Friday, April 16, 2010

Transparent Algebra: Assessment

As I noted previously, I’m going to be doing a series of posts sharing my thinking about my Algebra class for next year and soliciting feedback and ideas from my network. First up: assessment. (Note: I’m not quite ready to reach Dan Meyer’s level of magnificence in terms of assessment – this and this are well worth your time – but his ideas very much influenced my current plan.)

I’m starting with assessment for several reasons. First, I’ve always liked the idea of “begin with the end in mind,” so focusing on what I’d like the outcomes to be and working backwards seems to make sense. Second, in this accountability-obsessed time, assessment is a pretty important topic that I not only need to get right for the accountability folks, but most importantly for my students. And finally, I think it makes sense to start with assessment because if you guys give me some ideas that make me radically rethink this, it would be better to do that up front instead of after working through all my other ideas.

So, what are my goals for this class? Well, there are a bunch, but let me try to narrow them down to the most essential ones.
Content Goal: Learn the Algebra skills.
Habits of Mind Goal: Become better problem solvers by getting better at asking good questions, thinking mathematically and reasoning mathematically.

Collaborative Goal: Become better at working together to achieve a common objective.

Metacognitive Goal: Learn more about themselves as a learner (via conversation and reflection) and use that to become better learners.
The first goal is obviously much easier to assess than the other three, so I’m mostly going to focus on that one in this post. But if you have ideas on some more formal ways to assess the Habits of Mind, Collaborative, and Metacognitive goals, I’d love to hear them. (I have some teaching techniques and activities in mind to try to foster those three goals, but am not real clear on a good way to assess how well we’ve done.)

From my previous incarnation as a math teacher I remember being frustrated with my assessments. (Actually, I think I probably didn’t think too deeply about my assessments, but I was frustrated with how well my students did - that’s obviously a critical distinction and I hope to be a better teacher this time.) Students in Algebra often struggle because they accumulate both an understanding and a skill deficit – they only partially understand concepts and they only become partially proficient at skills and, eventually, they sink.

My previous assessment strategy didn’t do much to alleviate that, as their deficits were often masked by “just good enough” performance (as reflected in their overall grade) that made it appear as though they didn’t really need much intervention. So I’m hoping to implement a better system of formative assessment this time that will allow me – and my students – to stay on top of things better. I’m currently planning on having three categories in my gradebook (I’m not a huge fan of grades, but that’s a topic for another post): Preparation (10%), Formative Assessment (70%) and Summative Assessment (20% plus - more on the “plus” in a moment).

Preparation (10%)
I’ve spent a lot of time struggling with this one. In general, I agree with the thinking that the practice and responsibility parts of being a good learner shouldn’t have much effect on their overall grade. “Being a good kid” is something I respect and want to promote, but it shouldn’t be reflected in their grade for Algebra. Since it says Algebra on the transcript, the grade should be a reflection of how well they know and can do Algebra.

On the other hand, I do want to encourage students to practice (because it will help them learn), and be responsible, and generally be a good kid. And I realize that their previous (and often their concurrent) experience often includes this piece as a big part of their grade. So my compromise is to include this as a small part of their grade. It will be comprised of a combination of homework, warm-ups, and other in-class activities. (Much more on homework in my next post but, for now, suffice it to say that it won’t be 1-31 odd.)

Formative Assessment (70%)
This is the heart and soul of my assessment strategy and the part that I’d really like some constructive feedback on. While previously I relied heavily on chapter tests, this time I want my formative assessment to be much more, well, formative. As such, I want it to be more frequent, more targeted, and have a built-in process to try to give students a better opportunity to master the skills in a timely manner so they don’t accumulate those deficits I mentioned before.

Algebra is very much skill-based (although there is certainly a bigger-picture mathematical thinking/pattern recognition piece as well). As I talked about in my previous post, my class will meet four days a week (MWRF) for 59 minutes at a time, so my plan is to give formative assessments roughly once a week (although that will vary a little bit with the calendar, the particular Algebra concepts, and other happenings at school). These will be short, targeted assessments of six questions, covering only three concepts, with each concept having one relatively easy and one relatively more difficult question. The assessment would then get entered into the gradebook by concept (so three grades for each assessment). This should help both me and my students identify what they understand and what they need to spend some additional time on.

After the students turn in their assessment, I’ll have students come up to the board and immediately work through the questions. This will then be captured (it looks like I’ll be in a room with a Smart Board) and posted to the class website (mostly likely run through a blog, but I’m still thinking about that). Students should therefore have a pretty good idea right away about how they did, but I’ll also grade these assessments and get them into our student information system no later than that afternoon. I’ll not only record their grade, but will also use the comment feature to indicate which problems they missed on each concept. When students login they’ll therefore be able to see which concepts they need to work on and they’ll be able to refer to the actual assessment – and worked out solutions – from the class website.

If a student doesn’t show proficiency on a concept by getting both questions right they will then have the opportunity to retry the assessment (a different version) once each day until the next assessment. They will only have to retake the portion of the assessment that they didn’t get correct the first time. In other words, if they get both problems on a particular concept correct the first time, they won’t have to retake that part.

