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Standards-Based Grading: Math (1 of 7)

This is the first in a series on the implementation of standards-based grading in specific disciplines. A lot has already been written on this subject that I hope to expand on.  A bit of motivation before moving on: Ask yourself, “Does my gradebook mean anything to anyone other than myself? How can I use assessment tools to better communicate with  students about their progress?” Finally, can we use assessment to empower students to control their own formative behavior in the classroom?

Alright Math, you’re up first. I teach predominantly math, and use standards-based grading in all of my courses. My math classes used to be fairly conventional. A lot of me talking, students asking questions while I was talking, and then students practicing from a book. Then we had little benchmark quizzes. Once in a while we’d have big ol’ tests that would cover many chapters of material. The problem was that students weren’t breaking ideas up into their logical pieces. They’d see a grade in the gradebook like: “Quiz 5 – 23/30,” and think to themselves, “Oh man, I better step it up on the next quiz.”

Aye, there’s the rub. The NEXT quiz? What about the material you don’t know from this quiz? What about that? Huh?! To a student, a conventional quiz is just an opportunity to gain some more points. They don’t see it as a chance to demonstrate a specific skill set. Until now!

Standards-Based Grading!

Here’s the list of things I changed. I will explain the reasoning and logistics of the changes, and then I’ll run you through my typical experiences as students interact with the new system.

  1. Fracture Your Gradebook: I had to break my gradebook into much smaller pieces. Now there are entries for every single learning target (standard, concept, whatever), and they are all worth the same amount of points (10). The entries in your gradebook must now be super clear as to what they represent. These are formative and can be changed during the semester. There are also summative (traditional) entries, including tests, or anything else I want turned in on time (responsibility credit, more on this later).Here’s a screen shot of a part of my gradebook:
    Grade Booke Screenshot

    I have yet to give a summative assessment (test), mostly because we’re only a few weeks into the quarter. All of the grades represent specific items on a quiz (Q means Quiz and I means Item). The standards are clearly written for students and parents to understand in the description of each grade.

  2. Allow for the formative: I now have to give more opportunities for kids to demonstrate skills (yay!). This has really opened my classroom. I give quizzes about every 8 days that usually cover 1-5 standards. By listing quiz items separately we do not destroy any information by adding scores and reporting a block score for one quiz. Kids see a string of scores out of 10 (sometimes more or less depending on the importance of the standard), instead of one score out of a total. There are many other ways to assess a standard: class work, group work, talking with students, anything that you feel genuinely expresses their understanding counts. It is your discretion as to what counts as an assessment and what doesn’t.
  3. Allow for re-demonstration: Now comes the tricky part. This is the piece that will test whether you buy into this system or not. You now have to allow kids to show you that they’ve made progress. In my classes I allows students to come in no more than once per day to attempt one standard in any way I see fit. Sometimes I grill them interrogation style. Sometimes I just write a problem on the board. Sometimes I ask them to create a situation where the standard would apply in the real world and make them solve the issue. Many times I just put a question similar to one from a previous assessment on a more current quiz; that way I keep getting data about their understanding as we move away from a topic. All of these demonstrations are taken as a measurement of the student’s understanding, and the grades fluctuate in the gradebook, not new grades mind you. We no longer care what cryptically numbered quiz this is, we only care about what content each question is assessing.
  4. Less frequent Summative Assessments (Tests): I now am free to create true ‘Midterms’ and ‘Finals.’ These tests are now more trustworthy, because students have been informed all semester about the true status of their progress. These tests now represent a chance for a student to do many things that do not have to do with specific content standards: They can practice the skill of studying as needed for college, they can practice self-assessment, and they can develop a healthy confidence for an exam that they know can’t be reattempted. Those of you who are traditionalists will love this. Kids see the midterm and final as a kind of monster to prepare for, and they happily do it, explicitly checking off understandings as they move towards the battle.

