Ideas for the new school year

Alrighty, summer is over and students are back in school next week. These few days/weeks at the end of summer and very early start of term is my favorite time of the year because now there is so much energy, and so many possibilities to experiment with when the students finally arrive. So here's what I'm trying this year:

1.  Do more mistakes. Last year, I tried "my favorite no" but felt that it took a bit too much of class time. This year we will continue trying to improve "my favorite no" and also we will analyze the mistakes  at mathmistakes.org. Finally, we're definitely going to try the Mistakes Game, in which students collaborate to create and present problem solutions that contain realistic mistakes.

2. Get into whiteboarding. 10 mini-whiteboards sized 30x42 cm are on their way to my school, and I think they will be suitable at least for pair-work. I'll try to make my own larger whiteboards later on for larger groups. What are we doing with the whiteboards? Here are some ideas, many borrowed from Bowman Dickson:

• Mistakes game
• Rotating during a solution: groups finish other groups’ solutions during a gallery walk, or students take turns writing the steps to a solution within their group.
• Using color: chain rule, highlighting mistakes, rotating 
• Guess and Check: one person guesses an answer, the other checks if it’s right
• Solve and justify: one person writes the steps of a solution, the other justifies each step.
• Each group comes up with their own problem: then gallery walk with each team solving each other group’s problems.
• Each group comes up with an example of a concept, then shows it to the class.
• Each group writes an answer to a teacher-driven question – shows to the teacher
• Polling for multiple choice questions: can be summarized on the board
• Groups create multiple representations of the same idea: good for functions!
• Groups create different solutions to the same problem.  
• Groups create several problems related to the same information (information is given, or is asked for, or can be an intermediate step)

3. Let go of homework. I'm not giving up, exactly, on getting kids to do homework. It's just that this battle is one I've been fighting and losing for the past four years, so rather than continue to frustrate myself and the students I'm going to let go, lick my wounds, and take some time to build new strategies (which might involve the whole school culture rather than just my own classes). There is only one thing I'm going to try this year, and it's based on a suggestion from our physics teacher (thanks, Johan!). Students will receive one small quiz weekly, and this quiz will be on just one of perhaps 6 "model solutions" demonstrated during class the week before. So students will always have a very limited number of questions and solutions to understand or in the worst case simply memorize for the quiz. Not ideal, I know, but for many students even if all they do is memorize 6 questions and solutions then that's a huge improvement on their previous efforts.
What I'd like to add to this approach is to either include a question on the quiz asking students to justify a step of the solution, or to finish each quiz with a group/whole-class discussion of why the solution is appropriate.

Useful things to do in class

I have the distinct sensation of landing in math teaching. It's so nice to feel like the years of wonderful but chaotic experimenting are finally manifesting into a set of strategies that I trust and feel comfortable with. Here are some things (in no particular order) we do in my math classes right now:

1. Do Now - every day, class starts with a Do Now, so students start doing math the moment they come in the door (best case scenario, in reality they still need some prodding to focus on the task). The Do Now task always recaps something the students learned the previous lesson, and serves as both retrieval practice, feedback for students on their level of understanding and skill and opportunity to correct misunderstandings.

What I like the most about the Do Now task is that it can often bridge the previous lesson to the current one.
For example, a recent Do Now task asked

"What are the x-intercepts of f(x)=x^2+3x+2? Discuss in pairs."

The previous lessons, students had solved quadratic equations. This current lesson, the objective was to graph quadratics from standard form. The Do Now bridged the lessons by giving students the opportunity to help each other apply their understanding of equation-solving to graphing.

2. Traffic-lights - these are my cheap-o "clickers". They are just laminated cards red on one side and green on the other. I tried them a month ago and loved them immediately, as do my students. Every lesson, during the Do Now, I hand out these cards. Students use them throughout the lesson to signal their understanding or need for further clarifications. When I explain something, I typically ask students a specific question such as "Are you able to explain every step of this solution to a classmate who is absent today?". If a student holds up red, I ask what step(s) need further clarification, and then ask a student who holds up green to explain those steps. I find that when students are made to take a stand like this, students who would otherwise pretend to understand do speak up. Also, during individual work, students turn their cards red side up if they want my attention.

