mrdardy

Vocabulary

This post is inspired by a twitter exchange with the awesome Joe Schwartz (@JSchwartz10a) and by a running exchange with one of my Geometry students. Joe tweeted out the following picture:

The picture was accompanied by the question : “Do 3rd graders know the answer to this question? Truly curious.” It just so happens that my Lil Dardy is in 3rd grade. I showed her the question and (briefly) explained the equation written. I replied to Joe that she was not surprised to see it written that seven sixes is the same a five sixes plus two more. However, she did not know any vocabulary word to describe this. Joe replied, succinctly, “And she doesn’t need one…” It made me smile. It also made me think when I was reviewing for a test with one of my Geometry classes. We just finished a chapter on triangle bisectors and centers. Loads of vocab in this chapter. Very few new skills, just new words describing relationships. Thinking back to the exchange with Joe I found myself questioning my decisions in writing the book and in teaching this chapter. During the test review a student asked if there would be any vocabulary on the test. This particular student has asked this question before just about every test. I answered the way I do just about every time. I told him that he needed to know what these words mean to accurately interpret the questions at hand. For example, if I ask about altitudes to a triangle, he needs to know what that means. However, there would not be a question where I simply ask him to replicate the definition of an altitude. Thinking back on this exchange, and this way that I answer the question, I have a ton of questions that I need to ask myself and I will start by posing some of them my readers out there.

My guess (an uncharitable one) is that the student asking about vocabulary is looking to avoid committing anything formal to his short term memory before a test. Admirable in a certain way, but what does this question say about what he thinks his job on a test is? Why would students who have been working with words day after day express any serious concern about being asked what those words mean?
Real people have real vocabulary that they use in their studies, in their work environment, etc. I recoil at the suggestion that I should do something objectionable now because someone will do it to my students later. But, I am beginning to wonder whether I am cheating my students a bit. Should I be more emphatic in urging them to be careful about vocabulary now so that they will better understand what they read or hear later? Am I being lazy when I let them casually refer to the longest side of any triangle as the hypotenuse? [Note: I have written about this before. I DO correct them, but in a pretty gentle, nudging way. I remind them every time that the hypotenuse is a specific name, but this habit has settled in with my students for a couple of years now.]
What are we communicating to our math students if we mark points off or hold them accountable in some ways to formal language if they can get their mathematical ideas across through their work? Are these skills dependent upon one another? Is it okay that my students can swing into action and write the equation of an altitude of a triangle but be uncomfortable and vague if asked to write a definition for what an altitude of a triangle is? As someone who is so comfortable with these words, I struggle to understand how someone can write that line without being comfortable that they can write a definition, but I’ve been teaching long enough to know that this is a real thing.
Is this another instance where students have been trained to think that there is one right way to answer a question and their job is to make sure that they simply regurgitate (if they can decode correctly) what that correct answer is. I, of course, hope that my grading policies and the way that I communicate in class convinces my students that this is not the way life is in my classroom. However, I know that I am battling impressions that have formed over years.
More importantly – Does it matter that my students know things like the altitudes of a triangle intersect at the orthocenter? Is there ANY chance that they will remember this in a few months? In the past few years I taught the course, I pretty much only mentioned the word centroid and avoided talking about incenters, circumcenters, and orthocenters. I am not at all sure that I made the right decision then or that I made the right decision this year in explicitly defining them. In my text the words centroid and incenter are explicitly defined. Circumcenter and orthocenter do not even appear in the text. A mistake then? A mistake now? I’d love to hear some advice/opinions.

Gotta get dressed for school now. More thoughts swirling and I hope I am disciplined enough to get them down soon.

Thanks to Joe for prompting this post!

As always, you can reach me here in the comments section or over on twitter where I am @mrdardy

Greetings, 2018

A non-mathy post for this morning. I feel like I need to clear my head out a bit here.

