Add this to the list of reasons why the first year of teaching high school is so horrible: your body is not used to all the germs at a high school and you get sick all the time. I think this might be the 6th or 7th time getting sick since last September, even though I’m popping multivitamins, Coenzyme Q10, etc etc every day. I think I had the H1N1 flu last fall. (It wasn’t that bad.) But now my immune system should be super duper ready for anything!!

Here’s a question to you teachers:  How would you teach students to solve equations like

e^3x = 0.4 ?

I can think of at least three different ways to explain it to students.

Method 1. Use the definition of the logarithm.

log _b x = p       is an equivalent statement to          b^p = x.

So that means the equation e^3x = 0.4 is equivalent to 3x = log _e 0.4. Divide by 3 to finish solving for x…

Method 2. Take the natural logarithm of both sides of the equation.

ln e^{3x = ln 0.4}

Then use the power property of logarithms (log x^y = y log x) to get

3x ln e = ln 0.4.

Next, remember that ln e = 1, and divide both sides by 3.

Method 3. Take the logarithm of both sides of the equation as before. Use a logarithm with the same base as the base in the problem. Then, think of ln and exp as inverse functions so that ln e^3x = 3x = ln 0.4.

If you say “Teach all three methods!”, consider which one you would teach first. If you only had time to go through one explanation, which one would you use? Is there a better explanation that I’m overlooking?

This question becomes even more nuanced if you have an equation like 5^3x = 0.4, and you are interested in getting a numerical answer. In this case, if you use Method 1 or Method 3, then you end up with log ₅ 0.4 in your answer. Since most calculators only give the natural logarithm and base-10 logarithm, one would then need to apply the change of base formula to get a numerical answer. Method 2 might then be slightly less complicated as you could apply the natural logarithm or base-10 logarithm to the problem, regardless of the base present in the equation, and the final answer will be ready for use in a calculator.

Because of this reason and the fact we just got through learning about the properties of the logarithm (like the power property), I opted to show students Method 2 first. This method gets them to practice using the power property and to remember that log _b b = 1. I don’t think my students know about the idea of inverse functions so I’m probably going to skip Method 3 (even though this is the way that explain it in my college classes). If I have time, I’ll make sure to explain Method 1. Even though Method 1 is the most elegant, I’ve found that students have a really hard time going between log _b x = p  and  b^p = x, even if you have them write this out before trying to reshuffle the three numbers. If the number x is a complicated expression instead of a simple number, that adds adds to the cognitive demand.

I am raising this issue here not just because I think it’s an interesting question but also because I find that teaching relies so heavily on this special kind of mathematical knowledge for teaching (MKT). Think about all the knowledge that one needs to tackle this question: one needs to know what misconceptions students have about the logarithm, one needs to know what kinds of mistakes students frequently make, what they find difficult when solving exponential equations. One also needs to know that most calculators only give natural or base-10 logarithms. This information is specific to the work of teaching and is not required to actually do mathematics.

There has been some effort to describe and catalog this special kind of mathematical knowledge for teaching elementary school mathematics, but there is little effort at the second school level. I recently attended the National Council of Teachers of Mathematics Annual Meeting in San Diego and I didn’t see any talks on this subject. Why is this the case, if I am right that this kind of knowledge is so important to the work of teaching?

One of my biggest problems is that spend most of my time “teaching to the middle.” I would say that in my Algebra 1 and Geometry classes, about 10% are skilled and motivated, about 50% are somewhat motivated but lack language, mathematical, and learning skills, and about 40% are not motivated at all to learn and actively ignore the mathematics that is happening around them. I find I spend most of my time engaging that middle 50%. With enough practice and repeated explanations, I can get this group to learn most things. However, the problem is that I don’t move forward until this entire 50% “gets it” and so the whole class ends up going much slower than I would like. The 10% that are skilled and motivated are probably bored out of their minds.

One thing that helps is to get the skilled and motivated students to help explain things to the other students. Another strategy is to spend most of class time on individual and group work and less on lecturing, so I am freed up to walk around the class and give assistance to individuals. Yet another strategy is to create enough tasks to keep everyone busy, but that strategy only works if I can find enough energy and time to plan that much. Even with all of these strategies, I am not moving through the curriculum as fast as I am supposed to.