They will have the opportunity to review on their own and/or get help from me, other math teachers, or peer tutors, and then typically have up to four retakes before the next assessment rolls around. The strategy is that this is providing students an incentive to become proficient on those skills in a very timely manner, before those deficits start impeding their learning on future skills. Their new score (assuming it’s higher) will go in the gradebook and the comments will then change to indicate any problems they missed on the concept on this assessment.

I’ve struggled with the idea of only allowing those retakes until the next assessment (about a week). Philosophically I would like to allow them to continue to try after that if they need to, but practically I don’t think I can make it work. First there’s the simple management aspect of it, but there’s also the concern that if I extend that indefinitely, that invites procrastination which defeats the purpose of eliminating the understanding and skill deficits in a timely manner.

Summative Assessment (20% plus)
We give final exams each semester at my school, with each final lasting for 85 minutes, and the final is typically about 20% of the overall grade. The Algebra team that I’m joining gives a common final assessment each semester, so I will be giving that as well. This is a summative assessment that gives students a chance to demonstrate what they know and are able to do, and hopefully gives them a chance to coalesce their knowledge and make it more permanent (that’s the theory, at least).

I will make my summative assessment worth 20% of their overall grade as well, unless their performance on the final exceeds their existing grade, in which case the final will be worth 100%. In other words, if they can demonstrate they know more Algebra on the final then what their previous grade indicated, then I’m going with what they can demonstrate. In the long run, I’m not that interested in how much Algebra they knew in October or March, I’m interested in how much Algebra they know when they (tearfully) leave my class.

So there you have it. I have made some compromises due to the fact that I’m teaching just one section of Algebra instead of being a full-time math teacher, but I think it’s a decent start on a good assessment strategy that is actually doable given my other job responsibilities. Keeping in mind that limitation (excuse?), I’d love some constructive feedback.


  1. Looks good Karl. I esp like the summative's ability to wipe out the others... and I'm intrigued to hear how the going to the board after turns out.. maybe you could flip video some of those for posts..

    The formative sounds very similar to what Dan Meyer has developed with the grade per topic and 4 or so chances for a do-over. I really like that.

    What we tried this year - and what I would like more feedback on - is teaching the kids to grade their own tests. Sounds ridiculous and simple and perhaps trite at first. But for us, turned into a difficult yet extremely rewarding feat. It came about from the thinking of kids owning their own learning - and the need for them to self-assess. Ended with them having a much better understanding of how to present themselves.. and their feedback gave me a better understanding how to improve my test writing.

    Looking forward to the comments that follow. I feel we have botched the assessment piece of learning terribly. Typically kids think of them as a tool to tell them if they are good or bad, and then they are trashed.

  2. There are a couple of professors out of Cornell who have been doing a lot of work with thinking skills.
    When you consider how to assess the metacognitive piece their work might be worth considering. They have a youtube channel that might be helpful to:

    Our district (at least parts of it) has done a lot of work with them. In our creation of a new progress report for K-6 the thinking skills were included. Sadly, our leadership has changed and those are now gone from the report. But a lot of us had done some serious thinking and working together on how to assess thinking skills. It was time well spent.

  3. Karl--

    I like the balance of scoring categories, and the inclusion of the preparation component, noting your comments. Especially for 9th graders, it's not a bad idea to make being a responsible, prepared learner an accountable item.

    I'm thinking about a content area writing section to your gradebook for a couple of reasons. It supports building/district goals, it could be an indicator to your class collaboration and metacognition goals,and will, at least in the students' minds, validate the blogging that I anticipate they'll be doing.

    Also, I think that you are such a good writer yourself, you'll do a fine job sharing that with your students.


  4. Our building policy for assessment is summative 80%/formative 20%, and as I read your descriptions I am thinking about ours...we have described formative as practice, homework, classwork, warm-ups etc. similar to your "preparation" and the summative as the the end of the learning assessments for each concept, similar to your "formative". Our semester grade is then comprised of 45% marking period 1, 45% marking period 2 and 10% exam. The exam is truly the very end of the learning, so is that the only real "summative" assessment?
    The question I have is, does what we label these categories really matter or is it just semantics? Is your formative category the same as our summative category? Is this ok or is it confusing and further muddy the waters of assessment and grading practices? Just thoughts I have in my head...

  5. Karl, I like your thoughts about assessment (as I'm an Algebra teacher also). What do you think of Dan Meyer's idea of checking on the same concept twice? If I recall, the first time a student encounters a particular concept on a quiz, it's worth up to 4 points and the next time, it's worth 5, under Dan's arrangement.

  6. @monika hardy – Thanks. I thought about having students formally self-assess and, in the end, decided I didn’t know how to do it well and I wanted to make sure I was in the loop so that I could help more. I am planning on having them self-reflect and self-assess fairly often, just not on the formal assessments. Perhaps I can grow into more self-assessing if I continue to have Algebra sections after this year.

    @Jenny – Thanks for the links – I’ll check them out. Sorry to hear about the effect the leadership change had, but hopefully the work has helped your team with their teaching.