Implementation

What does this actually look like? Let me give you a narrative of a day in my life as a Calculus teacher:

  1. Today, I showed up before school and kids followed me up to my room. After some greetings and mild cajoling, students began to ask if they could “make up a problem.” Which is how they see it: getting points back for screwing up. Not the way I would hope they think about it, but students rarely fit your theory.
  2. I then decide how I will accept each student’s re-attempt. Some I give another problem to on the board, some I just talk to, and some I make write their own problem (depends on the kid).
  3. The first-block bell rings, and I’m left to grade a bunch of re-attempts. I alter scores in the gradebook based on new performances, and the day begins.
  4. I teach a few classes, presenting ground-breaking and highly-engaging lessons that are both rigorous and relevant. Hopefully.
  5. Students come in over the lunch hour to re-attempt questions. (See 2)
  6. I teach a few more classes.
  7. Students come in after school (See 2)

It’s all in how you do step 2. Are you hard-nosed? Do you coax? Do you prod? Are you aloof? Do I change my behavior based on the particular student? Yes, yes, yes, maybe, and uh-huh. The best part is when a kid says something like this:

Cherub: “Mr. Cornally, I was looking at my grades, and I see that I don’t really understand how to draw the graph of a function’s derivative, I have a 5/10.”

Cornally: “Did you review the concept with your notes, the book, and or try some by yourself?”

Cherub: “Yes, I tried a few from the book, and I think I get it now. Can I show you?”

Cornally: “Sure.” The student draws a function (simple parabola) and then draw its derivative fairly accurately.

Cherub: “Is this correct?”

Cornally: “Yes, but I need you to show me with a function that may not have already been in your head.” Cornally draws complicated function. Student draws derivative fairly well. “Ok, you didn’t quite get this part . . . but you’ve definitely shown improvement on some of the basic ideas behind this standard, I will change your score to a 7.5/10, a ‘C.'”

Cherub: “OK, thanks. I’ll be in tomorrow morning to try again.”

This. Actually. Happened.


  1. Standards-Based Grading: Math (1 of 7)
  2. Standards-Based Grading: LA – English (2 of 7)
  3. Standards-Based Grading: Science – Qualitative (3 of 7)
standards based gradingteaching

Shawn Cornally • February 16, 2010


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  3. Jemal August 16, 2010 - 6:51 pm

    OK. The gauntlet has been thrown down, the gloves have come off, the toast has popped up, etc. etc. I am sold and I WILL implement SBG THIS YEAR! Now, how? More specifically, I am still a little lost on your gradebook system. I’ve used teacherease in the past and will probably switch to snapgrade this year, but my question is rather mundane. As students reasses, how do you keep a record of their previous results? IE, if Tony had a grade of 5/10 on standard 4a based on a quiz, but then I use an in depth conversation with Tony as proof that his grade should change to an 8/10, is that 5 still recorded somewhere?

    I ask as a procedural issue. If I create an assignment called “standard 4a” in my gradebook, that assignment can only retain 1 grade. In order for me to track Tony’s progress and see previous grades, I’d have to create a whole category for standard 4a and then create multiple assignments within that category (ie, quiz, conversation, impromptu question, etc.) Is this what you do?

    • Shawn August 16, 2010 - 10:02 pm

      @Jemal: The dilemma you’re describing is exactly what lead me to program my own gradebook. Feel free to sign up: http://SBGradeBook.com

  4. matt greenwolfe June 16, 2010 - 8:51 pm

    I’m trying to wrap my head around how your system works, and I’m getting a different impression here from what I gathered from reading your more general descriptions of the system. Specifically, I was surprised to see grades associated with a particular assessment (quiz) and problem. I had assumed that the only thing in the gradebook would be the standard, for example “limit definition” and that this standard would be tested several times in the course of a year, sometimes by you on a quiz or test, sometimes initiated by a student asking for a reassessment. Whatever the assessment, the grade for that standard would change. So why is the standard associated with a particular problem on a particular quiz in the gradebook?
    Related questions: What if a particular problem tests several standards? What if it’s impossible to make up a reassessment that tests only a single standard? Do you change the grades on all the standards covered, or just for the specific one requested by the student?
    Sorry if this is a bit long. Hope the questions make sense.

