Derek Bruff had a post today talking about digital distractions and wondered briefly at the end about note taking. I wrote a comment to his post, but it is advice that I want to make sure I follow myself so I am posting it here for my own record.
In the interactive engagement world I think that note taking is one of a suite of reflective/feedback practices that we can help our students with. After a typical “one correct answer” clicker question, you will have some combination of students that were correct/incorrect for the right/wrong reasons. after some sort of sequence (revoting, class-wide discussion, instructor explanation), the students have now all heard the correct answer. But we know that some of them still don’t understand the answer enough to do anything else with it so it is time to get them to do some reflection or feedback. Options include: writing their own understanding down in their notes, answering a follow-up clicker question, collaborating with a group to answer a question on a worksheet, etc. I see note taking as one of many options in this type of cycle and if we are not getting them to do some other type of feedback/reflection activity then we should at least be giving them a minute or two to reflect in their notes. Of course, I do a terrible job of this AND many students are highly reluctant to take notes so, in general, I prefer the other type of reflection/feedback activities.
This is part 2 (part 1 here) of my post discussing feedback I got from a couple of my students after the conclusion of my Advanced Lab course. This went long again so it looks like there will have to be a part 3.
The 8-hour work-week and filling out time sheets
The combination of this having been only my second time teaching the course and my policy on student experiments always building on, but not repeating, the project of previous groups made it very hard for me to figure out projects of appropriate scope. So my solution was to ask that the students put in a minimum of 8 hours each week into the course and then I had to make sure that the projects consisted of sequences of achievable milestones. With that in place, I was happy to accept however far along each group made it with their projects as long as those 8 hours each week were actually spent working productively on the course.
So I got them to fill out and submit time sheets. I was worried that they would perceive these as being too strict or beneath them or terrible in some other way.
Interview feedback: No complaints. The time sheets were fine and did a good job of encouraging them to dedicate an appropriate amount of time to the course even when they felt like doing something else. Yay!
Future plan: It looks like I will continue to use these time sheets. The thoughtful-assessment part of my conscience doesn’t really like having to use these, but for the most part these students have never had to budget time for a longer project and they really need the scaffolding so that they don’t fall on their faces.
Oral and written dissemination
One of my major guiding principles in this course was (and continues to be) to try to make sure that the communication of their project was directed toward authentic audiences. For the weekly group meetings, they were bringing me up to speed on their project as well as informally presenting to their work to people with less project-specific expertise (the rest of their class-mates). Since projects are always meant to build on previous projects, their formal reports are going to be part of the literature used by the next group building on that same project. Their formal oral presentations were targeted at peers that lacked project-specific expertise (again the rest of the class).
The first time I taught the course, I had the students write journal-style articles (each partner wrote one). There were two problems. First, the partner that was not writing the article ended up contributing very little to the analysis and usually didn’t dig deep enough into how everything worked from either the theoretical or the experimental side of things (which is part of why I implemented the oral assessments into the course). Second, the background and theory sections often lacked an authentic audience for multiple reasons: (A) they were often vaguely repeating the work from a source journal article; (B) if they were building on a previous group’s work, writing their own background and theory sections would be mostly redundant; and (C) the topics were often deep enough that it was not reasonable to expect them to develop the project-specific expertise to do a very good job these sections.
So in this second incarnation of the course I decided to split the journal article up into two pieces, one for each partner: a LaTeX “technote” and a wiki entry for the background and theory. The idea was that future groups could add to the wiki entry, which would eliminate the redundancy of recreating essentially the same theory and background sections for future groups working on that research line. With the theory and background stripped out of the journal article, all I asked in the technote was that all equations and important terminology be clearly defined within the technote, and no other theory was needed. I thought this would have the added benefit of having both partners invested in a writing task for each project. But the whole thing did not work very well. The technotes worked fine, but the wiki entries ended up being so disconnected from the technotes that partners often didn’t even use the same notation between the wiki entry and technote.
It is worth noting that between a time crunch and the technote+wiki not working as well as I liked, I got the students to team up and write something a bit closer to a journal article for their second project.
In addition to their technote and wiki entry, each student gave a 12-15 minute formal oral presentation on one of their projects (each partner presented on the project for which they wrote the technote) instead of writing a final exam.