Thanks to Meg Craig and the #Fitbos gang for helping to keep me motivated this past year. I set two goals for myself with my trusty fitbit flex. I wanted to accumulate an average of 30 minutes per day at an ‘active’ level. I compiled a total of 207.45 active hours. Last time I checked my multiplication, this exceeds my goal! I also set a goal of walking 2017 miles in 2017. I ended up at 2050.16. I am pretty pleased, but time is still working against me, despite this level of activity I am more achy and a bit paunchier than I was this time last year. Have to ramp it up to fight against Father Time.
Thanks to connections that my wife has at her college I was able to score a gig as a DJ at the local college radio station. Almost every Thursday since June, I have had the great pleasure of spending two hours (from 4 – 6 PM ET on wrkc.kings.edu) playing pretty much whatever music amuses me on terrestrial radio. I have been compiling playlists over at Spotify where you can search me up as mrdardy. It has been one of the real joys of my life this past year.
Mostly a consequence of my DJ gig, I have listened to more new music released in 2017 than any year since the birth of my son in 2003. It feels great to be reminded of the pleasure of discovering new music again. I still feel a bit overwhelmed when I read Best of lists at the end of the year, but there is a better chance of me knowing a number of items on these lists than I have had in years.
At work we have had a couple of important changes. We moved to a new, rotating schedule. We have 7 periods, 5 of which meet each day. In a seven day cycle each class meets five times. Four of the meetings are 50 minute classes (every once in a while an assembly moves that back to 45 minutes) and meets once for a 90 minute block. This has been a great change in our daily lives.
In our department we adopted a test correction policy where all students are allowed to earn back points by reflecting on their work. We ask them to submit corrections in the form of pointing out where/what went wrong in the problem’s work and then correcting said problem. I am super excited about this project and I see students being really thoughtful and attentive in submitting these corrections.
My life at school has been a bit more hectic than I’d like, despite the change in schedule. I have five classes this year (more often than not, this has been my standard work load here) which is especially manageable in this new rotation. What has been tiring is that I have four different class preps. Keeping all these trains running in my mind, especially since my two Geometry classes are rarely ever aligned anymore, has been a tiring challenge. I think being 53 and having a 14 year old boy and an 8 year old girl in the house has an impact as well!
I was able to attend TwitterMathCamp for the fourth summer in a row. As an added bonus, this past year did not conflict with my daughter’s birthday. Another bonus was that Atlanta is the home of an old high school buddy who was also my first college roommate. I had not seen him in years and had a lovely night with him and his family on a warm southern night, hours spent on his porch catching up was a delight.
My time at TMC was followed by a trip to FLA that included a couple of nights catching up with friends in my old hometown of Gainesville. Had not been there in a few years.
My school is a PK – PG school on two campuses. We live on the upper school campus, the lower school is about three miles away. My son is in 8th grade and he and his pals will be in my hallways in 8 months. Exciting and scary at the same time!
My wife is nearing the end of her Master’s Degree program. It’s been fun listening to her talk about her school experiences. It has been ten years now since my course work last ended. I’m a bit jealous, I think.
Off to face the new day, the new year, I guess, now…

A Tale of Two Questions

This past week I had a quiz for each of my Geometry sections. The two sections are out of synch a bit due to our rotating schedule. They typically assess on different days with different versions of whatever quiz or test I recently wrote. This week’s quiz had two different forms of the final question. I present them below:

In the diagram below you see a triangle ABC and you see what are called the exterior angles of the triangle marked. What is the sum of the measures these exterior angles? Be careful to carefully show your reasoning. Mark any angles clearly that you want to refer to in your explanation.

The problem above was presented to my class on Thursday.

In the diagram below you see a triangle ABC and you see what are called the exterior angles of the triangle marked. The sum of these exterior angles is 360⁰. Write a proof explaining to me why this is true. Mark any angles on the diagram that you refer to in your proof.

The problem above was presented to my class on Friday.

Both classes had the same first problem on their quiz. They were asked to prove that the interior angles of a triangle sum to 180 degrees. This proof was explicitly presented in class and in their text. My thought was that this challenging fifth problem should be a (somewhat) natural consequence of the first problem on the quiz.

