Data Literacy is a Necessity in High School Mathematics

With the volume of disinformation available across all forms of media - television, newspapers, social media, etc. - it's paramount that we teach all students how to be critical thinkers when consuming data. I go into further detail in this article.

Think Critically When Interpreting Statistics

NCTM Seattle Reflection

As I sit here at the Seattle airport waiting to board my flight back to MSP, I’m deep in thought about the past three days. I had the privilege and honor of attending and speaking at the NCTM regional conference here in Seattle.

I am thinking about a buffet of items: the sense of relief knowing that I no longer must prepare for my presentation, the validation and motivation I feel sparked by the sessions I attended, and what my classroom is going to look like tomorrow when I go back to school and prepare for class.

I also can’t help to think about the sense of déjà vu I’m feeling right now. As I gathered my thoughts to compose this blog post, I happened to read my post from November 2018 in which I reflected on the first time I spoke at an NCTM regional conference.  In that post, I was proud of meeting a professional goal of presenting at an NCTM conference. The referenced déjà vu occurred when I clicked over to my website and noticed that I had met my goal of speaking again at an NCTM conference.

I wish I could blog about all of the items swirling through my mind, but unfortunately I don’t have time for that. The main purpose of this post is reflect on one specific session I attended, titled “Leading from Within – Improving the Instructional System While Remaining in the Classroom.”

The session began with Thomas and Brent asking us (session attendees) to think about which of our colleagues had the biggest impact on us professionally. They followed up with asking us to think about what roles those individuals were in when they impacted our lives. Their hope was for us to understand that we can positively impact those around us, as well as the greater system of education, without leaving the classroom.

Thomas spent a few minutes talking about a diagram that he found in the book called "Instructional Rounds in Education". The diagram summarizes "the instructional core" and focuses on the relationship between teachers, students, and the content. Thinking deeply about systems and changing the complex system of education took me back to the SDMath/Sci leadership trainings I attended. 

Thomas and Brent then spoke at length about the "four act drama" for effective teacher leaders.  It was sort of a checklist of things that really motivated me.

1. Fearlessly commit to best practice.
2. Operate with a leadership mindset.
3. Build collective teacher efficacy
4. Embrace courageous lifelong learning

A bit more detail about each bullet:

Fearlessly commit to best practice.

Thomas and Brent served up a beautiful reminder of what ambitious teaching requires: student-centered environment, problem based, collaborative, rigorous, 8 SMP, 5 Practices, high quality formative assessments, culturally responsive teaching, equitable instruction and grading practices, and social-emotional learning. They summarized this list by reminding us that these practices are "critical for some, but beneficial for all".

Operate with a leadership mindset.

Thomas used a compass analogy very effectively. North: as teacher leaders, sometimes we push our administrators to help drive change.  East/West: as teacher leaders, we need colleagues and other teacher leaders we can lean on for guidance. I think of the many professional colleagues nationwide I have via Desmos Fellowship, Twitter, NCTM committees, etc.  South: as teacher leaders, we must be ready to help the new teachers and veteran teachers who are ready to be led.

Thomas and Brent mentioned a few more things that hit home to me. First, we should always be monitoring our team's progress.  Data should be driving our thinking. We should be using our MAP testing data and celebrating our successes while also analyzing our non-successes.  Second, we must be strategic and know how to push the right buttons to drive change. Thomas mentioned that there are some colleagues in his department that don't care to hear any ideas for change from him. Therefore, he has to be strategic and enlist the help of others to pitch the ideas to those who don't care to change.  Last, teacher leaders must remember that "We are always learning. We are always leading. Teaching causes learning."

Build collective teacher efficacy.

We (as teacher leaders) believe that positive change to the system can happen.  But does our team?

Brent referenced Hattie's research and that teacher efficacy has and effect size of 1.57. 

He also mentioned the idea that "wins lead to wins", in other words success breeds success.  Our team should be trying to emulate the schools who are experiencing success right now.

I can't help to think about the data showed to the school board at a recent school board meeting and how we are doing on the state exams relative to the other ESD schools.  (Not well)

I am confident in saying that our teacher efficacy has not been good these past 3-5 years.

Embrace courageous lifelong learning.

Thomas called on teacher leaders to stay up to date with the newest research, to attend and speak at conferences, and find peers who dare to do great work. He reminded us to continually spread the word because we never know who we will touch and we never know who is ready to learn.

This was a motivating session because it reminded me that I have a plethora of leadership skill and knowledge and we (as teacher leaders) always need to be leading. I sometimes forget that because leading is hard work. 

One final thing that I was reminded about in this session was that in order to truly drive change in our own districts, everyone on the team needs to be working together... the school board, administrators, curriculum directors, TOSAs, teachers, TAs, etc.  Our district has some work to do in order to achieve this goal.  