One of the things I wanted to discuss in the interview was what sort of improvements could we make to this dissemination process. I had some ideas in my head to discuss such as poster presentations and articles written for a lay-audience.
Interview feedback: for each project one person would write a journal article and the other would prepare and present a poster at a research symposium. The logistics of this still need to be worked out and we discussed a number of combinations of dissemination methods before coming to a consensus on this specific one. The interviewed students saw communicating science to a lay-audience (the research symposium attendees) as an important thing for them to practice.
Future plan: I really like how this combination makes each member of the group responsible for communicating all the important pieces of their project. Targeting them at different audiences means that they will be able to work together while still ultimately having to produce their own non-redundant (relative to each other) work.
Their are a lot of logistical issues to work out here. Our university has a student research day a couple of weeks before the end of our winter term and that would be a perfect place for them to present their poster. The problem is that, with a proper revision cycle for their poster, they will essentially have had to have completed both projects a month before the end of the term. I’m not certain I can make that work. We can always have our own research symposium, but it seems ideal to get involved with an existing one that already has an audience.
The other piece here is that I will probably ask them to keep the journal articles closer to the technotes than a real journal article (meaning bare-bones theory and background).
A vague notion of a plan dawned on me while proof-reading this post. I could probably get the timing with the student research symposium to work if I reduce the scope of each project by roughly a week and then in the final month of the course I could ask each group to revisit one of their experiments and push it a bit further forward. There are all sorts of problems with this plan, such as how they will disseminate this additional work and the experimental apparatus probably having been torn down, but it is still something to consider.
The timing of peer review for their lab reports
Each technote and journal article was allowed as many drafts as needed to get the paper up to “accepted for publication with minor revisions” standards (a B-grade) based on a very picky rubric. After that, they were allowed one final draft if they wished to try earn an A grade. A typical number of drafts was 3 or 4, but there were exceptions in both directions.
For the first report, I had each person do a peer review of another student’s submission. One of the questions I had on my mind for the feedback interview had to do with the timing of the peer review in the draft cycle. The first draft of the first paper is always an extremely rough thing to slog through, even those written by very strong students. Thus, asking them to do peer review on a first draft is asking them to do something very painful. But, having to critically apply a rubric and provide constructive feedback does wonders for getting students to pay much better attention to the specifics of the writing assignment and the sooner that happens in the course, the sooner that I see those improvements in their writing.
Interview feedback: not too sure if it is best to do peer review on a first or second draft. We discussed this for a bit, decided we could see both options as equally valid, and never came to any real conclusion.
Future plan: dunno yet. I could sign my course up for the Journal of the Advanced Undergraduate Physics Laboratory Investigation tool. They have peer review calibration tasks and the added benefit of anonymous peer reviewers from other institutions, but since JAUPLI is still small, the timing all has to work out magically well.
This is a quick follow-up to my previous post on my research related to the effect of immediate feedback during exams.
I love it when I’m going through my “to read” pile of papers and realize that there is something in there related to one of my own research questions. There is a paper from last year (Phys. Rev. ST Physics Ed. Research 7, 010107, 2011) by Fakcharoenphol, Potter and Stelzer from University of Illinois at Urbana-Champaign that looked at how students did on matched pairs of questions as part of preparing for an exam.
There’s a lot of interesting stuff in this paper, but the result which is most relevant to my own research is the following. They developed a web-based (voluntary) exam preparation tool where students would do a question, receive feedback as just the answer or as a solution, then do a matched question where they received the other type of feedback. They divided the students into four groups so that every student had equal access to answer vs. solution feedback on the questions. For each matched question pair (let’s call them questions A and B), the grouping of the students allowed half of the students to answer question A first and the other half to answer question B first. Within each of those groups, half of the students received answer only feedback for their first question and the other half received solution feedback for their first question.
They called the first question of a pair answered by the students their baseline and the students scored 58.8%±0.2% on those questions. Keeping in mind that they had many pairs of questions, the average performance of the students on the follow questions was 63.5±0.3% when only the answer was supplied after answering the first question and 66.0±0.3% when the solution was provided after answering the first question. There are statistically significant differences between all of these numbers, but the gains from receiving the feedback are not overly impressive. More on this in a moment.
