Wednesday, November 28, 2007

Carnival of Education

As we approach the end of the year will we be inundated with end of the year awards, and of course not so recently we had the Nobel Prizes. Named after Alfred Nobel the inventor of TNT (Boom!). Well here at the 147th Carnival of Education we are going to have our own prizes. The Noble Prizes, these prizes will celebrate that noble avocation to which we have all taken the time to consider: Education. Education is defined by Miriam-Webster to be:

... 2 a: to develop mentally, morally, or aesthetically especially by instruction b: to provide with information : inform ...

So to all of you that have made Education your passion even for only this week, Thank You, here is a prize.

The first Noble Prize goes to one of my favorite teachers growing up, Brinda Price. She was my ninth grade English teacher. As I remember her she was about 4 feet 10 inches, and she was tough as nails. You had better get your Romeo and Juliet packet in on time or else she wasn't going to take it. On the other hand, she loved us. She loved us with a mother's love. She wanted us to grow, and live, and enjoy life. She had a passion for her subject that was only exceeded by her passion for her students. She is one of my heroes, in the sense that I want to make my students feel the way I did in her class. I want my students to see me as a force of nature that cares about them and about math and mostly about their education. I desperately, forcefully, passionately want them to develop mentally, morally, or aesthetically especially by my instruction.

So the first Noble Prize goes to Brinda Price of Columbus Alternative High School circa 1984. Thank you, and I am sure that the lack of prize money will not surprise you given your choice of profession.

The Noble Prizes are given to recognize the sacrifice that we all make to our charges, you are all working to make the future better, to make each child's lives better.

The Noble Prizes for Valiant Effort go to Henry Cate for Public schools - a Gordian Knot or a Sisyphean activity?, Mike Cruz for Losing Difficult Students - Blessing or Loss?, and Mrs. Bluebird for Playing Principal.

The Noble Prize for Growing and Becoming More Zen so that Your Students Can Have a Good Day goes to Siobhan Curious at small tasks.

The Noble Prize for Improving the Profession goes to Bill Ferriter for The PLC Mandate. . ..

The Noble Prize for Literature goes to Rebecca Wallace-Segall for Schools: Celebrate Teen Writers and Lessons from the Newest Generation of Writers (& Thinkers).

The Noble Prizes for Teaching Resource Coordination go to ms. teacher for Sharing!, Joel for 50 Classroom Management Tips I Have Learned This Month, and Ryan for Bridging the Research-Practice Gap.

The Noble Prize for Opening a Window into a Soul goes to IB a Math Teacher for The Book of Me.

The Noble Prize for Unschooling goes to Laureen for Bucket-Free.

The Noble Prizes for Elementary Education go to What It's Like on the Inside for The Sad State of Elementary Science, and Ryan for How Kindergarten Has Changed.

The Noble Prize for Anti-Telepathy goes to Mr. Pullen for Hey, Kids: Guess What I'm Thinking!.

The Noble Prizes for Public Policy go to Judy Aron for Tax Credits For Homeschoolers - Bad Idea!, EdWonk for EduDecision 2008: Obama's $18 Billion EduFix?, Joanne Jacobs for Defining dangerous down, and Dave Saba for Math: there is no substitute | American Board for Certification of Teacher Excellence.

The Noble Prizes for Math go to Denise for Fraction models, and a card game, Tony Lucchese for A Letter to a Young Mathematician, and Matt (that's me!) for Lessons on Lessons While Cooking Mashed Potatoes.

The Noble Prize for Art goes to Scott Walker for Some sketches during a staff development session.

The Noble Prize for Thoughtful Homeschooling (is there any other kind?) goes to Dana at A workable solution for American education.

The Noble Prize for Film Advertising goes to Matthew K. Tabor for his highlighting of a Screening of 2 Million Minutes.

While the Noble Prizes for Film go to Larry Ferlazzo at Math Movies and More, and Adam for The Academic Schools.

The Noble Prizes for Humor go to mister teacher for Helpful or Harmful, Smellington G. Worthington III for Welcome, and Carol Richtsmeier for Lists, Parents & Paperwork.