3. Weekly homework-checks - this could be just looking through students notes during the Do Now, or even a mini-quiz integrated in the Do Now, or a much larger hand-in assignment. In whatever way I do it, I need to check student homework not because I'm so very interested in whether they did it, but because students are much more likely to do it if I check it. In my psychology teaching, every time I've done weekly homework checks, student test results have risen dramatically. Jury is still out on whether the same improvement is seen in mathematics, but I'm hopeful. I'm still working on a system to check homework comprehensively yet quickly.

4. Investigations, where appropriate - I've tended to overdo or underdo investigations in class. Now, I ask myself before each lesson whether the goals for that lesson are reachable by investigation within the time constraints of that lesson. Next lesson, the goal is for students to understand and apply function transformations. This is superbly appropriate for an investigation task, and most of the lesson will be devoted to it. Investigations are great, because of too many reasons to list here. However, investigations are time-consuming and don't always lead to the intended understanding and skill, so they need to be balanced with more structured teacher-led demonstrations.

5. Direct instruction, where appropriate - some goals, such as applying trigonometric relations or log rules, are more skill- than understanding-based and require lots of practice. In such cases, I prefer a structured teacher-led approach in which I or we together solve an exercise, then students practice individually or in small groups, and then the process repeats. This can also work well with less teacher scaffolding and more group-work, such as examining and evaluating solutions of problems.

6. Problem solving, where appropriate - sometimes, when solutions require many different steps, such as when students graph functions from first and second derivatives, or use derivatives in optimization problems, I prefer to put students in groups and have them figure out the solution to a problem themselves, before writing a structured summary on the board. I find that this type of scaffolding, students helping each other, helps students feel confidence and ownership of the solution method. If instead I was to simply present such a complex multi-step solution, students would be more likely to feel intimidated and to try to memorize all the steps instead of trust that they can construct a solution based on their understanding of relevant concepts.
I'd like to integrate WCYDWT style problems, but so far I have never felt that I have the time to do so.

7. Closing summary - OK, I'm not actually doing this one every time or even most of the time, but I'd really like to! When I do have the presence of mind to recognize that 5 minutes remain, I like to ask students to summarize either orally or in writing what they have learned during today's class. Sometimes, rarely, I use exit slips as well.

So - this is my math teaching in a nutshell. Lots of room for improvement still (and I hope I'll always feel that way!), but at least and at last there is some stability with strategies that promote quick and efficient feedback, confidence, understanding, mastery and fun.

As always, I welcome all feedback.

Starting students on logical thinking

Every year, I make time in the first class of mathematics to introduce my students to the kind of thinking I expect from them throughout the year. That is, the inquisitive and logical thinking that'll prompt them to require and enjoy logical soundness in everything they are asked to learn in math class. I do this by showing them Lewis Carroll type syllogisms, such as this one:

• All babies are illogical
• Anyone who can handle a crocodile is not despised.
• Anyone who is illogical is despised. 

I ask the students to form conclusions based on these premises, and every year I find that students are completely stumped by this task. 

Most of them start by questioning the premises. 

"It's true that babies are illogical, but you can be a pretty horrible person and still be able to handle a crocodile..."

I can work with this. After all, it's important that the premises are sound, or else everything is on shaky ground indeed. However, what the students are clearly telling me that they are not able to either do, or understand, the task I am setting before them. In short, they are failing the standard Piaget formal operational stage test.

