How rigorous mathematics should be taught

Following up on the discussion on f(t) about how to teach log laws, I'd like to share the primary school video I mentioned in the comments.
Actually, there are at least two videos, and this brilliantness is produced by always brilliant Deborah Loewenberg Ball.


1. Sean numbers*: Ball improvises a lesson about even and odd numbers, wherein her third-graders derive the precise definitions. 


2. Betsy's conjecture: wherein third-graders explore proof. 
You can also find transcripts and teacher notes for these videos. 


I first read about them in this NYT article, which contains many other ideas worth thinking about.


*Side note: I LOVE how Ball engages her students by giving students' names to conjectures and numbers! 

My intro to Logarithms

This year was the first time I taught logarithms, and since logarithms is a tricky subject for many students (it was for me, for many years) I wanted to make sure to get it right from the start. In my school, it is very common that students decide to switch to the lower level math when they encounter logarithms. I was determined that this not happen this year.

There were several awesome resources I found when browsing the blogosphere, including James Tanton's approach of making a riddle and changing the name. I also loved Dan Greene's idea of having a symbol, L, instead of the word log. 

However, the idea that really struck me was in a comment by Mr H to JD2718's post on logarithms. Mr H suggested that we start with logarithmic tables, and go from there, and that's kinda what I did.

Table of Bases and Powers

This is how the class actually went:

1. Give the class a few exponential equations which they can solve by converting to the same base. Intermix with these a few equations which cannot be solved that way. The students must graph or be stuck. 
2. When a sufficient number of students are bewildered, give them the table above and ask them without further explanations if perhaps this table can be of some help. Let them figure out how to read the table and use it. Most of my students were able to figure it out quite quickly. 
3. Ask students: what are you doing, how would you explain it to someone who missed today's class? Think pair share one minute per step. 
4. Divide whiteboard in two, on one side write Logarithms and on the other write Square roots.  Compare and contrast different features, like how both are something you do to something else,   how square roots and squares are inverses, logarithms and bases are inverses (they "undo" each other if students haven't done inverse functions yet) and how they are used in specific examples.
5. Ask students to return to table and solve 10^x = 50. Show them that calculator can do it as well, but that for now base 10 problems are the only ones calc is useful for. 
6. Give a few more exercises using table and calculator. 
7. Show youtube logarithm song.

Homework: students received a link to a google-form to answer some questions about logarithms, so that I could check their conceptual understanding. Only about half of the students did this but their answers showed good understanding. 

Next class, I started out by giving students this handout:

Using Logs for Effective Graphing

The students were quite impressed with this new tool to fit different size data on the same scale, and were even more intrigued once I showed them the xkcd graphs depth and height. 

I also showed them pictures of the richter scale such as this one and students were intrigued and showed good understanding such as "Look, the Chile earth quake was about 10^3.5 times stronger than the Hiroshima bomb!". Music to my ears. 

Then I introduced the laws of logarithms. I butchered that one. Check out Kate Nowak's recent post for a killer idea instead.

Next class, we looked at logarithmic functions. I started out by giving students a tiny review of inverses ("they undo each other, reflected around y=x, x and y are reversed") and then gave them this worksheet. 

Logarithmic Functions

I really don't see the need to go into detail with log functions so I didn't. 

Natural logarithm was introduced as inverse of natural exponential function and little else was done with it. 

Test results were decent, no one treated logs as an object, and no one switched to the lower level. :)

Scratch

A friend today told me about Scratch as a possible fun and simplish move towards that Conrad Wolfram vision of computational math in schools. I'm currently too groggy from post-surgery morphine to dig in and create any teaching material using scratch right now.  Has anyone used this for teaching yet?

Friday article summary - what is "good" teaching?

When Good Teaching Leads to Bad Results: The Disasters of "Well Taught" Mathematics Courses
Schoenfield, A. Educational Psychologist. (1988)

Original article is available here.

Judging from how often I've come across people referring to this article, everybody but me must have read it long ago. Probably during teacher training? Teacher education in Sweden is ridiculous and at least mine did not include any research articles of any kind. So I read this article today for my own sake, just to figure out what it is everyone else is talking about.

