Tiered Assessment for Geometry

A wonderful colleague Ms. C. (now at a different school, alas) and I were inspired four years ago by a guest speaker at our school (I believe his name was David Holden) to create an assessment system for our Geometry classes based on a 4-point scale. I believe this would fall under the umbrella of SBG but it’s a bit different from what I see most folks talking about on their blogs. For one thing, we don’t use a rubric. It reminds me of a workshop I attended long ago on the topic of tiered instruction so I refer to it as tiered assessment, to differentiate it from rubric-based SBG. I don’t claim that it’s better, just different.

We decided to make each page of a test deal with just one objective, listed at the top of the page. We created four questions, one at what we felt was a D level (student was awake at some point during class), one C (if student leaves my class only knowing this much, I won’t be a complete failure), one B (student knows how to put some pieces together), and one A level question (student has this objective down cold). We tried to keep Bloom’s Taxonomy in mind when writing the questions, though Bloom could have been more accommodating by only having four levels for us to work from.

Here is one version (we always made two) of one of our assessments, which roughly corresponds to the first four sections of chapter 6 in our old (Glencoe) textbook.

We decided not to have each question worth individual points. Instead we marked each question with a +, √, or -, which roughly correspond to full credit, half credit, and no credit. Then we created a decidedly complex grading system whereby each page (i.e., each objective) received a single score from 0 to 4, based on the combination of marks on the page.  The simple examples are these: If a student only got the D question right they earned a 1. If they got the D and the C question correct they received a 2. D, C, and B correct yielded a 3, while a perfect page earned a perfect score of 4. Other combinations were given intermediate scores, and half-credit on a question dropped the overall score a certain amount based on which level of question had the half-credit. You can see, I’m sure, how the entire grading scale turned out to be rather byzantine. I should also mention that we made it very clear to our students that a B or A question would never receive even partial credit if there was no work shown. (There were probably a few exceptions, but none come to mind presently.)

These took forever to grade at first, but I noticed one day that Ms. C. had memorized the grading scale and I was inspired by her example to do so myself. This sped things up considerably, though in truth grading these assessments never got to be a brief task for either of us. Nonetheless we liked the results we were getting. A student taking one of our quizzes always knew what we thought was deserving of an A or any other grade, and the grades they received lined up nicely with how well they appeared to know the material in our formative assessments. You couldn’t get an A by answering 90% of a bunch of B and C (and D) level questions, which is what the tests that came with the book were like. After implementing this system I had less A’s in my class, but also less grades below C level. (That’s a pun–ha ha!) The students who only wanted a C–just to pass the class–knew exactly what they had to do to get there. And the B questions were always right there in front of them, hopefully to entice them to try and reach a little higher. And again, you couldn’t get an A just by working hard; you had to really learn the concepts deeply. (I don’t claim that every assessment we wrote meets this standard, but it was always our target.)

One other thing we did was to not give an overall quiz score. Instead the students received all of their scores filed under specific objectives. There was a 0 to 4 score in our grade book for all 70+ objectives we had for the year. We also allowed students to retake any page (one objective, all four questions, in a different version from the one they had taken before) one time and didn’t penalize them if their score went down. Score reductions didn’t happen very often though, as most students who bothered to re-test made a point to try and understand the material better beforehand. There was very little chance that they could improve their grade significantly otherwise.

Ms. C and I decided to start making our homework look much like the assessments, so that the students would have a better sense of what we thought “A” work looked like before they were tested on it. (We did try to avoid having the homework questions look just like the test questions but with different numbers, especially the B and A questions.) Two and a half years later, we finally had tests and homework for every objective. And for what it’s worth, I wrote my own class notes and posted them online (in particular for students who missed class), so my students and I ended up hardly ever using the textbook at all.

As you might expect, parents were often confused about how our grading system worked, since we weren’t working by percentage but the grade-reporting program the whole staff was required to use would still show a percentage no matter how hard we tried to avoid it. We had to adopt a grading scale that looked bizarre by normal standards, with 25% being the lowest D, 45% for a C, 65% for a B, and 85% for an A. These percentages were a never-ending source of wonder (to put it nicely) that we got to explain to parents ad nauseam, but when the end of the semester came and you paired up the grades with the names, our students all seemed to get what they deserved. It felt like we had done something right.

