Category Archives: Elementary

The Only Math App You Need

When I received my first iPad, I, like many, engaged in the process of rapid app downloading. After attending a workshop or seminar around using iPads, I would hastily download any and all apps recommended by the presenter. And like most that go through this process, I found that I used few (if any) of those apps taking up space on the device.

Sure, we’ve curated a modest list of recommended “educational productivity and creativity apps” for mathematics in St. Vrain, and we’ve purposefully stayed away from any apps that push content to students or resemble any type of rote practice or digital worksheet. So if apps aren’t the way to go, how can these learning devices be used productively in the teaching and learning of mathematics?

Looking at the Standards for Mathematical Practice, these three stand out with respect to communication of mathematical ideas:

  • MP1: Make sense of problems and persevere in solving them.
  • MP3: Construct viable arguments and critique the reasoning of others.
  • MP6: Attend to precision.

No doubt, if we want to assess students’ understanding of mathematics, we must focus on their mathematical communication and transparency in their thinking and processing; yet one key barrier appears to exist, repeated by most teachers: “How can I possibly meet with every student one-on-one to do this regularly?”

This is where technology can facilitate capturing these moments and providing meaningful data to teachers beyond paper and pencil means. As part of our District-provided apps, Explain Everything is pushed out to all iPads for students and teachers. Forget all of the other math apps you wish your students could access, as this one allows you to hear and see what they are thinking as they work out a solution to a problem. That is, if they are given a worthwhile task that allows them to use the Standards of Mathematical Practice listed above and not simply practice rote procedures.

Are you leery of Explain Everything or see potential limitations in its functionality? Then simply capture a video right off the iPad using the camera. What better way to record real time student thinking and gain more formative information than any paper-pencil assessment.

Making the Most of Adaptive & Individualized Learning Tools

Digital tools can help streamline the process of assessing students, identifying perceived student skill gaps and strengths, and prescribing appropriate resources (via algorithms) to help students “fill” those gaps or extend learning. These forms of adaptive learning and individualized learning efficiently curate resources for each student in a matter of seconds, versus the time and energy a teacher would spend to complete the process manually and search for appropriate materials. These digital tools also allow for students to continue learning and make progress at anytime and anyplace, not just in a classroom or at school.

These tools, however, can present a conundrum for teachers and students with respect to time management and how minutes during the school day are utilized. But let’s not confuse these tools and their intentions with digital practice or homework assignments. David Wees and others have offered criticisms and drawbacks of such digital practice, which is presumably taking place outside of the school day to practice or apply what was learned during the day’s lesson. This conundrum is about taking instructional time during the school day for adaptive and individualized learning.

Michael Fenton gave an impassioned ignite talk in 2014 on this issue, cautioning us to resist the temptation of connecting students to devices in the classroom for consumption in an isolated environment and, instead, using classroom technology to promote collaboration, conversation, and creativity.

Here are some questions to ensure we using these digital tools for adaptive learning and individualized learning in the most productive ways:

  • Are students asked to keep a written record (i.e. math notebook/math journal) as they progress through assignments/lessons that can be referred to later?
  • What opportunities does the student have to set goals, identify strengths, identify areas of challenge, and reflect on their learning when engaging with these tools?
  • What opportunities do students have to create something that demonstrates their learning from the assignments/lessons they complete?
  • What role does the teacher play as students are actively completing assignments/lessons? How does formative assessment look in this setting?
  • What peer-to-peer or student-to-teacher conversations about learning are taking place while completing these assignments/lessons?
  • How does the content of the digital assignments/lessons connect with the class learning goals and current unit of study?
  • Even though a digital dashboard may show that students have successfully completed a series of assignments/lessons how do we know they learned anything? How do we know what misconceptions still exist and what questions they still have?

This isn’t meant to be a repudiation of these adaptive learning and individualized learning tools. After all, they offer a utility that schools that schools and teachers are seeking, especially when students are not performing at grade level or need additional challenge to stay engaged; moreover, these resources are consistent with the “on demand” and customizable nature of content we enjoy in many aspects of our lives. These tools can serve as valuable assets to student learning, as long as we use them in conjunction with teachers and human-to-human interactions, not in place of them. This is a question about balance and ensuring the human relationships and interactions that are at the heart of dynamic learning environments are never replaced by artificial intelligence algorithms and data dashboards.

On the Road to Modification and Redefinition

“Through the Learning Technology Plan, students and teachers have the tools they need to investigate, communicate, collaborate, create, model, and explore concepts and content in authentic contexts.”

Based on this vision of the St. Vrain Learning Technology Plan, how are we doing? Yes, we have the devices in our students’ hands (1:1 at secondary), ensuring access and equity. Yes, our new instructional materials adoptions are more digitally-based, packaged with dynamic content. Yes, we have Schoology as a learning management system for workflow, communication, and assessment. And yes, we have the Google apps suite available for staff and students. We have checked the boxes that earned us the distinction of a top 10 district for digital curriculum and integrated technology use in the country two years in a row. The question, however, is around how these devices and tools are being used: Are our students digital consumers or digital creators? (This can also be described as the digital use divide, as defined in the National Education Technology Plan.)

True, digital instructional materials in mathematics offer features and supports that no print textbook will ever provide. Seeing animations of concepts and relationships is much more likely to stick that arduously performing the same tasks with paper and pencil. Students getting instant feedback and help supports with digital assignments provide on-demand help and reteaching opportunities instead of having to wait until the next class period. But these value-added features still describe a student consumption-based model of approaching content; we’ve simply substituted print, static resources with digital, dynamic resources (remember the SAMR model?). So how do we move up to Modification and Redefinition and how might we support our students in transforming the school experience? The answer is not quite that simple in practice: have them become self-directed content creators using the devices and suite of tools at their fingertips.

