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.

Click here to view the list of pilot teachers for 2017-2018.

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.

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.

Support Articles:

Resources for Creating Dynamic Mathematics Learning Environments

NCTM’s Principles to Actions describes effective teaching practices, conditions, and structures to ensure mathematics learning environments that benefit all learners.  A simple assessment of whether or not a mathematics classroom is learner-centered is to ask the following 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?

These curated resources, produced by reputable experts in the field, may help you create a dynamic learning environment that promotes mathematical understanding, thinking, and student voice.

I.  Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools (Ron Ritchhart)

II.  Mathematical Tasks

III.  Promoting Student Discourse

IV.  Using Questioning to Elicit Student Thinking & Promote Discourse

Some thoughts on questioning in the classroom:

  • If you want to know what a student is thinking, ask a question you do not know the answer to.  This avoids the “guess what’s in my head” type of questioning which can typically be lower-level recall.
  • Ask a questions that require students to respond with a second sentence (or more). How many of your questions require students to answer with a second sentence (or even a single sentence)?
  • What questioning sentence starters could you post in your classroom to encourage student-to-student questioning and explanation of thinking?  How can these questioning sentence starters become a classroom norm among teacher-student, student-teacher, and (ultimately) student-student interactions?

V.  Classroom Norms

So often, it is teachers that design the learning environment and set up the classroom rules and expectations.  But do students have a say in designing their learning environment that meets their needs?  Some questions to ponder:

  • Do students have choice and voice in your classroom?  
  • What is “tight” and what is “loose” in the classroom?  Who decides?
  • How often do students have the opportunity to provide you with feedback on how the class is running and what should be improved?

The Science Behind Classroom Norming

Some teachers have even adapted the Seven Norms of Collaboration from Adaptive Schools (or had students adapt these norms with their language) to build common collaborative school norms for students and adults alike.

“Mathematics” vs. “Tricks”

I recently came across an article on Twitter, “Nine simple math tricks you’ll wish you had always known.” My immediate response to was suggest renaming the article to “Nine simple tricks you’ll wish you had always known,” since these tricks are isolated in their utility and provide little-to-no understanding of mathematics and how operations work.  To me, the most troubling part of this article was the opening line, “It’s not entirely your fault if you’re terrible at math — maybe you just didn’t know the tricks to make any math problem a piece of cake!”  While these types of “tricks” can be introduced to students with positive intentions of helping them attain procedural fluency, do they actually help?  Do they promote a growth mindset in learning and understanding mathematics?  Do they build conceptual understanding to support procedural fluency and transparency behind algorithms?  And finally, are these tricks flexible – can they be applied in a variety of situations, or are they strictly dependent on the digits that appear?

Even though most of these tricks are elementary-based, they exist within secondary mathematics, too.  Ever heard of “copy dot flip” for division of fractions or “cross multiply and set equal” to solve a proportion?  Even the “F.O.I.L. method” for multiplying binomials attempts to illustrate the distributive property, but this trick fails when multiplying a binomial and a trinomial together. (See Phil Daro – Against “Answer-Getting” video and a recent blog post by Dan Meyer for additional examples.)

With all of this said, is there value in teaching these kinds of math tricks to students?  is there a way they can become “mathematics” and not just “tricks?”

Rethinking Practices with Assessments & Instructional Materials

Following Unit Plans and adopted instructional resources are necessary to help our students master standards, but we must also be mindful of the updated standardized assessments students will take each spring. Whether we’re referring to PARCC, PSAT, or SAT expectations, we must ask ourselves, “Do our assessments and instructional tasks hit the right target of expectations?”

If we’re loyal to The Teaching & Learning Cycle, we start with looking at the standards (i.e. Unit Plans) to determine what our students are expected to know, understand, and be able to do. But it’s the next stage that typically get short-changed: Creating the post-assessment that will determine what/if students have learned. Instead of focusing on developing the end-of-unit assessment and matching instructional lessons & tasks to the rigor of that assessment, we typically fall into the pattern and habit of marching through the adopted instructional resources (with little-to-no modification at times) and using the publisher-created assessments that mirror the format and rigor of the lesson examples and homework problems.

To see if these publisher-created assessments are hitting the right target of expectation, let’s compare assessment items and tasks to the PARCC/PSAT/SAT released items. If they are falling short (which they most likely will), how can we use these released and practice items (and other test specification resources provided to us) to raise the expectations for classroom assessments? Paraphrasing a colleague of mine, if PARCC/PSAT/SAT is the hardest assessment students take all year, then our classroom assessments do not match the appropriate expectations. Once the end-of-unit assessment is intentionally designed, only then can we ensure our lesson design and choice of instructional tasks will help students prepare for the assessment. It may require adaptation and modification of the adopted instructional resources and tweaking tasks slightly. Ideally, our lessons and tasks would regularly include and incorporate the Standards for Mathematical Practice & Colorado 21st Century Skills and Readiness Competencies in Mathematics where students are doing the thinking, talking, and the mathematics, not the teacher. (Mark Chubb has some nice blog posts around this – check out A Few Simple BeliefsFocus on Relational Understanding, and Aiming for Mastery? Dan Meyer’s April 2016 keynote “Beyond Relevance & Real World: Stronger Strategies for Student Engagement” is a good reference, too.)

For St. Vrain teachers, we’ve curated some other resources to assist with designing classroom assessments that meet the expectations of student mastery of the Colorado Academic Standards. Check out Illustrative Mathematics and our assessment tasks linked to applicable Unit Plans for ideas and prompts. The i-Ready Standards Mastery component can also be a great tool for PARCC-like items in a digital format.

Summary points to remember:

Have fidelity to the standards. These define and describe what our students must know, understand, and be able to do. There is room for prioritization here, identifying the conceptual “big ideas” and focusing instructional time accordingly.

Assessment communicates expectations. Are your classroom assessments communicating the right expectations?

Adopted instructional resources are tools. One of the Standards for Mathematical Practice is “Use appropriate tools strategically.” You are the highly-qualified teacher that builds relationships with your students, builds a classroom culture of mathematical discourse, and designs dynamic lessons to actively engage students in the content. Don’t outsource yourself to the script of a textbook or go on “autopilot” by blindly covering lessons without knowing the assessment expectations!  Check out Elementary Mathematics Achievement by Design or Secondary Mathematics Achievement by Design for more ideas around structuring your math class/block.