    @Mike Porter – I thought about including writing as a separate piece but decided I just wasn’t good enough to figure out how to do it well. Similar to my comments to monika above, perhaps I can grow into that in subsequent years. At the moment, I’m not sure blogging is going to be a big part of what I do (at least not this year). More on this in a future post, but I’m going to be using Google Apps for some reflective pieces and that will be included in the Preparation category.

    @Michelle Herring – I’m not sure the labels matter that much as long as we’re doing things that help our students (although it’s helpful for us to have a common vocabulary to talk about it). For me, I think of these assessments along the way as formative because it helps me and the student identify what they need to work on, and then hopefully they work on it. The final is the only summative exam in my mind because it’s at the end (the “autopsy”), and there’s no particular intervention associated with it (at least not in my class, although theoretically there could be intervention in the next class).

    @RichTCS – I love what Dan Meyer is doing, I just don’t think I can pull it off this year for two reasons. One, my already stated excuse of only teaching one section and having another full-time responsibility. Two, because I just don’t think I’m good enough yet to do it as well as he does. Perhaps after this year, with a year back in the Algebra classroom, I may be back in the swing enough that I can try to take it to the next level (which is what I feel Dan has done).

  7. Hi Karl. I, too, have read a lot of Dan Meyer's blog and have created a system somewhat similar to his for my high school math students. I think your assessment scheme is on the right track. It may be semantics, but I'm not sure your "formative" assessments are really "formative."

    What type of feedback will you be giving students in between your instruction and these "formative" skill-based quizzes? I'm wondering if the in between opportunities might actually be the "formative" assessments while the skills-based quizzes are just repeated summative assessments. Again, it may just be semantics. Kudos to you for your transparency.

    I wrote up a post and left myself open for some criticism, too:

  8. @Matt Townsley - Yeah, I worry about that as well. The feedback part happens when the kids come in and get that help (before/after school, during their unscheduled hours, or even on their own if they're up to doing that), but it's certainly not as good as I would like to get it.

    I guess I still think of the quizzes as "formative" because it allows us to diagnose what they don't understand and then respond to it, as opposed to "summative" which doesn't allow for that. But I may not be using the terms correctly.

  9. Karl,
    I'll push you a bit farther, but only because I'm sensing that you're open to it. Feel free to say, "hey, Matt...cut me some slack here!"

    From my experience, students see anything we enter into the grade book as permanent. Ask your students, my hunch is they'll agree. With that in mind, why should ever quiz go into the grade book? Could there be an ungraded "pre-quiz" or maybe call it a "checkpoint." Every student would get this feedback rather than relying on them to come in outside of class. Use these formative assessments to help you decide what and when you need to re-teach. Do it. Then, give assessments that go into the grade book. This small little step might make your assessment/grading a bit more formative, at least from my perspective.

    You could still allow for re-assessments as you stated in your previous comment.

    Take it or leave it as you see fit. It's your blog. :)

  10. @Matt Townsley – Hey, Matt, cut me … kidding. This is good stuff, and I’m going to have to think more about it, but my initial thoughts are:

    As soon as I have these students, then I can ask them that question! I guess I’m hoping that if I explain this correctly, they won’t see that grade in the gradebook as permanent. (Acknowledging, however, that it can become permanent, which I don’t like, but see aforementioned nod to I’m not sure I can take it to the next level just yet.)

    I think I agree philosophically that anything formative shouldn’t go in the gradebook. But my problem is I don’t see where formative ever ends (other than death), so we all have to pick a line somewhere. (Again, my preference would be to eliminate grades altogether, but I doubt I’ll be able to singlehandedly accomplish that in my school with my one section of Algebra next year.)

    My thinking was (and perhaps still is, pondering), that their ability to retake multiple times, with opportunities for help in between, is a decent middle ground. I guess I sort of feel like my initial, whole class quiz (I’m actually trying to avoid the ‘q’ word, hence the awkward use of “formative assessment” as my category title) is the “pre-quiz” or “checkpoint,” and then the feedback/reteach comes one-on-one (or one on few) both in-class and outside of class. The grade in the gradebook is a placeholder, but I admit that I’m using that placeholder as a little bit of leverage to get them to do the retakes. I think your system is better, but I’m not sure I can pull it off with the limited number of times I see these kids, and with my rustiness as an Algebra teacher. But I need to think a little bit more to see if that’s really true or not.

    Keep it coming . . .

  11. Convince me we need a summative test...

  12. If there are opportunities for reassessments, is it really summative, Matt? Summative, formative, differentiation, inquiry...all these terms seem to take in their own meanings depending in whose classroom we are talking about.

    Could the inclusion of a summative be driven by the need to see if students have retained the skill and/or can apply them in problem solving situations, Monika? It seems like this is what the summative part of the grading scheme hopes to accomplish. Or did I misread you, Karl?

