Algebra and a Mixed Convergence

The title probably makes no sense, but algebra has been in my face lately in a multi-faceted way.

FIRST I'm following a friend's post and comments on facebook about her experience wanting to be an RN or at least get an AA degree, but intermediate algebra is stopping her in her tracks.  She's in her forties; she's a bright, dedicated, mature person and very, very frustrated.  There is even more frustration in the comments of others to her post - about how algebra is from the devil and that no one ever uses it and how awful and how hard it is and that it is the one thing that keeps people from their educational and career goals.  (My first thoughts and feelings on this are that I am desperately sad for her - and that I both love and hate being a math teacher.  I love it that I am in a position to help people who have never liked or been successful with math before - it's quite a high to be part of helping someone succeed where they never thought they could! - but I hate it that I as a person am often disliked or even hated at face value because of the topic I teach.)

SECOND In my Math for Liberal Arts Majors class today my students were giving presentations on mathematicians as part of a math history unit.  One person presented on Rene Descartes whom we have to thank for the Cartesian Coordinate System on which we do most of our graphing.  The student mentioned how by creating this system Descartes put geometry and algebra together, thus making algebra EASIER.  (Yeah, of course, because now you can make a picture of your equation, and you can see how it behaves and where the maximum and minimum points are and so on.)

THIRD In my position as liaison to the tutoring center I was in the center evaluating tutors today.  One thing I observed was multiple tutors trying to help a student who was working on finding graphical information about a quadratic equation.  He needed to find such things as the axis of symmetry and the vertex.  It was an uphill battle to say the least, and it just struck me how this thing (graphing) that was supposed to make things easier (according to my student earlier in the day) had now become its own intense struggle.  The contradiction between seeing and hearing these two things in the same day really struck me!

So I have a lot of thoughts roiling in my mind tonight about all this.  Partly it confirms me in my desire to pursue the history of mathematics.  I think that if we could teach math in a more historical context in order to show students how and why concepts were developed and how each additional topic makes the quest easier rather than harder - that perhaps this would be a good way to go.  I think it would help too for students to understand what the big question that was being answered was back in the day when each new tool was developed, because it would shine light on the meaning and use.

It seems that if the student in the tutoring center had been given a real-life question about a business owner wanting to maximize his profits and an equation that related to that and then the student was allowed to struggle with that for a while - trying to make his way in to the problem - plugging in numbers, trying to find the biggest value, wondering if he had found the largest for sure or not, figuring out how to be certain - and THEN the student was introduced to the idea of graphing, which would give him a picture of the equation - a picture that would allow him to see where the highest point on the graph was, which would also be where profit was greatest - that then a graph,which is a picture of what's happening, would be seen as the help it is rather than as just another topic that was thrown at him.

If I didn't need to earn a living I think I'd like to take a few years off and try my hand at writing a textbook that would allow students to see the benefit of what they are learning and how all of it relates to all the other topics they are learning - where it fits in the scheme of things - what it's good for - and why and when and how it was developed.

Maybe in my next life  .  .  .

For now I'll just do the best I can to help those who come my way and try to change their worlds, one student at a time.

A Good Teaching Day

Well, I was just hitting the slump I usually do a few weeks in to the fall semester when I realize summer is truly over - and when the adrenaline of the first couple of weeks of the semester wears off - and when some class management issues start to pop up.  Yesterday I was not a happy camper, but today was a whole different story!

After teaching one of my favorite topics (which is not always a favorite for students, especially at this level), three students stayed after the 2-hour lecture and talked with me for another 45 minutes about the topic at hand - set theory - particularly the continuum hypothesis and different "sizes" of infinity.  They were very excited - had lots of questions - lots of thoughts.

THIS is what it's all about!  THIS is what I live for!
THIS is what makes teaching worthwhile and exciting!

Made My Day

There is a YouTube TED presentation put on by math teacher Dan Meyer that describes in the opening sentences the plight in which math teachers often find themselves:

"Can I ask you to recall a time when you really loved something - a movie, an album, a song or a book? And you recommended it whole-heartedly to someone you also really liked, and you anticipated that reaction, you waited for it, and it came back and the person really hated it. So by way of introduction that is the same state in which I've spent every working day of the last 6 years. I teach high school math.