The Great Divide

An April post on Research in Practice caught my interest, and made it very clear what's been going on for me the last year and especially since I started daily reading all the great math teaching blogs out there: 

There is a growing divide between my students' and mine perception of what good math teaching and learning looks like. 

Simply put, I'm breaking the math-class contract. 

Teaching theories in psychology - how?

In my teaching of psychology, I am keen that my students should have knowledge and understanding of the science behind the concepts, models and theories that I ask them to learn. I do this by giving the students tons of research study summaries, and then ask them to use this research to back up whatever they are arguing in their essays. This is going very well. The students enjoy the studies, are getting very proficient at evaluating them critically, and are recognizing the need for scientific support for whatever claims they are making.  Here is the problem: the students are not learning theories.

Friday article summary - concrete vs abstract?

The Advantage of Abstract Examples in Learning Math
Kaminsky, J. et al. Science. (2008)

When learning new mathematical concepts, for example the concept of mathematical groups of three objects, is it better to start with a concrete example or with a generic representation? This is the question Kaminsky and colleagues at Ohio State University investigated in their 2008 article.

Time

Dan Meyer has his important ratio no. 1 which deals with the worth of instructional decisions as a ratio of instructional value and time expended in class (or out-of-class homework).  It's student focused: the time expended is student time, and rightly Dan wants to maximize value per minute.

I think I need to develop my own important ratio, and it needs to deal with my time planning and evaluating lessons and student performance.  It could go something like this:

Friday article summary - is cooperative group work bad?

This term, I'm actually supposed to being taking a course which entails me to read and think about research articles in the teaching of mathematics. So to help me stay on track I'll use Fridays to read and comment articles.

First out: 

Relationship between student and instructional factors and algebra achievement of students in the USA and Japan: an analysis of TIMSS 2003 data. 

J. Daniel House and James A. Telese. (2008)

Formative assessment

This week was a conference week, with several general and subject specific conferences for all city-employed teachers in Stockholm.
My knee-jerk attitude to these forced events is somewhere on the extreme end of the disgust-scale, but during the event it usually shifts to anger and then to desperation. Now that I have an iPhone things have gotten much easier.

This particular week, however, had at least one worthwhile presentation. Lena Göthe, the principal at a local school, talked about formative assessment, why and how to use it in secondary classrooms.

On the Use of History in Calculus Education

My colleague is currently struggling a bit with how to introduce and motivate the Fundamental Theorem of Calculus. So I looked in the essay I wrote for my teaching diploma and found a quote by Leibniz, which is useful, and some less than useful "intuitive" arguments based on distance and speed.
While it's worth remembering that I had done virtually no teaching at the time of writing, the essay does still manage to provide some insights into teaching concepts in calculus.

Maybe someone else will find it useful - it's available here.

Current status - a reflection overdue

There is this itch inside me that signals it's time to stop and reflect. Not just over individual lessons, the way I always do, but reeeeaaally think about what's going right and wrong. It feels a bit weird to air these thoughts in public, but I'm hoping for some reactions and comments from people who can relate.

Motivating e

Here's the background: my honor's class has done sequences and series, some basic function stuff (domain and range, composition and inverses) and some descriptive statistics. During the last week, we've also handled exponent rules, equations and functions. My textbook and syllabus are pushing for me to introduce e as "the natural exponent" next week, but how do I, at this stage, motivate that it's "natural"?
Continuous compounding is not a nice fit right now, and we're not even touching calculus until maybe late spring.

What other options are there?

Edit: oh, and I found this, and it's fantastic and I got the whole e-book and it's making me consider compounding after all.

Edit: I decided compounding may work and made this worksheet that students get as homework (for later class discussion).
I'm also giving something very similar to my regular class (seniors) who are doing financial math. Given my previous less-than-perfect (crash and burn) experiences with doing investigations with this class, I'd really like to get this right. It'll be optional, as e is not in their syllabus. Even so - any suggestions will be highly appreciated.

As always, google docs kills equations, so download the documents for best effect.

No time to think - the IB way of examination

This is an example of a recent test (with correct answers attached) given to my senior class. The time limit was 60 minutes.