The Noble Prize for Zoology goes to Ms. Cornelius for Wanted: One case of mouse-sized Depends Undergarments.

The Noble Prizes for NYC Education go to Norm Scott for UFT Candlelight Vigil Snuffed, Woodlass for Sacrificing the learning years — Why?, and NYC Educator for his accounting of 7.2 million dollars in What A Bargain!.

The Noble Prize for Statistics goes to Edwonkette at Lies, Damned Lies, and NAEP Exemptions.

The Noble Prize for Networking goes to Pat for Networking is Important for All Teachers.

The Noble Prize for Civics goes to Matt Johnston for Not Every Education Problem Begins and Ends at NCLB.

Then Noble Prize for Homework goes to michele lestage for Too Homework Much Help Can Result In Failure.

The Noble Prize for Identifying Hypocrisy goes to Right on the Left Coast for Teachers in My District Say Teachers Don't Care About Students.

The Noble Prize for Foreign Language goes to Maria Fernandez for Free Spanish online lessons on mp3.

The Noble Prize for British Education goes to oldandrew for The Two Discipline Systems.

The 148th Carnival of Education will be at So You Want to Teach. Entries are due at 5pm Central on December 4th.

Reading the submissions was a great honor. I learned a lot, and that is what is all about. Thanks to all our contributors for their thoughtfulness and their giving hearts.

Thursday, November 22, 2007

Lessons on Lessons while Cooking Mashed Potatoes

I hope you had a wonderful Thanksgiving. We had a wonderful time here. We didn't go anywhere we stayed home, my Mom came, and we cooked here. As often happens, while discussing making mashed potatoes with my wife, I had a little epiphany. Maybe it isn't an earth shattering discovery, but I love a metaphor and I think this is a good one, so bear with me and I think you find a story that all teachers can use with their students.

As I said we made Thanksgiving dinner here this year. My wife picked the recipes, and it just happened that we picked all of our recipes from a cookbook we have from America's Test Kitchen. One of the things that we love about this cookbook is that it tells you their theories about "why" they do things. My wife were talking particularly about the mash potato recipe. I am sure that many of you know this, but it is important to add the butter BEFORE the milk. This has to do with "coating starches, etc., etc.", and my wife noted that it was not something she knew about. I told her that I had shared that same information, about the butter before the milk, with my wife's mother. My mother-in-law seemed to think that this was not interesting news of any sort, but something everyone knew. Of course, her own daughter didn't know it.

What does this have to do with teaching math. We often show students what to do. And often we are surprised by the ways they fail to do what we show them. But in this example my wife watched her mother make mashed potatoes many times, but because she didn't know why the butter went in before the milk she didn't know that there was any importance to the order. Since then, my wife has been mixing mashed potatoes, milk, and butter all at the same time. The Horror.

Well the mashed potatoes were great, and I believe that my classes will be improved because I have been reminded one more time that the model of teaching math where you ask students to just do what you do, and not show why is forever deeply flawed. Why is it flawed? Because it leads to lumpy mashed potatoes.

More secrets of mashed potatoes here.

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Tuesday, November 20, 2007

Carnival coming!!!

The Carnival of Education will be hosted here next week. Can't wait for the submissions. Check back here on November 28th to be shocked, entertained and amazed by how much can go on in world's finest, absurdist institution: the school. Entries are due Nov 27 at 11:59 pm EST. Send entries to mbardoe (at) att (dot) net.

Project Redux

So my fun projects/presentations are almost all in and on the whole they are: alright. I thought about classifying them as "not bad", but that sounded too good. I thought about classifying them as shockingly medicore, but I realize now I should not be shocked. Though I tried, I did not do a good job of giving feedback and guidance along the way. So in the end I think that each student group in their own earnest way did really work hard to learn about the things that I had asked them to learn, but what they didn't do was learn it well enough to explain it to others. There explanations were on the whole subpar. I found myself sitting there wondering if this is what they experience everyday from me. I think I will delude myself with some of my student evaluations from previous years to stop that thought. As usual my students failed in planning their presentations to think about one important thing: the audience. Most talked directly to me the whole time, or to the board. They expected me to say things like "right", "wrong", "yes", and, "I see". I ended up saying some of that stuff, oh well.

