Thinking about educational and curriculum reform for a rapidly changing world, these are questions I often wrestle with and don’t have clear answers.

- What is the appropriate balance between using district adopted, commercially produced instructional materials vs. teachers curating and creating their own lessons collaboratively with colleagues that are aligned to standards? What should we encourage as a system to ensure access and equity
**and**recognize the amazing talent and professionalism of teachers? - At what point should we give simply give students calculators in upper elementary and middle school to ensure they can access grade level tasks and content and not get bogged down in potential computational fluency shortcomings?
- At what point do we “give up” on having students master computational fluency in elementary school (and long division in 6th grade) in order to focus on critical thinking, understanding, and problem solving (i.e. things that require human capacity and cannot be optimized with technology)?
- How do we cultivate curiosity within students around mathematics when most adults learned mathematics in a prescriptive, procedural way?
- Do teachers and students see mathematics as a place for creativity, beauty, and joy?
- What would happen if we thoughtfully and strategically eliminated a third to a half of the standards at each grade level (or high school course) to spend more time developing mathematical reasoning and problem solving with non-routine, large-scale problems and tasks? Would we get the same or better outcomes than we’re getting today based on how academic achievement is measured?
- How much symbolic manipulation is required for demonstrating “mastery” of algebraic concepts? When do we unlock the power of graphing utilities and other technology tools to explore relationships and solve relevant and interesting problems using algebraic concepts?
- Is there a compelling reason why
**all**students must study polynomial, rational, logarithmic, and trigonometric functions in high school? Would we be better off focusing on linear, exponential, and quadratic functions for all students, and leaving these other families of functions to advanced mathematics courses based on student interests and postsecondary goals? - What if we retooled the “traditional,” required high school course sequence of Algebra 1, Geometry, and Algebra 2 with the following sequence of one year courses:
**Algebra 1**(focusing solely on linear, exponential, and quadratic functions),**Geometry**, and**Statistics & Probability**? Would this better serve all students for a variety of contexts beyond high school? - Why is practice-based homework a quintessential part of learning mathematics in school? How might changing the role of homework drive changes in instruction and uses of class time?

What are your thoughts? We always have to work within constraints beyond our control, but that doesn’t stop the design thinking process from creating a better outcome or tweaking part of the whole. What questions do you wrestle with? What would you add to this list?

]]>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.

]]>**enVisionmath2.0 Adoption Training – Summer 2018 (Choose ONE set)****May 30 & 31****August 1 & 2**

Register by grade level through the St. Vrain CourseWhere system

**Desmos.com – A Deeper Dive**(Wednesday, June 6; 8:00 AM – 4:00 PM; Lincoln ESC Spruce Room; 0.5 credit)**You: The Mathematician**(Wednesday-Friday, June 6-8; 8:00 AM – 4:00 PM each day; LSC Evergreen Room; 1.5 credits)**MQI Coaching Workshop**(Monday-Thursday, July 23-26; 8:00 AM – 3:30 PM each day; Learning Services Center; see description for credit & compensation options)

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

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,

*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 21 ^{st} 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?”*

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.)*

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.

Resources:

]]>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.)*

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?*