But it’s not just feeling rushed to cover content. I wish I had time to really go deeper. I feel like I barely have enough time to get students to master the basics of each topic and then it’s time to move on to the next thing. I don’t have time to make connections to other topics or to tackle more interesting, deeper questions. For example, I recently taught my Geometry students how to use trigonometric ratios to find the lengths of sides in a triangle. I hate that I was only able to teach them a procedure for doing it and that we didn’t get to do anything deeper with the trigonometry. Most of my students will probably remember SOHCAHTOA and little else.

Recently, one of my bright Geometry students asked me for extra work. She told me she doesn’t like Geometry, so I’ve been giving her work that I’m preparing for my Algebra 2 class. She’s been eating that up, but I wish I more time to prepare work to help her change her mind about Geometry. Geometry is so cool!

I’m trying my best to meet the needs of all my students but I’m nowhere near the perfect ideal of differentiated instruction, in which everything is tailored to students’ interests, skills and learning styles. I need way more time–more instructional time and more time to plan. This is one the biggest reasons why I find this year of high school teaching to be so frustrating.

What a frustrating day.

1. Three days so far of concentrated effort on learning how to find the areas of rectangles, circles and triangles in my Geometry classes. Still very little progress from large portions of my classes. Ugh! What am I doing wrong?

2. Some kids took candy from my candy stash today. I am angry at those #@!%#! brats. Really makes me want to give up on helping them learn math.

3. So very tired. Not in body but in spirit. I wonder how some teachers can last for decades in the classroom.

Our district union has voted to accept a deal to accept five furlough days this year and seven furlough days the next academic year in exchange for fewer job cuts. So that means the school year will end one week earlier this year. The school board still has to approve the deal.

I’m not surprised that most teachers in our district voted to accept the deal. I’m tired and want the year to be over. A friend told me that I’m starting to think like a teacher now.

One lovely thing about my school is that it is new and has lots of shiny features. For example, the rooms were all recently set up for wireless sound distribution. Each teacher received a wireless teardrop microphone and a wireless handheld microphone. The microphones connect to a receiver in one of the closets and then transmits the sound in speakers mounted in the ceiling of the room.

I was a little skeptical at first about the microphone system, but I am now a believer–the system helps in that I can speak normally (with the teardrop microphone hanging from my neck) and my voice is then is nicely distributed around the room. I don’t have to raise my voice and can speak in a more even tone, which I think helps the students feel more calm.

The handheld microphone is also very handy as I can use it to allow different students to speak. I’ve told the students that when someone is trying to speak, everyone needs to be silent. It’s a nice combination of the talking stick idea and technology.

So far, the students seem very receptive to the system and they seem to get quiet quicker when I want their attention. I also don’t feel quite so worn out at the end of the day because of having to PROJECT MY VOICE FOR AN EXTENDED PERIOD OF TIME–it’s surprising how tired that makes you feel.

The end of the school year is in sight. Though I’ve complained in this blog and to my colleagues, friends and family about my experiences this year, I know I will be completely depressed at the end of the year. What I like (and will miss) most about teaching high school mathematics are the challenges and joys of working with students of this particular age group.

To my colleagues at my home institution: Don’t worry–I’m definitely not considering a career change. I know now that I lack the stamina, perseverance and patience to work with high school students. However, I am starting to see why I love working with high school students. It is so rewarding to be around students who are right in the midst of developing their self-identity. These students are starting to struggle with some of the most difficult issues of lives RIGHT NOW and I have a chance to push them in some direction to their benefit or detriment. This makes it thrilling to go to work everyday. And, high school students can be pretty entertaining; they say and do some of the most ridiculous/funny/dumb/intelligent things.

Many of my students have gone through very difficult circumstances and done all sorts of things that I never did at their age, but they still have an innocence that makes you want to coach/cheer/mentor/scold/encourage them. A lot of them don’t yet know how to interact well with each other and with adults, haven’t yet had the thrill of accomplishing something they thought was impossible, don’t yet know what possible paths lay in their future. So, the thought of being able to affect the life of a young person while he or she is in such a formative stage is both scary and inspiring. As my colleague says, these students almost need us to be life coaches more than they need us to be math teachers.

Of course, I will miss my colleagues at this school and I’ll miss the students themselves–I feel like I’ve made a strong connection with many of them. But then again, I have many more colleagues and students at my home institution that I miss terribly now. And while the math that I’m teaching now is profound, I am really itching to teach mathematics that richer and more complex. So, what have I learned after this year? The job of a high school teacher is bewilderingly difficult and rewarding at the same time.

Every time the district’s periodic assessments come around again, I always get so stressed out. I am reminded by these tests that my classes are way behind according to the district’s plan for what needs to be taught and when.