    • Shawn June 17, 2010 - 9:08 am

      @Matt: You seem to have it right. Sorry if my post was confusing. Individual assessments are only used as separate trials of the same information. What I’m developing now is a system to keep track of each assessment by standard, so that multiple pieces of information can be easily wrapped up into creating the assessment of one standard.

      Your second question is a bit more difficult, but again the emphasis is to be taken away from the assessment and moved towards each individual idea. One test item can receive any number of grades, depending on what standards it assessed. I’ve had a single quiz item get four separate scores before, one for an algebra skill, one for vertical asymptotes, one for use of the product rule, and another for finding critical points. All separate skills. You may argue that they need to be wrapped together to show a truly cogent understanding, which is true, but in the end reporting grades should be about telling a student what they know. If they get 10/10 on everything except finding critical points, they now know what to work on. It’s impossible to do well without getting the big idea, grading this way just scaffolds the process a bit more.

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  7. Jessica April 29, 2010 - 1:54 pm

    I’m late to the party, but I just wanted to come out of the woodwork and say that I sincerely appreciate all you’re doing and sharing on this blog. I’ve recently ventured into Standards Based Grading and many of the “logistics” you discuss are the hurdles that my colleague and I are facing. Thanks for hashing out your thought processes and successes here.

  8. Marc March 31, 2010 - 8:42 pm

    I like this method, as it gives the student incentive to continue learning, even past the inevitable quizzes they cram for.

    However, isn’t this method quite subjective? Is there a standardized difference between, say, an 8/10 and a 10/10 ? Does 10/10 imply that they can solve an advanced problem in that learning standard, or does it just indicate basic competence?

    In the example you give, you change the students grade from a 5 to a 7.5, but the way you do it seems quite flippant. Do you have a standard for determining what each grade is worth (e.g. 8 – Understands intermediate problems, but has problems with advanced problems)?

    Also, have you run into administrative difficulties by implementing this system?

    • Shawn March 31, 2010 - 9:06 pm

      Marc:

      Your first statement is the reason I switched to this system. Grades now mean more to my kids, and change their behavior instead of just making them sad or happy and forgetful of previous material.

      You worry is a good one. Yes, but I think you have to admit that there’s subjectivity in all grades. My solution to the easy, medium, hard debate is to put a standard in for each, so that the kid knows which level of problem they’re good at. The gradebook is digital; who care how big it gets!

      My administration loves the idea. With a more conservative administration, you’ll want to have the definition of 0-10 defined and on file.

      Thanks so much for the comments, please let me know if I can clarify anything else!

      =shawn

  9. Jeny March 14, 2010 - 3:13 am

    I grade almost exactly like you do! I wish I would have found this blog about a year and a half ago when I was talking to myself constantly and trying to make sense of how to do this accurately! I felt like I was reinventing a wheel that seemed so obvious. My grade book looks just like yours does and I do a 10 point system as well.

    I would like to know how you convert you assessments and scores into your 10 point scale.

    My colleagues and I have all adopted a method we agree on but I wonder if there is something better out there!

    Thanks for sharing everything you have done – I just found your blog and will continue to read!! LOVE IT! :)

  10. David March 2, 2010 - 5:45 pm

    As I am sure has been your experience also, many students struggle with the algebra of a calculus problem. How do you account for this in the assessment of a particular skill? For example suppose that in a problem to assess understanding of the quotient rule, a student correctly applies the rule however makes an algebraic error in simplifying the result. How do you score the skill?

    • Shawn March 2, 2010 - 5:54 pm

      David:

      That is a fantastic question. Generally, I separate the two concepts into two separate grades: one for the rule and one for the algebra. This can be tricky, and really depends on the skill level of your kids. To my kids, the description: “Student can complete the algebra necessary to simplify a fraction including algebraic expressions” usually guides them.

      If they perform the quotient rule perfectly, but mess up the algebra I would assign a 9/10 just so that they know (wholistically) that they need to work on the quotient rule. The algebra component of it would then be assessed based on the magnitude of the error. (5/10, 4/10…)

      I’ve had some classes that were quite deficient algebraically, so I had to include very specific standards relating to things like common denominators and rationalizing roots. Some class just needed one general algebra standard to keep track of their skill level. Hope that’s clear.

