George Woodbury’s Blogarithm

Final Exam Review Videos

I have two sets of final exam review videos on YouTube.

The first playlist, for Elementary Algebra, can be found here: http://www.youtube.com/playlist?list=PLCE703DB5743508D7

The second playlist, for Intermediate Algebra, can be found here: http://www.youtube.com/playlist?list=PL15AA6E8E21593D3D

Please feel free to share with students and instructors who can use these.

George

December 5, 2011 at 5:40 pm 2 comments

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.

November 21, 2011 at 2:55 pm 1 comment

Should We Be Teaching Polynomial Factoring In Elementary Algebra?

I was at a conference last week, and factoring polynomials was under fire from several instructors. Is this still a valuable skill? Is it important to cover factoring? I’d love to hear your thoughts.

(I’m putting together a blog with my thoughts on the issue.)

-George

November 17, 2011 at 12:00 pm 4 comments

2011 AMATYC Wrap Up

This year’s annual AMATYC conference in Austin just wrapped up and, as usual, was very inspiring. One topic that was presented several times was the idea of incorporating simulations to teach hypothesis testing. I have used this recently in my own statistics class, and I can report that it helps students to conceptually understand hypothesis testing in general and p-values as well.

I will be trying to recruit several AMATYC presenters to share their work by writing a guest blog. Stay tuned!

-George

November 14, 2011 at 4:35 pm Leave a comment

Doing Homework

Sorry for the lack of blogging – this has been a busy semester. One project I have been working on is incorporating elements of game design into the grading policy for my intermediate algebra class. Students earn points on each exam based upon their exam score and their homework/quiz status.

Students who have scores of at least 90% on each homework assignment and at least 70% on each quiz are said to have satisfactory scores. (I probably need a better name for that than ”satisfactory”.) Students in this category have done pretty well when it comes to passing the exams.

Test 1: 33 out of 38 (87%)
Test 2: 24 out of 28 (86%)
Test 3: 19 out of 25 (76%)

Students who do not have satisfactory scores have not performed as well on the exams.

Test 1: 7 out of 9 (78%)
Test 2: 13 out of 19 (68%)
Test 3: 11 out of 21 (52%)

So, students doing the homework are doing better on exams. Now I understand that I cannot go on to claim that doing the homework leads to better performance, but there is an association here. One thing I am concerned about is that the number of students earning satisfactory scores is dropping as the semester progresses. However, the fourth exam is today and it appears that there will be 32 students with satisfactory scores and 14 without – a good sign.

In upcoming blogs I will try to explain the game design dynamics, and include student thoughts on the policies of the class.

- George

October 26, 2011 at 9:43 am Leave a comment

What I Learned About Teaching Math – At Starbucks

Last night I dropped my daughter off at her dance class, and as is sometimes the case I went to Starbucks to work on the new Statistics project I am involved with. I sat down next to a 9th or 10th grade kid and his college aged tutor. They were working on Algebra I.

Over the 5 minutes that they stayed, I found myself getting more and more frustrated. Here are words that came directly from the tutor’s mouth, followed by my thoughts.

“Just do it like the one I did a minute ago.”
Having students mimic your work will definitely not lead to understanding. I need to keep this in mind every day during class. “I’m the expert. Just do what I do.” is not an effective way to get students to understand. Students can watch me work one hundred problems in a row and not be able to work problem 101 without me helping them to find some level of understanding first. I’ve watched Tim Wakefield throw thousands of knuckle balls, but that doesn’t mean that I can do that too.
What’s wrong with you? Don’t you get it?
I’m pretty sure that making the student feel inferior doesn’t help at all. If you want to help a student to understand mathematics, you need that student to be motivated and inspired. You cannot motivate and inspire a student by making the student feel bad. I think the tutor might want to pick up a copy of Daniel Pink’s Drive.
You know those 20 problems you did on that worksheet earlier? Just do it like those.
This just screams to how ineffective “drill & kill” can be when there is no understanding tied to it. Mindlessly going through those 20 problems did not increase the student’s level one bit. I need to remember that fewer, well-chosen problems can definitely make for a better assignment that 1-157 odd. When students view homework assignments as a tedious, meaningless task that must be accomplished, then I am afraid we have lost that student.
Any number times 1 is … one. Followed shortly thereafter by Any number divided by 1 is … one.
Yes, both statements were wrong. I know the tutor probably meant to finish both sentences with “the number itself”, but he didn’t. We have to be careful with what we say, because a confused student will just get more confused when we misspeak.

The night didn’t end as badly as it started. Two older women came in and sat next to me – a 40-ish woman with her 60-ish French tutor The tutor was warm and positive. When her student made a mistake, she was supportive and tried to help her student understand why what she said was incorrect. I was fascinated as I listened to them. That’s the type of teacher we all need to be. One more point – the student was trying to learn French so she could travel to Paris. When students understand why the material they are trying to learn is important, they will be more motivated to learn.

Have any lessons that math teachers should learn? I’d love to hear them.