I SELL A PRODUCT TO A MARKET THAT DOESN'T WANT IT BUT IS FORCED BY LAW TO BUY IT."

That's a tough spot!

I shared a couple of posts ago how I try to work to "create wonder" in my classes. Given the above, it's a tough prospect, but when it works it is a joyful thing for me and my students.

Today, in two situations, I saw the wonder, and it was beautiful!

One of these things was a student coming to office hours today who is in my Math for Liberal Arts class - basically a Math Appreciation class - just as there are Music Appreciation classes and Art Appreciation classes. We do something similar, sampling a wide variety of topics. I try to bring in the topics that are the most beautiful (tessellations and fractals) and that are the most different from arithmetic and algebra but are very applicable (topology/graph theory and logic). It was so cool talking with this student today, because he saw how much of the purpose of class was to develop a new way of looking at things, a mathematical vision that lends itself to a bigger picture of life and also new problem solving techniques that stretch the brain to envision things and approach them in new and productive ways. YES!!

Another situation was with another Math 101 student. One aspect of this class is math history, and the students will be teaching that part. They each need to select a mathematician on whom to do a presentation. This student had interacted with me a bit, debating which one to choose. She finally decided on Piet Hein. That's a name that appears regularly on my blog because he is a poet as well as a mathematician (as well as a scientist, inventor, game creator, and designer). He created a type of poetry that is one of my favorites: Grooks. For example, one Grook that has come to my mind a lot lately (in the face of the dilemma described in the opening lines of this post) is one that I've posted before:

THE CURE FOR EXHAUSTION

Sometimes, exhausted
with toil and endeavour,
I wish I could sleep
for ever and ever;
but then this reflection
my longing allays:
I shall be doing it
one of these days.

In sending me her official request to have Piet Hein as her mathematician, this student wrote the following:

I'll speak of Hein's mathematical psalms,
Although, I must add this caveat:
I admit to having some qualms
About choosing the teacher's favorite!

KH

I've had a smile on my face all evening from these two interactions. Teaching math is not easy, but sometimes it has some pretty rewarding moments, and today I had not just one but TWO of them that made my day!

Get Off the Ladder!

There is something wrong with the way mathematics is taught. Most people claim to hate math, but I don't think most people ever got a chance to know what Math (captial M) really is. Here is what often happens when I meet someone new:

Other person: What do you do?

Me: I teach math. :-)

Other person (after involuntarily jumping backwards 2 feet): UGH! I've always HATED math!

Me: Well, if it makes you feel any better, let me assure you you are not alone. Most people respond to me that way.

Other person: You know, I do remember a problem my teacher showed me in 3rd grade, though, it was about getting stuff across a river, a fox, a goose and some grain, I think. I've always wanted to know how to do that problem. Can you help me with it?

Me: Well, let's think it through . . .

Fun and joyful conversation about mathematics continues at length from here.

ALWAYS.

There should be more opportunity IN the math classroom for fun and joyful conversations about mathematics, which will help alleviate individual anxiety and perhaps benefit our nation as more students might decide they like math and want to pursue it!

Math is not linear on Prezi

For All of Us

Professor Edward Burger of Williams College addressed this comment to college students, but I think it applies to anyone who cares to be a lifelong learner - and more -

You.

. . . your education centers about the most important creative feat of your life – the creation of yourself. It is about you – creating your effective mind, a mind enlivened by curiosity with the intellectual audacity to take risks and create new ideas, a mind that sees a world of unlimited possibilities. So I ask again: Have you failed recently? If not, then what are you waiting for?

Formula for Failure

Calculus of Friendship

As soon as the semester ends, and I have time to read again, I am looking forward to reading this book that recently came to my attention. It is about mathematics and relationships - specifically the relationship between a math professor and his high school math teacher, the change in role from teacher/student to former teacher/professor. Calculus is a study of change, so the title reflects both mathematics and relationship.

