Algebra 1 is hard

Today was a good day in Algebra 1–my students are on the way to factoring simple trinomials of the form x2+bx+c. Students did an incredible amount of math today, much more than normal. They seem to like it when I prepare a series of worksheets and they get to work at their own pace through them. We ran out of time so not all of them were proficient by the end of class, but we will definitely work on it more next Monday.

But, learning about factoring was not my original plan for today’s Algebra 1 class. I had originally planned to teach students how to find the point of intersection between two lines in a plane (algebraically and graphically). I had to scuttle my lesson plan because I realized just how hopelessly behind we are. My students are probably going to do very badly on the Algebra 1 California Standards Test (CST). There is just so much content in Algebra 1, and there is not enough time to get to it all. Even I feel overwhelmed about how much content there is to learn in Algebra 1.

Here is what we’ve learned so far in Algebra 1 this year:

  • arithmetic operations on positive and negative numbers
  • arithmetic commutative, associative, distributive properties (Standard 1.0)
  • simplifying an algebraic expression by collecting like terms, and using the distributive property (Standard 4.0)
  • substituting values into expressions and evaluate them using correct order of operations
  • determining if a point is on a line (Standard 7.0)
  • graphing linear functions (Standard 6.0), identifying positive, negative and zero slope
  • knowing what the slope of a line means and how to calculate the slope of a line from a graph (but I’m not sure they know how to calculate the slope between two points)
  • looking at a table of numbers in a linear pattern and being able to predict terms in the pattern and describe it with words and algebra (sort of related to Standard 7.0)
  • making connections between a pattern, a graph, an equation and a table of numbers
  • solving linear equations (Standard 5.0)
  • expanding the product of binomials, maybe polynomials too (Standard 10.0)
  • graphing quadratic functions (Standard 21.0)
  • square roots, identifying the perfect squares up from 1 to around 144
  • solving quadratic equations using the quadratic formula, but right now the quadratic formula is just something that came out of thin air (Standard 22.0)
  • factoring simple trinomials (Standard 11.0)
  • and, we also have spent time learning how to answer multiple-choice questions well

Here is what my students have yet to learn to be able to do well on the Algebra 1 CST.

  • finding the reciprocal or additive opposite of a number; calculating roots, raising numbers to a fractional powers; understanding and using the rules of exponents (Standard 2.0)
  • solve equations and inequalities involving absolute values (Standard 3.0)
  • solving multi-step word problems (Standard 5.0)
  • sketching the region defined by linear inequality (Standard 6.0)
  • understanding the concepts of parallel lines and perpendicular lines and how those slopes are related; finding the equation of a line perpendicular to a given line that passes through a given point (Standard 8.0)
  • solving a system of two linear equations in two variables algebraically (Standard 9.0)
  • more generalized factoring: finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials (Standard 11.0)
  • simplifying rational expressions by factoring (Standard 12.0)
  • add, subtract, multiply, and divide rational expressions and functions, meaning that you have to know how to find the common denominator when adding two rational expressions (Standard 13.0)
  • solving rate problems, work problems, and percent mixture problems (Standard 15.0)
  • understanding the concepts of a relation and a function, determine whether a given relation defines a function, determining domain and range of a function (Standard 16.0, 17.0)
  • determining whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function (Standard 18.0)
  • using the quadratic formula and being familiar with its proof by completing the square (Standard 19.0)
  • knowing that the roots of quadratic functions are its x-intercepts on a graph and being able to find them (Standard 21.0)
  • using the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points (Standard 22.0)
  • applying quadratic equations to physical problems, such as the motion of an object under the force of gravity (Standard 23.0)
  • knowing the simple aspects of logical arguments: difference between inductive and deductive reasoning, identifying hypothesis and conclusion in a statement, using and recognizing counterexamples (Standard 24.0)
  • using properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements (Standard 25.0, not really sure what this standard means)

As I look at the complete list of state standards in Algebra 1, I feel completely overwhelmed. No wonder students find Algebra 1 so difficult! And as I look at the list of topics yet to be learned this year, I feel totally depressed. There is no way we can get to all of these topics by the end of the year, let alone in the remaining two weeks before my students take the CST. All I can do at this point is just try to find the topics that appear most frequently the CST and tackle those. That is why today I skipped finding the point of intersection between two lines–since the CST is a multiple-choice test, for now I will just tell them to take the four choices and determine if each point is on both lines (they already know how to do that). If the point is on both lines, it must be a point of intersection. On the other hand, factoring seems to appear quite a bit on the test, so that is why I have changed suddenly to the topic of factoring polynomials.

When you these lists of what my students know and don’t yet know, it will seem like we’ve been playing around in our Algebra 1 class. But, I promise you that I’ve been very efficient with my instructional time. We are doing math most every minute of every class, except for short breaks. And yet, we are so far from where we need to be. How did this happen?

Excuse #1. I’m a new teacher and I suck at this. My pacing is all wrong. I’m very reluctant to move on until most of my students are proficient with something–perhaps I need to move on to the next topic sooner and hope instead that accumulated review will solidify students’ skills and understanding after the initial exposure to a topic. Also, as I wrote before, I still have a lot to learn about differentiating instruction. I need to switch topics faster and find other ways to help students who need more time and practice.

Excuse #2. We’ve spent time on prerequisite skills and concepts that aren’t on the state standards, maybe too much time. For example, we’ve spent time developing fluency with positive and negative numbers, order of operations, learning how to the area and perimeter of a rectangle, calculating the slope of a line, substituting values into expressions, knowing what a square root is–these are things that aren’t on the Algebra 1 standards.

Excuse #3. We’re using the CPM curriculum for Algebra 1 in this school. Our district’s periodic assessments don’t match up with the curriculum, and the curriculum includes things that aren’t explicitly on the state standards: focusing on patterns, making connections between graphs and patterns. CPM encourages students to construct their own solution methods and concepts before the teacher tells it to them–this takes more time than direct instruction. (I firmly believe more of my students are becoming proficient because of the student-centered instruction, however.)

Excuse #4. We lost a whole month of instruction at the beginning of the year. I’m really wishing for that extra month right now.

A good friend of mine gave me some advice: Have the courage to intentionally skip topics that are on the state standards. She’s not saying that we should throw out the state standards (even though I’m not crazy about them). I think what she is saying is that I should trust my intuition on what I think is best for my students and not feel bound this list of topics. If I try to cover all of the material, we would be going so fast that most of my students would probably not get it (at least not with my current teaching skills) and so I’d be setting them up for failure. I should instead focus on a smaller set of topics and make sure my students really know it well and feel confident about themselves.

Argh! There is so much that I would do differently if I had the chance to start the year over.

One positive thing–last year, only 2% of students at this school were deemed “Proficient” by the state in Algebra 1. There’s no way my students will do worse than that, I’m confident of that.

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By the way, who in California decided that the pinnacle of knowledge in Algebra 1 is knowing how to derive the quadratic formula by completing the square? When you look at the released questions for the CST, it certainly feels that way.

Schools are full of germs

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!!

Logarithms and MKT

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

e3x = 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          bp = x.

So that means the equation e3x = 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 e3x = ln 0.4

Then use the power property of logarithms (log xy = 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 e3x = 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 53x = 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  bp = 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?

My teaching lacks differentiation

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.

Brats

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.

Furlough days (update)

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.

Wireless sound distribution

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.