Back to my own research. During an exam, I used matched pairs of questions and gave the students feedback on their first question (in the form of just the answer) before they answered the second question. I saw a statistically significant improvement from the first question (65.3±6.8%) to the second one (77.5±6.0%), but due to low statistics there was not much to conclude other than it was worth pursuing this research study further. The results from the UIUC folks set the magnitude scale for the effect I will see once I am able to improve my statistics (58.8%±0.2% to 63.5±0.3% due to solutions only feedback).
I’m really not certain if I expect to see less, equal or more improvement for my “during an exam feedback” design as their “preparing for an exam feedback” design. In their design, the level of preparation of their students when using their study tool is all over the map (they look at this in more detail in their paper) so it is not known if the learning effect due to the feedback also depends on when during their overall study plan they were using the tool (e.g. as a starting point for their studying vs. to check their understanding after having done a bunch of studying). Since both our designs use multiple-choice questions (but preparation vs assessment conditions) I am not certain how guessing would play into everything.
I have to admit that if my future research into the effect of feedback during an exam finds that I am getting only a 5% gain (like UIUC did for their solution only feedback) from this intervention that I doubt that I would continue with the practice.
Once I started writing this it got pretty long so I will call this part 1 and work on part 2 another day.
A month ago I took a couple of my students to a local coffee shop, filled them full of treats, and poked their brains for a couple of hours about my Advanced Laboratory course that ended in April, 2012. I’m summarizing their feedback here to make sure I have a good record of their feedback and, of course, to share. For any given piece of feedback from the students I will try my best to explain the context and by the end you should have a pretty good idea of how the course looked and where it will be headed in the future.
Let’s start with my definition of an Advanced Laboratory course from an earlier post:
This type of course, a standard course in most physics departments, is a standalone lab course without any associated lecture course. There is an amazing amount of variability from one Advanced Lab course to the next and they range in format from one experiment per week with everything already set up and cookbook procedures ready to be followed, to a single student-developed project over the entire term (or year!).
In my specific incarnation, we spend the first month doing some introductory activities to build up some foundational skills which are mostly related to data analysis and presentation. For the rest of the course pairs of students work on two month-long experimental physics projects. The students are guided to work on projects that can be viewed as being part of a larger research line, where they build on the work of previous students and future students will build on their work. Thus no two groups will ever perform identical experiments.
Onto the feedback!
Weekly research group meetings
Each week we had a research group meeting where each group was asked to post a couple of figures or tables to the course wiki and quickly bring us up to speed on what they had done the previous week as well as what they planned on doing the next week. A very small part of their grade was based on how much they contributed to these research group meetings. My expectation was that, averaged over multiple meetings, each student would ask at least one meaningful question to the presenters per meeting or contribute in some other way to the conversations surrounding the projects of the other groups. I had twelve students enrolled so I split the class into two larger research groups so each research group consisted of myself and three pairs of students.
I was quite happy with how these meetings worked and felt it was really valuable for the presenters to have to frame things so that people other than me and the presenters actually understood what they were up to. Anecdotally it felt like students spent more of their initial time learning about the basics (experiment and theory) of their projects than in the past because they were going to have to explain things to somebody else.
Interview feedback: the meetings felt too long. Their initial suggestion was to make the meetings every other week or to time-limit them somehow. A research group meeting took somewhere between a half-hour and an hour each week. The actual presentations were reasonably concise, but there were always lots of questions from the other groups and feedback from me. This was also the place where we hashed out a lot of the gory details of what they should try to do in the coming week. But the thing was that the other groups often contributed to these discussions on what to tackle in the coming week so I felt like the whole process was extremely valuable for all parties. But it could probably be tightened up.
Future plan: Partners will alternate being the main presenter each week (previously they were both expected to present each week) and will be asked to present 1 or 2 tables of figures. The feedback was that it sometimes felt like a stretch to find that 2nd table or figure to present. The actual presentation will be limited to 5 minutes for a total of 10 minutes per project group between the presentation and the discussion afterwards. I won’t be strict on the time limits, but will be mindful of the clock to help prioritize which discussions to have at that moment and which ones can be saved for private discussions later. One of the students also suggested that having time limits on their presentation time would serve as good practice for their formal oral presentations later in the course and these did have strict time limits.