  13. Somehow I knew David would crash the party with a pragmatic view on the situation. (That's a complement, Mr. Cox!)
    Here's where I still see some controversy in the formative vs. summative argument: If the *only* opportunity students have to be re-assessed is outside of class (I realize this isn't the case in your proposed model, Karl), then is it truly a formative opportunity? If we leave the re-assessment to chance and motivation, I don't think it really can be considered formative. James Popham in Transformative Assessment said, "Formative assessment is a planned process in which assessment-elicited evidence of students' status is used by teachers to adjust their ongoing instructional
    procedures or by students to adjust their current learning tactics."

    If teachers are providing minimal in-class purposeful feedback and then using the assessment-elicited evidence only to encourage students to come in *outside of class*, I don't know if this truly fits the assessment FOR learning model. I see it as a "use assessment to see how many kids we can motivate to come in outside of class" model while our formative assessment techniques inside the classroom have room for improvement.

    Disclaimer: I'm bringing all of this up to question my own practice just as much as anyone else's assessment scheme.

    The bottom line I'm trying to mull over: once a teacher has taken the dive into this type of system, what is the appropriate balance of in class vs. outside of class feedback and re-assessment?

    (Karl, you have established a nice middle ground. Keep your head up, man. :)

  14. update to previous comment: see Jason's thoughts

  15. Thanks everyone, this is really helping my thinking. Let me take a moment to share some more information about my school that will perhaps address some of the questions above, then respond to individual commenters. This will be somewhat lengthy.

    I’m guessing we’d all agree (at least I hope so) that each school/classroom has some unique characteristics that are going to affect the teaching and learning. A huge one at my school that impacts this discussion is our variable schedule and the culture that goes with that. As I’ve mentioned previously, our school has a college-like schedule, with some classes meeting 5 days a week, but others only MWF or TR or four days a week like our Algebra classes. Built into this schedule is the idea of unscheduled hours, with freshmen having between two and four each week, sophomores between four and six, and juniors and seniors between six and eight (although many upperclassmen do end up scheduling more hours based on their interests/needs) – this is out of thirty hours total (six period day).

    Students can use those unscheduled hours as they see fit – working on their own, going to the library, getting help from teachers, or just hanging out with friends (they can even go off-campus if they wish). But part of our culture is the expectation that students use those unscheduled hours to come in and get help from their teachers when they need it. So students have three options for help during their unscheduled time. They can come see their teacher if their teacher is also unscheduled. If their teacher is not unscheduled, they can visit the Math Department Office where there is always at least one teacher (and usually more) who are unscheduled each period (and part of our culture as teachers is that we are to help any student that comes into our department offices – our unscheduled hours are not designated as “planning” only). Or they can visit the study center, which is always staffed with a few teachers (although not necessarily always a math teacher) and with peer tutors (upperclassmen who are doing well and want to help – think National Honor Society type students).

    For my students this will be complicated somewhat by the fact that I’m “unscheduled” all the time when I’m not teaching them, but I have other full-time responsibilities (and my desk is not in the department office). I’m planning on trying to address this by having them first try to schedule with me if they are planning on coming in (so that I can put it on my calendar if I’m available), and then they can always drop in and see if I’m available. (And, of course, they still have the Math Department and the Study Center.) Some of how they utilize this may even factor into the 10% preparation part of their grade, but I'm not sure about that yet.

    Part of the job of teachers who teach freshmen at my school is to acculturate students to this idea of using their unscheduled hours to go in and see teachers, to get that extra help and to build relationships. So the part of my assessment plan involving coming in to get help and to take the retakes is intentional - taking advantage of our schedule and building on our culture. (I will also use class time to help them – and have them help each other – more on that in a future post about what I’m planning on doing during class time – one downside of choosing assessment as my first post about this.)

    So, hopefully that provides a little bit of context for some of this (not saying it addresses all the issues brought up above). Now, on to your comments in the next comment (4096 character limit).

  16. @monika hardy – I’m not sure I can convince you we need a summative test, but I have three possible answers to that.

    First, and this is the one I cling to so that I can sleep at night, I don’t really have a choice. It’s an expectation at my school to give final exams, and the Algebra team I’m joining already has a common assessment for the final exam as a result of our district PLC work (based on DuFour). I’m not going to be able to change that teaching one section of Algebra.

    Second, and perhaps a more palatable answer, is that I do think there is some value in having students periodically review and coalesce all their knowledge. I think this not only helps them retain the knowledge, but also provides an opportunity for them to perhaps see some relationships and connections between the mathematics that they didn’t see along the way. I sort of relate it to a performance in art, music, drama or athletics, where students have to bring all their skills to bear in a single moment. (I know that’s not a perfect analogy, and I also know that a typical final exam like the one I’ll have to give is a far cry from those other performances, but I still think there’s something there there.)

    Third answer – I don’t really know.

    @David Cox – Thanks for joining in. See my second answer to monika above for what I was thinking in terms of the value of the summative assessment.

    @Matt Townsley – See my lengthy description above about unscheduled hours and the expectation that students come in for help. I think that at least partially addresses your concern about outside of class, motivation, etc. – that’s actually an intentional piece of this based on the schedule and culture of my school. Having said that, I am also planning on using class time to address some of this (although probably not the actual retakes) – again, more in a future post on my plans for in class. And this is helping me think a little bit more carefully about that part of my in class plan, so thanks a ton. Thanks also for the link to Jason’s blog – very helpful.