Here's the problem: while I have some liberty in designing the test, I am preparing the students for a final examination and so far I've found it easiest to choose questions and time limits from previous IB final exams. 

The students, of course, hate the extreme time pressure. I don't like it either. When at university, my exams were 5 hours long and 6 questions large. Sometimes I left after 1 hour, sometimes after 5 hours, and more often than not I was able to use that extra amount of time available to dig deep enough in memory to find what I needed. Sometimes I re-derived formulas and above all, the type of thinking I engaged in during the extra hours was a good learning experience and added to my understanding of the topics. 

So on one hand I'm preparing students for time-pressured examinations, and want to give them practice in such settings. On the other hand, the exams are frustrating, very procedure-oriented, and not especially conducive to learning. At this point I'm welcoming any suggestions on how to proceed. 

Dealing with a job that's both meaningful and fun - without burnout

When reading Sam's touching post about his work recently and in the immediate future, I realized that I too have been struggling with managing the work load and my own attitudes to work. This post will be about some ways I've found that help me deal with being a teacher.

Winding down: an update to what worked and what didn't and dealing with test results

Today I met my seniors for the first time since their disastrous test. I mean Disastrous. I think only 20% passed, and believe me the passing boundaries are low on this thing: way under 50%.  I asked them to consider what they could do differently, and also how I can improve teaching strategies. As I suspected, the students said that they wanted more "traditional" teaching with me explaining new material and them practicing a few problems. Less investigations, less open-endedness, more of me just showing and them repeating. So I did that, on Geometric Sequences, and they loved it. "I understand something for the first time this semester!" was one memorable exclamation from a usually sullen student.

I hate this. On one hand, fulfilling their request will save me 90% of the time I usually put on planning their lessons. I'll have more time for my honors students, many of whom enjoy the challenges they get in class.
On the other hand, this feels like existential suffocation. What's the point of me spoon-feeding the seniors stuff about trig functions, logic and all the other interesting topics we before us, if all they are doing is trying their best to put in least amount of effort to pass the exams? Teaching loses its meaning and joy.

Redesigned

Dan Meyer has had a huge influence on my life since I first discovered his blog mid-July. So far, thanks to him, I have:

• Developed the habit of reading 20 other good blogs
• Started this blog
• Spent way too much of my free time considering (and, currently, rejecting) the WCYDWT strategy of teaching (of course it's great! but who has the time needed to make it happen?)
• Experimented with open-ended questions in mathematics
• Given much more thought to applying for the ph.d. programme next year

This weekend, however, has been devoted to changing an aspect of my teaching to which I had previously given very little thought: my powerpoint presentations. 

Fractions Mnemonic

My students can invert function and tell me for which values of x a geometric series converges, yet they can't do squat with fractions. It's not for lack of "teaching for understanding" - I made sure every step of the way was solid when we did fractions last year. But since then lack of practice and reflection has pushed the understanding and skills far back into awkward recesses of their long-term memory (at best!).

So I've been searching for a good mnemonic, and haven't found anything that covers the whole topic. In high school, I too had trouble remembering the rules and often used the 1/2 fraction to remember. This week, I saw a student use it the same way I used to, without me or anyone else having shown him, and so I decided that together he and I could introduce this method to the rest of the class.

Here is the worksheet I've designed. Google Docs mangles equations, so download the file for best effect. Please offer suggestions for improvement. I'll hand this out to the class on Monday.

Why do students love this?

Last year, with my PreDP class (think Algebra I and II) I gave a class the Riddle of Diophantus and asked them to solve it. The students, with just a basic understanding of fractions, loved this problem and spent a good 40 minutes or so working on it. It allowed me to introduce algebraic expressions, substitution, and simple equations - as well as give the students so well-needed practice with fractions. This year, my colleague used it in her class with equal success. But what makes it work and engage students in a way that smaller problems rarely do?

Dealing with test results

This week, I've given three tests - a test in each of my three math classes (Math Studies, Math Standard and Math Higher). Inspired by all the brilliant SBG posts I've read lately, I want to find a good way to deal with the students' results and reactions.