Why is all of this written in the past tense? Because, as I mentioned in an earlier post, we have new books this year and they are very different from our old ones. I haven’t made a final decision yet on how I want to formally assess my Geometry students this year, but I don’t think I’ll be able to just re-tool all that Ms. C. and I have created. I have high hopes for our new curriculum (CPM) but it’s a shame that so much previous work will probably just end up in the academic museum. So it goes.


24 comments on “Tiered Assessment for Geometry

  1. Jean Redus says:

    Hi Steve – It’s super fun to read your blog, which is both informative and entertaining. I’ll be sure to keep on reading, so I hope you keep on writing. As a math tutor (and past math teacher), I find this relevant and riveting material 🙂

    • Oh, Jean, thanks so much for your kind words. One day when I rule the MathTwitterBlogoSphere (ha!), you can say that you were the very first person to add a comment on my blog! (I have had one link, but not a proper comment until now.)

  2. jnewman85 says:

    Wow, I would totally use this idea if only there were assessments like this already created for my subjects (I teach Precalculus, Chemistry, and Math). This would definitely motivate students and help them see (and us see) how well they understand the material.

    Have you ever seen anyone other than you (and Ms. C) use this type of assessment? Did it catch on to anyone else at your school? I suppose I’m just hoping for that magical bank of assessments for my own classes.

    Oh, and keep writing, by the way. Love your posts!

    • Jonathan, first let me thank you for the kind words and for apparently reading all the way to the end of this post. When I compare my blog to other math blogs, I worry that my posts must be seen as long and boring. The answer to that is: Don’t compare myself to other bloggers. Either my posts will get shorter and more like the rest of the crowd or perhaps I’ll pick up a small niche market of followers who somehow like this kind of thing. I never in my life expected to quote Popeye, but it feels appropriate here: “I yam what I yam.”

      You asked if anyone else has used these assessments. Well, when Ms. C. got involuntarily moved to another school for the 2011-12 year, I got a new Geometry partner, Alpha. (She had also been “bumped” from her previous position–a frustrating process that happens too often in our district. I myself ended up at my current school as a result of being bumped after one year at a school on the other side of town.) It took some time before she “converted”, but eventually we were giving nearly identical assessments. (Alpha sometimes cut and pasted in problems she liked better than what I had, more power to her.) She also started using the same grading scale, which you more or less have to do. I have also had the occasion to show these documents to other teachers (and administrators) now and then–to quite positive reviews–but no one from another school has ever actually asked to use them. (I would have said yes, I swear!)

      I taught Algebra 2 last year for the first time in a while, and I decided not to try to use this system because starting all over with the creation process was very daunting, all the more so because Ms. C. wasn’t around to share in the brainstorming and the grunt work. As it happens I had a handful of students who had been through my Geometry program the year before who were hoping I would do the same thing in Algebra 2, but alas I really felt like I didn’t have enough steam for the project. This year I am teaching nothing but Geometry, so I won’t be working on anything new along these lines for the foreseeable future.

      As you must have already surmised, there is no such system of assessments set up for any other curriculum, I’m sorry to say. (Unless someone out there has created something similar outside of my knowledge. You never know.) If you get nothing else out of this, please know that your comment means a lot to me and will help me continue writing. I don’t know if I would keep this up if the only audience was myself and the crickets, but if even a tiny group of people find it interesting or (wonder of wonders) useful, I will endeavor to keep going.

      Thanks again.

  3. […] for this year.  What can I do?  Well, I came across this cool idea from Steve Grossberg called Tiered Assessments.  As he explains, it’s not better than SBG, just different, but with a similar […]

  4. jnewman85 says:

    Steve, just wanted to let you know that I tried this out in Physics and just blogged about it. You can check it out here:
    Thanks again for your post!