So how do I feel about the experiment. I think that I have to do it again. It can be better, and they can do a better job of communicating their thoughts. If they can't they desperately need the practice. What was also interesting was how the students went about using technology. Some eschewed it completely. That didn't necessarily help their presentations. They argued while they gave a team presentation, they said "I don't really understand this". They didn't rehearse. All in all there was room for improvement. But I did help one group make a mathcast of an argument to show that the square encloses the largest area of any rectangle of a certain perimeter. And after we were done the student was so excited to see what they had made. It was pretty dry mathematics, but they were excited to see what they had done with it. Sort of made the whole thing worthwhile.

Thursday, November 15, 2007

Big Idea

Check out the posting on ASCD Express of my essay about the "Big Idea" in math. I come off a little more strident than I realized at first, but I do believe in these ideas. Check it out at ASCD Express.

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Friday, November 9, 2007

Fractions Quiz Redux

As a teacher, who teaches some middle school I am fascinated by this post by Denise at Let's Play Math. The questions here are tremendously important, and I don't think most of us don't have good answers to these. Every middle school teacher must have good answers to these questions.

ASCD's journal Educational Leadership has an article this month by Lynn Arthur Steen about math, with a significant section on fractions. In this article is a section about students having to estimate the sum of two fractions on a standardized test, and basically the results show that students have no idea in general. Steen then makes a very strong case for why we need this skill, even though few people will ever actually have to add unlike denominators in their "real" lives.

It is my belief that we owe every student the opportunity to learn how to think "mathematically". This is for the most part a birth right of most students. My most challenged students are the ones that try to dissociate their "natural" math from their school math when they are in the classroom. Many students will tell you that they are terrible at mental arithmetic, when they can do complex calculations with nickels, dimes, and quarters. I encourage you to test this with any one who tells you that they can't add in their head.

In speaking with a student this week we were speaking about 9/2 and how to calculate that number to some decimal. I asked what she pictured in here mind when I said the number nine. She replied that it was a picture of the numeral, "9". I asked her to picture again, and she drew for me a 3x3 grid of circles. This is a big improvement, but not as helpful (and this probably only my own perspective) to this problem, as being able to picture two rows, one with 5 circles and one with 4 circles. It is this lexicon of images, pictures, etc. That allows me to speak and think with fluency in mathematics. This is why the "Round-Food Model" that Denise at Let's Play Math talks about in her most recent post about fractions is important, and why it is important that we construct a powerful picture and story about the rest of the rules of fractions.

Monday, November 5, 2007

New courses?

Today was the deadline at my school to propose a new course for next year. I have proposed a few courses in the past and some of them have been accepted. This year I am putting my most ambitious proposal out there. I am proposing a course in mathematical modeling. This course would hopefully be able to handle students who have had Calculus AB or BC, or possibly even for students who are interested in waiting a year to take calculus. I think that there might be a lot to gain for the students who have not had calculus yet. I would help them have sense of the "why" of calculus before they learned the "how".

One of the biggest hurdles I see right now is how to get around the current course that we offer after Calculus BC, which is Multivariable Calculus. The major reason I can see for offering this course would be to reinforce the calculus skills that students have already learned. It would seem that the students would have retake this course in college if they are going to continue in mathematics. Taking a course over is not a great idea in my opinion. It definitely has to be course with enough meat to it, to be presented in a significantly different way. I am not sure that Multivariable Calculus meets that.

So we will see how my math department colleagues feel about this. I am not sure what they would think.

Also, I am looking at trying to build part of a one week course on global warming for middle schoolers. If anyone has good resources, please let me know.

Sunday, November 4, 2007

Classroom Projects

So late one night last year, I had a strong desire to change the way I teach. In many ways I see myself as very traditional. Some people tell me not so much, but I think at least philosophically I am very much in the land of I have knowledge; their minds are empty; must put my knowledge in their heads. Despite this I definitely see myself as a constructivist. A bad word to many I am sure (I know the spell check doesn't like it).

