Browsing Textbooks

Our textbook adoption committee has moved on evaluating different textbooks from various publishers and it’s been more fun than I would have thought. Previously in my teaching career I’ve only seen the textbook we’ve been assigned, so comparing and contrasting dozens of textbooks at the same times has helped me clarify my thinking about what I like, what I don’t like, and what I think a good text book offers.

I’ve noticed that I have some biases. For example, I’m sour on a textbook with too many technology activities. Common Core calls for using tools and technology appropriately, but my district is rather poor and we don’t have much of the technology that’s being referenced. A textbook that assumes all students will have a TI graphing calculator just get thrown in the metaphorical bin in my mind. Too much of the book is taken up with activities my students can’t do.

I love a book with projects and open-ended lessons. I have many resources for practice problems, so having if a textbook doesn’t have a ton, I’m not worried. I’m not a fan of “consumable” textbooks – that is, the kind students write in. I feel that they will get destroyed to quickly, and in my experience it is difficult to have a consistent supply of them.

Finally, I don’t understand why a textbook manufacturer would submit their text for consideration without including a Teacher’s edition. If I have a question like, “What standards does this lesson address?” or “How can I support English Language learners with this lesson?” the answer might be in the teacher’s edition, but I’d never know.

Textbook Adoption

I’ve been selected to be on my district’s textbook adoption committee as we select new materials to be used for the implementation of common core.  We were given High School publisher’s criteria to help us implement the standards as we review the materials.  We may use new materials, modify or combine existing resources, and/or analyze existing materials for strengths and weaknesses.  We will also be investigating digital and online materials to make sure that they follow the standards.

It is very important that the materials follow the standards as written and track the structure in the standards.  Connections and clusters that are in the standards need to be preserved, even when tackling specific learning objectives.  Content standards are designed to support focus and coherence. It is important to recognize that the standards are not all of uniform size and will require different amounts of time to master.

The idea is that fewer skills will be targeted, but those skills will be taught in a more in-depth and open-ended way.  There are three aspects of “Rigor” that are equally important:  (1) conceptual understanding, (2) procedural skill, and (3) applications.  These three aspects don’t always have to be together, although two or more may be.

It is important that all materials provide support for special populations, such as English learners or students with disabilities.  This would also include low skilled or remedial learners, who should deal with incomplete understanding of concepts taught in earlier grades inside the course level work, rather than in separate remediation.

There is a great emphasis on preparing students with prerequisite skills for postsecondary work, which, interestingly, can come from Grades 6-8.  These skills are developed and extended in high school-level work and form the backbone of student’s conceptual understanding and problem solving abilities.

The materials should have a variety of problems, such as conceptual problems with low computational difficulty and those that require identifying correspondences across different mathematical models and representations, leading students to observe structure.  Students will be spending considerable classroom time communicating reasoning, and as such, the teaching the language of mathematics and argument is supported in the materials, even for English learners or remedial students.

The materials need to support both problems, which teach specific mathematics concepts, and exercises, which build fluency and are performed in deliberate progressions.  It is also essential that the amount of both problems and exercises be balanced for optimal focus, concentration and coherence.  There should be a regular linking of conceptual understanding and exercising for proficiency.  Students should produce a variety of output such as explanations, diagrams, and models.

The teacher materials should support modeling of new methods, prompting broad student reasoning, anticipating a diverse array of student responses, and what mathematical behavior is desired from students, in addition to designing lessons that flow logically from unit to unit with strong correlation to the standards.

So overall, an immense task, but one that I am very interested to see unfold. I feel very grateful to have this opportunity to be a part of this process.

Open Ended Questioning

In our Math department meeting yesterday we talked about what how warm-ups are a great opportunity to introduce common-core thinking and the eight standards for mathematical practice.

In our school, students must begin working as soon as they enter the class – whether the bell has rung or not.  Ideally, a good warm-up should have three qualities:

1)      All students can do it.  This means it needs to be review, to some extent, or otherwise accessible.

2)      Content related. In my class that means my warm up needs to involve math, although it doesn’t necessarily have to be about what we’re studying now.

3)      Open ended.  Since students come in the room at different times, preferably it should be something that everyone can have a response on after a short time, but also can be explored more fully.

 The biggest challenge is making sure our warm-ups open ended.  For comparison:

 Closed question: “Graph the equation 𝑦=(2/3)𝑥+3 on a coordinate plane.”

 Open question: “Write down everything you know about 𝑦=(2/3)𝑥+3  and be prepared to compare your thinking with a partner.”

With the first question, a student who is behind might simply say, “I don’t know how to do that,” and give up, while an accelerated student would be done in under a minute.

But with the second question, even a student who lacks confidence in math could say something like, “2/3 is in front of the x.  We are adding a 3.  There are two letters.”  And an accelerated student would have many, many things to say.

“Write everything you know about (blank)” is a great open ended structure.  Others are error analysis (spot the error and correct it), compare reasoning (give an example and ask students to explain how we got there), or a really fun one, give the answer and have the students come up with the questions.

Right now, my warm-ups are focused on getting the students prepared for their CAHSEE, so I probably won’t get a chance to incorporate these open-ended warm-ups into my class until mid-March.  I’ll use the time until then coming up with a “bank” of open ended questions.

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

Summer Math Institute

We had a Math Institute at my school which took place the first week of our summer vacation.  (So, yeah, that sucked, but my goal is to BE POSITIVE and so I shuffled some things around and went.)  For several years now, my school has focused on Whole Class Processing, finding ways to get all students thinking and participating.  Now we want to find ways to go further, and get students to start reflecting on their own thought processes (metacognition).

Our department head had taken an online course through the University of San Diego called “Math is Not Just Numbers” that she recommends as a way to start facilitating classroom discussion.  Now, of course, in order to have effective classroom discussion, there must also be effective and consistent classroom management, but that is a discussion for another day.

The articles that we studied came from the National Council of Teachers of Mathematics (NCTM) .

Lesson planning goals:

Goal #1: Building Lessons
Creating lessons that focus on exploration rather than lecturing and that encourage students to visualize and discuss larger concepts.

Goal # 2: Modes of Response
Teaching students how you want them to think about things and encouraging group or partner discussion.

Goal #3: How You’re Asking for the Answer
Types of questions to ask students:  Explain your reasoning:  how did you get there?  Can you model it? Can you explain it?

Goal #4: Student Discourse
Engaging students in discussion and getting them to critique each other’s reasoning, encouraging them to look for counterexamples.

Some other ideas for encouraging students to think critically:

“Socratic Survivor.”  Points are given for each question level, and there are two groups of students arranged in a “fishbowl” design, where the inner group discusses, and the outer group evaluates the discussion.

A,B,C level responses on homework assignments

C level:  Do the problem; give the answer
B level: Show what formula or theorem is used.  Show all work.
A level:  Justify each step.  Explain how you got the answer.

This allows small and simple assignments that encourage whole class participation, while rewarding students who put in extra thought and effort.

Building the knowledge base of fundamentals so that the students have the working ability for higher-level thinking.  This can be done by introducing these higher-level thinking concepts with lower-level mathematics.

Resource for effective questioning: Developing Mathematical Thinking With Effective Questioning

Resource for interactive lesson plans:

Resources for test writing:  Smarter Balanced Assessments