What worked and what didn't

This week, I've tried a few things with my two math classes (honors juniors and standard seniors). One, a scaffolded proof of the Sine and Cosine rules, went horribly for some very mysterious reasons. The other, a Binder check procedure I've copied from another blogger, is going very well.

Fumbling with inverses

My IB Mathematics course has started out well, with a successful introduction of functions, domain and range and composite functions. So maybe I started getting too cocky and didn't spend enough time on today's lesson - on inverses. Whatever the reason, it felt like hitting a wall. While students happily solved questions such as "what's k(x) if h(x) = 5x and h(k(x)) = x?", they could not even start on the very similar question "what's the inverse of h(x) if h(x) = 5x+3?" Even with me explicitly explaining the connection, by definition, to the previous question, students were dumbfounded by these developments.

Now I'm the one who's a bit stuck. Do I give an extra class on inverse functions - this would break my planning and mean I'm lowering my standards a bit. Or do I send them to the math study hall and go on with my planning, hopefully managing to fit in inverses now and then again? I'm leaning towards the latter but would really like to figure out what went wrong (and how to fix it). Maybe the language just got to abstract? Riley at Point of Inflection has a series of posts I probably should have read more carefully.

Ah well. "Where you stumble, there lies your treasure", is this nice saying I've always liked. Maybe there'll be something superbly useful in this mess somewhere.

Standard Based Grading, the IB-system, and something in between

Since I started teaching two years ago, I've been teaching in part the Swedish (standard based by law) system and in part the IB (final exam) system. It's therefore very interesting and funny for me to read about the issues US teachers are having with SBG, and the arguments people have against it (no, students do NOT come and demand a retest infinitely many times - they're too lazy and anyway the teacher can limit the assessment opportunities). In this post, I'm going to share my experiences and thoughts about the two systems I know, and propose a compromise.

Bonding Day

Yesterday was my first Bonding Day. In my school, Bonding Day basically means teachers orchestrate a series of team-building activities for students, usually in some park in Stockholms. The purpose is for our 150 students to have fun together, and, by cooperate in teams, build friendships and a generally nice atmosphere school. 

The investment for this is as follows: teachers spend one and a half full day planning, and students miss one full day of classes during Bonding Day. 

As I enjoy the teaching part of being a teacher, I felt a bit frustrated that all this energy went into an activity that doesn't directly further students' understanding of school subjects. Even though I did not have to participate in the planning, I was indignant on behalf of my colleagues. But that was before Bonding Day. 

Finland versus Sweden

Finland rocks, at least when it comes to education, and at least as testified by several international surveys such as TIMSS. It confuses the hell out of me how our neighbouring country can score at the very top while Sweden comes in somewhere barely above the median. The fact that Norway and USA score even lower is not a comfort. So much money is spent on educational innovation in the US, and so many inspired and devoted teachers do their utmost to make every lesson a marvell of creativity, that it's exasperatingly frustrating that results are so poor.

This fall, I'm taking a course on mathematics instruction. It's very loosely organized, permitting me to choose my own area of interest and sources. So far, I'm leaning towards investigating these international differences, and hoping that I'll find that at least part of the explanation is differences in teaching methods. That'd be nice. That'd mean we (teachers) have at least some power to influence the results, not just so that our kids score higher than average for our school district or nation, but on an international scale.

So many theories, so little time: Direct Instruction

Action research, systematically investigating different teaching strategies, is an important part of my work - and one I enjoy very much. Usually, I find inspiration for teaching ideas in the different psychological theories of education - be they about cognitive development, motivation, or behavior management.
The problem I face right now, at the beginning of a new term when opportunities seem limitless, is to decide which theories to experiment with this year. This post if the first of several, each of which will focus on one theory or strategy of teaching.

Sources of Inspiration

The idea for this blog came one day when I had happened to visit a few particularly interesting or otherwise impressive websites or blogs (and was somewhat bewildered by the variety of ideas I found there). Here are some of them.

http://blog.mrmeyer.com/
http://blog.mathsage.com/?tag=two-step
http://www.teachingasleadership.org/
http://www.khanacademy.org/

This post will be updated continuously and will probably ultimately serve as a collection of sources for more inspiration.