This came to my attention on NPR's Science Friday a couple of weeks ago on which the author was interviewed. It was a pretty fun interview with interesting comments from callers about mathematics and the teaching of mathematics - kind of got me excited again in the midst of this overwhelming semester in which I was kind of losing my passion. If you'd like to listen in too, the link is:

http://www.sciencefriday.com/program/archives/201002262

Upside-Down Math Education

The conference was GREAT! (As well as getting fired up again for teaching, which I find conferences always do for me, and as well as learning about math and education, I played roulette for the first time. I did so because a student of mine left class Wednesday saying, "Bet on black." Had I placed only one bet I would have been fine!)

Anyway, a while back I "promised" to post reasons to learn math. While I did write this up for my students I found it too long to post here. Now, however, I have a whole different vision.

At one session Edward Burger (pictured above) turned everything upside-down for me with regard to this question in a very exciting way that makes a lot of sense. He asked:

WHAT WILL YOUR STUDENTS REMEMBER FROM YOUR CLASS IN 10 YEARS; WHAT WILL THEY TAKE FROM YOU FOR LIFE?

He went on to list 10 life skills that are also at the heart of learning mathematics:

1) Just do it.
2) Make mistakes and fail but never give up.
3) Keep an open mind.
4) Explore the consequences of new ideas.
5) Seek the essential.
6) Understand the issue.
7) Understand simple things deeply.
8) Break a difficult problem into easier ones.
9) Examine issues from several points of view.
10) Look for patterns and similarities.

I wish I could expand on every one of these, but I try not to get TOO long-winded on here. I will include some related quotes of his, though:

A teacher's goal is to touch lives, to change them, empowering students to think more effectively and to have them see and understand thier lives in a clearer more focused way.

As a community we don't teach our students the greatest lesson mathematics has to offer: HOW TO FAIL (and how to learn from failure).

Five percent of the grade in my class is failing well. If someone says something wrong it moves us forward, and the student doesn't have to be embarrassed about a mistake because it has contributed to the discussion.

Creativity can be taught. Teach students to look at the world, to think, to create . . .

POST SCRIPT: Part of the upside-down aspect of this is his suggestion to invert homework and classwork - for instance by having the lecture online so students take notes outside of class but THINK in class (rather than "recording minutes"). In class we should provide "opportunities for anxiety" - something that can prepare students for tests better than doing homework in relaxed situations. He had lots of suggestions for what to do in class: 1) STOP LECTURING, 2) Use activities like: "Speed Rounds," "Show No Work," "Round Robin," and "The Shuffle." 3) Provide opportunities for students to fail (safely) and show them how to learn from that failure - what specific lessons that failure teaches about what path not to take and why.

MOST Frequently Asked Question

A few years back Mattel got in big trouble for one of the statements its "Teen Talk Barbie" made: "Math class is tough."

Historically women are under-represented in mathematical fields. Often I've heard female students of mine who are coming back to school in their 30's say that when they were in high school a counselor told them not to worry about math - that girls aren't good at it and don't need it. That comment terrified them and kept them from taking math, which barred them from many professions and put their lives on hold to some degree until they were able to overcome that fear and return to school (and nearly all of these women did very well - typically scoring at or near the top of the class.)

So, anyway, of course Barbie's message is not the one we want to send out. This is not just a problem for females, though. I have found through 22 years of teaching mathematics that many (or most) people, BOTH male and female DO think "math class is tough." When things get particularly tough the following INVARIABLY happens. A hand goes up; I call on the person; the question comes: "Why do we have to learn math; when am I going to use this in my real life?" as if on cue.

This is so predictable I can almost tell what week of the semester we are in by when this question comes up. It is interesting to me that it never comes up at the beginning when we are doing review, and the material is relatively easy. It seems people don't care how something is going to be useful to them as long as it is not too hard, but when it gets hard that is the MOST FREQUENTLY ASKED QUESTION. (I have actually begun addressing that question during the first week of class since I know it is going to come up, but I'm finding people seem to forget by week 7 what I said during week 1). What I share with them is this FANTASTIC one-page article written by a middle-school math teacher.

SO, after 22 years, I have finally decided to handle this (M)FAQ by WRITING the answer and putting it up on the internet rather than answering it on the fly each time I get the question.

Please check in again soon. I will have MY answer up and hyperlinked within the next couple of days as to WHY IT IS IMPORTANT TO LEARN MATH. (I'm not being sarcastic. I do see it as a legitimate question, and it makes sense that people would ask this of something into which they are investing a lot of time and effort. And it is because I see this as a legitimate question that I want to give a substantial legitimate answer.)