So about a year later, I am doing something with my late night ponderings. In my 8th grade Algebra 2 class, we are doing a unit on quadratics with an introduction to complex numbers thrown in for good measure. I have done a few teaching to the whole class days, but mostly we have days for the kids to work on a variety of projects. Some examples you ask? Why sure

* Hardy-Weinberg Equations from Biology
* Understanding how complex numbers increase the range of quadratic functions
* Deriving the quadratic formula
* How do the a, b, c in ax^2+bx+c=y affect the graph of the function
* Real-life applications of parabolas

There are more, but you probably get the drift. Each student will have to make a "presentation" of some kind. Not every kid can make an oral presentation to the class, we don't have the time. I am hoping that technology will come to my rescue, and some kids will make little videos that I can assign for homework. Students will have to critique each other's work. These are teaching problems I haven't worked out before, but I am enjoying it so far...

Thursday, October 18, 2007

Linear Regression and Matrices

So I know a good amount of math, more than the average guy on the street, but I love to learn new things. This week I passed out copies to the book Supercrunchers to my two kids in math club, and to get the ball started we went over how linear regression is calculated. I had no idea, and I found the following link that does a really great job, but it does almost the whole thing with matrices. I had never seen this before, and I was basically stunned. One nice thing about it is that it gives a good application of the matrix transpose. The fact the calculation of the linear regression numbers can be written as a fairly short matrix equations is shocking to me. This of course reminds me of determinants, which continue to amaze with their power and breadth of application.

Tuesday, October 16, 2007

Chat about work with physics teacher.

I never really understood work. The physics kind too. So I asked a fellow teacher to sit down with me an explain it. I recorded it with ISight, and now I have something to share with my students, and you...

Sunday, September 23, 2007

Small Math

The idea of "small math" has been bopping around in my head for about two months now. What is "small Math"? Technically "small math" is a neurological syndrome named mathminuitis. I affects roughly 75% of all students ages 8-21. Symptoms include

* A desire to learn the minimum amount of math necessary to pass the next test, period.

* A belief that roughly 70% of the material studied in math course is superfluous, and can be more easily solved by cross-multiplying (whatever that means).

* A belief that all math is a series of fun mnemonic devices such as: FOIL, and "Don't ask why, just flip the second and multiply".

So far, there is no known course of treatment. Though many believe that by focusing on understanding what math is truly about, by including greater and greater amounts of applications within the mathematics curriculum, we can begin to stop the spread of this syndrome.

The most worriesome part to the trend for me is the number of math teachers that seem to show signs of coming down with mathminuitis. These teachers believe that the by shrinking the curriculum, by focusing on rote algorithms that have cute sayings attached we can create some worthwhile learning clearly are suffering from mathminuitis.

Friday, September 21, 2007

Determinants are calling me...

Good Story about a class, and my continuing fascination with determinants.

I go to class today, and it is Algebra 2, which I am never really sure how to approach. We are doing a chapter on lines. Now I had all this students last year for Algebra I and it is an honors course, so they are all pretty strong. I come in to class and put the following question on the board:

Determine if the points (-2, 1) (0, 3) and (7, 12) are collinear

And do you know what all the students asked (Go ahead test yourself, see if you can put yourselves in the mind of an 8th grader)

I hope you guessed "What is collinear?" Cause that would be right.

Now we are at the tony private school, so the smartboard is still on from the last class. I login in and go to and look up collinear. I let the students read the first couple lines, and now they are off and running. Get lots of good answers/approaches. Some students find the equation of line using two points and check the third point in the equation to see if it works. Some calculate the slope two different ways and see if they are equal. Some calculate the equation two different ways and see if they are equal. It is all very good.

While the students are working I can help but see the following information just a little bit lower down on the mathworld entry for collinear (and this is not a direct quote)

You can check if three points are collinear if with determinants. For points (x_1, y_1), (x_2, y_2), and (x_3, y_3) create the determinant.

|  x_1  y_1  1 |
| x_2 y_2 1 |
| x_3 y_3 1 |

if the determinant equals zero then the three points are collinear.

I had totally forgotten about this, even though I had written a post about determinants two months ago.