  5. […] has a blog named It’s all math. The first post for the Blogging Initiation is titled Tiered Assessment for Geometry” and the author sums it up as follows: A colleague and I wrote our own tiered assessments for […]

  6. […] actually ‘cheating’ a little bit because I didn’t comment on this particular blog post this week (it was perhaps a few weeks ago) and furthermore, I’ve already blogged about the blog post […]

  7. jnewman85 says:

    So I’ve tried the Tiered assessment out again a few times, Steve, and I blogged about my 3rd attempt. On my 2nd attempt, I really understood your concern about “what do you do if a student gets B correct but C incorrect?” I ended up just giving them a score I felt was right, but I probably was not 100% consistent, especially because I didn’t grade them all at the same time. The problem was I kind of made B and C cover different topics, so many students got what I thought was the easier problem wrong. I’ve fixed that for this next quiz, but I just wanted to thank you again for giving me this new format of quiz which I’m really starting to like!

    • I’m glad the Tiered Assessment has been working for you, and that you’re finding a way to make it your own. It’s so hard to imagine one is going to help anyone, as a new blogger, given all that is out there already.

      I subscribe to your blog and keep up on what you’ve been doing. I really enjoy it and encourage others to read it as well:


      • jnewman85 says:

        Thanks again, Steve. Yeah, it does seem like its hard to help anyone when there’s so much out there, but for me it feels totally different knowing there are others using this material this year or last year, and it’s working for them, AND they’re able to talk about its pros and cons. I really appreciate your blog and will continue to read yours as well!

  8. […] quizzes.  As long as my quizzes test for understanding, which I think they are starting to thanks to ideas from other bloggers, then I should allow students to retake quizzes as often as they can.  However, as I’ve seen […]

  9. […] I’ve been using the tiered assessments more and more these days, and I am really, really liking it.  Perhaps it’s just a taste of […]

  10. […] group of people, I’ve re-learned to love mathematics and teaching mathematics, I’ve completely changed every single quiz I’ve ever given, I’ve done creative projects with my Precalculus students that I would […]

  11. Misty says:

    I like that this makes it clear to students what the different levels are. They can see where they fall and see what they still need to work on.

  12. […] Oh, and why do I have #70, 80, 90, and 100 on the quiz, instead of #1, 2, 3, and 4?  See this post (original idea is from this guy.) […]

  13. Alicia says:

    “We had to adopt a grading scale that looked bizarre by normal standards, with 25% being the lowest D, 45% for a C, 65% for a B, and 85% for an A.”

    I hate to be “one of the parents” but I am a first year teacher and have no idea where these numbers came from or what they mean. Does this mean that students with an average of 85 earned an A?

    • As I was explaining to a student just today, if they do B-level work on every assignment and assessment, they would have all 3’s in the grade book, for an average of (you guessed it) 3. Now, with 4 being the highest possible grade (i.e. 100%), it wouldn’t make sense for a student with all 3’s (again, B’s, not C’s) to get a C using the “old school” percentage scale where 75% is the middle of the C range. Such a student should be getting a B of course, and we have set 75% as the middle of the B range, +/- 10%. 3 out of 4 simply does not mean “C level work” in a 4-point grading scale. I’d like to simply call it a B and leave it at that, but as I said our grade-reporting system didn’t seem to want me to do that.

      Look at it this way. Everyone’s GPA is reported on a 4-point scale, and we all know that a 3.0 is a B average. Yet if we forced everyone’s GPA to be reported as a percentage instead, the 3.0 student, the B student, would have a 75% by their name. It doesn’t mean they didn’t earn their B average. It doesn’t mean their grades are being inflated. It simply indicates that grades on the GPA scale doesn’t map directly onto grades earned by percentage. In a word, GPA is not based on a percentage scale and shouldn’t be reported with one. The same goes for my grading system. It’s not based on percentage so it doesn’t make sense to look at it through the lens of percentages.

  14. […] started creating Tiered Assessments for these skills, which I first read about at the It’s All Math blog but have since […]

  15. defiancehacks.wordpress.com says:

    Thanks for sharing your thoughts about highschool. Regards

    • Goodness, thanks for reading. I have been such a slacker about this blog since about halfway through the school year. It was a tremendously busy one, and I just found that I had to put the blog aside for survival reasons. Perhaps I shall pick it up again now. Who knows?

  16. […] they earn a 4. This post was really influential in the way I think about these conceptual skills: https://itsallmath.wordpress.com/2012/08/23/tiered-assessment-for-geometry/. The rest is all subjective and based on the context of the assessment. In these situations, I […]

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