  11. SHaluck March 1, 2010 - 7:18 pm

    So their grade on the final would be an 8.3/10 (all scores added up and then averaged, entered into a new column) or are they entered as specific grades on that standard, just in a new column with less weight than the formative assessments? If the midterm and final receive their own columns, what percentage of the grade are the midterm and final? If they were 10% and 15%, you are saying that the formative grades on standards would make up the other 75% of their grade.

    • Shawn March 1, 2010 - 10:58 pm

      Haluck:

      The final is a summative grade. The score from each question is added up and reported as a number out of the total.

      The midterm is also a summative grade. Their score is added up and recorded out of a total, but it is reported back to the kid as individual scores, so they can use it to inform their studying for the final.

      Any work in between those two days is considered formative. For example: there is one grade in the gradebook for being able to find common denominators. Anytime that kid uses that skill in any problem or whatever context you decide, that’s a measurement of that skill, and it can change the value of their grade on that single item. These are Standards and are weighted at 80%. The midterm and final are weighted at 10% each (this is a college readiness thing, I’m honestly not too sure how I feel about it).

      Thanks for the great comments, I really appreciate it.

  12. John Threlkeld March 1, 2010 - 3:43 pm

    I see a great deal of merit in what you’re arguing for here. I would love to incorporate more of this into what I do. I find myself wondering how students’ mathematical thinking and processing is assessed. It seems to me that bu giving kids another problem to try, you’re suggesting that the standard is that kind of problem — which strikes me as pretty restrictive. What happens when a student must use an idea in a novel situation. Are they ready? Any thoughts about this?

    • Shawn March 1, 2010 - 4:56 pm

      John:

      Excellent comment! Comments like this are why I wanted to do a blog. You are totally right, if you just keep throwing the same problem at a kid just with different numbers, not much gets done in the ol’ thinking department. Generally, my reassessments are richer than that, and come at the concept from a totally different perspective. Sometimes I just want them to practice a specific algebra skill that is weak, so I change the problem up entirely. Also, the work done in class is devoted to these novel situations. SBG is really just a way of making your gradebook more useful. It doesn’t supplant great pedagogy.

      The best part about this approach is that the teacher decides what is a demonstration of improvement or regression. Sometimes, in class, when I notice a kid is working with a manipulative in a way that they’ve never done before, I will just silently change a grade of theirs because it shows me their improvement implicitly. In short, reassessment does not solely rest on coming in to do another quiz problem.

  13. SHaluck February 28, 2010 - 11:13 pm

    Shawn,
    Thanks for all this great work… it’s rare that you can get teachers excited about next year, but you’ve accomplished it. I just hope that my department will let me try it; they’re a little old school and conformist.

    One question: you give the first small quiz on a few standards. The students gets a 6 on Item 3. They come in and work with you and get a 9 on Item 3. Item 3 shows up on a later quiz on other items and they get a 8. Then the midterm they get an 9 on that item. Then the final they get a 7. Their final score on Item 3 is a 7/10 because that’s their latest score right? Or do you weight the midterm and finals differently?

    Sorry, that was a really long question, but I want to make sure I understand the logic.

    • Shawn March 1, 2010 - 4:51 pm

      Haluck:

      The system you describe is 100% formative, and would definitely work given the correct implementation. The current system I use is just to the right of this. My calculus midterm and final are intentionally summative, and all of the work in-between is formative (SBG). The summatives are my way of saying: “Hey, you’ve been working on these ideas, reassessing, studying and all that, now it’s time to prove you have the academic-responsibility component.” The score on the midterm is reported by standard, but it is graded summatively. It represents a much smaller percentage of their final grade than the standards do, so if they bomb it they don’t drop to an F. If they bomb it, chances are they aren’t trying to improve their grades on standards anyway… The midterm and final are a way of reporting out a standard related to responsibility and college readiness.

      As for reassessing and giving future formative assessments, yes, their grade changes based on their most current performance. I intentionally bring ideas back, which can sometimes result in grades going down.