Now I am standing in front of my class while they work the collinearity problem wondering, should I go for it. Should I show them determinants right now? Should I use this moment to introduce this strange monster of mathematics? It then occurs to me that we can use this to find equations of lines as well, and that in the next chapter I am going to introduce Cramer's Rule. Cramer's Rule is nothing but determinants, and so I go for it. The kids really liked it, and it is a most amazing thing that the series of calculations does this.

The way to find the equation of the line is to take two points and create the determinant:

|  x    y    1 |
| x_1 y_1 1 |=0
| x_2 y_2 1 |

this works because you are looking for all the points that are collinear with the other two.

Another way to look at it is the fact that the determinant
|  x_1  y_1  1 |
| x_2 y_2 1 |*.5
| x_3 y_3 1 |

Gives the area of a triangle with those coordinates therefore the three points are collinear if and only if the three points define a triangle with zero area.

So the class went well, and they really had an appreciation for determinants. They liked the way this got them pretty close to standard form. They may even remember it. I showed them two by two determinants as well, we will see if they recognize it when it comes up in about two weeks.

So, after another foray with determinants, I left pondering. I wonder "What is a determinant?" What is it's core? It does so many things: linearity, area, invertibility, etc. I just don't know.

Saturday, September 15, 2007

Delta/Epsilon in HS

I was tutoring a AP calculus BC student today that goes to a different private high school. I saw in her notes there the definition of the limit as it regards the definition of the definite integral. It was complete with delta and epsilon, and of course mesh size and all the rigmarole. I am teaching Calc BC at my high school and we had gone over such stuff this week, and I didn't mention this topic at all.

I know why I did this. The short list is (and in no particular order):

* Not on the test so why burden the students
* Delta/Epsilon definitely not the test at all
* I have previously tried to explain Delta/Epsilon arguments to people and found that every method/analogy I have is ultimately more complicated than the actual argument. People clear agree that Delta/Epsilon is complicated, so why try to make what you can't make simpler simpler.

The only thing I have found that helps explain Delta/Epsilon proof is the following diagram:

This image is meant display that the function maps from the real numbers to the real numbers. What we are trying to do is show that a for a every little region around what anticipate will be the limit, it is possible to find a region of the domain that maps into the chosen section of the range.

This might be what Alfred S. Posamentier is talking about in this op-ed.

Thursday, September 13, 2007

Three 3's Competition

It was parent's night at the school that I teach at this week. There is generally a lot of waiting around for the teachers. So based on the brainteaser over at Text Savvy I decided to give my colleagues a little competition. Just as at Text Savvy the rules were:

1. Write a mathematical expression that evaluates to 9.
2. Use exactly three 3's and no other numerals.
3. Use no plus signs.

The idea was to be the "Most Creative", probably as easy to judge as "learning" so why not. I have to say that I was tremendously impressed with the results. With in the first half hour I had the following entries:

I really thought they were great. Some quibbling could be made with the sin(pi) and the units in the first, but they were creative.

Of course, word spread and the chemistry teacher at the school, who drove home with me on Wednesday came up with the following three in the car, which I think are wonderful too.

All in all a great success.

Thursday, August 23, 2007

The Scrap Heap of Mathematics

My nomination for the scrap heap of mathematics, the rational root theorem. It is a lovely pile of math that is collecting all of the math that is no longer relevant, no longer valued to be a part of the K-12 experience. Upon inspection of the heap we find: calculating square-roots by hand, manipulating a slide rule, and if not on the heap lying right next to it balled up like a piece of used kleenex, long division.

I remember learning the rational root theorem in my Algebra 2 class in high school. I loved it. I wanted to factor everything I could, and I needed something that would help me factor higher degree polynomials. Today, I would hopefully have a great understanding of the connections between zeros of a polynomial and the roots. I would graph the polynomial and use that to find the roots. And moreover, I don't think that the proof/reasoning behind the theorem is so enlightening that students will understand mathematics less if we never speak of it again. The reasoning, I believe, boils down to the fact of divisibility of the lead term and constant term. Students get this by factoring quadratics.

Am I missing something important here, or is this a slam dunk onto the scrap heap of mathematics?

