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Make Your Own Normal Distribution Questions

Today we were building our familiarity with the normal distributions. I had a scan of a textbook page with lots of normal distribution sketches, like the one below. I copied them onto yellow card and cut them out, discarding the textbook’s instructions and numbering.

Each student got a sketch and I asked them to make up a normal distribution question to go with it. Here are my instructions. Students were working on their mini whiteboards.

The students got out of their seats to solve other their classmates’ problems. There was a lot of collaboration and those who found writing the question hard got plenty of input from their peers. I was free to circulate and able to clarify some important ideas about continuous distributions and the appropriateness of the normal distribution as a model.

I was impressed by the questions’ variety and ingenuity.

I don’t think I emphsized thoroughly enough that students need to specify that the data they have chosen to write about is normally distributed. However, students were able to solve a wide range of questions, some more difficult than others.

This was definitely much more successful than a page of textbook questions!

Collaborative Talking in Math Class

For the next eight weeks I am participating in Exploring the MathTwitterBlogosphere, a project designed to help math teachers meet others in the cyber community we call home. (It is still a good time to join in, if you haven’t yet.)

Mission 1 is to write about what makes my classroom my very own. One thing I prize and make sure I develop in my students is their ability to communicate with each other about their math. I have been doing more and more discussion-based activities in lessons. I want them to talk about their conjectures and developing ideas. It is rare that my students are sitting in silence. They are usually discussing with the person next to them. Often they are moving around the room, talking to others. Even when we are doing “boring” practice questions they are talking.

Here two grade 12 students are discussing probability statements that may be never true, sometimes true, or always true. They had to go meet as many others as they can, discussing the statements on their cards, and each time trading cards. Then they go off and meet another person. (This is a quiz-quiz-trade activity.)

To help students communicate with each other, I have mini whiteboards (MWBs) and pens on each group of tables. Students love using them because they can quickly explain their thinking. They feel more free to make mistakes on the MWBs and to help and comment on each other’s work. Since having them always available, I have noticed a big increase in how much students help each other and talk about their thinking.

With the MWBs it is also easy to share the thinking of one or two students with the rest of the class. In another probability lesson, I asked students to visualize and then draw what they thought a certain probability distribution would look like. Then I brought six of the MWBs up to the front to discuss with the class their common and distinctive features. In the end, our discussion focused on just two of the graphs made by students.

All this constant discussion helps my students clarify and solidify their developing ideas. This makes my classroom unmistakably mine.

What makes your classroom unique?

“Naturally Occurring” Algebra

My grade nine extended class (similar to GCSE top set) have an algebra unit at the beginning of the year. I am assigning all the skills work for homework: simplifying, expanding brackets, the mechanics of solving equations, and so on. In class I am trying to give experiences that answer the unit question, “What did algebra ever do for anyone?” I’m trying to show how algebra’s power to solve problems is naturally arising from good problems.

Background: We have spent two lessons talking about types of sequences and formulas (such as 4k +12) that can lead to sequences.

Today I asked them to watch as I demonstrated something on the board in silence. I did the same thing you see in this video. In one minute, I demonstrated in silence how to find the sum 1 + 2 + 3 + … + 9 + 10.

Then I asked them to describe to their partner what they saw. That took another one minute.

Next I asked them a series of questions. First, could they adapt my method to find the sum of the first 100 positive integers?

Then, I put this list up on the board. What I like about this list (also from nrich) is that there is no time wasted on easy repetition. Each item is a little harder and provides a lot to talk about.

After about 20 minutes, I asked two students to come up to the board and put their solutions to the first two problems. As the lesson ended, some students were starting to attack the last part. What a beautiful and naturally occurring use of algebra!

Do you have any classroom problems that show how powerful algebra can be?

Email Workflow for Teachers

Our lives are so busy. Teachers are working to a deadline every hour of every working day. Keeping on top of emails is important, but I find that I only have time to read and write emails a few times a day – in the morning, in between lessons, and in the afternoon.

I keep my email inbox empty. Every time I open the inbox, I open each message and read it. A lot of messages just need to be read and filed away. I decide if anything in the message needs further attention. If a message has action that I can do in under two minutes, I try to do it immediately. (That’s a Getting Things Done [GTD] mantra.)

I have three main labels for things that require further action: @reply, @review, and @to-do. I use @ signs in front of these labels because it ensures they always stay at the top of the labels list.