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  18. A couple quick points.

    Read this brief work about assessment by Grant Wiggins.

    Now go read any other assessment pieces by both Grant Wiggins and Rick Wormeli that you can get your hands on.

    Both authors state explicitly in almost all of their discussions about feedback that formative assessment should not be graded or entered into a gradebook. They make the association with a driving aptitude test. We are not graded on how we did all along the way as we learned how to drive with our permit. Rather, we are graded in a final, summative assessment that demonstrates our proficiency at driving. Our final grade was based on our mastery performance.

    The idea of averaging performance over time is near absurdity. A strong statement to be sure, but yet one I feel is accurate.

    It would be as if I were to give your blog a grade right now from the time of its inception up to this point. Well, Karl, you got a 93% on your blog.

    93% of what? Even with subcategories, how would that inform your audience if I posted that on your blog for them to understand your proficiency as a blogger?

    Also, grading for effort is problematic at best. Which raises serious questions about homework.

    Our grading system needs serious help. I give you great credit for begin transparent about your practices and seeking to learn more through this experience.

    If only all educators were as willing.

  19. I feel a bit like someone jumping into a swiftly moving river - I've done my best to read the previous comments to ensure I'm not repeating and giving credit where is due, but apologies if I mis-step.

    I'm with Ben (and Grant, Harry, and Susan, etc. - all of the authors who advocate against grading formative assessment) but recognize the need to include more than the summmative grades in the book. It's going to take a while to wean parents, students and teachers off our collective addiction to grades and I respect the need for slow change.

    As an alternative of "counting" quizzes, what if you presented your students with the option of choice? You may still given them formative feedback in numerical terms (i.e. 7 out 10 correct) or only given them feedback on one item. Their task as a self-assessor is to identify ten formative assessments that reflect their growth as a learner. They will receive the full 70% if they accomplish two things. 1. Present you with 10 assignments (connects to your goal of helping them learn responsibility) 2. present you with a compelling written argument why those ten pieces demonstrate their growth as a mathematician AND their plan for the upcoming marking period for improving their mathematical understanding (NOT improving their grade). If you provide them with a rubric that articulates your expectations for said compelling argument and workplan, there is no reason why a student wouldn't easily earn all of the points available to them (or the 70%, however it works itself out). Arguably, a student who is not "good at math" could still pass the course through the strength of their self-assessment but when considering long-term life skills, a learner who recognizes their strengths and weaknesses is more likely to do better than one who is good at Algebra (with all due respect to Algebra).

    Your grading and documetning responsibility would be next to nill as they will responsible for collecting the components and organzing them. The main responsibility you'll face is giving them quality formative assessments that allow them express growth - which it sounds like you're already doing.

    Monika - I'll do my best to persuade you on the summative point by pointing to this blog. As authors, when we click "publish your comment", we are summarizing our learning - in effect, submitting a summative assessment for evaluation by others. Summative assessments happen all of the time in real life. To me the challenge isn't the need for summative but rather limiting summative to "test". No where is this challenge more felt than in the math and sciences (as Matt identified). In an ideal situation, students would demonstrate their mathematical understanding by solving a real-world problem but since we silo mathematics understanding, this is nearly impossible.

  20. @Ben Grey – Thanks for chiming in, even if you do doubt the reading I’ve done :-)

    I agree to a certain extent, which is why my summative assessment is 20% or 100%. If they can drive the Algebra car at the end of the semester (especially if the car leaves Denver traveling at 60 mph at 2 pm, and your car leaves Barrington at 3 pm traveling . . .), then I’m all for slapping that stellar grade on their transcript. (Even if I could make the final 100% for everyone – which I can’t, more on that below – then we’re back to their performance on one task at one moment in time, which I think you would agree also is problematic.)

    As far as homework, I think I pretty much covered that in the post (and I will discuss what I’m doing with homework in the next post). 10% for preparation (which includes homework, but other things as well) was my compromise. Perhaps if I was better than a 93% blogger you wouldn’t have asked that question. Nothing else to see here – move along.

    And I agree that our grading system needs serious help, and that averaging grades is far from ideal. As I’ve stated, I would choose to do away with grades altogether but, hello Mr. Grey, I’m teaching one section of 32 or so kids in a school of 2150 in a society that for the most part is unwilling to agree with you and me. So, I have to find a way (for now, at least) to do the best I can for my students within this system. And that includes grades, and not just at the end of the system because, first of all, their heads and their parents’ heads would explode, and second of all we have weekly eligibility which is pulled directly from our electronic gradebooks. Sure, I can game that system, but I’d rather spend my time trying to provide feedback to my students. As I said in the post, the topic of grades is for another post – please go ahead and write it.