The Mother's (or Teacher's) Curse

There’s something called The Mother’s Curse, which goes like this, “When you grow up, I hope you get children just like you!” This is typically said by a mother when a child is having a temper tantrum in the grocery store.

I don’t think my mom cursed me. I don’t remember ever hearing those words. In fact, I think I’ve been very blessed in terms of my children. Of course, I may find out about things later in life as my parents are finding out about now - at times when their adult children sit around and reminisce!

No, it wasn’t my mom.

I think it was my piano teacher who cursed me.

I love to play pieces on the piano that I know well and that sound beautiful, but I hated practicing. This is really sad to be admitting as a math teacher, but I particularly resisted learning to count. Oh, I know how – I know HOW to count in music (one-ee-and-a-two-ee-and-a-three-ee-and-a . . .). I just never did it. I would ask my teacher to play the new piece for me before I went home so I could hear it. I had a very good ear, and I would piece together the notes I saw with what I’d heard, and I’d basically play by ear.

That made life easy.

That also crippled me.

Now I would like to play the piano really well, and I can play, but I sound just like I did in junior high, and I am limited in that I can only play songs that I’ve heard before and can play partly by ear. (I do plan to rectify this soon and begin lessons again and break my bad habits, which is going to be harder than it would have been to put in the work and do it right the first time around!)

OK, here’s where I lose all my readers, I fear.

TODAY AS I WAS TEACHING MATH . . .

(Anybody still with me?)

I was trying with all my heart, soul and guts to impress upon my students how important it is to learn the MEANING rather than a random assortment of rules. Some (though not all) want to say to me, “Just give me the steps; just show me how (and leave me alone).”

We are working with adding and subtracting positive and negative numbers. As I’ve walked around the class over the last few days checking their work, I see that some of them know just enough to be dangerous (kind of like me on the piano). They’ve learned a random assortment of steps but not what they mean or when to apply them.

I am there ready, willing AND ABLE to make sense of this WITH them. I’m basically doing gymnastics and turning myself inside out in front of the class to explain WHY, but I can see in their eyes and body language that they (SOME - not all! - of them) just want to be left alone to stumble through with what they’ve got - hoping that a random guess, like a roll of the die, is going to miraculously be correct (and possibly fearing that digging deeper will make it more confusing instead of less - I understand that).

I DO understand that math is something that many people fear more than public speaking, snakes or death. I DO understand the aversion. I just wish I could have gotten them sooner before they got enough tricks to feel they can (should?) tune out now and hang onto the patchwork they've got and be OK.

The thing is, this is college, and they are in my class because they have a goal, whether it is graduation, transfer, a specific vocation . . . by their own choice of direction in life at this point they HAVE TO get through not only this class but 2 or 3 succeeding math classes that all BUILD on THIS material before they can get to their goal, and I want to help them reach those goals! BUT I can't want it FOR them and make it happen any more than my parents could want piano skills FOR me and make it happen.

Today I shared with them my story of piano lessons (and 2 other similar stories). I'm not a teacher who tells personal stories in class, so this was WAY out the ordinary. I shared that learning STEPS is easy, but steps are just as easy to forget as they are to learn. Learning meaning is hard, but once you have it, it stays with you, and you can even apply it to new concepts you hadn’t seen before (just like I would be able to play piano pieces I’d never heard before if I’d taken the right route instead of the easy route as a piano student).

So, now I'm the "adult" saying, "PLEASE believe me. I know what is in front of you." As you can imagine, this is going over as well with SOME of my students now as it did with me 30+ years ago to be told that if I quit taking piano I would regret it later. (I do). I didn’t believe the adults then; what child does?! SOME of my adult students don’t believe me now either – but SOME do. Kudos to them‼ They are going to make it!

(The first time!!)

And now back to me and fixing the error of MY ways:

"One-ee-and-a-two-ee-and-a-"

There's Gotta Be an Easier Way! (updated)

Click to enlarge:

PS: Happy Pi Day!

What special property does the following sentence have?

"I prefer pi."

To visit a site dedicated to Pi Day, click here.