I tag @reply messages and make sure to reply to them before I go home that day. Our school has a policy that we must reply to parents within 24 hours.

The @review label is for information that I know I need to spend some time digesting but I can’t do it in under two minutes. Things like an article I want to read or a link to a blog post. I use the @review items for those times when I have a few odd moments waiting for something or at the end of the day in my reading and thinking moods.

The @to-do label goes on things I have to get done. I go through the messages every morning and afternoon and try to attack the biggest thing that I have time for each time. Or if I’m just between lessons or meetings I try to attack three little things. Usually I keep the list manageable so there is no need for excessive prioritizing. When the list gets too long, I dedicate some evening or weekend time – but this is rare.

After all the label tagging, I archive everything in the inbox. An empty inbox keeps my stress down. And knowing that I have seen all my “open loops” and categorized them allows me to focus on jobs at hand instead of worrying about what else I will need to do later.

What is your email workflow?

Number Sets Activity

Which numbers are real? Rational? Natural? Whole? Here is an easy activity to help meet the number sets objective.

I made this set of cards with numbers from the above sets. As students enter, I’ll give them a card. Then I will ask the students to walk around the room, see what numbers others have, and organise themselves into groups based on their number. The instruction is purposely vague enough that many possible groupings are possible. When students have grouped themselves, I’ll start a group discussion about the way they are grouped.

I anticipate (I haven’t tried this activity yet) that I may have to prompt the students to think again about the groupings and ask questions to help them explore the idea of number sets. And then maybe ask if there are some numbers that are “purer” than others. I think students have a sense that whole numbers are more “number-y” than items like 2.4 or negative four fifths.

I think it would be interesting to put students holding whole number cards in the centre of the room and then build the number sets outwards around them. I think this would elucidate the idea of natural numbers being contained in the set of integers, and rational numbers being contained in the set of real numbers.

I think it’s my responsibility to introduce the names for the number sets once students have developed the concepts behind them. I’ll write the names and symbols for them on the board to conclude our discussion.

How do you teach about number sets?

Quiz-Quiz-Trade: Ways of Getting Students Out of Their Seats

My new school’s lessons are usually 80 minutes long. I think I shall try to get students out of their seats (at least) once each lesson – otherwise it is too long for them to be sitting. I also think it’s too long for me to plan in a single block. I can’t imagine any lesson where we work on the same thing for 80 minutes. My attention span isn’t that long and I can’t imagine any teenager’s is either.

Thus begins this series of posts on ways to get students out of their seats. Quiz-quiz-trade was an activity I learned about in Kagan Cooperative Learning (a brilliant book, by the way, which I thoroughly recommend). I have used quiz-quiz-trade regularly for years and adapted it in a few ways.

Preparation:
Print out some cards with questions on them. These could be taken from a textbook, revision sheet, made up by you, or made up by students.
Cut them up.

Activity:
Give one card to each student. While they are still sitting down, ask them to work through the question and verify that they know the correct answer.
Ask students to get out of their seats and meet someone new. Student A quizzes B, then student B quizzes A. They thank each other, then swap cards and move on to meet someone else.
After several trades, each student has met many other students and has also answered many questions.

Works Well With…
1. Questions that aren’t too long to solve. Or let students take mini whiteboards with them (pictured above).
2. Worksheet or textbook questions that you think are too boring as a worksheet. Just cut the sheet into strips and hand the questions out. Make sure you have enough for each student to get one.
3. Revising for exams. Use past papers cut into questions.

Adaptions:
1. Before quiz-quiz-trade, ask students to make up questions to demonstrate their understanding of a topic. Then you can use this questions for quiz-quiz-trade, either immediately or later.
2. You can print the answers on the back (for safety!).

What are some ways you get students out of their seats?

Check back next week for the next post in the Out of Their Seats series.

Better Group Work

I just finished reading an article called “When Smart Groups Fail” (Barron, 2003). It’s a study of groups of three sixth graders solving maths problems. Barron divides the groups into less successful and more successful at solving the problems and then analyses what features of their group work were significant.