Rapid fire reactions about other aspects of the conference:

• Peter Liljedahl's "Building Thinking Classrooms" is quickly becoming a must-do in mathematics education. Not surprisingly, there are still many who are not yet aware of the research or who are not implementing the BTC strategies.  I am happy with how we are implementing it in advanced algebra 2 this year. But I still need to become better at the fine details of implementation.
• As I have seen with other past NCTM regional conferences, there were many time slots that had multiple sessions I wanted to attend and was forced to choose. If only I had Hermione's time turner, I could attend more than one concurrent session.
• A number of sessions I attended had excellent presenters, whereas a few had presenters that were sub-par. Six years ago, Robert Kaplinsky blogged about things he's learned from being a presenter. The post detailed some of the nitty-gritty details of creating a quality presentation.  A lot of what he says holds up, but there almost needs to be an updated post about this due to improvements in technology.
• This comes with no surprise, but there is also a high level of skill in creating and delivering a high quality presentation in conference settings like this. One of the best tips I can provide is to borrow presentation moves from other presenters who you view as high-quality.  I was fortunate to listen to Graham Fletcher at the SD STEM ED conference just over a week ago, and he absolutely crushed both presentations I attended.  I borrowed three or four presentation moves directly from him.
• Speaking of my presentation, I was happy with how it went and that is mostly due to how happy I was with my preparation. My session was in a very large ballroom (capacity ~400).  Honestly, I was fairly nervous heading into it, mainly because of the size of the room. I didn't know what to expect for attendance.  The attendance turned out to be somewhere between 50 and 60 people, which wasn't so overwhelming.  I received positive feedback from a few participants who stuck around after the presentation ask questions.  

Mathigon: A must-use resource

Okay math teachers... let's have a chat.  Mathigon is something you need to look into using today. 

I have a confession... since COVID teaching, I've been much less energetic about looking into the "next cool math tech tool".  But during this past winter break, I was able to spend a little time getting a bit more intimate with Mathigon, and I have fallen in love. (S/O to David Poras [@davidporas] for your help!)

Then, a few weeks ago at the SD STEM ED conference, I attended a presentation by Dr. Kevin Smith (@kevinsmithsd) and there was a lot of buzz from the attendees about the awesomeness of Mathigon.

I'd still call myself a Mathigon rookie, but I'd like to share a few things that I do like about Mathigon.  I wrote about a few of the things for the SDCTM winter newsletter. 

Recently in advanced algebra 2, we were working on extracting nth roots of variable expressions.  While Mathigon didn't help with the variable parts of the expression, it was very useful when looking at the integer values.

Example problem:

Rewrite each expression by extracting all possible roots.

Mathigon provided a very useful visual with the prime factor circles.

If you've never worked with prime factor circles, I invite you to open a blank Polypad in Mathigon and play around for a few minutes. In less than 1 minute, I was able to show students how the prime factor circles work and had them thinking about why we could extract a factor of 16 (2^4).  Something that has always been fairly abstract for students now has a very clear, neat visual model.

I'm super excited to see where else I can use Mathigon.  

P.S.  A recent update allows users to split algebra tiles.  WHAT?!?!!

Go check out Mathigon today!

 

2021 - 2022 End of Year Reflection

Full disclaimer: I’m writing this blog post for very selfish reasons. I’ve already highlighted the positive effects of blogging. This post serves as somewhat of a scrapbook for me. 

As my 20th season of teaching wraps up, I finally found a moment to breathe and reflect. I’m a big believer in the mentality of “what you put into life is what you get out of it”.  It’s been an extremely grueling, yet rewarding, year. I am thankful that we were face-to-face for the entire year (during peak-COVID mid-year I believe we were close to shutting down). I am also grateful that the COVID numbers dropped to a level that allowed our masks to be “strongly recommended”.  It continues to amaze me how different students look when have their masks on/off. There were a number of times this year I saw a student who was in my class last year (face-to-face with masks) and they look completely different than what I expected them to look like with their mask down.  I speculate they feel the same way about me!

This season was the first season since my rookie year that I didn’t teach a geometry class. Two recent retirements in our department created openings that needed to be filled. I taught three sections of advanced algebra 2, one section of algebra 1, and one section of AP Statistics. By our department absorbing those positions, our class sizes ballooned to volumes higher than ever before. At the start of the year, my algebra 1 section had 31 students, my AP Stats section had 24, and my advanced algebra 2 had 31, 33, and 33. I only had 30 desks in my room, so I had to borrow 3 desks from a colleague down the hall just so everyone could have a seat. Increasing class sizes from 24 to 30+ may not sound like much, but I can’t overstate how much more of a challenge it is to be an effective teacher. 