    So, while I’m still trying to figure out how to incorporate some of the excellent pushback in these comments into my assessment strategy, I think I’m on the right track to help my students learn Algebra, become better learners and people, and learn more about themselves and the world. I think providing them timely feedback on specific concepts, as well as the ability to improve on those concepts (and, yes, by enticing them with a better grade in the gradebook), is a decent start. I’ve done as much as I think I can feasibly do given the various restraints on my particular situation to pretty much remove grades from the equation. (It’s going to be really hard to get a bad grade in my class – my hope is that after a few weeks the kids are going to figure that out and – gasp – focus on the learning. We’ll see.) After a semester or two back in the saddle, then I may be able to tell you how much of this actually makes sense, and what I’ve got totally wrong.

  21. @Jennifer – I’m going to have to spend some more time thinking about your suggestions because I like some of where you’re heading (I feel like I’m still very weak on their self-assessment of their work/learning). The part of your comment that I struggle with is:

    “Arguably, a student who is not "good at math" could still pass the course through the strength of their self-assessment but when considering long-term life skills, a learner who recognizes their strengths and weaknesses is more likely to do better than one who is good at Algebra (with all due respect to Algebra).”

    While I agree about the power and importance of self-assessment and life skills, I’m still tasked with them learning the Algebra skills. I’m not interested in them “passing the course” if they don’t know the Algebra, because I don’t think that’s going to help them. I’d like to find a way to do both – learn the Algebra skills and get the self-assessment/lifelong learner skills.

  22. Karl - You raise a valid and reasonable concern. Consider for a moment the flip side of the coin. As a special ed teacher, I had at least three students I can name off the top of my head who could do the math. Without a doubt, they could talk me through the problem solving, balance equations and do what was required of a 13-year old mathematician. They "failed" math because they couldn't/didn't do the work. It was that simple. When report cards came out, comments went to their lack of studying, failing to do homework - all items that had to do with playing school, not math. I'd imagine you've failed at least one student because they didn't do the work. So - does passing always mean they master the math? (I passed enough math to get to college. It wasn't until my second year of statistics before I realized I didn't know how to speak math and needed some serious tutoring to get through the heavier doctoral-level stuff and now embrace DataDiva as my Twitter name).

    Given the concern you raised about knowing the algebra, I'd advocate for changing your percentage break-down. If what matters to you at the end of the day is their ability to do the Algebra (as best assessed through on-demand Algebra tasks), what about a 60/40 split? 60% on-demand (summative tasks), 40% formative learning reflection?

    Ha! My capcha is "radness" - apparently Google approves of this topic!

  23. I work from the assumption that all reassessments will take place in class, Matt.. I know that's not realistic for many classes, but I agree with you that in order for a system to be equitable, all students need access. I have the luxury of 94 minutes per day/ 5 days per week for entire year. I have to build this time for reassessment into my schedule. Oh, and I definitely took the party crashing comment as a compliment

    Karl, if you are giving two questions are concept, how are you arriving at a score for the given concept? If a student nails the more difficult problem, does that validate them missing the easier problem? How are you delivering your exams and retakes? The reason I ask is because I started to deliver my formatives via our LAN and it has made keeping track of things a snap. Retakes are a breeze as well. I described the process here. I agree with your procrastination concern, even though I have allowed them an unlimited number of retakes.

    Thanks again for your transparency and for allowing me to "crash the party." ;-)

  24. @Jennifer - I dunno, I think those 3 students you describe would have a pretty good chance of not only passing my class, but getting an A or a B (for whatever worth those letters are). If they can do the Algebra, they'll do just fine, even if they never do homework (and, again, for those of you counting at home, my next post will be about homework). They would have to do pretty poorly on the lots of the initial assessments, and choose never to to do a retake. (And, if they're special ed, they also have an additional support system built in, which also gives them additional opportunities to take those retakes - at least in my building).

    Now, to answer your question, "does passing always mean they master the math" - my hope is that in my class, yes, yes it does.

  25. @David Cox - You're going to have to give me some time to digest your post and look at ExamView (yikes, so much to learn). Without really looking at all yet, my first question is likely to be about multiple choice versus free response with any kind of automated system.

    94 minutes a day for 5 days a week for an entire year? Um, yeah, that would help. I'll have them for 59 minutes a day, 4 days a week, and some kids will change at semester. Based on our calendar this year (and we are talking possible furlough days in my district next year due to budget issues), I'm estimating I'll have about 60 days with them first semester, and 64 second semester (and 5 of those each semester are shortened classes due to late start for our PLC days).

  26. Karl,

    Well I'm a bit late to the party, but I hope to add a bit of decent content.

    First I would suggest reading this blog.

    A great example of implementing standards based grading into high school math class.

    I know you can't do standards based grading. Why not concept based grading with trends?

    If you haven't heard of that it's because I just made it up. I'm thinking define concepts (no longer than a week of classroom time). Grade those using a formative/summative system. Taking the idea that students can retake or in some other way prove that they learned a concept later. (I like the idea of letting students explain what they did wrong and why, or writing and solving their own problems) Add a bastardized trending system from Marzano and give more weight to the most recent assessment.

    Basically the idea is to break the course down into the essential concepts (questions if you want) assess those questions and allow students the opportunity to prove that even though they didn't understand the concept the first time, they did eventually pick it up later.

