Author Archives: Greg George

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.

Mathematics Acceleration by Design

In a recent NCTM President’s Message, Matt Larson’s “Mathematics Learning: A Journey, Not a Sprint” is a reminder that students should be considered for accelerated mathematics coursework (which is defined as skipping a course or two, mainly in middle school) if they must demonstrate “significant depth of understanding of all the content that would be skipped.” This is much broader than a narrow definition of mathematical ability through mere speed and accuracy with numeric computation and/or symbolic manipulation. From the post:

We must emphasize to parents, teachers, counselors, administrators, and students that the goals of learning mathematics are multidimensional and balanced: students must develop a deep conceptual understanding (why), coupled with procedural fluency (how), but in addition they also need the ability to reason and apply mathematics (when), and all while developing a positive mathematics identity and high sense of agency. All four goals are critical components of what it means to be mathematically literate in the 21st century.

In St. Vrain, we have a District goal of increasing student access and successful completion of Algebra 1 in 8th grade, and we also believe in our tagline, “academic excellence by design.” It’s the by design component that is worthy of our attention and efforts, ensuring students are on a pathway that encourages them to access and successfully complete a sequence of challenging mathematics courses throughout high school. For some, this sequence might allow access to college mathematics courses. That’s why our Guidelines for Recommending Advanced Middle School Math Students offer some guidance, yet are purposefully vague. We want to look at a comprehensive body of evidence (achievement data, classroom performance, student interest & goals, etc.) when recommending students for advanced mathematics courses, keeping in mind this is a multi-year decision and course trajectory that must continuously be reevaluated.

Beyond course recommendations, access and equity issues can arise when examining an accelerated mathematics program. Matt Larson offers the following questions from his post for reflection:

  • What are the demographics of students in eighth grade algebra? Do they match your district’s overall demographics?
  • What are the demographics of students in calculus or AP Statistics?
  • How do the demographics change from eighth grade algebra to AP Statistics or calculus enrollment?
  • Was the instructional climate not supportive of each and every student?
  • Was the instructional focus not on developing depth of understanding?
  • Were students accelerated into eighth grade algebra on the basis of computational proficiency, but without the conceptual foundation necessary to be successful in the long run?

One additional question to add: How many students that take algebra in eighth grade also take AP Calculus or AP Statistics (or beyond) in high school?

And two more questions to add for middle school: How do the demographics of students in advanced middle school mathematics courses (in preparation for algebra in eighth grade) match those of your school and our district? How are students identified to access these advanced courses?

As the recommendation and registration season for next year will soon be upon us, how would you and your school (or feeder system) answer these questions? What changes should be or could be made for next year to continue our St. Vrain pride of “academic excellence by design?”

St. Vrain Math Professional Development – Winter/Spring 2018

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

  • Multiplication: A Deeper Dive Grades 3-5 (Saturdays – January 20, February 10, March 3; 8:00 AM – 4:00 PM each day; LSC Timberline Rooms; 1.5 credits) 
  • K-5 TEACHERS – SAVE THE DATES!  enVisionmath2.0 Adoption Training – Summer 2018 (Choose ONE set)
    • May 30 & 31
    • July 25 & 26 OR August 1 & 2 (depending on 2018-19 school calendar)

    Registration will open by the beginning of March.

A Few Thoughts on Digital Homework Assignments

Digital homework assignments within digits and HMH A-G-A provide value-added supports to students such as instant feedback and on-demand help. No print textbook has ever been able to provide this kind of just-in-time support, so that is a benefit of using technology as a tool for learning mathematics.

When interacting with a digital interface, it is common for students to neglect paper and pencil and perform (or attempt) all calculations in their head. While the digital interface can entice students to forget about paper and pencil and diligently documenting their thinking and processing, we must insist this practice remains a core component of learning, doing, and practicing mathematics. Students should always record their mathematical thinking and processing in some form of a math notebook, regardless of whether it’s scored for completion or accuracy. If you are thinking digital assignments will save you (the teacher) time by not having to collect, grade, or review homework altogether, this is a misconception. An unfortunate possibility of digital homework is that students may end up learning more about “gaming” the algorithm behind such assignments and learning to cheat on assignments vs. learning and practicing mathematics as intended. Also, the help features can become a significant crutch for students, essentially walking them through similar problems where only the numeric values are different and the key becomes to learn the structure of the problem and where to put the numbers. In this case, students could score 100% on assignments without knowing any mathematics whatsoever. That’s why there must be a balance with paper-pencil and digital interface to collect accurate formative assessment data. In addition, do digital assignments provide the types of items that are worth students’ time and effort? Are the items mere rote practice exercises requiring low-level recall or simple procedures? Do these items allow students to apply the Standards for Mathematical Practice (particularly SMPs #1 and #3)?

So, why insist on having students write down their work and processing with digital assignments?

  • It provides a complete written record of exercises and tasks that can be referenced later. Digital platforms, at best, only archive the answers submitted, whereas a complete written record can be referenced at anytime with a useful amount of detail.
  • It allows for error analysis and precise feedback, whether scored for completion or accuracy. Digital platforms will only report right or wrong answers and cannot diagnose where errors occurred or the potential misconceptions that may exist.
  • Benefits of writing for long-term comprehension. Even though this is referring to note-taking through writing vs. on a laptop, the same ideas can transfer to mathematics with the value of writing things down and the mental processing involved.
  • Opportunities for metacognition, self-assessment, and reflection. Being able to review work, identify strengths and weaknesses, and reflect on next steps are much more viable with written work where annotations and editing are possible.
  • A written record makes thinking transparent (especially for complex tasks and problems). Rote practice exercises can be done without needing to precisely record all steps and thinking, but complex tasks and problems that require analysis, reasoning, and synthesis require some recording of steps.

Overall, digital homework assignments are not bad, as there are valuable benefits for students as they complete the exercises. Perhaps the most limiting aspect of digital assignments are the item types that must be used for the digital scoring. These items may pose multi-step problems that hint at using the Standards for Mathematical Practice and actually “doing mathematics,” (see the Mathematical Task Analysis Guide by Stein, Smith, Henningsen, & Silver, 2000) but the digital interface can reduce the student experience to multiple choice, filling-in-the-blanks, selecting from drop-down menus, or simply being asked to enter a numeric answer. In some cases, students may be asked to complete a specific line of thinking and reasoning for a multi-step item that may feel forced or inauthentic, not honoring alternative strategies or respecting how the student would approach and solve the problem. When the goal is for students to engage in multi-step items that require perseverance, problem-solving, justification, and applications of concepts (hopefully students have the opportunity to engage in these kinds of tasks regularly), simply give these tasks in a paper-pencil format where students must construct their own mathematical reasoning and communication of ideas. The resulting mathematical discourse in the classroom will be much more valuable and meaningful to everyone than viewing a data dashboard of right and wrong responses.

(Want more? David Wees shared some additional thoughts in a recent blog post.)

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.)

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?