To be fair, more recent research (after Piaget's) is implying that a disappointingly small percentage (about 20%) of the adult population can do this kind of formal operational task. Also, to be fair, it's probably not that students or adults are "not able", but rather that they don't understand what's expected of them because of a lack of practice with these kinds of tasks. Still, when I showed this video to my psych class (16-17 year olds) a few years back, a large majority of students agreed with the boy instead of the girl. 

So, back to my first class lesson. Walking around the room, I explain to the students what they are asked to do. "Suspend reality for a moment, let's pretend that these premises are correct. Then what can you conclude?" Once students had understood the task, they could solve it with only a tiny bit more prodding.  

I then show them a simple equation, say x + 5 = 7. Students are happy to conclude that x = 2, but now I hope that they are understanding that they are making a logical conclusion based on a premise which might or might not be true. 

I follow this up with the best example of mathematical problem solving that I know of: a Sudoku. Not only is Sudoku great because students are making logical conclusions every step of the way, and in a way such that the reasoning is their own and the "logicalness" of it is readily visible to them, but Sudokus also illustrate many important principles about mathematical problem solving. I'll write about that later.

I think it's important that students get this initial "feel" for logical reasoning. In fact, I ask them to feel it - for me, when something is logically sound, I have this satisfied calm feeling in my stomach. And when it's not, there is a worry, almost like an unpleasant itch. I know some students feel this more than others. Last year, suddenly, a student in the regular (non-accelerated class) simply refused, for weeks, to learn the basic rule of differentiating polynomials because she had missed the class where it was explained why this rule works.  I've encountered this many times already, often in students who are considered to be "weak" at math. Sometimes, when these students' need for logical soundness has been satisfied, they become much "stronger". I like and respect this resistance to learn things without understanding them. 

Throughout the year, I'll be coming back to this initial lesson on logical thinking. I will frequently ask students to explain why something is true, and not just to show me that they know how it is applied. I hope my students will understand, and appreciate, that this insisting on logical reasoning and understanding is not something "extra", something added to the already considerable pressures of their studies. Instead, at least in my experience, 20 minutes of effort at understanding why something is true pays off in exam scores better than hours of practice with more or less routine exercises. After all, in order to fully understand why the rules of logarithms are the way they are, you must fully understand logarithms. In order to solve routine exercises by looking at solved examples, you need only be able to use a formula. 

Of course, this introduction to logical thinking leaves many finer aspects unexplored. When students were attempting to draw conclusions based on the above premises, many offered that "Anyone who can not handle a crocodile is despised" and other illegal moves. Ideally, I'd like to spend more time on these kinds of issues, and in the Mathematical Studies class there is even a section on logic which I really like. Also, students may not be aware what constitutes a logical conclusion in all cases. As Sue recently pointed out, many students are not able to distinguish between an example and a mathematical proof of Pythagoras rule. Such issues can be overcome with counter-examples, but also necessitate a discussion of the difference between inductive and deductive reasoning (and why deductive is so superior! :)). 

Overall, I'm happy with this start. We'll see how it plays out over the coming year. 

First attempts

It's Friday night, and what better way to spend it than read through nrich and ncetm resources? (I don't know whether to take a lengthy think about my life or just be happy that I love my work).
Anyways, I found a few ideas that I've been meaning to try out in my teaching but haven't gotten around to just yet. With a trig unit coming up, and the ncetm resources being so very well-written and inspiring, I decided to jump in and design my own resource based on two (for me) novel ideas.

1. Always, never, sometimes true classifications: Supposedly great for promoting conceptual understanding and constructive discussion.

2. Matching activity: I like this one because it can so easily be differentiated by creating more options and/or leaving blanks.

Here is the actual file, which will probably be changed a bit before use in class on Monday.
Matching Basic Trig

(I do not know why Scribd likes to minimize letters and rotate pictures. The original document looks neater.)

Something I'm not very happy about is the increase in paper copies these kinds of activities seem to require. I'll probably project the classification task on the board instead, and am grateful for any other suggestions.

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.

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.

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.

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.

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.

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?

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.

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.