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Sunday, August 12, 2007

Fraction Friction

I was speaking with a friend that teaches adults preparing for their GED’s. They were describing the moans and groans heard as the teacher announced they would begin to review fractions. It reminded me how many of my middle grade students hate fractions and love decimals. There is something very reassuring about decimals to students in grades 6-8, and possibly even older.

I have a theory about why fractions are so hated among this crowd. These theories have no research to back them up, so I hope that readers out there will feel free to call me to the mat on this. First lets do a little comparison between fractions and decimals.


So the ways in which decimals are better than fractions are the addition, subtraction, and comparison (by which I mean being able to compare the magnitude of numbers). I would say this is at the heart of my middle school student’s love of decimals. These are the operations which students feel most confident with. In fact, I would venture to say that students often don’t really feel the “know” a fraction. If they must give answer they like to give a decimal. Students love to give answers like 3.291098094. At this point the reason for this is almost beyond my grasp.

I will say of the three areas I think that comparison is the most important. I believe that this is because each student has a sort of internal measuring tape. They feel they know a number when they know where it goes on the measuring tape. It can be difficult for a student to place with a degree of certainty 4/13 on such a mental measuring tape.

How do we help students feel more comfortable with fractions? I propose the following steps:

➢ Emphasize fraction’s strengths. They solve division problems easily 3 divided by 23 is 3/23 for example. They can be multiplied easily. A good example is .125 * .75 is 3/32. Division too.
➢ Help student to understand the meaning of fraction in many different ways. As rates, as ratios, as the solution to division problems. As a number with a unit e.g., 3 eighths. Eighths can be thought of as a unit, like feet or inches. This is why we must have a common denominator to add.
➢ Help students find ways to estimate the size of fractions. This may help them place the answer on their mental number line. For example, 4/13 is a little smaller than 4/12 = 1/3 so 4/13 is just a little smaller than one-third.
➢ Help students understand through pictures and a variety of situations why multiplying fractions can lead to smaller numbers and division can lead to bigger ones. This should include a thorough discussion of the meanings of multiplication and division. Too often students just see these as ways to make numbers bigger and smaller.

It really is an interesting question. Why do students dislike fractions so? Fractions predate decimals by a good 600 years. I often wish I could watch a middle school math class prior to Simon Stevin to see what the kids were bitching about then.

Wednesday, August 8, 2007

Aha Moment, trapezoid and the series

Math is wonderful. Especially, when it all connects. Last spring I was teaching arithmetic series to my honors 8th grade class, and I made a connection that I had never made before. For those who don't know a arithmetic series is the sum of (generally) a lot of numbers that increase by steady amounts, such as 5+9+13+...101. The goal for our us is to find the sum with out actually doing all the dirty work of adding up all the numbers (easy for Gauss, but hard for many of us). There is a formula for this which is to add the first and the last (5 and 101 in my example), multiply by the number of numbers (25 in my case, harder to figure out than it sounds; the famous fence-post problem) and divide by 2. For those of you that look cool formulas it looks something like this:

The standard way of seeing this is with some pictures of bar graphs each bar representing one summand. The sum that we are trying to find is like adding up all the lengths represented by this bar graph

One nice way to see the formula is to take a copy of the graph and place it on top of the first but running backward from last to first as in this picture

Then each bar has the same height, the sum of the first and last, and we can multiply by the number of bars to find the total. We doubled what we are trying to find so to find the sum of the red columns I take that answer and divide by 2.

How do trapezoids fit in?

We if you look carefully, you will see the that the original picture is pretty much a trapezoid.

How do you find the area of this trapezoid? You take the length of the parallel sides, in this case 5 and 101. Multiply by the height between those sides, I guess this would be how many bars there are 25, and divide by 2. And in fact to proof that this formula works for trapezoids is exactly the same. Double the trapezoid to create a parallelogram that you already know how to find the area of.

So arithmetic series are isomorphic to trapezoids. Who knew?

