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Boyd at Northampton

Locally Sourced, organic, GMO-free mathematics


Northampton High School, Northampton, MA



The brief and pertinent biography:

I grew up in Needham, a suburb of Boston where I lived from 1st grade through the start of my freshman year at the high school. At that point I relocated to Vermont - the other state I call home - where I finished both high school (Woodstock) and college (Middlebury). I moved to Seattle and worked for a few years on sewing machines and heat presses, manufacturing kayak equipment, until I returned to school for teacher certification. I came close to certifying as a science teacher but the draw of mathematics was too much and I earned my Masters in Teaching in 2001 from Seattle University. I returned to my (first) hometown and taught at Needham High School for 6 years. For another 6 years, I explored a career in research and statistics, but I missed the stimulation and excitement of the classroom so and in 2012 I moved to the Pioneer Valley and started looking for teaching positions. Hired in August, I scrambled a bit to get ready for my first year back and it's been wonderful to be back in the classroom ever since. 


My upbringing had me outdoors - a lot. I've been well steeped in plenty of "adventure" sports, and played on a traveling club ultimate frisbee team or three for close to a decade. That last habit had me living in densely populated urban areas ever since leaving bucolic Vermont in '97. I was living back in the woods in Huntington for a while but my family moved last Thanksgiving to Florence. I'm off of Ryan Rd with my wife and her two children. She is a beautiful dancer by training and a world class teacher and coach of finding your artistic voice - a talent she is putting to use in her own studio, Ascendance Inner World Arts in Florence. She also sat opposite me in Spanish back in 6th and 7th grade.

I love food, and learned early on that good cooking is mostly born from not being afraid of it. Consequently, we try lots of experimental dishes at our house (though I've had to tone down the spice for the sake of the younger ones). Quiet water sea kayaking is the new thing for us, though. We're working towards an overnight trip somewhere once everyone is big enough to paddle their own.

The dream is to one day, many years from now, cash in on the bountiful salary of a public school teacher so we can retire to live on a 40' sailboat named "Elena" and we'll literally sail off into the sunset.

What to Expect from me as your teacher

I thrived under the older and more "traditional" methods of high school math instruction, but I appreciate that math is a terrifying subject for many students. My first objective in the classroom is to make it a safe place and one where the mindset of "I'm not good at math" amended to include the growth mindset twist of "not yet!" You can expect me to work hard to make our classroom a place where we are challenged, play with math, and get better at making mistakes that teach, not mistakes that shame.

I am not your answer key.

I will challenge you to *think* above all else. This means that I am very cautious about answering too often the question "Am I doing this right?" Do students find this frustrating? You bet - but that's not necessarily a bad thing. A mentor I had early on made the wise point that as a teacher, you want to keep the students in that low level of frustration that comes from not quite understanding something - a sweet spot of not so much that they get upset or defeated and quit, but just enough so that they are really dying to reach that insight and "get it." I won't leave you adrift, though. I will coach you to find confidence (or skepticism, as appropriate) for your ideas and encourage and support you to collaborate and consult with your classmates. More often than not, collectively we will move in the right direction, though I am always present and attentive to nudging us back on track or over the taller obstacles as we work together.

This is amazing.

Contradictory to the experiences of many - especially those of my generation - I strive to make our work fun. And math *is* fun! Math is not a long, scary list of skills that need to be mastered. Math is a language of thought, and a world full of curious wonders Asking the right questions can spark surprising conversations, with genuine problems that demand a surprisingly high degree of creativity to solve. It is often asked of math "When will we ever use this?" and to that I reply "Who cares! It's amazing!" We don't ask that question of art history, critical reading, the study of the causes of warfare, or dance - these all have obvious benefits to a healthy, connected, and thoughtful life. So too with mathematics, if we just approach it with a better attitude (and lots, and lots of pedagogical training in the teaching of the stuff, to be honest!)

To boil it all down, I will exhort you to Think Hard, Think for Yourself, and Take Chances - I'll be doing my level best to make sure no one falls too far and that they will always be helped back up.


What I expect from you, the student

Often in the first week, as part of building the culture of the class, I ask students (you) to reflect on the following quotation:

"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where explorers often get lost."   - W.S. Anglin

I don't agree with much of Anglin's writings about math, necessarily, but what he captures very well here is my attitude as I approach the classroom. To a certain extend, I will want you to adopt this attitude as well.

We are on an adventure!

