Teaching Algebra

Julie O’Connor is an ESL teacher with this helpful math video on TeacherTube.

Do you teach students who are already familiar with the math concepts themselves, but don’t know enough English to talk about them? Or are they being exposed to the math concepts for the first time? What’s the situation usually?

Many times, they are familiar with the concepts - some students from the Middle East and Asia have been exposed to more complicated math. However, the word problems are a major challenge for the ESL students, so I try to give them some tools.

Do they get frustrated or impatient since they know the math, but not how it’s being communicated (like through word problems). How do you handle that?

Yes, there is frustration, but their actual math class teachers play a big role in this. Some of the math teachers are very understanding and don’t count the word problems as full credit, while others are not as patient about the challenges. So, that teacher compassion is a big factor in student frustration - I try to support the children when they are with me and help them in their other subjects.

Do you have any tips you’d like to share with math teachers who have an ESL student in their class?

I think that teachers should count the word problems as extra credit. They also should use a lot of visuals to represent the lesson objectives. There are also ESL strategies and games such as Vocabulary Jigsaw which is a vocab. game - there are many fun ways to learn and then reinforce math vocab. Obviously, they should partner students with others of more language abilities to help the ESL child to communicate.

Juan is a private tutor in San Diego, and he’s got a blog about math tutoring.

Why is May the high season for math tutoring?

May is a high season for math tutoring because many students are preparing to take their final exams before the summer vacation. Suddenly they realize the once seemingly remote possibility of failing their course is now very much real, and fast approaching, so they frantically look for help, and my phone rings more often than it does during many other months.

How do you help them? Do they still have a chance of passing if they’ve neglected studying for so long?

Each student is a unique learner, in a unique academic situation. Everyone is different. I help each student to the best of my ability. How I help them, and how much I help them, that varies from student to student according to each one’s particular circumstances.

As to whether they have a chance of passing or not, or how big a chance they have, that also varies widely from student to student. My long personal experience in private tutoring tells me most of them have a chance, and many do have a pretty good chance of passing. I would not be getting any referrals if that was not the case.

Besides, avoiding a failing grade is not the only reason students look for tutoring help near the end of their course. Many students who are not in any danger of failing their course want, nonetheless, better their chances of getting a higher grade.

Could you give an example of a common, faulty thought process students can have and specifically how you help them with that example?

He gave me a great example from his site. Check it out… here’s a excerpt:

A very common mistake test takers make when they have been out of school for a while is that they automatically try to expand the square of a sum as if it was the same as the plain sum of the squares of the individual terms.

Algebra2Go creator Larry Perez talks about his online algebra resource and how he’s helping hard-to-reach students.
(Part 1 of 2)

How did you write the scripts for your videos? Did you try out your material on live students before recording? I’m imagining it like a sitcom, testing out your material with a live audience first, or did you just go for it?

The videos themselves are unscripted. Doing it this way seems to be more realistic to an actual classroom setting. (Before we go on, I want to make sure that you are aware that I myself take on the role of Charlie.) The first video I tried making was filmed at my home in a small room and I just went for it!

Prior to shooting this first video in June 2006, surveys and interviews were conducted with students regarding the lecture classroom environment. I found that some students did want to be called on during the classroom session as many became very nervous when I did this, and in some cases experienced tremendous fear of not knowing the answer. In addition, while studying math videos it was noted that in most cases everyone was talking directly to the viewer, similar to what instructors do in the classroom. The question then arose: do students experience a certain level of anxiety when they view a math video where the lectures speaks directly to the viewer? With this in mind the student character Charlie was created to allow me to re-direct the projection of information away from the viewer. Next, it was realized that Charlie could be used to break up the rhythm of the video by injecting elements of humor into the presentations.

There where mixed comments made regarding my early videos. Some students found them to be too long saying that it took me 5 minutes to do one problem. But some students just loved them and found the video to be very entertaining while simultaneously teaching them something. Future videos where refined using this student feedback. The videos that are produced today are still critiqued by a number of students with again mixed results. But what I have found today is that the students with the highest level of fear are the ones who enjoy the videos the most. These are the students that the Algebra2go project targets as these are the students that appear to have the lowest success rates in developmental math. The strong algebra students generally do not find the videos useful and sometimes see the student character Charlie as an annoyance.

My first student surveys back in 2002 revealed that many struggling students had difficulty trying to capture lecture notes while focusing on my lectures. It was then that I began to purchase lecture notes from my top students to see how they actually took lecture notes. I found that some students added their own personal annotations that I found to be useful in my effort in improving my own lectures. The following semester I was used their notes to model my lectures. Finally after a few years of this, I hired a former student to capture my entire semester of lecture notes for both pre and beginning algebra and decided to make them available to students. It was the students who suggested that I put them on the internet so they could have convenient access. This is how I ended up extending my teaching philosophy to the online environment.

Initially, I had no intention of using the internet to supplement my course. Student need is what drove me to the online environment. The videos themselves came after the fact where students suggested that I tape my video lectures so they could watch them at home. The pedagogical content within the videos mirrors the lecture notes, hence, mirrors my in class lectures. I never use the videos in my classroom, their intended use is to supplement my classroom instruction.

Kristine Mackowski is a special education teacher in New York.
Check out her videos at Teacher Tube!

Are there any tips you’ve accumulated from your experience with teaching algebra that you’d like to share?

I find that the use of manipulatives work well for my population of students. For example, when teaching one and two-step algebra equations, I use “Hands-on Algebra.” It provides my students a chance to see how to “balance” out the two sides of an equation by using number cubes and pawns on a paper scale. We then transfer to pencil and paper and they can still visualize the steps and the importance of keeping the scale balanced.

