Write Your Own Textbook

One week to go before school starts up again and I am beginning to get nervous.  I am also getting depressed because I feel like I am a poor teacher and I don’t know how to teach.  I know, POSITIVITY, please.  So, I am looking through back issues of Mathematics Teaching in the Middle School, which I was delighted to find my NCTM membership gives me access.

I picked Mathematics Teaching in the Middle School because as I stated before, it seems much more relevant to the level of mathematics that I teach.  And I’m hoping to find some ideas to use for the whole class discussion we have been studying in our Summer Institutes.  There is an online flipbook edition, and individual articles can be downloaded as pdfs.

Producing a How-To Book by Julie A. Mallia, Don Pawloski, and Peggy Daisey is from the February 2012 edition of  Mathematics Teaching In The Middle School. I tried to do this last year with my Geometry classes, having them write their own textbook for each unit.  But it was just a list of things to do, with very little original writing (put in your own words…) and most of it was just transcribed verbatim from the textbook, with very little evidence of understanding.  This article proposes a project that would be a summing up at the end of the year (or in my case of my Algebra double-block, end of semester).  Ideally, it would be done in concert with their English teachers, so they would make sure it was original writing.  Also, the students get to pick their topic (from a list I provide) so that they can select something they are confident that they understand. 


Summer Institute 2: Electric Boogaloo

We had the back-to-school edition of our Math Institute this week.  Our focus, appropriately enough, is on what is our administration wants to see when they are observing our classes. These are the four goals introduced in our June institute, namely:
Engaging Lessons
Modes of Response
Teacher Discourse
Student Discourse

We wanted to work on how we are planning our lessons, with emphasis on making an engaging lesson plan and what students look like during our lessons.

Previously, the majority of us planned our lessons in a teacher-led fashion where the teacher is walking the student through the class.  These classes tended to be more algorithmic, focusing on steps to take to solve a problem, and less concept-driven.  We concentrated on teaching “this problem” rather than problem solving strategies. Going forward, we’d like to make lessons more engaging by utilizing guided discovery lessons that illuminate larger concepts through student-driven explorations.  I interpret this to mean the students will be doing more thinking about what they’re doing and justifying it.

We also wanted to redefine what a “perfect” classroom looks like.  In the past we might have said that the ideal student would be looking at the board, paying attention, taking notes, and asking questions.  But now we want to put more emphasis on group discussion, so we want to encourage students to support each other (teamwork) and get excited to debate and defend their ideas.  I would add that I want to foster an environment where making mistakes is okay, and admitting a mistake is seen as a way to further explore and learn about a topic.

Some of us admitted to being apprehensive about group work because one student always seems to dominate while the other group members just sit back.  Possible solutions are to make sure each member knows that they are responsible for the task, and perhaps even asking the perceived “weakest” member to be responsible for explaining the solution.  But the bigger issue is to structure lessons where the journey is the issue, rather than putting all the focus on the solution.

In short, we need to be conscious of how we’re leading in the classroom.  Modes of Response have to with the actions we’re asking student to take.  Which individual are you calling on?  Are students to write down their answers or pair-share?  Are they supposed to brainstorm on their own or have a group discussion?  Teacher Discourse is about what questions we are asking to keep the focus not on the answer, but on the journey – making students become conscious of their thought processes.  (Resource for question types:  Developing Mathematical Thinking With Effective Questioning )

Strategies for implementation:

  • Assign roles for group work:  Manager, Liaison, Recorder, Quality Control.  Make the Liaison the mouthpiece of the group and only talk to that student.  At the end of the lesson have the students grade each other on how well they did each job and why they gave that grade.
  • Make a poster to hang in the classroom listing effective questions, or tape a list to the back of a clipboard.
  • Don’t ask the questions.  Give the students the set up and then let them pick the question (i.e. what do you want to find?)
  • Use a Discussion Rubric to let students know how they’re being evaluated and how they are getting points.
  • Teach students to explain wrong answers, or even give them the wrong answers and have them explain how that answer might have been reached.
  • Let kids know explicitly what we expect a discussion to look like.
  • BE PATIENT with yourself and your class.  There will be awkward silences, or classes won’t listen, or there may be management issues, but remember that it is a process.

There are concerns that we won’t be able to cover as much as we have in the past as we switch to this kind of discourse.  Yes, it may be slower, but as the students start to get more comfortable with these classroom conversations eventually the pace will pick up.  Also, remember that the goal is that the students will learn how to think and problem solve, so when they encounter something unfamiliar or more difficult they will have the skills and the confidence to tackle it.

