## Posts filed under ‘Math’

### Online Teaching – As Satisfying As Face To Face?

I recently overheard an instructor say that he does not want to teach online classes because he would miss the interaction with his students. My first thought was that if you are not interacting with your students in an online class then you are doing something wrong. The level of interaction betyween myself and my students has really skyrocketed over the last two years, to the point where students comment on how effective I am as an instructor despite the fact that we do not see each other on a daily basis.
I have started to use Facebook group pages as a way to increase interaction. It’s a great place to post supplementary information, photos, and videos. My students use it to ask each other questions and to provide support. It’s also nice to have a backup site to communicate through in case the course server crashes.
I chose Facebook because an overwhelming majority of my students use Facebook. I have found that students are much more likely to check their Facebook account than they are to check their email.
My class has an in-person orientation that I use to help build a connection with my students, and I also host a face-to-face review session prior to the midterm and final exams. The level of interaction in those sessions is very high, often much more interactive than a traditional classroom.
So, if you are unsure about teaching online because of the lack of contact, please consider it. The level of contact and interaction depends completely upon you. The contact and interaction are there, just in a different format.

– George

### Mastery Learning – Grading System

As I mentioned in the previous blog, I have started using a mastery learning based grading system in my developmental math courses.

### Tests

I give 6 tests in each course.
These tests are worth 0 points if the student scores below 70% or 1 point if the student scores 70% or higher.
If a student has satisfactory online scores (90% or higher on each exam and 70% or higher on each quiz) then they can earn bonuses on their test score. In that case, a student earns 2 points for a test between 70% & 79% and 3 points for a test that is 80% or higher.
A student who earns 0 points on a test but has satisfactory online scores earns the option to retake the test the following week for 1 point.

Take note that the online scores do not directly count to a student’s overall grade in any fashion.

There are 6 tests, so a student can earn up to 18 points from their tests.

### Bonus

For one exam (the one that students historically struggle with) I double the points for students with satisfactory online scores. That means that there are an additional 3 points up for grabs here.

### Review

I have a 34 question review quiz associated with a 136 question personalized homework assignment that serves as a review for the final exam. Students who score above 90% on the homework and 70% on the quiz earn 4 points to their total.

### Final Exam

The final exam is worth 100 points.

There are 125 points available in my course. Students need 100 points for an A, 88 points for a B, and 76 points for a C. I set up the C grade by reasoning that a student who did not do the online homework but managed to pass all 6 exams and the final exam should pass the class: 6 x 1 + 70 = 76. I then escalated the grades from there.

### Last Semester

Last semester I started with 47 students, and managed to keep 46 of them. 28 of the 46 passed (61%), with 15 of those grades being A’s. The pass rate at my college is in the mid 40’s for reference.

My students took a common final with four other classes, and the grading was shared between instructors. (I graded problems 1-7, another instructor graded 8-13, …). Students in my class had a mean score that was 12.5 points above the other classes, and the median was 14 points higher. 65% of my students scores 60 or higher (control: 37%), and 35% of my students scored 80 or higher (control: 7%). This shows that my students understood the material at a higher level than those not participating in the mastery learning approach.

In the next blog I will talk about some of the game design elements that really help make this work.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

### Mastery Learning – Doing Homework For The Right Reason

Homework has always been an issue of contention between math instructors and students. Instructors believe that students must do homework in order to learn the material, and I believe that statement is essentially true. Except I would change it to “If students want to learn, doing their homework with the appropriate approach can help them to learn and understand.” Many students view homework simply as a task that they must complete, they do not understand that it is supposed to help them learn.

We live in an age where it is easy to incorporate online, self-grading homework. There are all sorts of learning aids available to our students within these online homework programs, and when you also include online resources, videos, and tutors, we often assume that our students will learn the material by simply working through the homework. But if homework scores directly impact a student’s overall grade, students are encouraged to grind through the homework for the points rather than for learning. If 20% of your overall grade comes from online homework, students will try to earn as many points as possible in an effort to lower the scores required on the exams that make up the other 80% of their grade. (By the way, this is just as true if you use traditional pencil/paper homework.)

Some instructors decide not to even include a homework grade in their grading policy for this reason, but this will not motivate students to do their homework. Students do not do optional, even if we as instructors feel that it is necessary. Sure, we’d like our students to realize that it is necessary to do mathematics in order to learn mathematics, but few students will perform tasks that they do not understand the purpose of.

Last semester I began to work with a new grading system that incorporates mastery learning. Online homework and quizzes do not directly count towards my students’ overall grades. Instead, they do help them to earn bonuses and perks based on their test scores. I grade each exam out of 100, and students that score 70 or higher get 1 point for that test and students that score below 70 get 0 points for that test. 1 or 0, pass of fail. The perks?

• Students who meet performance benchmarks on the online work and score above 80 on the exam earn 3 points on their exam instead of 1.
• Students who meet performance benchmarks on the online work and score between 70 and 79 on the exam earn 2 points on their exam instead of 1.
• Students who meet performance benchmarks on the online work and score below 70 on the exam earn the opportunity to retake the test the following week for 1 point.

So, homework only impacts my students’ grades when they are passing exams. Now the goal of the homework is to learn the material, because if they don’t learn the material they cannot do well on the exams. And the exams generate all of the points in my class.