Part of the problem is that I have not been able to make up for the lost three to four weeks at the beginning of the school year. I constantly feel like if I just had another few weeks, we would be able to handle these periodic assessments easily. The other part of the “problem” is that I refuse to rush through material without giving my students lots and lots of chances to understand it and become proficient. I find my students need lots of time and lots of practice–that’s the way it should be with really learning something deeply, I feel.

Instead, I always feel a mad rush to cram students’ heads with facts. For example, I haven’t gotten to the quadratic formula in my Algebra 1 class. The CPM curriculum does not get to this until the very end, but the district’s periodic assessments don’t jive with the CPM curriculum’s pacing. What should I do? Should I just tell my students to memorize an arbitrary formula and use it? Or should I just tell them that the test will cover things that they don’t know yet and to skip them?

My apologies for the lack of updates on this blog. Today was our first day back to school after having a wonderful week of catch-up-with-other-responsibilities (a.k.a. Spring Break).

A colleague warned me today that this will be the longest and most difficult stretch of the school year. Students are tired, we’re tired, and there aren’t any more long breaks until the end of the school year, unless…….

Our school district and union have struck a tentative agreement to cut five days of instruction from this current school year and seven next year. I know it’s not a good thing–I want my students to learn more–but I am secretly hoping that the union membership votes to go along with the plan. Voting will take place this week.

I’ve been thinking about how to engage students in as many different senses as possible when learning math, so as to ensure that students have many distinct memories about a particular topic. The hope is that these different ways of knowing mathematics will create a more robust schema for the information that is acquired, which will then lead to better retention and deeper comprehension.

Ruling out the sense of smell (maybe “scratch and sniff” math books?) and taste, the remaining three senses that can be used to experience math are sight, sound and touch. Students hear math and see math frequently–that’s easy to do. Helping students experience mathematics using their hands or bodies usually requires a bit more creativity.

As I wrote in a previous post, I used Algebra Tiles and the “general area model” to help students calculate the product of two binomials using their visual and tactile senses. But since this approach can be problematic when working with negative quantities or higher powers of variables, it’s best to transition students to more general ways of knowing mathematics as they develop greater skill and understanding.

The ultimate goal is to get students to think of expanding products of polynomials as a generalization of the simple distributive property:  a(b+c) = ab + ac. (See note below about “FOIL.”) But how to get students to experience this visually or kinesthetically?

Here’s a silly thing that I did in my Algebra 1 class today: After showing students the “general area model” for expanding products of binomials, I asked two pairs of students to come up to the front of the room. The students’ names start with the letters A, S, E, and G. I purposely chose these students because A and S are close friends, as are E and G, but the two pairs don’t seem to associate with each other much. I wrote on the board (a + s)(e +g), then asked the four students to help me act out a short scene.

The scene: A and S, who are close friends, are going to a party hosted by E and G, who are also close friends. A and S walk in the door and are greeted by E and G. All four people shake hands and introduce themselves and each other.

I then let the students act out the scene in front of the whole class. Some students were shy, which made the whole thing a bit awkward, but they managed to do it just fine. I then asked the class what they saw–in particular, who shook hands with whom?

The students easily pointed out that A shook hands with E and G, then S shook hands with E and G. They also could explain to me that the reason A and S don’t shake hands is that they know each other already (which is the same reason why E and G don’t shake hands).

The point of all of this was to help students kinesthetically experience this product.

Each of the four terms in the answer matches up with one of four handshakes. The diagram (with colored arrows and terms) then became the visual reinforcement for the kinesthetic experience. Maybe a bit hokey, but I think this short demonstration made the idea of using the distributive law very concrete for some students.

Teachers: I’d love to hear what other activities you use to help students engage in mathematics using other senses. One of my close friends gets students to walk and run while graphing their motion, to help students learn the concept of slope.

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Footnote about “FOIL”: The “FOIL” acronym is often taught to students to help them remember how to expand products of two binomials.

(x+2)(x+3) = FIRST (x^2) + OUTER (3x) + INNER (2x) + LAST (6)

It’s a handy way to remember things, for sure, but I am consciously trying not to teach this to my students because it only applies to the product of two binomials–it doesn’t help if you are multiplying a polynomials with more than two terms, for example, (x+2)(x+y+3). It will be interesting to see if students can more easily make the jump to expanding (x+2)(x+y+3) because I’ve consciously avoided teaching  “FOIL” and instead focused on this idea of generalizing the distributive property.