Even though I’ve taught advanced algebra 2 and algebra 1 in the past, our district adopted a new textbook series last summer. Traditionally, I’m not one who follows the textbook step by step; rather, I use the textbook as a resource to help develop the lessons, activities, and tasks that students experience. The series we adopted at the high school is the latest Carnegie Learning series. I’m in love with about 80% of the series, which is a huge upgrade over our previous textbook. The lessons are designed to develop conceptual understanding by being inquiry-based and student-centered. Did we still need to tweak and supplement some of the lessons? You bet. Did we still write our own assessments? Yes, we did. Do I still need to improve my delivery to put more learning in the hands of students? Absolutely. The neatest thing about the series is that I learned a lot about the mathematics in those courses and some really mind-blowing connections between main topics. 

Using a new textbook series takes a lot of preparation time if done correctly. Analyzing the content of each lesson, planning the pacing, identifying where modifications should be made, creating / finding supplemental resources, building the Canvas modules, etc. all take time. Some of that work was done prior the school year starting, but many of those decisions include a reactive component that require me to know my students and their strengths, weaknesses, interests, and abilities. In other words, those decisions shouldn’t be made during the summer prior to even seeing my rosters. 

This was the first time I have ever taught AP Statistics. I felt like a first year teacher again. To help prepare, I attended (virtually) a week long course last July. It was very helpful and connected me with the right people to help me on my journey. Was it a challenge to teach something for the first time again? Absolutely. I’ve always believed that you don’t truly understand something until you can teach it to someone else. There were a number of nights that I was up late preparing (& learning) the lesson I was teaching the following day. Did I enjoy teaching the class? Immensely. I really enjoyed teaching seniors again (for the past 5-6 years, I have mostly worked with 9th & 10th graders). There were some 11th and 10th graders in the class as well. Overall, it was an excellent class of students who are far brighter than I ever was at their age.

I wouldn’t have survived without Stats Medic and the resources on their website. I love their philosophy of EFFL (Experience First, Formalize Later) and how open to sharing the AP Stats community is. I think it’s extremely helpful that all AP Stats teachers are teaching toward the same targets and that most of the major textbooks and resources are aligned on things like scope and sequence of the curriculum. Because a majority of AP Stats teachers are teaching the same units in the same general timeframe, there are relevant, rich conversations happening online. Each day I would skim the posts on the AP Stats Twitter and Facebook groups. I learned a lot and knew where I could go to ask questions if needed.

I enjoyed working with two student teachers this past fall. Mr. Venner and Mr. Pierzinski were awesome and will both be strong mathematics teachers. I love helping develop the next generation of mathematics teachers.

This year I also got back into coaching high school basketball. My oldest child was a freshmen and I was able to coach the freshmen boys basketball team this year. I hadn’t coached high school boys basketball since 2006 (I coached many years of girls since then). I loved being able to coach my son and be a bigger part of his high school experience. 

Being a high school coach takes a lot of time and commitment. Adding that to my plate meant that something else had to be removed. As a result, I scaled back in the realm of math leadership. I decided to pause my pursuit of National Board Certification for the school year. I was not able to attend the SDCTM conference in Huron this February. I attended only one (of four) regional math circles that I had intended to help facilitate. I have not written a blog post in nearly a year. The list of articles, journals, blog posts, etc. that I had planned to read continues to grow each day. I decided to “say no” to a number of opportunities that I might have previously taken.

I did have the opportunity to present at the TIE conference this spring. It felt to great to share my knowledge of Desmos in a face-to-face conference again. I did complete the written test portion of National Board Certification last June and scored extremely well. I’m excited to resume pursuing that goal. 

As I write this post, I am actually on my way back from the NCTM HQ in Reston, VA. I have served on the NCTM Classroom Resources Committee for the past 2+ years. We have done all of our work virtually since I began serving prior to this weekend. It was great to finally meet in person and gain a clearer vision of the work ahead of us. [This was the first time I was on an airplane since COVID began.] These past few years of not having in-person professional development & networking opportunities has truly made me realize how much I rely on those events to put wind in my sails. 

To recap, this has been a grueling, yet rewarding year for me for a number of reasons. 

Looking ahead, I’m excited for a number of reasons. Next year, it looks as though I will be teaching the same classes as this year. I’m excited for year 2 of AP Stats and how much more confidence I will have in my content knowledge. As our district begins to transition to a Competency Based Educational model, we will be implementing a form of target-based grading once again. I’m excited to bring that back into my classroom with my advanced algebra 2 students next year.

If you’re still reading, thanks for sticking with me. I invite you to put pen to paper and reflect on your journey as well!