  27. Ben
    If a teacher allows students to retake a particular assesent or part thereof and the new understanding replaces the old--it isn't averaged, then I the process of assessing and reassessing formative or summative? I guess I'm asking Matt this question as well.

    Shouldn't a student's grade reflect their understanding at that point in time? A formative may drive future instruction as well as future activities of the learner, but it's still an inication of what they know.

  28. Great post! I think your goals are are perfect and not too high for students to follow. I think your assessments are fair and will hold up strong in the classroom.

  29. David - I don't think re-assessments are the problem. As I stated earlier, making re-assessments accessible only outside of class and/or as the only means of "formative" (whatever that means these days) assessment doesn't seem to make sense though.

    Shouldn't a student's grade reflect their understanding at that point in time? Maybe this wasn't pointed my way, but I'll answer it anyway. Yes, I agree with you, David. I currently use the most recent score rather than averaging them.

    I don't know how long this thread is going to last, but I sure am enjoying it right now. I appreciate how so many involved aren't willing to settle for the status quo (my guess is that Karl's initial post/thoughts will be pushing the envelope in his school) and are searching for clarity beyond the obvious answers.

  30. Karl: Your unscheduled hours system is my dream. I would love to actually have a school day that matches what we teacher; that is, self-assessment and responsibility. Why should the kids have to do so much at home when they spend 8-10 hours at school per day? You don't spend that much time in a college lecture hall. Have you written about the advent of this system before? I'll go look.

    Maybe I haven't fully read the novel of comments here, but I have a slight issue with giving any credit for homework. Anytime credit is given, the students see it as an accumulation game.

    Assigning homework, but not "grading" it, sends the message that they have the choice to do it or not. Those who don't may do well, and good for them! Most likely, if they choose to ignore practice, they will fail. This should teach the message about the link of studying to understanding better than any graded practice ever could. I thought this line of logic was garbage, until I tried it and it worked. I've written about it in my room, at, which I see already has a link! OMG! I hope you guys can stand my typos...

    Great blog, btw. Welcome to my RSS Reader.


  31. @David and Matt - I'm guessing that Ben was talking about the fact that I would be averaging all of my formative assessments (to make up that 70%), not that I would be averaging the multiple retakes on a given assessment. I could be wrong about that.

  32. @Shawn Cornally - Welcome to the blog. Added you to my reader a few weeks ago due to Dan the man.

    Yeah, I struggled with the Preparation category lots. It's not just homework, but your logic applies to all the rest just as well. Like I said, I compromised (caved?) by giving it 10%.

    I'm not sure there's a great writeup anywhere of variable scheduling, but a brief description on page 5 of this (warning: large PDF file).

  33. Karl,

    Have you considered where pre-assessment fits into your grading? Will you provide students the opportunity to show you what they know BEFORE you begin instruction, so they can move forward without 'wasting' time on what they already know? If they show you mastery of a skill or concept (however you define that), could the pre-assessment take the place of the other grades you might give during your instruction on a given topic?

  34. Let me just comment on the your efforts to include a collaborative goal.

    I've been doing something similar with my classes albeit, they are not Mathematics but it shouldn't matter. In essence, I've had my students write a reflection on how they've contributed to the learning of others and how others have contributed to theirs. I have them track and provide specific examples. Early on I'll be very explicit when I observe it happen and cite it as an example of social learning which is the term I've used.

    Math may be more challenging but good teachers like yourself likely don't have a problem but the idea that students gather in a room together and don't expect to learn from and with each other is simply educational malpractice in many ways. I think we need to explicitly require students to see how valuable it is to learn from each other. I'll be interested to see how you incorporate that concept into your class.

  35. @Beth – That’s a great question. Yes, I have thought about pre-assessment. But, no, I haven’t figured out how to do it well. My thinking (excuse? cop-out?) is that, for the most part, this should be new material for students and therefore they will not know it to a level of mastery before we start. But, obviously, that may not be true for all students on all concepts. If I did figure out a way to do it well, then, yes, I would be comfortable putting that in the gradebook and allowing them to bypass the later assessment on that topic. Do you have any ideas of how to structure/manage this, and whether it’s likely in an Algebra class that students would truly have mastery (not just informed guessing) before the instruction?

    @shareski – Glad to see you figured out how to comment despite my flawed blogging platform. :-) That’s an interesting way to go about encouraging/assessing collaboration. I need to think a little bit more about how that might look in my class. My knee-jerk, oh-no-not-one-more-thing response is the old excuse: time. I’m worried about how to fit that into my limited time with students. I was planning on the reflective piece already (more on that in a future post), so perhaps I could include this idea as a formal part of that reflection. Thanks for making my thinking go there.

  36. Karl - I don't know if Beth was referring to CBM (Curriculum Based Measures) AKA OBM (Outcome Based Measures) but they address what she was describing - assessment with a goal of "non-time specific" mastery. The Fuchs from Vanderbilt have done a fair amount of research into it. I stopped adding to my library on the topic in about 2005 but I have about 20 articles I can throw up on my wiki that address the research and management of the process if you're interested. I had the chance to participate in research as both a student (in Middle School) and as a master's candidate. The best part was watching students engage in independent study during content they'd already mastered. Good stuff.

