the students to find two cylinders that were not congruent but that had the same surface area AND the same volume. I thought it was a pretty interesting question, but I realized I did not have a coherent strategy for discussing it other than playing with numbers. I threw the question out to Twitter and engaged in a terrific conversation with Matt Enslow (@CmonMattTHINK), John Stevens (@Jstevens009), Dave Radcliffe (@daveinstpaul), and David Wees (@davidwees) Some good math was tossed around, but what really has my brain bubbling is an exchange with David Wees. He said he thought it was a great question but he would not have used it as a homework question. When I asked him why he said something that reminded me of a conversation with a former colleague. My former colleague once said that he sees a difference between exercises and problems and that he liked to keep problems for times together with the students where they could work together. I find that I feel (hope) that meaningful conversations can happen in class more readily about a problem like this one if the students have had time to think about it first. However, his words carry some weight with me as do David’s. I feel as if there is some conclusion I want to reach, but I also suspect that there is no right answer to this. I would love to hear some opinions about this in the comments or through Twitter where you find me at @mrdardy

 

 

What Do Numbers Mean?

This week I wrote about experimenting with number base systems in my AP Calculus BC class. A question came into my head yesterday about repeating decimals in base ten and whether/how we could decide if that number is also repeating in different number bases. It was really hard and the calculations got pretty ugly. So, today I started class with the following idea. I wrote a repeating decimal in a different number base and then converted it to base ten. The calculations are clearly more manageable and I had a clear idea that this could link back to our conversations about infinite series. What excited me today was that my vision of the infinite series was different than that suggested by my student Megan AND it was entirely different than a suggestion by my student Elijah.

I started with the base 3 number 0.122122122… I saw this as three different infinite geometric series’ each with a ratio of 1/27 and I worked the problem this way. Megan saw this as one series made of the first three terms with a ratio of 1/27. We, of course, arrived at the same answer and I really liked the way that her techniques was only one series to calculate instead of calculating three different series’ the way I saw it. I will put a picture below here with a different example showing Megan’s technique.

image

The example above started with the base 5 number 3.021021021021…

Elijah had a completely different approach, one based on how we teach converting base ten repeating decimals into fractions. The picture below shows his approach.

image

A couple of notes here. First, Elijah is a terrific math mind and this is a really creative approach. Second, this approach models the approach that my students have already seen. Third, this tactic encourages you to actually live more thoughtfully in this different number base.

I just came away SO impressed by the thoughtfulness, the persistence, and the creativity of my students this morning.

Some Post AP Fun

My Calc BC kiddos took their AP test last Thursday and we still have classes through next Wednesday. So, I have some time to play with. This year is the first time through for me teaching a Discrete Math elective and one of the topics I ran through with that class was the notion of different number bases along with a little history about some counting systems and the symbols used. I decided that my Calc BC students deserved the opportunity to think about this as well and for the past two days we have had fun saying things like 5 + 4 = 13 (guess the base!) and things like 5 X 2 = A. My students have appreciated me joking that they should make sure to go home and tell their parents that I said 5 + 4 = 13. What I have appreciated is seeing the combination of discomfort and curiosity which turns into a bit of joy as my students wrap their heads around this topic. It is especially in testing to me to see that the BC kids, who are really the top math scholars here, are not inherently more comfortable with this topic than my Discrete students were. There is a pretty big gap in the comfort level with mathematical ideas between these two groups of students, but this notion of fundamentally reconstructing meaning for numbers is a great equalizer. In BC today I even threw out this question – convert the base 8 number 41.37 into a decimal number. Contextualizing the ‘decimal’ portion of this number was not obvious right away, but they were easily convinced once one of their classmates offered a rationale for it. I know that this is far from an earth-shattering ideas, but I also know that this is an idea that too many students are not exposed to in their high school experience and I am kind of pleased that I get to blow their minds a bit. Tomorrow we talk about the Mayans and the Babylonians and we wrestle with their numeration systems. A fun way to wind down the year.

Platonic Triangles

Too long ago I started a Geometry post by suggesting that I might have a two post day in me. Needless to say, it did not unfold that way and some combination of malaise, exhaustion, and the irresistible momentum of the end of the year has kept me away from this place of peace and comfort for some time now.

I want to share something from our Geometry class this year that was largely motivated by the work of Sam Shah (@samjshah) and his colleague Brendan Kinnell (@bmk2k)

At TMC Sam and Brendan shared boatloads of ideas and docs that they had created for their Geometry class and I am still in the process of digesting them. One that jumped out to me immediately was a document that they called The Platonic Book of Triangles that they were kind enough to share and to allow me to share in this space. Sam wrote about their process here and here.

What I did this year was try to de-emphasize naming the trig functions and just concentrate on the inherent similarities tying together right triangles as a lead in to discussing the inherent similarities relating all regular polygons and circles. Part out of a whole has become a mantra in my class these days. So, what I did was I went to a local copy shop and had them print out a class set of bound copies of the above referenced book of triangles. My students are referring to it as the magic book of numbers. We reference it regularly to set up proportions to solve right triangles. I had the book laid out so that each page had complementary angles on either side. So the students recognized – with a little prompting – that the side lengths on the triangle with the 38 degree angle marked matched up with the side lengths on the triangle with the 52 degree angle marked. I have been SO happy with how they have taken to this reference. In a way it reminds me of the trig tables I used to look up in the back of my book but this has a couple of major advantages. First, it is far more visual and helps the students orient themselves. Second, it does not rely on memorization of a mnemonic about the definitions of the cosine, the sine, or the tangent of an acute angle in a right triangle. I have been careful when I do use those terms to say as clearly as I can that for now they do not want to talk about these functions for anything other than acute angles in a right triangle. There is a whole world of trig excitement waiting for them after their experience in our Geometry class is a dusty memory.

