Secondary Mathematics Expectations of Practice

The following document has been presented to secondary principals, assistant principals, and deans as a vision for establishing some district-wide best practices in math education: Secondary Mathematics Expectations of Practice

These expectations of practice come from the presentation Secondary Mathematics Achievement by Design. For administrators, here are some guiding questions as you visit mathematics classrooms:

  • What does “consistent use” of our instructional materials look like in your building?
  • What do you want to see more of in your math classrooms?
  • What does a sensible balance of print vs. digital resources look like?
  • Who is doing the talking, the thinking, and the mathematics in the classroom?

Setting The Tone – Norms for Math Class

Here are some resources to help establish a classroom culture that promote the Standards for Mathematical Practice, growth mindset, and students’ confidence in understanding mathematics. How can we begin to shift the culture of our mathematics classrooms to ensure students are doing the talking, the thinking, and the mathematics in class (and not solely their teachers)?

What’s missing here?  What mathematical norms do you have in your classroom to create an inclusive, student-centered environment that promotes access, learning, and success for all students in mathematics?  What norms would students create?  What norms could come out of the question, Would I want to be a learner in my classroom?”

(Want to learn more about promoting growth mindset in mathematics? Check out Jo Boaler’s book, Mathematical Mindsets)

What Did You Read, Learn & Think About This Summer?

Summer is great to spend time with family and friends, go on vacation, play, and recharge as educators.  It’s also an important opportunity to purposefully reflect, learn, and think about continuous improvement with a new school year just around the corner.  Here are some highlights from my summer reading, learning, and thinking:

Becoming The Math Teacher You Wish You’d Had by Tracy Johnston Zager (@TracyZager, #BecomingMath)

  • Many students view being good at math as the ability to “answer the teacher’s questions fast, right, and easily.”  But when these students go on to higher mathematics and work as mathematicians, they quickly find the difference between true mathematical thinking and simply being able to follow directions.  Math is not defined as following directions.
  • We have to invest in our educators, not programs, to fix math instruction.  Our own experiences as math learners heavily influence what we do as teachers.
  • The math studied in school is “finished,” abstract, and known, which promotes obedience in teaching & learning mathematics.  Mathematicians continuously play, create, wonder, ask questions, take risks, test conjectures, fail, try new things, make mistakes, seek connections, reason, and invent what is currently unknown; obedience is not doing mathematics.
  • It is the classroom environment, language, and behaviors of the teacher that will instill the proper mathematician habits in students and cultivate a growth mindset for all.
  • Students need opportunities to engage in descaffolded mathematical tasks that promote multiple entry points, multiple strategies, and risk taking (makeover tasks/problems from your textbook or search for new ones from sites like, Illustrative Mathematics, or Dan Meyer’s 3-Act Tasks; routines such as Number Talks and “Which One Doesn’t Belong?” can serve the same purpose, too).
  • Children have innate mathematician traits and natural curiosities before entering school; it’s our challenge not to derail into obedience and turn them off to math.  How do we create curious teachers around mathematics as models for students and their curiosities?

The innovator’s Mindset by George Couros (@gcouros, #InnovatorsMindset)

  • If we want students to become innovators and creative thinkers, we must first develop educators to do the same.  Innovation is a mindset.
  • An important reflection question Couros offers promotes empathy with our students by asking, “Would I want to be a learner in my own classroom?”
  • Connect and network with others via Twitter and blogs.  There is so much great stuff being shared out there and so many great practitioners to learn from!  Start a blog yourself to share your thinking and the great things happening in your classroom.  Not only will blogging clarify your thoughts and improve your writing, but someone may stumble upon your ideas, too.
  • Inspiring and empowering students requires reflection and examination of  how we teach and design lessons – moving from compliant to engaged to empowered.
  • Removing the traditional classroom labels of teacher & student in the classroom and replacing with “learners” creates a culture everyone in the classroom is a learner (including the teacher).

It is refreshing that both of these books are written from actual classroom and school practitioners that share dynamic examples from their colleagues in classrooms.  In addition, both authors stress the importance of reaching out and connecting with other educators and their open resources via Twitter, blogs, etc.  There are so many resources available to us, and it’s about investing in teachers, not programs, to develop the facilitation of dynamic learning environments.  (A shout-out to my colleague Zac Chase [@SVVSDLA, @MrChase] since this book reminded me of several ideas in Building School 2.0: How to Create the Schools We Need, written by him and Chris Lehman.)

I came across a couple of intriguing posts on (@TeachThought) over the summer, too:

How can these notions of constructivism, connectivism, the suggested 12 Principles of Modern Learning, and questions to drive inquiry to form a vision of math classrooms that go beyond checklists of standards, high-stakes assessments, and how we approach homework assignments?  In other words, how can we innovate math instruction and our math classrooms (that productively leverage the iPads and all resources available) for our students?

So what?

I am now excited to start this upcoming school year with new perspective on the tools and resources we have been afforded by the support of our community and visionary leadership.  We have an amazing opportunity in St. Vrain to transform teaching and learning with the iPads available to our students on a daily basis.  Let’s not squander this opportunity to simply take our “traditional” teaching and learning paradigm and try to simply force-fit it into a 21st century learning model and continue the status quo.  Let’s move beyond substitution in the SAMR model to true transformation.