The students who took the quiz on Thursday struggled on the first problem and it bled over to the last. They generally performed better on the last problem than on the first. In part, this is due to my decisions about partial credit. I was definitely more generous with partial credit on the problem at the end of the quiz since they had not seen any explicit proof of this fact. My colleague who also teaches Geometry felt that I might be reaching a bit with this last question. My Friday class performed better on the first proof than the Thursday crew and they did a MUCH better job on the last problem. I am trying to sort this out and there are too many variables at play. First, the class who took the quiz on Friday has performed at a slightly, but consistently, higher level overall during the first trimester of our year. Second,there is always the possibility that information about the quiz was discussed in a way that gave the Friday class some advantage. Finally, the problem presented to them gave an answer and asked for justification while the problem as presented to the Thursday class did not provide the conclusion. I was more strict with partial credit with the Friday quiz class since the conclusion was given to them and the whole burden of the problem was the explanation.

The main reason I am writing about this is that I am trying to make myself think clearly about what my goals are in a problem like this one and to convince myself that I was trying to get at the same thing with both classes. Did I drastically change the nature of what was being assessed by presenting the conclusion already? I have thought out loud on this blogspace about a similar question here – https://mrdardy.mtbos.org/2017/09/22/a-quick-question-about-test-questions/

Did revealing the answer to the question fundamentally change the level of challenge inherent in the question? Is it THAT much easier to reason through the proof when you know what you are supposed to conclude?

Our Geometry course is the last course in our curriculum where there is no Honors option. Everyone who takes geometry takes the same course at our school. This means that there is a wider variety of interest and talent in this room than in my other classes. I think that there is a tendency in a non-honors math class to think that the students cannot tackle challenging or novel questions. I have heard several colleagues over the years say something along the lines of ‘I can’t ask that question if I haven’t shown them how to do it.’ These are terrific teachers saying this and they are coming from a good place, they want their students to succeed and they do not want them discouraged or dismayed by assessments. I think I am coming from a good place as well, it’s just a different place. I’d also say that in the case of the question above, especially in its first form, I do believe that I have shown my students how to tackle such a question. They know that the interior angles sum to 180 degrees. They see three supplementary pairs of angles so that sum is 540 degrees. The difference is the exterior angles. Half of the students in the Thursday group earned four or five points out of five on the problem. Those who earned four generally had sound logic with real flaws in the vocabulary explaining their answers. Maybe my docking them a point is an entirely different question about how I assess.

Another reason I am writing this is that I want to have a conversation with my department about questions like this one, questions that are not a simple transformation of what has already been practiced. I have students who imply that I am the first teacher they have who asks them questions that feel like they might be ‘from left field.’ I know that students (all people, really) will exaggerate their concerns in the face of feeling stressed. I think most of my students do a nice job of stepping up to challenges like this one, especially when points are riding on it on an assessment. But I also know that there is an instinct at times to simply dodge these situations. The same group of kids who took the quiz on Thursday were presented with a problem from Steve Wyborney’s website on Friday in class. I showed them the video of the duplicator lab problem. When the video ended I asked them to begin talking about the problem with their neighbors – in this class everyone sits in groups of three that get randomly reassigned every fifth day. I was met with mostly silence. To be fair, this was about 8:10 in the morning. However, when I showed them the comments section with teachers talking about their fourth and fifth graders solving the problem, they suddenly started talking. So, I don’t know if they were shamed into action or they simply needed to suspect that they were more than capable of solving the problem before they moved. I have to feel that the struggle with the problem on Thursday and their reluctance to engage with a novel problem on Friday morning are related. I also fear that I have not done enough yet to create a culture where they jump into these problems. I am interested in how the conversation goes with my department on Wednesday morning and I would love to hear from any readers as well.