In the next blog I will go through the point system for the entire course and share some data.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

### ICTCM & What’s On The Way

I just got back from the 24th ICTCM, and, as usual, I am full of great ideas that I picked up. There were two trends that I saw – Pencasts and Simulations.

There were several talks on using smart pens. I recently received one as a gift, and I think that it is becoming a very important tool for me as an educator. They can be used to communicate to students who email questions, post supplementary materials online, or even as a note taking tool. I will share my approach in future blogs.

There were also several talks that incorporated the use of simulations in introductory statistics. This is a BIG idea that will one day revolutionize the way we teach inferential statistics. Resampling and bootstrapping are effective ways to show our visual students what is really going on. I usually use StatCrunch for this purpose.

Matt Davis (Chabot College) did a great job, and he has some fantastic simulations. You can find him through a quick Google search for “Matt Davis Chabot”, and he seems pretty willing to share. (Tell him I sent you.)

I gave a talk on my new mastery based learning approach using MyMathLab, which incorporates elements of game design. I think that my students have changed their focus from doing homework to earn points to doing homework to learn and understand mathematics. I will be putting together a blog series on this new approach that will share last semester’s results, explain exactly how I set my class up, and share the elements of game design that I have incorporated, as well as some commentary from my son.

One other blog series that I will put together is one on my MyMathLab top 10-ish features. Now that most classes have gone to the new design of MyMathLab, it’s time to go over these features.

I’m looking forward to getting this blog rolling again. I hope you have all been well.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

### Start of the Semester

This semester I have been assigning reading homework. Students have to answer vocabulary questions and outline procedures that will be covered in the next class period. Essentially, I am having them read ahead. I also include a handful of warm-up problems that focus on the prerequisites for that section. (I get mine from a publisher workbook that goes with the textbook. On MyMathLab it is available under Instructor Resources. I am sure other textbooks/publishers provide similar material.)

I give my students the first 5 minutes of class to discuss their answers with their groups. I then take attendance, one group at a time, and ask for one of the answers. It is working out to be a great way to get students to warm up for class, and my classes seem to be more participatory this semester.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

### Geogebra – Graphing Absolute Value Functions

I added two new pages for graphing absolute value functions, complete with sliders for h & k.

You can find them here: http://georgewoodbury.com/geogebra.html

– George

### In Defense of Factoring

I must say that I was rather surprised when an instructor at AMATYC suggested that we no longer teach factoring of polynomials in elementary algebra. I mean, to me factoring is about as essential a skill as breathing. But I thought about it, and although I’m not in agreement, I believe that it is important that we critically examine the topics we teach and make sure that they are still necessary.

The number one reason instructors give me for not teaching factoring is that we can use the quadratic formula to solve ANY quadratic equation. While that is true, that’s not enough for me. Solving quadratic equations by using the quadratic formula is often not the most efficient technique. Many students make arithmetic or calculator errors when finding the solutions. If a student can factor the expression, the solutions flow without trouble. Would you want to solve the equation $(x+7)(x-5)=0$ by multiplying the two binomials and then using the quadratic formula? How about the equation $(x-8)^2=-20$? Square $x-8$, add 20, then use the quadratic formula? Hardly the most efficient way to do it.

It’s true that students taking an introductory statistics course will not have to factor a polynomial, but that is not a reason to remove factoring from the curriculum. Students who struggle in introductory statistics do not struggle because of their mathematical deficiencies, they struggle because they have not developed their critical thinking abilities. They struggle to determine which hypothesis test is appropriate because they do not know how to take a look at the facts presented and determine which tool can efficiently be used in this situation. Learning to factor polynomials, and knowing when to use factoring, help to develop critical thinking skills.

Could there be some compromise? Absolutely. Factoring trinomials whose leading coefficient is not equal to 1 is often tedious and time-consuming. When we reach intermediate algebra, I tell my students to use a 10-second rule when it comes to factoring (and jumping to the quadratic formula). I still think that this type of factoring helps to develop intuition, but I could understand leaving it out. Perhaps we could leave out sum/difference of cubes? My students seem to like them, but they wouldn’t shed a tear. I teach them because being able to identify polynomial “types” is beneficial to students.

One argument that does not carry as much weight with me is “We need to factor in order to work with rational expressions and equations.” I’m not so sure that rational expressions could be pushed off until college algebra. It is a great way to practice factoring (as many as 4 polynomials per problem). It is a great topic for developing critical thinking – do I need a common denominator, am I trying to get rid of the denominators, …

One comment that really hit me was one made by “footmassage”: “It (factoring) should be in the air throughout whether your solving, writing lines in point-slope form, rewriting fractions, simplifying rational functions, manipulating transcendental functions, etc.  It is an essential basic tool that should be developed over a course(s) for fluency.” (Check it out here.)

Another comment, from the AMATYC session, that summed it up for me was “If we take out all of the topics that we have mentioned here today, I’m afraid we will be left with students who will be unable to think at all.” (Paraphrased, speaker unknown) That’s what we need to keep in mind as we transition into mathematics in the 21st century. We are not just teaching mathematics, we are teaching students to reason and think. I fear the day that mathematics is taught to students in the same way that students are taught to use word processing software.

I do promise to keep an open mind, and vow to always examine whether topics should still be taught.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.