  14. Cassidy73 February 27, 2010 - 6:21 pm

    Right now I’m teaching Alg I & II. My gradebook is weighted with HW -5%, Participation -5%, Quizzes -15% & Tests -75%. I allow students to correct Quizzes & HW with a correction sheet, which they turn in and I regrade. Unfortunately a lot of copying occurs (which I think is a problem within our whole school), but I think using SBG will help deter that. Next year I want HW to have more weight, grade Quizzes with Standards and Skills and more weight, then Tests will still be weighted heavily still just not as heavily. Some HW might be turned in but most I would like to use @samjshah’s binder checks http://bit.ly/6jVJ3t to keep them accountable for it. Within that binder I’d also include a grade sheet so they can track their progress. I think I’d also like to do just 2 tests each quarter over the latest Standards. These are just my basic ideas so far. Do you grade your tests broken down by Standards as well? Are your students able to correct tests? I still feel I need formal tests with no corrections because that’s what they will see in college courses. Also what gradebook program does your school use? PowerTeacher?

    • Shawn February 28, 2010 - 4:21 am

      Mr. Cassidy:

      Thanks for the comment. I give very few rigid summative tests. Generally two per semesters (midterm, final). These tests are my way of emulating the collegiate environment, as you mentioned. My goal with SBG is to have my grade book scaffold for students the process of self-assessment during their preparation for exams. I break my midterm up into standards so that the kids can use that information (and so I can alter instruction) before the final. I give short single or double standard quizzes pretty frequently, and these really causes their standards grades to fluctuate.

      I do not grade homework, and I actually shy away from that word. I want the kids to see it as practice, and try to identify the most efficient amount that they need to do in order to master the concept. This is quite debatable, but it seems to focus them more on the concepts, and completely eliminates getting any credit for copying (barf).

      In short: students can attempt to show me their skill level for any standard at any time, and sometimes I provide those opportunities. Tests are summative, and I always make sure to tell them that it not only reflects their skills, but how well they’ve prepared. So that a low grade doesn’t only mean poor skills, but reflects study habits.

      We use PowerTeacher, yup.

  15. Cassidy73 February 22, 2010 - 2:33 pm

    I really like this idea and tried to implement a system a similar system but I didn’t do enough to get it working. Too late for this year so I’m prepping for next year already. Do you have any examples of your quizzes or syllabi for the classes you have used this system? If you do I’d love to see them. Love the blog and thanks.

    • Shawn February 23, 2010 - 3:41 am

      mr.bcassidy:
      Thanks for the comment! I definitely plan on posting syllabi and other course materials. What classes do you teach? What are your ideas about implementing standards-based grading? I’d love to provide any assistance I can.

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  17. Jami Packer February 17, 2010 - 10:07 pm

    I think the beauty of this approach is the flexibility for skills assessment to be so informal and on-the-fly (i.e. a problem on the whiteboard). The political climate where I am may not allow that but I think it’s worth pushing the envelope a bit.

  18. Shawn February 17, 2010 - 9:15 pm

    Jami-
    Thanks for the comment. If students show a decline in understanding later I generally lower the grade if it is currently a 10/10. I’ve been harsher, but it’s really up to the teacher.
    I’ve never had a parent or administrator inquire too deeply. Whenever I explain the system I emphasize “grading for learning” and both kids and adults seem enthused about it. Depending on your environment, keeping artifact may be necessary for a downgrade. I might be playing with fire a bit there, because most of my make-up work is done on a whiteboard.

  19. Jami Packer February 17, 2010 - 6:52 pm

    Logistics question…what if subsequent assessments show a decline in understanding (i.e. lack of retention of the learned skill), do you reduce scores?

    Also, have you had to justify a particular score to a parent or administrator who wants to see an artifact to represent the “grade”?

  20. Matt Townsley February 17, 2010 - 2:53 pm

    I’m especially looking forward to your thoughts on the Language Arts & Social Studies implementation of standards-based grading, Shawn. I wish the thoughts flowed as freely from my fingers as they do from yours! Keep up the great writing.

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