Surprisingly to me, the success of a group of students was not influenced by:

the amount of talk
the average achievement level of the students
whether anyone in the group had a correct idea

Instead, Barron found that group success was marked by:

accepting or discussing correct ideas rather than dismissing them
correct ideas were brought up when they related to ideas that were being discussed at the moment
group members paid attention to each other and if the others were paying attention to them
group members physically showed their togetherness by eye contact and standing around or pointing to a common workbook

This article’s findings indicate to me that my classroom culture can lead to better group work. I can help students learn to discuss a mathematical idea by building on each other’s thoughts. A well-managed whole group discussion can lead to more cooperative group work sessions. I want my students to learn to listen carefully to what is said and then agree or disagree or ask questions. I can request these responses in whole class time. Then students will see they are the norms that also should guide their group work.

Furthermore, the groups in Barron’s study were of mixed ability and their previous achievement levels did not correlate with their success as a group. This adds to my feelings about the benefits of mixed ability teaching. If students of differing abilities can be helped to communicate well, they can all achieve well.

The study also found that students in successful group went on to be more successful in individual tasks. Interestingly, students in less successful groups did as well as if they had worked on their own. Thus poor group work neither helped nor hindered their achievement. However, good group work improved the success of individual students in to a significant degree.

The school year is just about to start and I am looking forward to inculcating a culture of social, mathematically focused talk.

Barron, Brigid. “When Smart Groups Fail.” The Journal of Learning Sciences 12.3 (2003): 307-359.

Vital Behaviours

It’s orientation week for new teachers at my school and the head of school gave a welcome talk. In it he mentioned vital behaviours for teachers and students for success at school. Afterwards I did a bit of reading about vital behaviours. (Here are two posts that helped me.) The phrase is from a book called Influencer: The New Science of Leading Change. Vital behaviours are those actions that lead to our goals. They are the smallest possible actions that make the biggest impact towards meeting goals.

Our head of school identified three vital behaviours for successful students:

attendance
completion of homework
leadership outside of the classroom

He also identified three vital behaviours for successful teaching staff at our school:

collaboration
use of data and evidence to guide decisions
high expectations for all students

The idea of vital behaviours really stuck with me. And made me wonder if I can generate some of my own. First I would need to think of a goal and then identify the fewest essential actions to meet that goal.

And since it’s goal setting time for the new school year, I thought I would give it a try.

Goal: Coach my grade 9 students (two classes) to communicate meaningfully in their maths portfolios/journals. (I’m not sure yet what I am going to call these books.)

First I brainstormed the steps I might take to work towards this goal.
introduce journals
provide exemplars (I already have some pictures of these.)
examine with students: What is meaningful communication and reflection?
establish thinking routines
provide rich tasks
provide reflection time
give summative feedback on tasks
have students give regular presentations in front of class
peer assess communication
write a rubric of expectations

And then I refined these to a list of vital behaviours:

provide rich tasks
provide reflection time
examine with students: what is meaningful communication and reflection?

I’m sure that after school actually starts I will see if this goal and these behaviours need to be updated or changed.

Have you ever used the idea of vital behaviours in your planning?

What are your goals for this school year?

Maths Trauma

Over the next three weeks, I’m participating in a free online course called How to Learn Math. It’s offered by Jo Boaler from Stanford University. I recently read Boaler’s book The Elephant in the Classroom (published in the USA with the title What’s Math Got to Do With It?). The course consists of short (5 minute) video segments, interspersed with tasks for me to do.

One of today’s tasks was to watch a short video of university students who describe difficulties they had with school mathematics. This concept map (below) summarises their perceptions of maths.

Similarly, a lot of adults talk about the trauma they experienced at school related to maths. As I meet people and tell them I’m a maths teacher, the usual response by adults of all backgrounds and careers is “I was so bad at maths.” A lot of highly successful adults hated maths at school and still think they are bad at it.

I have become very concerned with the way parents talk about their lack of maths ability in front of their children. For example, on parents’ evening I often hear a parent say, “I can’t help her with her maths homework at all. I am so bad at maths.” I think this is a very irresponsible thing for a parent to say. It doesn’t help the child see that maths is something at which they can improve. I am still looking for a tactful and helpful reply to these adults!

Was maths traumatic at school for you?

How would you reply to an adult who was traumatized by maths?

Procrastination Postcard

I am taking part in a leadership course that runs through the year. One outcome is to make a postcard that is shared with the whole group. I wrote (here on this blog) about procrastination recently and adapted this into my postcard. I took the picture on the front after plastering the space above my classroom desk with sticky notes.