And, math teachers, we have to stop using the excuse, “But math is different…those ideas just won’t work in the math classroom with all the content we have to teach.” Especially if our adopted instructional resources don’t force students to engage in inquiry where they are empowered to own their learning and create, it is that much more important we do it on our own and create those opportunities.  We have the access to resources, we just need to make the time.

Those are some of my summer takeaways.  What did you read, learn, and think about this summer?

Matt Larson: Math Education Is STEM Education!

As policymakers, districts, and schools attempt to define STEM, it is important not to dilute the “M” or take away resources and efforts to continuously improve and change mathematics education for students.  NCTM president Matt Larson wrote is his President’s Message, STEM Education Is Math Education!  From his May 17, 2017 message:

…There is no universally agreed upon definition of what constitutes STEM education. This complicates matters and allows each entity to define STEM education in its own way to fit its experiences, biases, and agendas—NCTM included. In some cases this leads to math or science classrooms where students build bridges or program robots, but fail to acquire a deep understanding of grade level (or beyond) math or science learning standards.  

Could K–12 math classrooms fail to have students engaged and learning the mathematics content and practices necessary to advance in the curriculum, but have integrated some technology, engineering, coding activities, or connections to science and be called a “STEM Program”? If students are not equipped to pursue a post-secondary STEM major and career, is it really an effective K–12 STEM program? My answer is no. No number of fun activities or shiny technology will overcome this fatal shortcoming. 

STEM programming and opportunities for students to engage in engineering design challenges, using design thinking and productive uses of technology, certainly appeal to the Standards for Mathematical Practice & Colorado 21st Century Skills and Readiness Competencies in Mathematics, but math lessons should offer the same opportunities on a daily basis.  It’s all about defined learning goals, intentionality in planning for instruction, and a desire to think beyond the textbook.  Take LEGOs for example – students can use LEGOs in a very imaginative and innovative way to design, prototype, and problem solve (based on the open-ended task they are given); however, LEGOs can also be used to promote following directions and using prescriptive steps to achieve a predetermined result (did we all make the exact same spaceship?).  Which one sounds like a STEM opportunity, and which one sounds like the typical math class?  Unfortunately, most will answer this question the same way.

In St. Vrain, our team of STEM Coordinators crafted the STEM by Design document, which focuses on actions and attributes of a STEM program based on beliefs and vision.  Most notably, this document is grounded on the notions of integration in core content areas, direct connections to standards, and focus on Tier 1 instruction.  This work was funded through a four-year Race to the Top district grant, and like any good design challenge, it is a prototype that will keep evolving and improving.

Reflecting on the School Year

As another school year draws to a close, it’s a natural time for reflection, relaxation, and recharging.  As you look back on the school year, a few questions to consider:

  • Who worked harder in my classroom, me or my students?
  • What did I learn from my students this year?
  • What new risks did I take?
  • What opportunities did my students have to investigate, communicate, collaborate, create, model, and explore concepts and content in authentic contexts, with or without the use of instructional technology?
  • What feedback do I want or need from my students to determine next steps?

Considering changes for next year?  Here are some thoughts and additional questions:

“It is unreasonable to ask a professional to change much more than 10 percent a year, but it is unprofessional to change by much less than 10 percent a year.” – Steve Leinwand

“A goal without a plan is just a wish.” – Antoine de Saint-Exupéry

  • What excites me?
  • What don’t I know? What professional development or new professional connections do I need?
  • What SMART Goal(s) will I set for myself next year? How will I hold myself accountable to the goals I set?
  • What matters to me as an educator?  What can I control?
  • How will I push myself to take new instructional risks outside my comfort zone?




Our Words Matter

A recent article in EdWeek shared the results of a survey:

A survey last month of more than 2,500 parents found that they generally rank math and science as lower in importance and relevance to their children’s lives than reading. Moreover, 38 percent of parents, including half the fathers surveyed, agreed with the statement “Skills in math are mostly useful for those that have careers related to math, so average Americans do not have much need for math skills,” according to the survey by the Overdeck and Simons foundations.

Another key quote from the article:

“Nobody is proud to say, ‘I can barely read,’ but plenty of parents are proud to stand up and say, ‘I can barely do math, I didn’t grow up doing well in math, and my kid’s not doing well in math; that’s just the way it is,’ ” said Mike Steele, a math education professor at the University of Wisconsin-Milwaukee who was not associated with the study…”We need to shift the mindset that math is just some innate ability that has a genetic component, and you are either a math person or you are not, to a conception that everybody can do math with effort and support … and to understand why that’s important.”

Our words matter. Changing our own beliefs and the language we use with children around mathematics is important if we want students to succeed in this area. Jo Boaler has created and published several resources around the notion of “Growth Mindset” in mathematics to support this change. Instead of saying, “I’m not good at math,” what if we begin to say, “I’m not good at math…yet!”  It gives us as adults room to grow and learn new things, too. And perhaps that’s the best model we can be for our children and young learners.

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.