P.S. You can still vote on your favorite graph from my advanced algebra 2 Desmos art project!

Water the Lawn Project

 

Each year, my colleague Jarrod and I tweak our geometry curriculum with the intent to improve and build on the previous year.  Some years we make major changes, such as an overhauled grading system.  Other years we integrate new technology.  (This year, another colleague [Adam] is teaching geometry as well.)

One of the goals of our high school is to improve student engagement.  We (the geometry teachers) are trying to implement more problem-based activities into our classrooms.  We've had great success each spring semester with a Floor Plan project.  We wanted to create a project for the fall semester that met these checkpoints: low floor / high ceiling, open ended, (semi) real-world application, and connected a number of learning targets together.  Additionally, we knew that we wanted to build something that we could expand on in future years, specifically to be able to connect community experts / partners with our students.

In our HS geometry curriculum, we focus on writing the equation of a circle and finding areas of sectors.  In years past, we would not cover those topics until the spring semester.  However, last year we moved the target of writing the equation of a circle into the first semester.  In our first unit, we revisit the Pythagorean Theorem, derive the distance formula, and then write the equation of circles on a coordinate plane.  This year, we decided to also move the concepts of arc length and area of sector into the first unit.  (We treat circumference and area of a circle as prerequisite skills that students have learned in middle school.)

We were intrigued by a Desmos activity called "Water the Lawn".  In the activity, students are asked to place sprinklers at locations of their choice in order to cover a rectangular region that represents a lawn.  (Take a peek.)  The Desmos activity primarily focuses on finding the areas of circles.  We needed to make the activity more challenging and aligned to our HS learning targets.  Let's dive in...

Task:

Install a sprinkler system in the yard of the given diagram.  (Link to diagram)

Rules:

1.    You must water the entire yard.

2.    You must stay within the boundary of the yard.

3.    Overlapping areas are necessary.  The amount of overlapping area must be less than 3000 sq ft.

Requirements:

1.    Neatly sketch (using a compass, ruler, & protractor) your plan on the blueprint.

2.    For each sprinkler head, you must find the following information.

On the Lowe's website, we found three different types of sprinkler heads.  (Note: the two bigger sprinkler heads had minimum degree restrictions.  Also, none of the heads have a radius between 16 and 18 feet.  This created a bit of a challenge for students.)

We showed students an example (albeit, a bad example because of too much overlapping area) to help get them started.

And that is it!  We gave students 1 full block day (90 minutes) to work on this project in class.  It was then due one week later.

Results:

I was very pleased with the overall results of the project.  Here are some examples of what students created.

    

    

Student Reflection:

At the conclusion of the project, we asked students to reflect on their experience.  In an email to their parents / guardians & their geometry teacher, students had to answer the following questions.

1.  What are two things that you learned by completing the project?

2.  What was the most challenging part of the project for you?

3.  What piece of advice about completing the project would you give to future geometry students?

4.  What do you think the geometry teachers could do to improve the project for future classes?

Here are a few quotes from the student emails:

• "If I were to give advice to future geometry students about this project, I would tell them not to overthink the goal and try ideas first. When I started this project, I found myself thinking too hard and not trying the ideas I had come up with because I feared the area of overlap would be too great. After finishing the project, I was under the maximum requirement of overlap. I should have carried through with my other ideas instead of trying to make it perfect."
•  "Two things I learned from this were that building sprinkler systems are hard, and even when you think you can’t solve the problem, you can."
• "This project allowed me to learn quite a lot, and I enjoyed it.  It made learning more exciting."
• "It is also fun to be creative with this project. I found that it was useful to use other angles besides 90, 180, 170, and 360-degree angles."

The students gave us some great ideas on how to improve the project for next year.  One common suggestion is to have a budget on the cost of the sprinklers.

Future Years

We think it would be great if we could have a professional landscape specialist from the community be a part of this project in the future.  The community expert could potentially help launch the project by providing greater details about how sprinkler systems are installed and bring some of the sprinkler heads for students to see (and maybe learn about how they restrict the degree of spray).  Or, we could have students present their plans to the landscape specialist and they could provide feedback.

Final Thoughts

Overall, I'm happy with the first year of the project.  Students applied areas of sectors and wrote equations of circles, which met our learning targets.  They also improved their skills with a compass.  I look forward to improving this project in years to come.

I do understand the project has its limitations.  I mean, come on...who needs to design a sprinkler system that doesn't extend a little outside of the boundary of your yard?  What if the wind is blowing hard one day?  Isn't the water going to leave the yard a bit anyway?  

How would improve this project?  What sort of things are we forgetting that could make this better?

If you're interested in any of the files we used, enjoy!