The rest of the time was spent creating and teaching lessons in pairs and getting feedback from the rest of the group on our modes of response and teacher discourse.

Also of interest:
Interactive notebook reference

Styles and Strategies for Teaching High School Mathematics: 21 Techniques for Differentiating Instruction and Assessment by Edward J. Thomas, John R. Brunsting, Pam L. Warrick

Reading: Discourse that Promotes Conceptual Understanding

Dan Mayer’s  TED talk and blog, and his 101questions blog

A Mathematician’s Lament

Bell Work

I love this post from Mathy McMatherson about bell work.

We are required to do bell work at my school, and we are encouraged to make it as easy as possible – drilling previously taught skills.  And that’s okay, although I’d like to utilize some sort of A-B-C leveling on it to get to all skill levels.

What I like about this post is the way he scores the work.  “I take attendance, then I walk around and stamp students who have begun the bell work. If a student hasn’t done anything more than write their name on the bell work, no stamp… This is important because of how I grade bell work: each bell work is always worth 3 points. They get 1 point for the stamp – a motivation to not waste time and get started on time.”

He actually goes over the Bell work, which is frowned upon in my school.  But I may be willing to buck tradition if I find it works…

After I posted this, I went back to a post I had read last year from Square Root of Negative One that detailed her bell work procedure.  This avoids the going over the bell work by making that the responsibility of the early finishers.

So I’m going to combine the two concepts.  Here’s what I’ve got.

  1. Student picks up the bell work when she walks in the room. It’s right by the door.
  2. Walk around and stamp students who have begun the bell work. If a student hasn’t done anything more than write their name on the bell work, no stamp.
  3. Student works out bell work and raises a hand to check.
  4. I check the bell work and give out a green star. Student receives a green pen (with a green plastic fork or spoon taped to it!) of her own.
  5. Student checks in with partner to assist as needed and gives the partner a green star.
  6. Student checks in with people sitting nearby to see if green stars/assistance are needed.
  7. I continue passing out green stars and green pens until pretty soon, the green pens and green stars have branched out through the whole room and bell work is done.
  8. Direct students toward the review/extension problems. Could be guided questions to start thinking about the day’s lesson, or some ACT practice questions, or review, or extension/ challenge problems.


  • Make bell work a small percentage of the overall grade (in past years I’ve done 10%)
  • Worth 3 points per day
    • 1 point for turning it in (Seriously.  This is an issue with my students.)
    • 1 point for stamp
    • 1 point for attempting each problem (in pen or pencil)
  • The bell work is not handed back, since they have reviewed it with another student and the green pen.

Let’s see how it works.  It does mean that I have to photocopy bell work every day.  Previously I have used Kuta to put 4 questions on the overhead. I do have first period prep this year so if I forget to make them ahead of time I do have time before the day’s teaching gets started.

Also, they are supposed to start bell work as soon as they come in, so timing when I do the stamp could be problematic.  I could just give them each a 20 second “grace” period before the stamp.  But that’s a lot to keep track of for 25+ students.  It is possible that if I put in the effort early in the year, they will be trained and I can relax by November – maybe.

Another issue with bell work is finding the right number of problems – since they are supposed to start as soon as they get in to class, the earliest arriving students (usually the most conscientious students) have a seven-minute head start on the students that arrive just as the bell rings (usually the most disengaged students).  But I’m really not supposed to spend more than the first five minutes of class time on bell work, so bell work has to be something that takes the highest functioning students twelve minutes (7 + 5) but the lowest functioning students can do in five minutes.  (I don’t usually envy English teachers, but this is one that they have made in the shade, since their bell work is silent reading.)  I’m hoping that the peer correction with green pens will help with this problem.

Word Walls

The two articles about building vocabulary both mention word walls (they also mention graphic organizers, but that is a topic for another post).  Other than writing words on the board and leaving them there for reference, how do I implement a word wall?  I’ve seen word walls in English classrooms.  Usually there is a central topic (“New Orleans”) surrounded by related words (Hurricane Katrina, Mardi Gras, Jazz, gumbo, above-ground cemeteries, etc).  I always wondered if the teacher just wrote these words, or, if in a KWL, she had the students volunteer words that the central topic brought to mind.  It seems to me that there should be a larger activity to get students engaged with the words rather then just writing them up on the board and referencing them occasionally.