Twice during the intro sequence I tried to have a number of small groups working on different things and then had them report out to the class. The second time we did this I called them “parallel investigations” and sent them off to study goodness of fit, Monte-Carlo methods for fitting or Monte-Carlo methods for error analysis. In addition to orally reporting out their findings, I asked one partner to write their findings up in LaTeX and the other on the course wiki. These two write-ups were allowed to be identical and the reason that I used two formats was because one of the students was going to write up the background and theory for their first project on the wiki and the other was going to write up the analysis and results as a LaTeX “tech note”. Thus I wanted them to have some practice using these writing formats. Note that on the second project they were to switch who wrote on the wiki and who wrote the LaTeX “tech note”.
Interview feedback: this might not have been the best use of their time. Yup, I agree. In the end, for the investigations that they did not perform, each group was simply ont he receiving end of a mediocre lecture on the topic and never got a chance to actively engage with the ideas. This is the exact opposite of how I try to run my courses.
Future plan: I’m planning on completely restructuring how the first-month introductory stuff works and will talk about that a bit more later, but I think the parallel investigations idea as it existed is officially dead.
Introduction to LaTeX
This is a place where I have not offered my students a ton of support. I wrote a tutorial so that they can install miktex and texniccenter on their windows machines and gave them a couple of sample LaTeX documents that cover all the basics, but that’s it.
Interview feedback: they would like some coherent instruction and resources. For most of them this is their first time ever dealing with a markup language and the learning curve seems to be steeper than I have been admitting to myself.
Future plan: It looks like a crew of us on twitter are going to put together a LaTeX workshop for the Summer 2013 AAPT meeting and I am hoping that as part of this process we will have put together a straight forward introduction to LaTeX for physicists package that I can drop on my students like a big ol’ pile of awesomeness.
From Student-Generated Scientific Inquiry (Leslie Atkins and Irene Salter) I used their lab notebook activity which uses pages taken from different famous scientists actual lab notebooks. The students are asked, in small groups, to take some notes on how these famous scientists took notes and organized their information. As a large group we then built a rubric for their lab notebooks based on their observations of the pages from the notebooks of the famous scientists. The students were highly engaged in this activity and seemed to be supportive of the rubric that we developed from this activity.
Interview feedback: they thought this activity was great. But they didn’t find it in the way that I expected. The two students I interviewed both had some previous experience with lab notebooks in research labs and had, in the past, put way too much emphasis on maintaining an immaculate lab notebook. This activity had let them know that it was OK to have rough notes in their lab notebook.
Future plan: I hate marking lab notebooks. It is the worst. And with so much of the work they do being digital these days it is really hard to find a solution that fits into their work flow and doesn’t involve pasting umpteen print-outs into their lab notebook. I’m actually planning on backing off of trying to get them to keep a really good lab notebook and emphasize getting them to report at the beginning and end of the day what they planned on doing and what they actually accomplished (science fair style!). I will be checking it every class period and it will be graded as complete or incomplete. Once I feel that I can get a group of students doing a consistently good job of this, I will consider the next step to take.
I have three new-to-me courses that I am teaching this fall: comp-phys, digital electronics lecture and digital electronics lab. I am sharing the syllabus for my digital electronics “lecture” course below, but have removed a few things which are only relevant internally.
UFV Physics 362 – Digital Electronics and Computer Interfacing Syllabus (V1) – Section AB1, Fall 2012
About this course
In addition to learning about digital electronics, one of the main goals of this course is to help you develop as a lifelong independent learner. Robert Talbert puts it much better than I ever could (http://goo.gl/ZIh0R):
“As you move through your degree and eventually into your career and your adult post-college life, your main value to the rest of the world and to the people you love is your ability to learn and grow without needing other people around to make it happen. There are many times in life where you MUST learn something, and you can’t wait for the next semester at the local college to come around for you to sign up for a course. You have to take charge. You have to learn on your own.”
This course is structured around the idea that you will do some initial learning on your own before you come to class and then in class you will work with your peers to deepen your understanding. You will be doing the heavy lifting in class instead of just watching me do examples and derivations on the board (do you remember how proficient you became at sword-fighting by watching the Princess Bride?). Some students find this very disorienting and some of you will find that this course structure will take you out of your normal comfort zone. The best thing you can do is come into the course with a positive attitude and be prepared to tweak your normal recipes for success to be able to get the most out of this course.
Please note that this course has a corequisite lab (Physics 372) which will focus on the hands-on aspects of digital electronics as well as the interplay between theory and hands-on applications.