That sounds like a kindergarten teacher but it's still true in high school. To be in a state of learning, you must be on the edge of your understanding, pushing yourself into an unknown land. This is scary stuff, at times, especially if you've been burned in some way (bad grades, academic shaming of some sort, etc). I'll do my best to keep it exciting and not scary, but I will need buy in from you that I will make you safe but not necessarily comfortable - you must adapt to this, make friends with the discomfort, and apply yourself to find your way around, mapping out your own understanding, and thereby broadening your own world.

Here is what you need to do:

  • come to class prepared materially
    • with something to write with
    • something to write on
    • your work completed
  • come to class prepared mentally and emotionally
    • sleep adequately
    • eat breakfast and lunch
    • be open minded about what is going to happen, and game to participate

Grading Practices

I follow a practice called Standards Based Grading, a modern and research-supported system of aligning grades to specific Learning Standards - the nuts and bolts of what we are learning in the curriculum. The idea is that grades best work as feedback for students if they communicate current understanding and performance on specific learning objectives ("proficient" working with the pythagoriean theorem) as opposed to performance on specific learning instruments ("88% on the Unit 6 Test")

I go into detail on the Grading Practices page here.

Feedback and Grades

Grades are what goes on a report card. Grades eventually live in your transcript, which you might send to a college admissions board or a prospective employer, and eventually no one will ask for them anymore because your achievements in life after you leave high school are so much more interesting than what you did in high school.

Feedback is the information you and I exchange while we sort out what you have already learned and what you need to learn next. Feedback comes in many forms, and the most common is our day-to-day experience in class. You get feedback from me during class discussions when I am guiding, prompting and correcting. You get feedback when you ask a question, and you even get feedback while you are working with a classmate on a challenging problem. Some feedback I use both to communicate to you what you have learned and what should be next *and* I use it to create a grade for your report card.

Grades are for those outside of the classroom, to tell them where you ended up at the end of our work together. Feedback is for those inside the classroom and is a key part of the process of learning. In fact, in many ways, assigning a grade signifies the end of learning - "this is where the student ended up in their learning process." Feedback is ongoing, part of a process of growth. I will talk here mostly about what you can expect for feedback, and a little about how I turn some of that feedback into a grade for the report card.

It's not about points.

Rather than give scores in categories like Tests, Quizzes, HW, etc., I align scores with specific skills and concepts that are part of the course curriculum, called Learning Objectives. A typical course might include between 8 and 25 Learning Objectives, and naturally they would vary from course to course, with some overlap in the Integrated Math series.

As a student, you will receive feedback on these Learning Objectives first in comments and prompting questions provided on written work and also by rating your level of proficiency in each Learning Objective on a simplified scale of proficiency levels. There are four levels that could be roughly described as follows:

Level 1: Just beginning, not sure where or how to start

Level 2: Approaching Proficiency, can perform the task or use/explain the concept but usually requires help

Level 3: Proficient, can perform the task or use/explain the concept pretty consistently on their own

Level 4: Advanced, can perform the task or use/explain the concept with notable efficiency and clarity, even in novel or challenging situations

To determine what level a student is operating on at any point in time, we look not just at correct answers but at the thought processes, so many questions will ask students to explain or demonstrate their thinking. Each Learning Objective has a detailed rubric outlining the different levels of proficiency. This is provided to students and used consistently will all grading of student work. We will both keep track of your rating in each Learning Objective. You, the student, will keep track in the Student Learning Worksheet (I will help you with this), and I will keep track in my own master spreadsheet and in ASPEN, the school's grading and student database system. 

Learning is a process, and what you, the student, understand and can do is going to change over time, so I will often offer reassessments on Learning Objectives you have already demonstrated. If at any time you believe your proficiency has improved you may also request reassessment in any Learning Objective. The results of the reassessment completely replace the old scores in the Learning Objective. If you understand how to do the skill today, then I don’t care if you couldn’t do it last week - the most up-to-date score is used in determining the overall grade. Speaking of which…

How is my Overall Grade Calculated?

There are no quizzes and tests, only shorter and longer assessments, some in class, some completed at home - everything counts the same, with the same weight and importance. Some assessments or graded assignments will address multiple Learning Objectives, so students will often receive proficiency ratings on several objectives in the same assignment. The overall grade is calculated based on how many standards have been demonstrated to proficiency. The process can be thought of in the following manner:

Every grade begins with 50 points. Students are awarded up to 20 points when all standards are at least “Approaching” proficiency. They are awarded another 20 points when all standards are at least Proficient. This means if a student demonstrates proficiency in all standards, their overall grade at that point is considered to be a 90. Having a mix of proficiency scores means the overall grade will be between 70 and 90 - the more standards at proficient the higher the grade. Advanced ratings mean the score could exceed 90, and being Advanced in all Learning Objectives matches an overall grade of 100.