I got out my copy of Developing Readers and Writers in Content Areas by David W. Moore et al. to see what they had to say on the subject.  Here are the ideas that seemed most relevant to implementation in a Math classroom.

To select the list of vocabulary words, consider the unit and select the key words, making sure to include words with multiple meanings.  Students can be expected to learn ten words a week, so for a three-week unit, there could be a maximum of thirty associated words.  These are the words that will be displayed on the word wall.  (The blog Math=Love has a great Vocabulary Knowledge Survey that uses these concepts)

Have students do their own pre-assessment of the words using the following scale:
0 = I have never heard of that word in my whole life.
1 = I have heard it, but I have no idea what it means.
2 = I couldn’t tell you what it means, but I might be able to pick the right meaning from four choices.
3 = I can tell you a little about that word.
4 = I could put that word in a good sentence that would show its meaning.
5 = I could use the word correctly in discussion and writing.

When introducing the words, put the students in direct contact with the thing the word represents whenever possible, either with pictures or models.  I’m not sure how this would work with math words, although geometry students could perhaps make cut outs of different shapes and label them. The students can also create analogies for each word. Again, I’m not quite sure how this would work in the math classroom.

Another strategy is organizing words.  Put the words on the word wall and have the students put the words in to categories.  The categories can be provided for the students, but it seems to me a better technique would be to have the students group them on their own, and then justify their reasoning.

Have students write sentences that use the words. Have them share their sentences, then revise them if needed and share again.

August Journal # 2

I received the second of my two journals.  This one is Mathematics Teaching in the Middle School (a name that the NCTM is looking to change), and as I predicted, this had much more that was relevant to my classes.  In fact, I don’t think there is one thing in this issue that I couldn’t use in my classroom.  So I’ll just hit the highlights.

I-THINK I Can Problem Solve, by Sararose D. Lynch, Jeremey M. Lynch, and Johnna Bolyard explains a method to scaffold student discussion and to encourage metacognition.  It compares this method to Think-Pair-Share (a technique that I realized, upon reading this article, I’ve been doing wrong all this time.  Facepalm.).   I-THINK stands for:

I   –   Individually think about the problem
T  –  Talk about the problem
H –  How can it be solved?
I   –  Identify a strategy to solve the problem
N –  Notice how your strategy helped you solve the problem (I love this step!  Thinking about what you just did rather than rushing on to the next problem)
K –  Keep thinking about the problem.  Does it make sense?  Is there another way to solve it?

The authors also discuss how to implement this method, which I really appreciate.  It suggests starting with a classroom discussion on “Thinking about Thinking” and then moving on to a problem with no one solution.  Using this to introduce I-THINK and using visual aids and graphic organizers to reinforce the framework is recommended.  They also gave a rubric for scoring student work.  Even the authors said they only do a I-THINK lesson once per week.

Develop Reasoning through Pictorial Representations by Wendy P. Ruchti and Cory A. Bennett is another way to, as the article put it, “illustrate students’ understandings or misunderstandings.”  This reminded me of Twice as Less in that it provides a window in to the through processes of the students.  And again, it gets the students thinking about thinking, so that they can analyze their own thought processes.  The authors point out that this method is invaluable in teaching fractions, as it lets teachers know the flaws in student reasoning, and it gives the students a method for checking their own work.

There’s another article about building vocabulary through graphic organizers, but since it goes over the same ground as the article in Mathematics Teacher I won’t go in to it here.  There are a couple of projects and a nice open-ended discussion activity, plus a “palette of problems”  that could be used as openers.  Lots to use!

August Journal

Earlier this year, I bit the bullet and joined the National Council of Teachers of Mathematics , or NCTM .  I had resisted joining due to the cost, frankly.  And I wasn’t wrong.  My membership cost $115 for one year.  In all fairness, the cost was so high because I decided to subscribe to two of their journals rather than one.  I received my first journal this week.

Mathematics Teacher is geared toward high school teachers like me.  Unlike me, the teachers in the target audience clearly teach higher-level classes.  The majority of the articles relate to pre-Calculus or Calculus classes.  Generally, I teach lower-level classes:  Algebra 1 and Geometry.  But I still found some stuff to be excited about.