Math educator Michael Fenton did an ignite talk in 2014, Technology and the Curious Mind, urging educators move away from Indifference, Consumption, Competition, Isolation to Curiosity, Creativity, Collaboration, Conversation with use of technology in the classroom. In 2015, Rick Wormeli published Moving Students from Passive Consumers to Active Creators, where he claims, “this is a call for more project-based learning, integrated learning, and inquiry-method across the curriculum. These three methods provide more opportunities for true student creation than simply listening and repeating content.” In era where a student can simply Google information just-in-time instead of relying on textbooks and teachers in classrooms, students need to engage in tasks where answers cannot simply be Googled (trivial facts) or solved by Photomath (procedural, rote exercises). Curious about project-based learning and have no idea where to start? It’s okay, anything new can be scary and lead to more questions than concrete answers, especially since most of us grew up in a traditional educational setting from elementary school through college (I sure did!). But since Google and Photomath are here to stay, the old paradigm of just-in-case education needs to be transformed using just-in-time technologies and resources. Let’s figure this out together, brainstorm, fail, succeed, and learn from each other, just like we expect from our students on a daily basis.


Do We Really Promote “Growth Mindset?”

Many of us have read the trending books (i.e. Mindset, Mathematical Mindsets), attended a motivating keynote presentation, explored Jo Boaler’s, and posted the inspirational classroom posters reminding students of growth mindset language in learning mathematics. But as educators, do we really promote growth mindset in our classroom rituals, routines, and practices? It’s an attractive buzz term these days with the hopes of reversing a cultural trend of apathy toward mathematics and the notion that only “math people” are destined to understand, appreciate, and find usefulness in the subject (just like there are “reading people,” destined to be the only ones who can read and productively use literacy skills on a daily basis, right?).

Promoting a growth mindset in our mathematics classrooms cannot stop at simply putting posters on the wall or responding with “yet” immediately after a student claims, “I don’t get it” or “I can’t do this.” It’s about changing our paradigms about teaching, learning, assessment, classroom culture, rituals, routines, language, and grading. Basically, embracing a true growth mindset around student learning in mathematics means abandoning the teacher-centered and compliance-based classroom paradigm most of us experienced as students in K-12 schooling and in college lecture courses. It’s a tall order that requires great reflection and examination of core beliefs. Here are some questions for self-assessment, reflection, and conversation to gauge if growth mindset is really being promoted:  

  • Who is doing the talking, the thinking, and the mathematics in your classroom: you or your students?
  • How do adults perceive mathematics across the school? Do students interact with adults that claim they cannot do math or are not “math people?”
  • When students are chosen to present in class, how are those students chosen? For right answers, correct processes, a mistake, or an interesting idea worth discussing?
  • Is speed implicitly honored in your classroom?
  • Do students demonstrate stamina in wrestling with in-class learning tasks or do they wait for the solution after minimal effort.
  • How is praise given? For answers or for thinking and perseverance? Do you have students that identify themselves as “smart?” How does that label impact their behavior and academic habits? (See The Problem With Praise)
  • Do the phrases “This is easy” or “This is too hard” permeate your classroom? (Why might these be troublesome phrases?)
  • Are students given the opportunity to experience productive struggle to find relationships, make connections, and use multiple representations, or are the opportunities based on replicating procedures and getting correct answers?
  • How are mistakes handled in class? Are they viewed as opportunities for discourse or something to be “fixed” and admonished? How is feedback provided to students when mistakes are made or uncovered?
  • Is feedback to students asset-based or deficit-based?
  • Is student reflection part of the classroom, including opportunities for self-assessment and goal setting?
  • Do assessment practices represent a static snapshot of understanding based on a pacing guide or are students able to demonstrate their learning at any time over the course of the year, building a body of evidence?
  • Whose classroom is it? Is it teacher-centered with lots compliance-based rules, procedures, and routines or is it a student-centered, driven by their questions, ideas, and sense of empowerment?
  • Are the questions posed to students answer-driven or ideas-driven?
  • Elementary: Observe various flexible groups in your building and across a grade level. What do you notice? Describe the instruction and classroom culture in these classes. Are there implicit messages being sent to students?
  • Secondary: Observe “honors” and “regular” classes in your building. What do you notice? Describe the instruction and classroom culture in these classes. Are there implicit messages being sent to students?
  • What other questions should be added to this list?

I’m not going to pretend to be an expert here or judge other who are giving growth mindset principles a try in their classrooms. As a high school teacher, I was more fixed mindset that I wish to admit, and my classroom was much more teacher-centered than I wanted it to be. The influences and practices of my K-12 teachers, my college professors, and even my cooperating teachers when I student taught formed a strong schema about how teaching mathematics was supposed to be. Fortunately, through some professional learning opportunities early in my career, I was able to recognize that I was only teaching the students that learned like I do in my classroom and not all of the students in my classroom. That recognition and acknowledgement alone was the first step in improving my practice as a young teacher. It was challenging to give up some traditional beliefs around assessment and grades, and I didn’t make all of the progress I could have. That’s why growth mindset is more than just a buzz term, posters on the wall, or catch phrases. To do it well and for it to actively live in our classrooms, we all (teachers and students) have to challenge our schema and beliefs around mathematics, what it means to “do mathematics,” and the learning environments that best represent what mathematicians actually do.

(Want more? Mark Chubb shared some similar thoughts and resources in a blog post last year.)

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 2016-2017 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.