Tuesday, August 7, 2007

Math as Metaphor

Tony at Pencils Down has an interesting autobiographical post about the importance on metaphor in mathematics education. Both how it is essential and how it can be a hindrance to learning. It reminds me of the opening section of How Students Learn by the National Research Council. There they use the story Fish is Fish to show how all learning is based on previous experience.

I am currently trying to write a short piece for ASCD about the "The Value of Mathematics". All this analogy, metaphor stuff fits in nicely with the point that I will be trying to make. If ASCD doesn't want it, I will post it here.

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Thursday, August 2, 2007

Good Children's Book

As part of bedtime tonight I read The Three Silly Billies by Margie Palatini. It is a wonderful retelling of the Three Billy Goats Gruff with appearances by from many of your other favorite story characters (Jack, Little Red Riding Hood, etc.) The conflict is for the heroes to get the 1 dollar needed to pay the toll on the troll bridge. Slowly each character adds a little bit for their carpool.

This could make an excellent literature connection for a K-3 class which is studying the values of various denominations of money, or 2-digit plus 1-digit numbers. Check it out.

Wednesday, August 1, 2007


Determinants are interesting part of mathematics. They are an important measure of a transformation, they form a way to compute the cross product, my mathematical training says that they are connected with character theory. But how do I help student's understand their significance. Determinants have not been around very long, getting started about 1750. If I don't know much about matrices, transformations, or characters, then what are determinants. Until we can answer that question I don't know how to teach determinants. I don't want to just teach an algorithm to compute a number. We must have reasons for it.

Here is a reason that I found while reading mathworld. It is possible to create determinant equations that immediately give equations for lines, circles, parabolas given the right number of points. This emphasizes several of the important properties of determinants: linearity, and the property that if two rows are identical then the determinant has value zero. In fact, it is this last fact that is the key point of this. Let's see some examples. We start with equation for a line. (Bear with me, I don't know how to format the math yet.)

If I want to find the equation of a line through the points (3, 2) and (5, 6) then I can set up the following determinant:

| x  y  1 |
| 3 2 1 | = 0
| 5 6 1 |

First, this is a linear equation by the linearity of the determinant, and if we substitute x=3 and y=2 into this determinant then we get a true equation because of the property that if two rows are identical then the determinant is zero, similarly with (5, 6).

When you expand the determinant you get -4x+2y+8=0. This is not quite standard form, but close enough.

What is more amazing is that this trick works for other types of equations such as parabolas. In the case of the parabola you use a determinant of the form:

| x^2 x y 1 |
| a^2 a b 1 |
| c^2 c d 1 | = 0
| e^2 e f 1 |

where (a, b), (c, d), and (e, f) are points on the parabola.

There is even a version for a circle given three points and on mathworld there is the general case for any conic given 5 points.

I like the way that this approach unifies these different processes and emphasizes the important qualities of the determinant, but I am still left with my big question. What is a determinant? (Clear and Concise please)

Test Corrections Pro/Con

➢ Students must have a way to be successful in the class beyond the test.
o What if the student is not a good test taker?
o What if the student is going through some huge personal stress, a bad day?
➢ Teachers must have a way of documenting improvement.
➢ Test Corrections can be done in a quick and efficient framework. With out too much disruption to the class as a whole.
➢ Re-working one problem gives less assurance in general of mastery of a topic. There is a benefit to the “Test” structure. The randomness of the questions requires a greater level of mastery.
➢ A student’s ability to do math should be the primary reflection of the grade. The mastery of the subject is primarily calculated by tests.
o When a parent sees that their child has received a B in a class, there is a reasonable expectation that they have achieved some level of competency with the material.
➢ There is an ability to cheat the system. Tutors, other students, even over utilizing the teacher.
➢ There seems to be little in the way of research in the effectiveness of corrections.

Tuesday, July 31, 2007

Test Corrections

One of the hallmarks of independent school (private school) education is that almost every interaction must have the likely conclusion of a positive outcome. And why not, why not make success an ever present option.

This philosophy leads to my question: What to do when students do poorly on tests?

Currently at my school there are two basic schemes:

  • Test Corrections
  • Test Retakes
Both of these have pros and cons which I will lay out in future posts, but I am interested in other methods. Feel free to describe them in the comments...