Author Archives: Greg George

New High School Instructional Materials Adopted

On April 12, the St. Vrain Board of Education approved the adoption of new instructional materials for all high school mathematics courses (including middle school Algebra 1 & Geometry) after a yearlong pilot during the 2016-2017 school year:

For Algebra 1-Geometry-Algebra 2 (A-G-A), each teacher will receive the following:

  • Print Teacher’s Edition
  • Classroom set of 15 print student editions

All teachers and students will have a digital license (web and app access) to access the full program, including PDFs and interactive features.  The teacher license includes access to all ancillary teacher resources.

For the “electives,” (all courses above Algebra 2, including Intermediate Algebra), each teacher will receive the following:

  • Print Teacher’s Edition & teacher resources
  • Classroom set of 30 print student editions

Students will have access to an eText that can be downloaded onto the iPad mini for offline access.

St. Vrain Math Professional Development – Summer 2017

The following mathematics classes will be offered through the Office of Professional Development this summer.  A minimum of 10 participants are necessary for a class to run. 

Elementary Mathematics Pilot Selection for 2017-2018

St. Vrain Valley Schools will pilot Pearson’s enVisionmath2.0 for elementary mathematics during the 2017-2018 school year.  Since fall, a committee of District teachers and administrators have been reviewing and evaluating candidate programs, leading to this pilot selection.  Thank you to all of the St. Vrain staff, parents, and community members that participated in providing feedback in this process.  The complete timeline for the pilot and adoption is as follows:

  • 2016-2017: Review & evaluate candidate programs; determine one program for pilot
  • 2017-2018: Pilot selected program in 1-2 classrooms per school
  • 2018-2019: Implement selected program in all St. Vrain classrooms

Click here to see the adoption committee’s meeting notes and resources associated with this process.

Creating Classroom Cultures of Mathematical Thinking (Inquiry)

Last fall, I posted “Resources for Creating Dynamic Mathematics Learning Environments” that included a variety of resources and tools to create learner-centered environments, but that post mostly concentrated on teacher actions.  This post aims to set a vision for student actions in the mathematics classroom.  (Sorry teachers, you are not off the hook for this one. Your actions make this vision possible for students!)  Once again, the goal is to continuously reflect on these questions: Who is carrying the cognitive load in your classroom during mathematics lessons?  Who is doing the mathematics?  Who is doing the thinking and talking?

(1) The role of student-driven questions.  Teacher questioning is important in facilitating classroom discourse and making important connections, yet we oftentimes leave students out of the question posing process.  Fostering Student Questions: Strategies for Inquiry-Based Learning and Creating a Culture of Inquiry offer some ideas and protocols on how to develop a culture of student-driven inquiry based on questions, justification, and choice.  Of course, good questions come from good tasks and learning experiences crafted by teachers.  This is where the distinction between problems and exercises is important (see “Rethinking Mathematics Homework” post for the difference between the two). Good questions come from good problems where the mathematics needed to solve isn’t readily apparent.  Exercises can generate questions, too, but the nature of those questions is generally more procedural and structural in nature.  (Note that exercises can be turned into problems given the right context and perhaps some tweaking/editing.)  Yet, none of this will happen without classroom norms that challenge the established and status quo culture of many mathematics classrooms, which is a culture of correct answers.  To transform classrooms into a culture of questions, Jo Boaler offers some Positive Norms to Encourage in Math Class that challenge many of the preconceived notions of mathematics that permeate throughout the United States and our own experiences as learners of mathematics.

(2) Intentional Focus on the Standards for Mathematical Practice.  While all eight Standards for Mathematical Practice are important in terms of establishing habits of mind for emerging mathematicians K-12, particular emphasis should be made on two of them if we wish to transform our mathematics learning environments:

  • SMP #1: Make sense of problems and persevere in solving them.
  • SMP #3: Construct viable arguments and critique the reasoning of others.

If we make strides with these specific practices, our mathematics classrooms will feel and look very different.  In addition, the role of student questions fits perfectly with these specific practices.  But let’s not just limit these to the mathematics classroom, as these specific practices transcend all content areas and are good habits of mind for everyone within an organization.  As a school department/faculty, when was the last time the team made sense of a school problem and persevered in finding a solution?  When was the last time adult discourse occurred at a department/faculty meeting around critiquing reasoning and arguments (and not the people making those arguments)?  When was the last time a meeting was driven by questions posed by the participants?

If we want students to carry the cognitive load and do the mathematical thinking in our classrooms, then we have to model these actions.  As teachers, do we model posing questions around a problem?  Do we model perseverance in solving problems, even when a solution strategy might lead to a dead end?  Do we model complete mathematical arguments using proper academic vocabulary?  If students never see their teachers perform these actions, it’s a tough ask to demand of them without some frame of reference.