Launch Page

Blueprint

Student Work Page

Target Based Grading v1.0: 2019-2020

In this blog post, I will detail why and how I implemented a target based grading system in geometry this past year.  I will share my thoughts and reflections about what went well, what didn't, and what changes I'm contemplating for next year.

I've been meaning to write this blog post for many months.  I gave a presentation at the SD STEM ED conference in February about this topic, and there was a high level of interest about it.  Thanks to being stuck at home for the past seven weeks, I've found a bit of time to write this post.

I typically blog as a vehicle for self reflection.  When I have a lot going on in my brain, I find the process of writing something semi-coherent helps me organize my thoughts and think more deeply about things.

That said, I am very interested in hearing feedback about my target-based grading system as I begin to prepare for next year and target based grading v2.0.  Please do not hesitate to comment, question, or push back on this post.  I welcome the discussion around the topic of grading.

(Note: S/O to my colleague Jarrod for joining me on this journey.  We continue to work in tandem and mirror each other day by day and are both better for it.  A majority of this process was "we" instead of "I".)


Part I.  WHY

My decision to implement a target based grading system was influenced by a number of factors.

• I've always been intrigued by standards based grading, dating back to somewhere around 2014.  I implemented a standards based grading system as part of my Master's action research during the 2014-15 school year.  I didn't love the grading system I used for the action research, so I stopped using it after one year.  But I was still intrigued by standards based grading.
• In my school district, the K-3 elementary buildings began using standards based report cards in about 2016-17.  I enjoyed seeing my children's standards based report cards compared to a traditional report card and continued to be intrigued.
• In 2017-18, my district began to implement what we are calling "mass customized learning (MCL)" in the K-3 buildings.  In 2018-19, MCL began in the Intermediate (4-5) building.  In 2019-20, MCL began in the Middle School (6th grade class).  It is scheduled to arrive at my high school in 2022-23.  MCL is centered on students demonstrating mastery of learning targets, which are written in student-friendly "I can..." statements.  One of the components of MCL is assessing (and reporting) student growth on the learning targets.  In other words, MCL and target based grading go hand in hand.
• I read Jo Boaler's Mathematical Mindsets and was convinced that traditional grading and assessing has some negative consequences.  Boaler talks about assessment for learning and gives some advice on grading.  The target based grading system that I hoped to implement followed many of Boaler's guidelines.
• One recommendation that Boaler makes is to not include homework as part of a student's grade.  She makes a convincing argument that grading homework is an inequitable practice.
• In past years, I have graded homework a number of different ways.  Some years, I would have students do a self-check and report their score.  Other years, I would collect and grade the homework on completion and/or accuracy.  Either way, homework has been graded and counted between 15-25% of the student's grade.  Recently, I grew tired of students copying answers or being dishonest when they self-graded.  It wasn't all students each year, but a growing percentage of students.  I was ready to completely remove the homework component from a student's grade.
• In 2018-19, I participated in a leadership cohort (called SDMath/Sci) through the South Dakota DOE.  It was some of the best professional development I have ever been a part of.  We learned about a leadership theory by Simon Sinek centered on "Finding Your WHY".  Equity was a major component of the training.  Echoing Boaler's claims about equity, the cohort opened my eyes to some of the systemic structures that are a part of our educational system.  Traditional grading practices were something we discussed.  I had found my WHY: a target based system would create a more equitable way of assessing the students in my classroom. 

Part II.  Setup

Target Based Grading vs. Standards Based Grading
I mentioned earlier that I didn't love the standards based grading system that I had implemented for my action research.  At the time, South Dakota had adopted the Common Core State Standards.  I felt that certain standards were too broad to accurately assess student understanding.

Take Standard F.IF.C.7.B for instance:

Graphing square roots functions, cube root functions, and piece-wise defined functions (including absolute value functions) were all topics I covered in Algebra 2.  I claim that there are potentially students who have full understanding of graphing square root and cube root functions, but who struggle to graph piecewise functions.  And vice-versa.  In a standards based grading system, what sort of grade should those students get?  If I'm using a 4.0 scale, would that student earn a 2.0?  A 2.5?  Would those scores truly reflect what the student understands?

That ambiguity helped me decide that I wanted a target based grading system, where I assess students on each learning target.  So last summer I created the complete list of learning targets that I wanted to assess for geometry.  [Geometry Learning Targets]
Note: I am already in the process of tweaking this list for next year.  I realized that I needed to combine some targets together, especially in the surface area / volume unit.  I also plan to re-sequence a few of the targets.


Learning Target Quizzes
Each learning target was assessed on an in-class QUIZ.  Each quiz had between two and five learning targets on it.  The quizzes were designed to be completed within one 50-minute class period.

Example: This is Quiz #5, which assess learning targets 16-19.