I also think it is important to teach the vocabulary with each unit. I like to try to give my students a picture to go along with each word. We take a quiz on the vocabulary and I create a word wall in my room to refer to. It provides them a better understanding in the concepts of algebra.

What do you do when you make a small arithmetic error while teaching? Do you humbly acknowledge it and move on?

I definitely acknowledge it! It makes me, being a teacher of special education students, happy that they were able to catch my mistake. If they didn’t, I will still stop and use it as a teachable moment. I let them know that they may make mistakes like this too and discuss why it was made. My students know that making errors is part of math and that is why I work so hard at emphasizing the need to check their own work.

Have you noticed any characteristics of students who seem to understand algebra faster? What about slower?

Students who understand algebra faster are the students who have a full mastery of rational numbers. These students can concentrate on learning the algebra instead of learning or relearning the concepts along with the algebra.

Also, students who can reason abstractly.

Conversely the slower students have trouble reasoning abstractly and don’t have a mastery of rational numbers.

Another major difference between the students is a self-confidence in their ability. Students who have the attitude that they can learn the material, can and do.

What concepts are covered under a “full mastery of rational numbers”?

I would say Fractions, decimals and percents along with adding, subtracting, multiplying and dividing positive and negative rational numbers.

- William Gripentrog, South Dakota

Here’s ‘Heathen Mom’ from http://www.heathenhomeschoolers.com

After reading your blog some more, am I right in assuming you haven’t gotten to algebra yet in your teaching experience? (your kids are pretty young)

I was a classroom teacher in the past. I have taught prealgebra. I have also taught stats.

I am interested in stressing to parents that algebra doesn’t have to be something to fear. Algebra concepts are taught in elem level math.

Oh that’s great! Are you framing your elementary math lessons with a longview towards algebra? If so… how?

How do you handle a student who gets a little too excited about the subject and becomes disruptive? Maybe a little disruption is fine?

A little disruption is fine.  The trick is to not let the student take over the class.  It’s important to get some other students involved.  Ask other students for their input, or get them into small groups with an assignment to discuss an aspect of the subject, and then get back into a larger group to revisit, and then the class is more focused. 

Say, the subject is media literacy.  I would teach a class on sports, and ask them to watch a sports game at home and report how many advertisements there are on the field, and in the stands, and on hats and such.  You know, not commercials, product placement.  They come back, and most kids have done it, and are sort of interested, but a few really paid attention and take over.  I would break them into small groups, have them make a list of the advertisers according to product, and bring them back together.  So each small group has to report, and it is less likely for one or two students to take over.

There are other ways, it depends on the subject.

-Lynn from Alabama

Here’s Erika from Florida discussing practice and overcoming a topic’s bad reputation.

Do you give practice problems every night?

I do give practice problems almost every night.  I think it’s one of the most important things to do, because math, especially algebra, isn’t something you can learn by just looking at it or memorizing the steps.  To truly understand how to solve any given problem, you have to practice it.

Have you stumbled upon any tips for teaching algebra from your experience in the classroom?

Keep a track of how long it took you to teach each subject, and what specifically the students had trouble with.  It really helps for when you’re trying to pace out the course if you’re teaching it again in the future.

Which topic in algebra is the hardest to teach? How do you deal with that?

If you’re “lucky” enough to be teaching trigonometry, it’s definitely the hardest to tackle.  Not because it’s a difficult topic, but because the students most likely already have a negative view about it, and it’s hard to get over that first barrier.  Taking it really slow and relating it to previous topics, especially ones they liked, is the best way I know to push through that barrier and to get the students interested in trig.

To start things off here’s Anonymous from California.

Do you try to get into a certain mindset before starting an algebra class?

I try to bring an interesting fact from science or a joke or a personal story
to start the class with.  That helps to draw the students away from thier
conversations and to focus them on me and the class as a whole.  The students
are ninth graders and the class is a requirement for them so adding a little
fun and interest to the class helps.

I will have reviewed their homework and will have a list of the kinds of
mistakes they are making.  Often the mistakes are because they did not learn
prior material correctly or sometimes they get new material confused with old
material.  If I am introducing new material then I try to predict and cut-off
in advance misconceptions and common mistakes.

The students are accustomed to starting work on a warm-up problem at the
beginning of class.  If the mistakes or issues can be dealt with from the
warm-up problem I will use it.  Otherwise after going over the warm-up problem
I will try to explain the mistakes and misconceptions that I had been
witnessing in the student work.

Being a student teacher, I am being observed by a senior teacher.  She can be
pretty critical of my techniques and performance, so my main worries are with
how she will view my performance.  She does not accept as much conversation in
the class as I might so I have to make an extra point to keep the students on
task, quiet and focused.  Also she does not like more than two or three
minutes of class time spent on jokes and off topic subjects.  I have to edit
my connection time to this short amount.

Do you have an example of a common mistake or misconception you try to cut-off in advance?

Quite a few of the students did not realize that when a group of variables
and numbers are being raised to a power the parentheses in the exponential
expression show the student which factors the exponent will be applied to.
Also if there is no parenthese that means that the exponent only applies to
the thing that it is superfixed above.

 Some students do not think about what makes an equation an equation and not
an expression.  After learning how to use completing the square to solve
quadratic equations some of them try to use completing the square to factor
rational expressions with quadratics.  They should be using a diamond and box
routine to factor the quadratics because without the equal sign there is no
advantage adding numbers to be able to complete a square on a binomial.

So before the unit on completing the square I ask some students for the
difference between equations and expressions.