- George

October 6, 2011 at 11:06 am 5 comments

Setting Up Binomial and Poisson Probability Problems

I have uploaded 2 videos to YouTube that go over how to set up binomial and Poisson probability problems. (There are 8 binomial problems, and 6 Poisson problems.) I go over the steps for identifying the problems, as well as give the correct answers. If you’d like a copy of the actual problems, just drop me a line.

Binomial: http://www.youtube.com/watch?v=jLAePWjEZYE
Poisson: http://www.youtube.com/watch?v=GupBzWFL-KY

- George

October 2, 2011 at 8:06 pm Leave a comment

StatCrunch – Binomial Probability Calculator

I just finished up a short unit on binomial probabilities with my Intro Stat class, using StatCrunch as the primary method of calculating probabilities. To access the calculator in StatCrunch, click the Stat button, and select Binomial from the Calculator option.

Here is the interface.

Enter the number of trials in the box labeled n and the probability of success on one trial in the box labeled p. When it comes to the number of successes, you have several options: <= (for $\le$ ), => (for $\ge$ ), <, >, or =. Enter the appropriate value for x once you have selected the option you need and press Compute.

By the way, if you need to find something like $P(3 \le x \le 7)$ , you will have to do it in 2 steps. (StatCrunch does not have a “between 3 & 7″ option.) First, find $P(x \le 7)$ , and then subtract the result you get from $P(x<3)$ .

One of the features I like is the graphical display of the probabilities. The values of x that you are working with are displayed with red bars.

By making the actual calculations easier, I find that we can spend more time on challenging problems. My students also have a better understanding of the big picture, instead of getting lost in the weeds with their calculators and 20 pages of binomial probability tables.

- George

September 29, 2011 at 9:37 am Leave a comment

Solve This #1 – Wrap Up

This was pretty interesting. I posted a blog asking for solutions to a problem, and got a lot of feedback.

Colleen Young (@ColleenYoung on Twitter) in the comments & Colin Graham (@ColinTGraham on Twitter) on Twitter used a graphical approach. They graphed $y=|x^2-9|$ and noted where the function was between 2 & 9.

David Radcliffe (@daveinstpaul on Twitter) and Liz Durkin (@LizDK on Twitter) both used an algebraic approach that was right on the money.

Shawn Urban (@stefras on Twitter) used a unique algebraic approach as shown in the blog post Algebra 2: Solving Absolute Value Equations by Kate Nowak (@k8nowak on Twitter). I hadn’t really seen that before, and liked it a great deal. I may work it in next semester when solving absolute value equations/inequalities.

My Solution

What jumped out to me was a graphical approach as well. I thought about graphing the parabola $y=x^2-9$ , and then determining where the function was between 2 & 9 and between -9 & -2.

All that remains at that point is finding the endpoints of the interval algebraically.

$x^2-9=9$
$x^2-9=2$
$x^2-9=-2$
$x^2-9=-9$

Of course, I imagine many students might initially miss the problem at x = 0.

Second Solution

Another approach that came to me was to initially solve the inequality $|x^2-9|<9$ , and then taking out the solutions to $|x^2-9|<2$ . These two inequalities are quite easy to solve algebraically as they are intersections as opposed to unions.

In other words, I’d start with $(-\sqrt {18},0) U (0, \sqrt{18})$ and then delete the intervals $(-\sqrt{11},-\sqrt{7})$ and $(\sqrt{7},\sqrt{11})$ .

Summary

I think this can be a fun feature of the blog. I will try to work it in on a regular basis. If you have any ideas for problems to include, let me know. I also welcome comments on my two approaches to this problem.

- George

September 23, 2011 at 8:53 am Leave a comment

What Does Pythagoras Have Against the Sox?

It’s been a tough month in Red Sox Nation. Thanks to a 5-14 record this month, the Sox have dropped from a sure playoff appearance to clinging to a 1 game lead in the loss column. They have scores 113 runs in those 19 games and given up 123. According to the Pythagorean record calculation:

$\frac {(Runs Scored)^2} {(Runs Scored)^2+(Runs Allowed)^2}$

the Red Sox should have won 45.8% of those games, or 8.7 out of 19.

$\frac {(113)^2} {(113)^2+(123)^2}=0.458$

$19(0.458)=8.7$

I know that if the Sox had won 4 more of these games I’d be much more relaxed about our chances. C’mon Pythagoras – you owe us!

Why are the Sox underperforming their Pythagorean record? Check out the standard deviation in Runs Scored & Runs Allowed. For Runs Scored it’s 5.7 runs, while for Runs Allowed it’s only 3.1 runs. A little more consistency in offensive production is what we need.

- George

September 20, 2011 at 2:00 pm Leave a comment

Older Posts

George Woodbury’s Blogarithm

Final Exam Review Videos

In Defense of Factoring

Should We Be Teaching Polynomial Factoring In Elementary Algebra?

2011 AMATYC Wrap Up

Doing Homework

What I Learned About Teaching Math – At Starbucks

Setting Up Binomial and Poisson Probability Problems

StatCrunch – Binomial Probability Calculator

Solve This #1 – Wrap Up

What Does Pythagoras Have Against the Sox?

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