A couple of things are worth noting as you glance through the quiz.

• Students are scored on a 0.0 to 3.0 scale.  A score of 0.0 is given if a student does not complete the questions for that target or demonstrates absolutely no understanding of the target.
• Students cannot earn a score of 0.5.  I don't have any good reason why that score wasn't allowed other than I felt that if a student gave full effort and demonstrated some level of understanding, a minimum score of 1.0 was valid.
• The difference between scores of 1.5 vs. 2.0 vs. 2.5 varied by learning target.  Some learning targets had only one question to answer (see target 19 above), while others had up to five questions to answer.  
• A score of 3.0 demonstrated quiz mastery.

Retake Policy & Process

If a student earned less than a 3.0 on any target target, they were expected to retake the part of the quiz aligned to the target.  

There was the process to completing a quiz retake.

1. Error analysis & reflection: For each target needing re-assessing, students completed a Quiz Correction Form (see below).
2. Practice & study: Students were encouraged to review the practice problems and previous quiz before retaking.
3. Retake: Students could come into my classroom any time throughout the day to retake, even if I had other class in session.

Students stapled their quiz to the quiz correction form(s) and turned them into me.  I would check to make sure students had correctly completed the problems on the quiz correction form.  If they did so, I would use a rubber stamp to "stamp" their correction form, indicating that they had earned the opportunity to retake that learning target assessment.  "Stamped" forms were returned to students to be able to review and study from.  If there were errors made on the quiz correction form, I would return the form back to the student for revision.

When a student completed a retake of any learning target, the new score replaced the old score.  It rarely happened, but students could earn a lower score on their retake.

If a student did not yet earn a score of 3.0 after retaking the assessment once, then a second retake was available.  Before the second retake, I would have a one-on-one help session with the student instead of repeating the quiz correction form process.  The help sessions were typically fairly brief (between 3-10 minutes) and took place whenever there was enough free time to have one (before or after school, during my planning periods, or even between class periods).  

Quizzes were graded and returned to students the next day.  Students then typically had 10 calendar days to complete the quiz retake process.  I know that some standard-based grading systems call for students to be able to retake / improve their scores for as long as needed (or at least until the end of the semester or grading period).  I struggle to see the benefit of that theory for mathematics classes.  So much of mathematics builds on prior knowledge in a logical & sequential manner.  I need my students to gain understanding of the learning targets in a timely manner so that they will possess the prior knowledge needed for the next set of learning targets.  Therefore, I set a retake deadline for each set of learning targets.


Enrichment Tasks
After a student earned a 3.0 to demonstrate quiz mastery, they had an opportunity to earn a 4.0 and enrich their understanding.  For each learning target, I created an enrichment task.  Enrichment tasks were intended to deepen understanding and extend thinking.  Enrichment tasks were released (via Google Classroom) to all students on the day of the quiz.  Students could work on the enrichment tasks individually or with partners / in small groups; each student had to turn in their own copy of the task for credit.  Enrichment tasks followed the same deadline as the quiz retake.

Examples of the enrichment tasks for learning targets 16-19:

More things worth noting about the enrichment tasks:

• Students could earn scores of "Not Yet", 3.5, and 4.0.  (Recall: in order to submit an enrichment task, students must have first earned a 3.0 on the quiz.)
• Enrichment tasks were not a one-and-done deal.  If a student submitted work that wasn't correct, I would give feedback and allow students to fix their mistakes.  There were many occasions where a student earned a score of "Not Yet" or 3.5, I gave feedback, and they made revisions and ultimately earned a 4.0.
• The content on the tasks varied in nature, depending on the learning target.  Sometimes the enrichment task would include a problem or two that involved a little tougher algebra (see learning target 16 - solving a quadratic equation and considering extraneous solutions).  Other times the task might be to research some deeper idea that we simply don't cover in our regular instruction (see learning target 17 - considering spherical geometry).  

Grade Conversion:
In an ideal target based grading system, I wouldn't have to report traditional A-B-C-D-F grades.  Unfortunately, my district requires me to because we need to report things like GPA and class rank.  Boooooo!

I needed to create a way to convert the target based scores into traditional grades.

Before I forget, here is the grading scale for my school.

As you can see, semester grades were calculated by a weighted average of the quarter grades (40% each) and the semester final tasks (20%).

Quarter grades were calculated by a weighted average of the learning target assessments (80%) and miscellaneous tasks (20%).

Learning Target Assessments (80% of quarter grade):
Each quarter, all of the learning target scores were averaged to create an "Average Target Score".  The Average Target Score was then converted using this conversion chart.