Persistence and Creativity

I spent six hours yesterday watching students take final exams. Three two-hour shifts. Sigh…

My last group taking an exam was my AP Calculus BC team. They had 24 multiple-choice questions and four free response questions, so they had to be efficient in their problem-solving. One of my students asked me about a multiple choice question that troubled him. It was an infinite geometric series question. This student joined our school, and our country, last year as a freshman. He place tested so highly that he started in AP Calculus AB. However, it has become apparent that there are a few facts/skills that he does not have at his command. This is rarely a problem since he is so creative. On this question, he did not have the formula in his brain for calculating the sum of an infinite geometric series. He could have listed out a handful of terms to look for convergence. He could have shrugged his shoulders and guessed since it was one of twenty-four questions. Instead, he wrote a short program on his TI-84 that gave him increasingly good approximations until he saw one of the answer choices emerge. He did this in the middle of a two-hour exam! He asked me afterwards and just laughed when I showed him how easy this problem could have been. I am convinced that he will remember this formula forever now, but I am also convinced that I will remember this story forever now. His ability to problem-solve in this situation is SO much more powerful than having a formula at his command. It is interesting that this happened on the same day that I sent our a question on twitter about the use of formula sheets on assessments. I am disinclined to use formula sheets, but I could be convinced otherwise. Here is a story that would have never happened if a formula sheet was present.

I am going to cross post this over at the One Good Thing blog space.

Optimistic

Another quick post. We are in exam week here at my school. I have ALL sorts of thoughts about term exams and why we do them, but those are for another place rather than this public forum. I have written about our department’s decision to move toward a test correction policy. I am so so so optimistic about our exams this week. I really believe that we will see largely improved results because the students have been more actively involved with examining their tests and reflecting on what went right and wrong on their tests. They have been talking to each other and comparing ideas. They have been talking to their teachers about how to fix their problems solving approaches. Our department exams are mostly on the next to last day of exam week. This will work against our students as energy levels start running low. Despite this, I am hopeful that we will see a different level of engagement on these cumulative exams. I will report back either way in a week or two, but I feel good based on what I have seen in the past week of reviews in class and from watching and listening to kids in the hallway as they work together.

Another Fun Problem Debate

A super quick post this morning. On my last problem set for AP Calculus BC I included the problem below:

A 10 m rope is fastened to one of the outside corners of a house, which has the form of a rectangle, 6 m long and 4 m wide. A dog is fastened to the rope. What is the perimeter of the region that the dog can access?

I have asked a form of this question a couple of times over the years. One year I did not mention that the rope was on the exterior of the house and I had a student assume that the dog was tied to a leash inside the house. I fixed that mistake.

A GeoGebra sketch below shows the image I have in my head for this problem.

My answer to this question, and the answer that 11 of my 14 students had, was that the perimeter is 20π. Two students argued that the answer should be 20π + 20. Their argument is that the borders of the house, the sides of the rectangle in the drawing, are also part of the perimeter. I loved the debate that ensued and most students migrated to this point of view. I reflexively thought of perimeter as exterior, while these two students argued that perimeter is boundary. I think I agree with them and I LOVE the fact that they cared enough to debate this on a problem that counted for one point out of about 400 something for the term. I also LOVE that students who got their answer marked correct started arguing against the answer that they arrived at.

AP Calculus BC – A Lovely Debate

It has been WAY too long since I posted. Energy has been scattered in a number of directions lately…

Yesterday in AP Calculus BC their sixth problem set of the year was due. I heard quite a bit of debate among my young scholars about one question in particular. Here it is:

I wish I remembered where I found this problem. I know I have used it in the past but I do not remember much debate about it. A little background about how I approach these problem sets before I relate the conversation we had. We are on a seven day class rotation in our new schedule. This rotation sometimes even includes two weekends. I assign a problem set the first day we meet on each rotation and it is due the first day of the next rotation. My students have time to wrestle with these problems, to ask each other questions, ask me questions, look up ideas/clues/data/etc. These are sets of ten problems and they only count ten points. All of the problem sets together count about as much as one test in a term. By the time the term is up, each problem set will end up weighing about 2 – 3% of a term grade. I grade them pretty easily, looking mostly for thoughtful process and I am willing to deal with half points which feels a bit silly but it honors thoughtful work rather than whacking them for some little mistake. Anyway, the point here is that these should be low stress and they should encourage collaborative thought and work. I have been clear that I encourage them to think together. Anyway, when they came into class yesterday there were a flood of questions for each other about fine points of interpretation for this problem and students were asking me about two distinct interpretations of part b of the problem. I loved the debate, I just wish it had been happening more than 2 minutes before the work was due. The primary debate was whether to answer part b by interpreting the year as the interval from 0 to 12 or whether to interpret it as 1 to 13. Our friend GeoGebra helped us a bit. Three screen shots below:

Now, the curve with the integral from 0 to 12 highlighted:

Finally, the curve with the integral from 1 to 13 highlighted:

Nice, right? The area is the same according to GeoGebra. What followed was such a highlight of my day. Intense debate about why these seem to be the same. One student notes that the obnoxious coefficient for x in the function yields a period of essentially 12. Not a surprise, right? Nice observation. A nice debate about the limits. Is 1 the end of January or the beginning? I feel it needs to be the end of the month the way the problem is written, so I came down on the side of 1 to 12 as the proper integral. An observation about the symmetry of right Riemann sums where some months have overlaps and others have underestimates. Another nice observation. One student questions this model because he notes, as a northern city, Seattle should have more rain in the warmer weather months rather than have this mediterranean rainfall pattern. Wow. I pointed out that I know people who lived in Seattle and have visited there and that this is indicative of what I know about the area. I do not think that this would have ever occurred to me! A debate about whether we simply want to add f(1) + f(2) + f(3) + … + f(12) rather than integrate. One student mentions that he looked up the yearly rainfall in the Seattle area and the answer he found was pretty close to our integral. Good golly, that made me happy. A student questioned whether I wanted two answers to part a. I admitted that the word extreme, in this context, only made me think of a Maximum. I admitted that I should have been more careful than to make that assumption and I should include the minimum in the answer to part a as well.

We have a test today, so I pivoted the class conversation toward some practice problems but I felt that there was probably still more to discuss about this problem. Days like yesterday remind me of how spoiled I am that I get to do math with a group of scholars that are willing to engage in energetic debate about math. They are rarely interested in just knowing the answer, they seem to genuinely enjoy to process that gets them there. I am not going to pretend that all 15 students were actively engaged joining in with their ideas. I will say that there were all at least attentive and willing to let the conversation flow without pointing out, anxiously, that we need to talk about the upcoming test.

We decided a number of years ago to teach BC as a second year Calculus course at our school. We know that many of our students can accelerate through the BC curriculum in one fast year or by buying some Precalculus Honors time, but conversations like this one would be hard (not impossible) to find time for. Every year I hear a scholar say something along the lines of ‘I knew how to do this before but now I think I know why it works.’ Statements like that one, and conversations like yesterday’s make me feel good about this curriculum decision we made.

Improving a Lesson Plan

My Geometry classes have just finished a few days considering different translations on the Cartesian plane. We are working toward being comfortable with rotations (almost always around the origin), reflections (horizontal and vertical lines and the lines y = x and y = – x), and vector translations. Last week I had a particularly unsuccessful lesson where I tried to help my students discover a pattern for 90 degree rotations around the origin. I want to try and outline my thinking here and I would love love love any insight into why it did not go well and how I can improve this in the future, or even how to go back to cycle and revisit this with my current team of scholars.

