Rocks in a Jar

I was talking to a colleague recently about how we plan for a new school year, and the parable of rocks in a jar came up.

You’ve probably heard it – a professor holds up a jar with several large rocks in it.  “Is it full?” the professor asks.  Her students say sure.  She then adds gravel, which fills in the space around the rocks.  “Is it full?” she asks again. Her students, beginning to understand, say no.   The professor then proceeds to pour in some sand, filling the spaces between the gravel.  “Is it full?” she asks again.  The students are uncertain, until the professor pours water into the jar, filling the jar to the top.  “What is the moral of this lesson?” asks the professor.  A student yells out, “You can always fit more in the jar!” The professor shakes her head.  “No,” she says. “The moral is that if you don’t put the big rocks in first, they’ll never fit.”

Reflecting on this parable, I decided to select four big “rocks” – one for each quarter – that are the foundational ideas I want to cover this year.  Building lessons while keeping the four big concepts in mind would help ensure that they wouldn’t get lost in the day-to-day gravel and sand of the school year.

This year I will be teaching Math 8 and Integrated Math 1. So, with the California Common Core State Standards for Mathematics in hand, I picked my four rocks.

For Math 8, I picked these four:

  • Linear relationships, including their graphs and equations
  • Exponential relationships, including roots
  • Rates of change, including slope
  • Right triangle congruence and similarity, including the Pythagorean Theorem

For Integrated Math 1, I picked these four:

  • Arithmetic (linear) vs. Geometric (exponential) growth
  • Systems of Equations
  • Represent and interpret data sets, including bivariate data
  • Rigid motion in the coordinate plane

What do you think?  Do you agree with my “rocks”?


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.