Notice:

• Students needed an average score of 2.0 or higher to earn a passing grade in the "Learning Target Assessments" category.
• An average score of 3.0 earned a student a B+ in the "Learning Target Assessments" category.  In order to earn an A in the category, some enrichment tasks needed to be completed.

I used a Google Sheets Gradebook that automatically averaged and converted the score to a letter grade.  I also had to record scores in Infinite Campus.  I wasn't a fan of having to manage two separate gradebooks, but I survived.

Miscellanous Tasks (20% of Quarter Grade):

I did a lot of thinking about the types of assessments that I have used in past years and how they would fit into a target based grading system.  I wasn't ready to completely remove the variety of assessments that have been a staple in my classroom, so I decided to create a separate category and maintain the ability to assess students in a variety of ways.  The learning target average score counted as 80% of the quarter grade, while the "miscellaneous tasks" counted as 20% of the quarter grade.

One of my biggest concerns about implementing target based grading was that the assessments would become "silos" - too focused on one particular target.  I want students to see and understand the connections between the learning targets.

Some examples of assessments that went into this category included:

• In class group tasks
• Every couple of weeks, I have students work in pairs during class to solve review problems.  A majority of the problems relate to the geometric concepts and topics that we have learned about up to that point.  Some of the problems review middle school and algebra 1 topics; others focus on probability and statistics.  I pull a lot of questions from ACT reviews to expose students to those types of questions.  I also include problems that connect multiple learning targets together and/or include different solution methods.

• Choice Assignments
• To help try to increase student engagement in my high school, all teachers were asked to implement assignments where students had a choice of options.
• Desmos activities
• From time to time, I will assess a Desmos activity.  Typically, these activities are completed in class as part of the lesson for the day.  I will assess them when I want to ensure that students who were absent during class complete the activity.
• Projects
• Each quarter, students complete at least one project.  Some projects are in groups, others are individual.
• Reflection tasks
• I had students reflect on their learning with the help of Flipgrid.

• I also had students write an email each week to reflect on their learning.  The email went to their parents / guardians and to me.  Students would have to answer a series of prompts that helped guide their reflection.  See an example of my prompts here.

One thing that I removed from class this year was "Parent Tests".  In the past, I would send a multiple choice assessment home each unit for parents / guardians to complete.  The questions would relate to the concepts we were learning about in class.  There was a process that needed to be completed and students would earn extra credit if they completed the tasks and returned the paper.

I've learned that awarding extra credit - especially in way that involves a student's situation at home -  is an inequitable practice.  I have decided to not award any extra credit from this point forward.  I am in the process of researching ways to include parents / guardians in the learning process again.

Part III.  Implementation Details

I want to highlight and comment on a couple of things I learned during this first year of implementation.

I made three sets of each quiz.  Each set was intended to be equally difficult.  Students had the opportunity to take the quiz earlier that the designated quiz day.  Students taking the quiz early were given Set #1.  Set #2 was given to every student who took the quiz on the quiz day.  Set #3 was given to students who wanted to retake.  If a student needed a second retake, they would then take the quiz that they hadn't yet taken.


On the day following a quiz day, students would have "WIN" (what I need) time.  Students would get their graded quizzes back and do the following:

1. Write their reflection email (see above).
2. Begin the quiz correction process if they scored less than a 3.0.
3. Work on enrichment tasks if all scores were 3.0.

Some students would also use that time to work on quiz retakes from previous quizzes, assuming the retake deadline had not passed.

I would meet individually with each student to check in with how they were doing during this time.

Some students used the WIN time very effectively.  Students were able to ask questions of each other.  A lot of peer helping took place.

As the year went along, more and more students started abusing their time during WIN days.  I would see students working on assignments from other classes.  Others would treat the time as a social hangout.  I need to improve my classroom management a bit on those days moving forward.


Each class had its own tray on my desk for students to turning their work in.  This helped manage the steady flow of papers that were turned in each day.

Each student had their own hanging folder in a basket at the front of my room.  I would put all papers (graded quizzes, quiz correction forms, notes from class they missed, etc) in their basket.  It was their job to empty their basket as they entered the room each day.  Being able to "hand back" papers at any time was a huge time saver.

Speaking of grading papers... it was a daily job.  Each day, there was always something turned into the trays.  I felt the responsibility to grade papers and update grades every night.  If it was a quiz day, students needed their quizzes returned the following day for WIN time.  If it wasn't, students needed daily feedback on quiz correction forms and quiz retakes.  The process of retaking depended on it.

That said, in past years I would find myself grading every night as well - often times homework.  I felt much more motivated to grade quizzes, retakes, and quiz retake forms versus homework.  At minimum, I knew that the quizzes and quiz retakes were authentic evidence of what the students had learned.  I dreaded grading homework knowing that a certain percentage of students had copied their answers from someone else and learned nothing along the way.