Course Description (from the official outline)
This course emphasizes elementary digital electronics and interfaces. Topics include gates and Boolean algebra, Karnaugh maps, flip flops, registers, counters and memories, digital components, microprocessor functions and architecture, instruction sets, D/A and A/D converters, and waveshaping. PHYS 372, the laboratory portion of this course, must be taken concurrently. This course is designed to provide practical experience with the basic digital logic chips and how digital circuits can be interfaced with microprocessors.
Note that we will co-construct a proper set of detailed learning goals as we proceed through this course and those detailed learning goals will define what sort of questions can be asked on the quizzes and the final exam. The learning goals listed below, which are from the official course outline, are meant to be very broad and as such only provide a very rough framework in which we will fit all the fun that is digital electronics.
Learning goals from the official course outline: This course is designed to provide students with:
- the theory needed to understand the purpose and how digital devices function;
- an understanding and an appreciation of how a digital computer functions;
- the ability to design, construct and test simple digital logic circuits;
- an ability to program the common microprocessors;
- how information can be transferred to and from computers.
Tony R. Kuphaldt, Lesson in Circuits: Volume IV – Digital, http://openbookproject.net/electricCircuits/Digital/index.html
In addition to this online textbook, I will leave a nice big pile of electronics textbooks in A353 for your use. As a group we can sort out a reasonable scheme for lending out these books while making sure that they are still available to everybody.
Pre-class Assignments: The engine that drives this course is the collection of Socratic Electronics worksheets. For each worksheet, you will be assigned to research and answer a subset of the questions. In class you will present your findings in small and large groups. The goal is for you to learn how to locate information, problem-solve, collaborate, and clearly articulate your thoughts while learning about digital electronics. The answers to all the questions will be provided with the worksheets, so it is the solutions in which I am most interested and for which you are responsible in your preparation.
Class Periods: I run each class period under the assumption that you have completed the relevant pre-class-assignment and have made a genuine effort to make some sense of the material before showing up to class. We will use class time to help you clarify your understanding of the material and to build on the core ideas that you wrestled with in your pre-class assignments. In class you will mostly be working in small groups. Not all members of a group will have been assigned the exact same pre-class questions, so the first thing that you will do is present your own findings and come to group consensus on the solutions. In class I will also ask you to work on additional questions from the worksheets as well as other additional questions which I will provide. At appropriate times, I will provide mini-lectures to clarify ideas or to plant the initial seed for an idea which you will be studying on an upcoming pre-class assignment.
Peer and instructor assessment of pre-class and in-class work: Each class period you will be given a number of contribution points to spread among the rest of your group (not including yourself) based on how much their pre-class preparation and in-class work contributed to your group’s overall learning in class. The exact number is 8*(N-1)+1, where N is the number of students in your group. You can give any individual student up to 10 points and do not have to give out all of your points. I will average the points assigned to you by the rest of your group for each class period. If needed, I may adjust this average up or down by up to a couple of points if I feel that your class period average is a very poor match to my own observations of your pre-class preparation and in-class contributions. I prefer not to have to make any adjustments this way and will very clearly spell out for you what factors I have considered when adjusting this daily class period average. I will drop your five worst daily class period averages when calculating your final mark for this category.
Homework: Nope, but I will make sure that you have sufficient resources for quiz and exam preparation.
Quizzes: Roughly every two weeks we will use the entire class period to have a quiz, for a total of 5 quizzes over the course of the term. The quiz will be split into two pieces: a solo quiz and a group quiz. You will first write the solo quiz and then approximately 2/3rds of the way through the class period I will collect the solo quizzes and then get you to write the group quiz, typically in groups of 3 or 4. The group quiz will mostly be the same as the solo quiz, but will often have some extra questions. If you score higher on the solo quiz than the group quiz, I will use your solo quiz mark when calculating your overall group-quiz average.