Grading Logic Rules.PNG

The process of converting a collection of proficiency scores to a single, points-based overall grade loses some information and within the practice of Standards Based Grading there are many approaches to this conversion. A rough mapping from rubric scores to 50-100 point scores is shown in the diagram at right. Of course, the exact points-based score for a particular student depends on the exact collection of proficiency scores as demonstrated in their body of work. It works out roughly that a student who has achieved Level 3 in *all* of the learning objectives will have a 90 on the report card. A student who has not achieved Level 3 in *any*, however, only Level 2 or Level 1, would earn no higher than 70. The goal for every student should be to reach at Proficiency (Level 3) in as many Learning Objectives as they can.

This is explained in more detail in the following documents:

Feedback and Grading

Computing the Overall Grade

Frequently Asked Questions

I want to improve my grade!

Disappointed in your current overall grade? Think you got graded harshly? Think you understand something better *after* the test? Just want to improve your grade? You get a redo. 

I strongly encourage students that for anything less than a 3 they should ask to reassess. A reassessment is simply a request to have an opportunity to demonstrate further learning or mastery of a particular learning objective. Students can reassess as many times as they wish and the most recent scores are what get counted - old scores are noted for historical reasons, but do not count in the grade calculation. The sooner you take advantage of this, students, the sooner you will feel how much power you have over your report card for this class.

To reassess, you will have to complete a short reflective form so I know what Learning Objective you want to reassess, and so we can plan how best to make sure that the reassessment is an improvement and not just another stab at the same kinds of questions. Most likely we will schedule some time together after school to review your previous work, and then we will schedule another meeting on a different day to administer the reassessment.

Please note, however, that the grade is entirely determined from what the student has learned. There is no participation grade, no points to earn, no extra credit, no extra worksheet, no test corrections, not even doing more homework will have a direct impact on the grade in a report card. The overall grade at any point is strictly a product of what Learning Objectives a student has mastered or not and the only way to improve a grade is to demonstrate additional or more complete learning.

Why should I do my homework, then?

Why indeed. If it’s not graded, what is the point?

This question comes up a lot, and I submit that it’s the wrong question - the homework is assigned as practice, sometimes as an exploration into a new area, or as a priming exercise for the next lesson. You will make mistakes on the homework, and I do not want to grade mistakes - I want to arrange predictable moments and carefully crafted moments to assess your understanding. Not doing the homework assignments will not directly effect the calculation of your overall grade, but it robs you of important experiences essential to the learning process I have designed for the course. You’re not helping yourself by blowing off assignments because they seem hard or you just don’t feel like it.

Still - I have an answer for why you should do your homework:

If you don’t do your homework, I won’t let you reassess.

That just seems fair to me - if you’re not already doing everything you’re supposed to, then it doesn’t seem fair to let you take extra opportunities to improve your grade. How will I know you’re not doing it? I check it during class, and periodically I’ll plan to collect some of your work in portfolio assignments. I don’t check every single time, but enough that if you’re missing one, I will ask you to complete it before we can schedule any part of the reassessment process.

Who thinks of this stuff?

I didn't cook up this approach on my own. This approach is called Standards-Based Grading (or SBG), and its been around since the emphasis in education shift from a norm-based model to a standards based model - mid-1980's when the National Commission on Excellence in Education published the era-defining study, "A Nation At Risk." SBG is a departure from the system I studied under in high school, but is a common model in elementary schools (including some in Northampton) and several recent studies have found standards-based practices correlated with increased student achievement and is being adopted with increasing frequency by whole school districts across the country

For more information, I have included a very short list of links to articles - both published and online - that give other perspectives on the practice. I am more than happy to discuss this, or any of my practices, with you by phone, email or in person. Reach out and we'll set up a time.

Here are the references:

A recent dissertation summarizing current research on Standards Based Grading

An article from Educational Leadership, a publication of the Association for Supervision and Curriculum Development.

A published article from the National Council of Teachers of English (it's not just math teachers!)

A collection of entries on Standards Based Grading from a teacher who writes well on this broad and rich topic.