From this conversation about solving triangles and using this to lead into explorations of regular polygons I wanted to make sure to introduce the idea of radian measures to my young charges. I came up with what seemed like a clever idea. It was a chilly, drippy day here in NE PA so I called up the weather bug applet on my laptop. However, what I did before class was I changed its unit of measure to celsius rather than fahrenheit. A student mentioned that it was unpleasant outside – with a little prompting – so I called up my weather bug and expressed surprise that it was only 13 degrees outside. Students quickly pointed out that this was simply a different way to measure the same thing, that there is a way to jump from one representation to the other. Aha, the hook was baited! I then launched into a pretty unexciting, standard representation tying together radians and degrees, relying on my mantra of part out of a whole over and over again. I am not fully convinced that they are buying in and there is evidence that many of my students seem to think that attaching pi to a degree measure is simply some sort of stunt. I am also seeing evidence that simplifying fractions, especially those where the numerator is already a fraction, is a serious challenge to too many of my students. However, what I am convinced of at this point is that a seed has been planted that has a better chance of blooming in precalculus than for those students who did not see the concept of a radian presented to them before. We have our unit test on Monday and I hope not to be disappointed.

 

A Geometry Explanation Idea

Today will likely be a two post day after too long of a layoff from this space. It’s funny, I know that this writing relaxes me and lets my brain breathe in important ways. However, I am too prone to let busy-ness in my life prevent me from taking advantage of that.

In Geometry we are examining relationships between angles and arcs. You know the drill, right? Central angles = intercepted arc measure. Inscribed angles are equal to half of the measure of their intercepted arc. Interior angles (other than those at the center) are equal to half of the sum of their intercepted arcs. Exterior angles equal half of the difference of their intercepted arcs. As I am explaining this yesterday in class I am emphasizing this half relationship over and over but I just kind of gloss over the fact that the central angle does not fit this pattern. I have gone through this explanation in the past but I do not think I was as explicit about the developing pattern with the one half scale factor in the past. So this morning I was thinking about how I can best help my students focus on this detail. I was motivated, in part, by a lovely blog post by Michael Pershan (@mpershan) which, in part, is about managing the processing load of our students. I was struck when reading by the fact that I was asking my students to juggle seemingly different rules yesterday. So I had this idea and it probably is not revolutionary in any way. But I think I want to simply say that central angles are interior angles. They happen to be interior angles that form two congruent arcs. This way I can ask my students to think about the one half scale factor for every angle/arc question that they think about.

Does anyone else out there do this? Do you agree with this idea? Do you think that there is a downside here that I am not seeing yet? I’d love to hear some thoughts in the comments or you can find me over on twitter @mrdardy

 

More Calculus Fun with Series

Many many thanks to the wonderful Mike Thayer (@mthayer_nj) who sent a link to this lovely video https://www.youtube.com/watch?v=XFDM1ip5HdU in response to yesterday’s post.

We started each Calc BC class today by revisiting the rational function that caused me so many problems yesterday. Innocently enough, I decided that it would be interesting to examine 1/(1-x) first by long division where the divisor is 1 – x and we got the power series 1 + x + x^2 + x^3 + … as the quotient. Then, we used -x + 1 as the divisor and got the series -1/x – 1/x^2 – 1/x^3 – 1/x^4 … as our quotient. Class ended too soon and I was not able to answer the question of how we could consider these two very different looking series as being equal to each other since they were each results of considering the rational function 1/(1-x).

So, after sending out my call for wisdom in yesterday’s post I went to GeoGebra and discovered something lovely. Something that I was sure my students would be able to discover for themselves. I created a GGB file and I planned our day today. Can I tell you how proud I am of my students for how they handles today? Well, I won’t wait for your permission, I will just come out and tell you how thrilled I am.

I started the day by quickly revisiting yesterday’s two division results and then I called up the GGB file with only the rational function showing. They saw that I had each of the other two series expressions typed in already (out to x^5 for each) and I asked them which of the two they wanted to see first. My morning class wanted to see the series with x as the ratio first, my after lunch class wanted to see the one with the ratio of 1/x first. In each case, after unveiling the function of choice and noticing the relationship between the rational function and this new series expansion my students made the following observations:

  1. The graphs only seem to match over a certain set of x values
  2. If I were to add more terms, that match would improve
  3. If we look at the graphs of both series then we will have a nearly complete match to the graph of the rational function

What my students realized – what I realized last night – is that the two series we can generate have completely opposite intervals of convergence. It was absolutely lovely to see geogebra help this intuition along and it was fantastic that they made this realization before seeing the second graph to confirm it.

After this breakthrough we watched the video together and all of our brains were a bit achy by the end. Amazing that Mike found/knew this link so quickly when I blogged last night.

Some other notes – A student in my after lunch class made this observation about the complementary nature of the intervals of convergence before we even looked at geogebra. I gave one last example, f(x)= x^3/(x+5) and the student who was bothered by -1 = 1 + 2 + 4 + … quickly converted x + 5 into a form of 1 – r so we could interpret the rational function as an infinite geometric series. Another student converted it to x^2/(1 + 5/x) and we, once again, had two different series’ that had complementary intervals of convergence. I have taught this course five or six years before this and never had this ‘discovery’ pop up. Now, we cannot avoid it. It provides a wonderful context for our upcoming conversations about Taylor series and it gives us the opportunity to be more aware of convergence expectations. A pretty great day!