Quiz Averages: I will use your best 3 of 5 group quiz scores when calculating your overall group-quiz average. Things are a little more complicated for your overall solo-quiz average. In addition to the three-hour final exam, I will be creating five different half-hour-long re-tests, one for the material covered on each of the five quizzes during the term. You can choose to write two of these re-tests as part of the final exam and your mark from each of those re-tests can replace your earlier mark on the corresponding quiz (including if you missed the earlier quiz completely). The catch here is that I will only allow you to write a given re-test if you demonstrate to me that you have put in a reasonable amount of effort to learn that material. I will expect you to make your case by presenting me with the specific things that you did to learn the material and that you did to learn from your mistakes on the initial quiz.
|Peer and instructor assessment of pre-class and in-class work:||
Tentative Course Schedule
The numbers Sxx indicate the worksheet number for that day’s worksheet. The worksheets can be found at http://www.ibiblio.org/kuphaldt/socratic/doc/topical.html
|Sept. 3||D01 – Numeration Systems (S04)||No class||Classes begin Sept. 4.|
|Sept. 10||D02 – Binary Arithmetic (S05)||D03 – Digital Codes (S06)||D04 – Basic Logic Gates (S03)|
|Sept. 17||D05 – TTL Logic Gates (S07)||D06 – CMOS Logic Gates (S08)||No class|
|Sept. 24||Quiz 1||D07 – Trouble Gates (S09)||D08 – Boolean Algebra (S13)|
|Oct. 1||D09 – Sum-of-Products and Product-of-Sums Expressions (S14)||D10 – Karnaugh Mapping (S15)||No class|
|Oct. 8||Thanksgiving. No classes.||D11 – Binary Math Circuits (S16)||Quiz 2||Wednesday Oct. 10 is last day to withdraw without W appearing on transcript.|
|Oct. 15||D12 – Encoders and Decoders (S17)||D13 – Multiplexers and Demultiplexers (S18)||No class|
|Oct. 22||D14 – Latch Circuits (S21)||D15 – Timer Circuits (S22)||Quiz 3|
|Oct. 29||D16 – Flip-flop Circuits (S23)||D17 – Counters (S26)||No class|
|Nov. 5||D18 – Shift Registers (S28)||Quiz 4||Remembrance day. No classes.|
|Nov. 12||D19 – Digital-to-Analog Conversion (S30)||D20 – Analog-to-Digital Conversion (S31)||No class||Tuesday Nov. 13 is the final day to withdraw from courses.|
|Nov. 19||D21 – Memory Devices (S34)||D22 – Optional Topics (see notes)||D23 – – Optional Topics|
|Nov. 26||Quiz 5||D24 – Optional Topics||No class||Potential topics include digital communication, micro-controllers, state machines and electro-mechanical relays.|
|Dec. 3||D25 – Optional Topics||Monday Dec. 3 is the last day of classes|
My 6 (nearly 7) year-old son and I made up a new game today. It’s a good math skills game. The game is based on putting tiles down so that the numbers on the tiles add up to multiples of 5.
Skills your child needs to have to play this game
- Adding 2-3 small double-digit numbers
These are the rules that we found made a game that didn’t drag on, but also meant that the winner wasn’t simply the first person to play. The winner is the first player to get rid of all of their dominoes.
- Place all the dominoes on the table face down.
- Each player draws 9 dominoes. Draw a single domino and put it face up in the middle of the table (the 10/10 in my example picture). The youngest player goes first.
- To play a tile, there are two options.
- The first option is to play a single side on a single side already played where the sum of these two sides needs to be a multiple of 5. Examples in the picture include the 0 played next to the 10 on the right-hand side or the 2 played next to the 8 on the middle of the left-hand side of the picture.
- The second option is to play both sides of a new domino (“sideways”) on a single side already played, where the sum of all three sides needs to be a multiple of 5. There are two examples in the picture: the 7/8 played on the 10 on the left-hand side and the 7/11 played on the 2 at the top.
- After a player plays a tile, it becomes the other player’s turn.
- If a player can’t play a tile on their turn, they draw a single tile from the face-down tiles and then it is the other players turn.
- A sideways domino cannot be played on a sideways domino. For example, a 2/3 tile could not be played sideways on the 7/8 tile at the bottom left. A 3, 8 or 13 could have been played on the 7, or a 2, 7 or 12 could have been played on the 8 as was done.
- The winner is the first player to play all of their tiles.
- Beginner rules – Get rid of all rules that involve sideways placements. That way you are only ever adding two numbers together
- Sums other than multiples of 5 – Instead of multiples of 5, try multiples of 7 or some other base that is worth practicing with your child.
- Double sideways - Allow a sideways tile to be played on a sideways tile so that the multiple of 5 comes from summing all 4 sides.