  37. @Jennifer - Yes, please, I would appreciate it.

    Do you have any thoughts on my question to Beth re: the likelihood of students in my Algebra class already possessing mastery level knowledge before instruction? In my previous incarnation as a math teacher, I certainly felt like I never saw that, but then again I didn't specifically look for it.

  38. Sent you a tweet with a link to my wiki with a bunch of articles on CBM.

    Re: previous knowledge. To me, many of these conversations circle around the idea of teaching as an art versus a science. As an artist, it's your sense that students didn't have mastery prior to your instruction. As a scientist, you'd know for sure through diagnostic or pre-assessment. At worst, you'll know exactly where their issues are. At best, you'll know the content you can skip and be able to demonstrate to higher ups why you made that call.

  39. Wow- and I can say I knew you when...

    Here are some thoughts from your friendly neighborhood English teacher:

    1- Like Mike Porter said, you need to put writing into their work. This could be as simple as a reflection on their learning, writing out their thinking to solving a problem, extensions of their learning (where math is used in the real world or real world applications of what they are learning)etc... but I think writing is that important. (see goal 2, 3, 4)

    2- I would like to see your kids scribe. We talk about this all the time, but this gives kids ownership over the class, and contribute something meaningful back to the class. (see goals 2, 3, and 4)

    3- where is the real world context? Where are you showing them or helping to show them the applicability of what they are learning? Think about all those AWNM discussions with Hannah and others.

    4- I like your grade breakdown, but I think you should see what the kids think before you nail it down. Giving ownership and responsibillity creates a collaboratively learning environment (see Goal 3)

    5- No matter what, remember to teach with passion, This is NOT education as Usual, it is ok to make mistakes, and CHANGE THE WORLD

  40. @Anne - Thanks. Remember, this post was just about assessment. Future posts will expand on this, but I'll be bringing the real world into class as I can. It's just not a separate, delineated part of my assessment. And writing (both reflection and as part of their formative assessments) is also part of the plan, just not a separate category in the assessment.

    I'm considering scribing, but it's a little more problematic since I won't have laptops and since scribing math is a little trickier (and, yes, I know Darren did it with his students, but not freshmen). At this point my thinking is I'd rather their writing focus on the reflection piece, and the problem solving piece, not the scribe piece, but I may change my mind about that.

    As far as having the students determine the grade breakdown, I considered it, but in the end decided that wasn't going to work. They don't know enough in this case to make the best decision (IMO), and will likely revert to what they are used to which, again IMO, is not the best method to use.

  41. Very cool comments you guys ... made me pen a post - thank you for that - this is all such a refining process.

    Only sharing a link here - within Karl's post - because your comments and thinking have been so valuable to me... want to share back if I can.

    I think my beef with summative is just semantics... In my head, if it's worth testing it should be worth refining... formative.

  42. Not knowing the make up of your students and your vertical math articulation makes it hard to guess if students will have much algebraic knowledge before they reach you. I work specifically with gifted students, so I always take that angle when I look at things. I can tell you that I have a significant number of students SUCCESSFULLY taking Algebra 1 (the same text, course objectives, and summative assessments at the 9th grade course) in 6th grade. All because we started to pre-assess and place them more effectively when they leave fifth grade, as opposed to having everyone take the same math courses all the way through. And before anyone flips out, they aren't rigid 'tracks' like the old days - they are fluid...students move up and down as necessary, and they are subject-specific (we have advanced courses in all subject areas, not just math; students who are advanced in math, may not be advanced in other subjucts, which is okay!).

    That said, I would not assume it is appropriate for all students to take Algebra 1 early (that will lead to a whole different debate!!), but, given my experience, it is certainly within reason that you will have some students come to you with a vast amount of algebraic reasoning, understanding, and intuition.

    How to pull it off? Create strong pre-assessments (they don't have to be long, but they should be enough for students to clearly show they have mastery of the concept you are teaching), and then be prepared to have alternate learning experiences for those students. Tomlinson, Kingore, and Heacox are all great Differentiation experts to check out. The book "Developing Math Talent," by Assouline and Lupkowski-Shoplik is also great. Be warned though, it will change the way you think about your top math students!!

  43. @Beth - Thanks for the info. For the most part, the students you describe will have been identified and tracked into Algebra before they get to me in high school. We have a large number of 7th and 8th graders who take Algebra because they were ready (or thought they were ready at least) to take it before high school.

    That doesn't mean, of course, that there won't be some that have slipped through. If I figure out a way to do this, would you suggest that pre-testing be over the same concepts (about 3 at a time) as I described for my assessments, or one big pre-test over say the first semester's worth of concepts?

  44. I would suggest going concept by concept. It will be easier to manage and will fit well with your other assessment strategies. Even if only a handful of students are able to show mastery with a pre-test, you will have a better idea how to focus your instruction if you have seen how students tackle the pre-test (thus making it a powerful formative assessment tool!).