One grading note:
I did have a few students average between a 1.75 and 2.00 on their target based assessments, which converted to a 64% in the learning target assessment (80%) category of the gradebook.  Their grade on the miscellaneous tasks (20%) was fairly good, so those students did pass the semester when the weighted grades were calculated.


Communication to parents / guardians about the grading system was important.  There were some parents who asked for clarity on the grading system at the start of the year.  I included a detailed explanation in one of my newsletters. 

I started the year with a quiz over the prerequisite skills.  The quiz didn't count towards the student's grade but I scored the quiz using the target based system.  It served as a good example for students to see how they were going to be assessed.


Part IV.  Reflection / Next Steps

I felt the target based system was effective in helping develop growth mindsets in students.  I liked that students didn't get their graded quizzes back and immediately saw a score / total points or a percentage.  Instead, they saw scores for each target.  In the case where a student didn't do very well on one or more targets on a quiz, it was still easy to shine positive light on the targets they did do well on.

I had to train myself to not be frustrated when students struggled on the first attempt on the quiz.  I had to remember that this was their first attempt at each quiz and that they did have the opportunity to retake.  I needed to focus on the final scores after the retake deadline had passed.  After all, those were the final scores that students were earning.

Opponents of a target based grading system have a worthy argument centered around that previous paragraph.  What's stopping a student from not studying / preparing at all for the first quiz when they know they always have an opportunity to retake?  What type of bad habits and lack of work ethic / preparation does that develop?

My counter to that argument lies in my belief that if we teach students to be mindful of their learner agency and the 16 Habits of Mind, students will begin to make make the correct decision when it comes to preparation and work ethic.  Yes, I believe that the mathematics I teach in geometry is important for students to have future success.  But teaching students to be lifelong learners and to possess a growth mindset and self confidence is more important. 


In the future, one long term change that I'm interested in looking into would be the idea of "mastery" learning.  A student would need to demonstrate mastery on each standard before they would be allowed to move on.  In other words, a student must earn a 3.0 on each learning target (or group of targets on a given quiz) before they move forward.  A big change in the structure of our class schedule would need to take place in order for this type of change to be implemented.  It is something our district is looking into pursuing and would help fight against students doing the bare minimum to pass.


I do welcome all feedback and discourse around this system in an effort to improve for future years.  I hope to convince my colleagues that target based grading is something we should be doing throughout our entire department. 

Algebra 1 Project

As part of a first semester project for Algebra 1 this year, students completed a two-day activity called "All Knotted Up".  I first found this activity in the September 2013 edition of NCTM's Mathematics Teacher publication.  (Credit to Jamie-Marie L. Wilder and Molly H. Fischer for the great activity and supporting article.)

Materials needed: poster board, marker / colored pencils, meter sticks or tape measures, a variety of ropes or string.

Working in pairs (or groups of three if needed), groups chose one type of rope (or string) and cut a piece that was at least 100 cm long.  (More about the length in a minute.)

Students measured the length of the rope and recorded the length.  (It was helpful to have tape measures instead of meter sticks or rulers.)  Then students tied one knot in the rope and re-measured the length.  They repeated this process until 10 knots had been tied.
I provided groups a table to organize their data.

Groups then made a scatterplot of their data on the poster board.  I asked that groups be sure to include their data table on the poster.

To finish the first day, students reflected on the following questions:

A)        Looking at your graph and table of values, what trends do you see?

B)        Suppose you were to tie three more knots in your string.  How long do you think the string would measure?  Explain how you arrived at your estimate. 

To prepare for the second day, I chose three groups that used different rope and had started with different lengths of rope.  I created a Desmos graph that included the data from three groups.  I calculated the linear regression equation for each set of data. (https://www.desmos.com/calculator/it3v9a0efu).  I also hung three posters on my board, along with the rope they used.

When students entered the room, they naturally started looking at the board and making observations.  I asked students to individually write down "What do you notice?  What do you wonder?"  After two minutes of individual thinking, students engaged in a large group discussion.  I let their observations and wonders drive the discussion.

Here is a partial list of things students noticed, discussed, and analyzed:

• The y-intercepts of the graphs are the initial length of rope.
• The slope for each line is different; the slope for the thickest rope is steeper than than the slope for the thinnest string.
• The data is fairly linear, but not perfectly linear for any of the graphs.  We discussed why that is the case.

One tip with the initial length of the rope: The rope must be long enough to be able to tie 10 knots in it.  If the rope is too short, the knots end up knotting together and the data becomes a lot more variable.  I had a group that chose a very thick piece of rope and by the 7th knot, their rope was just one huge chunk of knot that was impossible to measure.  That group had to restart with a longer piece of rope.