So, my idea was this – try to pull together what we know about perpendicular slopes, our developing ideas of vectors as a physical object similar in nature to a line segment, and developing an intuition about the fact that a 90 degree rotation should result in a move of one quadrant in a certain direction. I asked for three coordinates from my students and drew a triangle. I asked them to predict where this triangle would end up after we moved it 90 degrees clockwise. Two of the coordinates given were in quadrant I and one was in quadrant IV. It seemed that my students were happy/comfortable with the idea that the two quadrant I points would live now in quadrant IV and that the quadrant IV point would move to quadrant III. This may have been a tepid agreement in retrospect. Next, I focused our attention on one of the quadrant I coordinates and I drew a segment from the origin to that point. We talked about the slope of the segment, we compared this segment to a vector, we talked about the length of the segment. I then asked the students to imagine a wheel and I told them that when I think about rotations I think about a bicycle wheel. In my mind I saw this segment as a spoke and I thought about distance from the point to the center of the wheel. Here is one place where I know I failed my students. I did not explicitly stop at this point and discuss distance the way I could have/should have. I also have a handful of students who are still struggling terribly with the idea of calculating distance. We have been talking about it since day two, I have coached them to think about Pythagoras, we have practiced it repeatedly. The combination of squaring, of square roots, of subtraction in one piece of ‘the distance formula’ and addition in another piece of it, comfort with mental arithmetic, all of these factors are working against my students being unanimously comfortable with calculating distances. So, the next step in my plan was to ask them what they recalled about perpendicular slopes. They all should know this and most recalled it pretty quickly. We had a segment in front of us with a slope of 4/3 and my students quickly agreed, maybe passively maybe enthusiastically, that a segment perpendicular to this would have a slope of -3/4. So, the question at hand was now whether the fraction was in the form or -3 over +4 or in the form of +3 over -4. I was convinced in advance of this lesson that this string of conversations would be a positive path to take. I felt that the combination of recalling past slope ideas, looking at the physical Cartesian plane, tying in ideas of line equations, etc. would gel together to make a lasting learning experience. I was wrong. When I prodded them toward the conclusion that -3 over +4 was the conclusion we wanted I saw some uncomfortable faces. When I mentioned the idea of a spoke as a visual to hold on to, I saw blank faces at this point. I got a bit frustrated and asked for my students to describe to me what they were thinking of when I mentioned the word spoke. Nothing. I pulled up a google image of a bicycle wheel and asked them to tell me what the spoke was in the image. By this point their reluctance to engage in this conversation was building, my frustration was increasing, and any positive momentum in building this process was falling apart. My fault for showing my frustration. My fault for stacking up too many ideas at once, I think. When I spoke to my Geometry colleague she felt that adding on the layer of talking about perpendicular slopes was the tipping point of discomfort for my students. I trust her instincts on a number of levels in part due to her experience in teaching our Algebra I course. She knows these Algebra kiddos and knows not only what they know but how comfortable they are knowing it. So, at this point it was clear to me that this was slipping away. We limped to the end of the conversation. Most students were willing to agree that the point (4,3) would end up in quadrant IV. They were split on whether it would land on (4,-3) or on (3,-4) and it honestly felt like many were mentally tossing coins to make this call. I showed them the conclusion on GeoGebra and we sort of ran out of time by this point. We have since gone back and tried to reinforce the conclusion we reached and I think that most of my students can reliably answer this question, but I am completely uncomfortable with how we got there. I would love any insight/advice about how to best structure this info. You can certainly drop a comment her or over on twitter where I am @mrdardy

A Quick Question about Test Questions

One of the classes I teach is a non – AP Calculus class. Many of my students are showing a reasonably firm grasp of the ideas of Calculus that we are studying, but they are plagued by algebraic struggles and fears at times. I was thinking about the following two ways that I could ask them the same question:

I asked this class today how they would feel about these two forms of the question. Most of them quickly agreed that they would feel more comfortable answering question B than question A. I followed up by asking whether success on question A might feel more like success and a handful agreed that this probably was true. I am not sure what to make of this informal poll and I am bouncing this idea around in my head. I’d love to hear what you think and push back with some questions about my motivations/hopes in (possibly) restructuring my assessment in this direction.

P.S.

I have been urging them to have GeoGebra open nearby when they do their work so that they can look at graphs and ask for derivatives to verify their own work.