Vocabulary beyond the Definitions by Nancy S. Roberts and Mary P. Truxaw has some solid recommendations on incorporating techniques such as word walls and graphic organizers into the math classroom, along with some hints about typical trouble spots for English learners.  My Algebra students tend to do fairly well with vocabulary, but my Geometry students struggle mightily with the volumes of new words thrown at them during the year.  In my classroom this isn’t a phenomena restricted to English Language Learners, but as the saying goes, SADIE teaching is just good teaching.

Improving Student Reasoning in Geometry by Bobson Wong and Larisa Bukalov details a method that allows for differentiated learning in the classroom that is student-directed and allows the teacher to address one concept at a time.  (This is similar to the A-B-C technique discussed at the Summer Institute)  The authors acknowledge the challenges involved with creating such a curriculum, namely the large amount of time it takes and how to properly assess, but there is so much potential with this method.

I also found even more opportunities to spend money, namely in some interesting books advertised or referenced in the articles.
I’m intrigued by Danica McKellar’s books. I wonder if my students would find them relevant and/or interesting.
I’m curious about the advice in Success from the Start: Your First Years Teaching Secondary Mathematics, by Rob Wieman and Fran Arbaugh.  Even though I’m mid-career, I got very little subject-specific training in my credentialing program.

I don’t have a hyperlink for the second title because I found the NCTM online store difficult to navigate.  There is not a search function (so I couldn’t just put in the title or the author), and categories can not be filtered (for example, “Alphabetical by Title” or “Alphabetical by Author’s Last Name”).  I can’t understand why NCTM would go to the trouble to advertise a book they clearly want to sell, but then make it nearly impossible to find on their site.

Twice as Less

Twice as Less: Black English and the Performance of Black Students in Science and Mathematics, by Eleanor Wilson Orr

I picked up this book based on its title.  Having noticed that my black students often use different phrasing than other students (in particular, “take away” rather than “subtract” or “minus,” which to my way of thinking is a rather arcane usage), I thought it might help me close the comprehension gap.

The first examples given in the book brought recognition and a ray of hope:  I had seen papers like this and thought the students in question were hopelessly confused, or perhaps even suffered from some processing issue.  Indeed, the book brings up the fact that black students are inordinately put into special education classes when compared with their white counterparts.

Unfortunately, I found the book to be extremely dense and hard to follow as it parsed example after example, and I began to get discouraged that I could get through the book at all, much less glean any sort of useful information from it.

I felt much like my students must, reading page after incomprehensible page and feeling over my head.  But rather than give up, I went back to the table of contents.  I decided to start at the end, with the discussion of Black English usage and the conclusions that the author drew.  Once I read those, I went back and re-read the examples to have a better context for them.

The main issue seems to be the use of prepositions in Black English when compared to standard usage.  For example, some words are used interchangeably:  at/to, in/to, on/in, by/from.

It is harder to correct this kind of dialect than it would be to teach a whole new language, since in the latter case a foreign language speaker just has to substitute on word for another, whereas a dialect speaker must often dismantle their whole language and then build it up again.  This may account for why black students are seemingly “left behind” while immigrant students often are not.  The biggest difficulty is that we as educators may think we know what students mean when they try to explain their reasoning, but we may be missing the meaning entirely.

This is further complicated by the fact that students tend to combine nonstandard usage with standard usage, for example, using a standard usage given in class in a situation where it is not appropriate, or mixing standard usage for a generalized concept with nonstandard usage when asked to explain their own reasoning. The author diagrams these usages and calls them compound sentences, that is, sentences that seem to have two meanings happening concurrently, but are difficult to parse without essentially diagramming the sentences.

It would seem like this would be a job for these students’ English teachers to correct, but as the author points out, usage in Math and Science differs significantly from usage in other subject areas.  In other words, as Math teachers, we can’t just leave this for someone else to solve.  As we get in to the common core standards, particularly asking students to explain their reasoning, we will run into these issues more.  In fact, we may join our English teacher colleagues in taking home piles of student writing to parse.

Going forward, the goal is to create engaging lesson plans that seek to get students thinking in larger, problem solving ways.  Part of this is due to the upcoming change in California to the Common Core standards.  But a larger piece is trying to be more effective at getting students to retain content, think rationally, and not to give up.

I like the exercises that the author used for sample work.  They emphasize thought process over mathematical operations and thus get students to start thinking about why they do something, rather than relying on rote memorization, or worse, random association.  I have adapted some of them to use in the classroom: Twice as Less Exercises