Still not sure what this is all about or what the vision looks like?  Mark Chubb (Instructional Coach in Niagara) offers some ideas in his blog post “Quick fixes and silver bullets…” that illustrates common practices that suppress mathematical thinking.

Enactive – Iconic – Symbolic Concept Development in Mathematics

Check out this “PLC Byte” on the development of mathematical concepts through the Enactive – Iconic – Symbolic progression.  Some discussion questions as you view the short webinar:

  • What are the concepts of Enactive – Iconic – Symbolic?
  • How do these concepts relate to other educational theories?
  • How can the concepts of Enactive – Iconic – Symbolic support the development of mathematical understanding by students?

Graham Fletcher’s Progression Videos

Graham Fletcher, a K-8 mathematics specialist in Atlanta, GA, has created a series of videos illustrating how concepts are developed across the elementary grades to promote conceptual understanding and procedural fluency.  His current videos include the Progression of Addition and Subtraction, Progression of Division, Progression of Multiplication, and Progression of Fractions: The Meaning, Equivalence, & Comparison.

Click here to view Graham Fletcher’s videos.

There is also a document, Development of Number Sense & Computation in Grades K-5, that provides more detail with concept progressions across grades K-5 with specific representations and references from the Common Core State Standards.

 

Best Practices in Mathematics Education

In their recent book, Balancing the Equation: A Guide to School Mathematics for Educators & Parents, Matt Larson and Tim Kanold summarize research-based instructional recommendations from the National Research Council, which are also reflected in NCTM’s Principles to Actions: Ensuring Mathematical Success for All.  The recommendations have been paraphrased and added to the presentations Elementary Mathematics Achievement by Design and Secondary Mathematics Achievement by Design.

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Rethinking Mathematics Homework

The typical rationale for assigning mathematics homework is for students to independently practice what was taught in class.  After all, practice makes permanent.  While the opportunity to practice must be present for students to gain procedural fluency, we should rethink mathematics homework as an opportunity to change mindsets, too, around mathematics and perseverance with problem solving.  Don’t get me wrong, skill development with repetition is important.  Just ask any basketball player about practicing their on-court drills or a musician practicing their scales as part of their craft.  Yet, it isn’t the drills or scales that are the end goal for the athlete or musician; instead, the goal is to use those skills to compete against another team on the court or perform music with an ensemble.  So why do computation and rote skills become the goal for learning mathematics?  Instead of sending students home to complete 10-20 practice items that have no intrinsic value whatsoever, what if we sent home only 4-5 items with a couple that specifically focus on applications, problem solving, building arguments, and perseverance (i.e. the Standards for Mathematical Practice & Colorado 21st Century Skills and Readiness Competencies in Mathematics).

In other words, assignments should have a mix of problems and exercises.  What’s the difference?  From the High School Publishers’ Criteria for the Common Core State Standards for Mathematics, “the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Problems are problems because students haven’t yet learned how to solve them; students are learning from solving them.”  If students are truly asked to grapple with problems, then we shouldn’t expect right answers coming to class the next day (in contrast to exercises, where correct answers should be expected).  This means documentation of an initial strategy may be sufficient for homework, knowing more progress will be completed in class through a small-group or whole-group discussion.

What are the potential benefits are rethinking mathematics homework in this way?  

  • Students may actually complete the assignment if it is shorter in length.
  • Assigning a few purposeful exercises can reduce the chance that students may repeatedly practice incorrectly without timely feedback.  The assumption is that timely feedback would come in class through daily formative assessment and opportunities to practice.
  • Reducing the assignment set focuses the homework review routine in class the next day to meet specific learning goals.  Adequate attention can be given the to skills along with appropriate discussion of the exercises.  This is much more purposeful than teachers asking, “which ones would you like to see?”  The teacher gives up all control and the opportunity to build connections is lost when homework review is left up to going over randomly-requested problems from the assignment set.
  • Having a couple of contextual application problems that incorporate the Standards for Mathematical Practice help students investigate non-fiction texts in math class, which is a focus of the English Language Arts standards and all other content areas.  Breaking down a problem to identify what it is asking is more cognitively complex than performing the computation(s) to solve the problem.  Close reading and citing evidence have a place in mathematics, too!

Instead of treating mathematics homework as a right of passage that generations of mathematics students have endured, let’s focus on quality assignments that reflect rigor, as defined within the Colorado Academic Standards (conceptual understanding, procedural skill & fluency, and applications with equal intensity).  Less can be more, and we may find shifts in mindsets and attitudes toward mathematics in the process.

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