Checking in on a new Policy

I recently blogged about a commitment made by our math department. As a reminder to you, dear reader, here is the statement from one of my course syllabi:

Beginning in the 2017 – 2018 academic year, our math department is adopting a policy of expecting test corrections on all in-class tests. The policy is described below.

When grading tests initially each question will get one of three point assignments

Full credit for reasonable support work and correct answer.

Half-credit for minor mistakes as long as some reasoning is shown.

Zero credit (in very rare cases) when there is no reasonable support shown or if the question is simply left blank.

When grading tests, I will not put comments, I will simply mark one of these three ways.

You will be allowed to turn in corrections. Corrections will be on separate paper and will have written explanations of errors made in addition to the correct work and answer. This work is to be in the student’s words but can be the result of consultation/help. These corrections will always be due at the beginning of the second class meeting day after the assessment is returned. You will return your original test along with your correction notes. I will remind you of this every time I return a graded test to you.

It is not required that you turn in test corrections.

The student can earn up to half of the points they missed on each individual problem.

This policy does not apply to quizzes, only to in-class tests.

There are a couple of items in the learning curve to report on here. I will lead with the positives –

We have an after school conference time that many students take advantage of for extra help. My room has been more crowded this year than it has been for years. I take this as a plus sign that students are committed to seeking help and investing in their math studies. Many of them are there talking with me and with their classmates about test corrections.
I have had a number of students turn in their corrections the same day that they received their tests. It is a rare thing for students to turn in work days early, it is happening regularly right now.
I walked out of my classroom today with a new student who was talking excitedly about how she is really thinking carefully about her work and she is sure that she’ll remember material better because of this.
It is faster to return tests since I am simply marking them 100%, 50%, or 0% for each problem.

Now a few negatives, followed by some philosophical pondering about this whole endeavor.

I anticipated that there would be very few zero scores on problems. There have been more than I thought but I hope that is a factor of students learning to show some work. This may be a positive as a zero stings a bit and they may be more inclined to be careful in their explications.
Some students feel stung by relatively minor mistakes that initially result in a 50% on an individual problem. These minor mistakes turn into 75%. I am trying to point out that relatively major mistakes can also end up at the 75% level but some students feel a bit cheated.
Early in the year averages fluctuate quite a bit anyway, but the fluctuations are exaggerated in this system. I see already that overall averages are a bit higher than normal and there is much less variance in scores. However, some students are scared since they have a hard time seeing the long game as clearly.
Explaining this to folks outside the department has been a bit of a challenge.

When I was working on my doctorate I had a professor (my thesis advisor) who had a policy that every paper will be rewritten, not just every paper can be rewritten. The way he did this was to return our first drafts with no comments, just hash marks in the margin at certain points of the paper. These hash marks might be there to point out a flaw in our argument or our paper’s structure. They might also indicate a highlight. They might indicate a misspelling or a simple grammar problem. He was willing to discuss these hash marks in his office hours as long as it was clear that we had sound questions about them, in other words we had to prove to him that we had reflected on our writing. I have never thought so much about my own writing as I did in that class. I do not expect my 14 year olds to do this kind of self analysis, but I know that they ARE capable of careful reflection if they are given the time, space, and motivation to do so. What I am seeing when they come to me is that they have looked over their test, they have referenced notes and their HW. They have done careful thinking and they can usually explain their mistake on their own. This is not universal, but it is happening more often than not. Many students who come to my room to work on corrections have almost no question for me. They are berating themselves for ‘stupid mistakes’, they are laughing at silly things they wrote, they are even saying ‘I have no idea what this work means’ I am pretty convinced that this can be a huge growth opportunity for my students. They are being responsible for their own error analysis here and they are writing thoughtful reflections in the form of ‘On problem 4 I made this mistake, I should have done this instead’

We have midterm grade comments looming and as department chair I know I will have some questions coming my way about our reasoning and the long-term impact on grades. I will try to steer the conversations to long-term impact on learning and self-sufficiency. So far the policy has exceeded my hopes in my classes. I will be checking in with my department to get other points of view soon and I plan on sharing some of those conversations as well.