Dear Teachers
May is fast approaching, and with it our benchmarking season.
As I work with teachers throughout our division I am often
asked questions about benchmarking. I am providing this
resource in an effort to answer some of those questions and to
support you as you lead your classes through this process.
“Score student work collaboratively
against agreed-upon criteria. The power
of collaborative scoring should not be
underestimated. It helps teachers
internalize what quality looks like and to
arrive at more consistent professional
judgments.”-Davies, Herbst, Reynolds, 2011
Just to clear up some terminology: “Anchor” refers to an example of student work. We sometimes us
anchor papers to show students different levels of success. The best quality example of student work is
an “exemplar”. This is sometimes confusing because “Exemplars” ©is also the trade the name of the
assessment program we use for benchmarking.
Q: Why do we benchmark students’ writing in math? Why not just a standard test that focuses on
basic skills and procedures?
An important part of assessing students’ mathematical understanding is understanding their reasoning.
Being a proficient at mathematics requires more than computational skills and procedures; real
mathematicians and proficient students can communicate their argument and reasoning.
Research into learning tells us that having students explain their reasoning forces them to learn more
and more deeply, and store learning in long term memory. Students are given opportunities to
communicate their reasoning, solidify their arguments, construct meaning, reflect, receive feedback,
and consolidate understanding. By sharing ideas and strategies, learners also become more flexible in
their reasoning. Problem solving allows students an opportunity to draw on prior knowledge and apply
appropriate skills. It creates learning that goes beyond procedural and computational skill development,
to reasoning and synthesizing math skills for application and transfer.
The Saskatchewan curriculum states that we must
teach curriculum outcomes through the processes
of communication, reasoning and proof, problem
solving, mathematical connections and
representation. Therefore benchmarking students
on computational and procedural skill alone would
not support our practice.
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Q: What are some ways I enhance my students’ communication skills in mathematics?
Here are a few ideas
 Provide opportunities for students to communicate verbally and in writing in your class. Coach
them about deliberate communication during these tasks. Encourage clear reasoning, logical
discourse and appropriate mathematical vocabulary.
 Use and refer to a word wall. Model use of correct mathematical vocabulary.

 Model clear and logical communication
and writing, even when you are writing notes and
explanations on the board. Point out the way you
write and communicate as an important way of
sharing mathematical ideas. Have students notice
the importance of clear communication.
 Use other writing tasks such as journal
entries or written answers during class or
homework to discuss and assess students’ communication.
 Involve students in the assessment process. Brainstorm what clear effective math writing looks
like. Make a list of characteristics of good math writing. Sort these characteristics into criteria
and use it to self and peer assess writing tasks.
(note that teaching students to understand criteria and assessment, involving them in
assessment, allowing them to self and peer evaluate provides them with skills that extend
beyond mathematics and beyond their grade level. These are skills that promote engagement in
learning and lifelong learning. Such practices are time well spent!)
 Depending on your grade level, you may decide to let students examine anchor papers as
examples of a range of writing/communicating criteria. I have anchor papers for every grade
saved from last year’s benchmarking. You can also find anchor papers and assessment criteria
here: http://www.exemplars.com/ (Click download samples, click summative assessment tasks
with anchor papers)
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This rubric refers to
work on previous
page
Q: Connections are tough to understand. It seems when we assess benchmarking papers we change
our understanding of connections. How can I give my students a clear idea of what we are looking for?
Like all educators and researchers, our understanding of our curriculum, the learning of mathematics,
and the mathematical processes is growing, and our criteria for success is getting better. Understanding
how to identify students’ mathematical connections has been difficult, and communicating to teachers
assessing at benchmarking how to interpret the rubric has been challenging. But we are getting better!!
The only way we acquire any new knowledge is to connect it to what is already known. All teachers
introduce new topics by building on what students already understand. For instance, we introduce
multiplication as repeated groups or repeated addition. We draw groups and we show mathematical
models and symbols. We give examples of where multiplication is much more efficient than addition.
We show area models and connect multiplication to area. Later we use that area model to teach
multiplication of fractions and decimals, and even later polynomials with algebra tiles. We will use a
basic knowledge of multiplication to introduce exponents, graphs, areas, volumes, sequences and series,
geometric progressions, and many more advanced topics. Another example: teaching fractions of a
whole leads to fractions of a set, which leads to ratios, proportions, percents, money, and probability.
Each of these new learnings is predicated on the
understanding we built before. But do we ever stop to
highlight those connections for students?
During instruction, we need to stop and point out to
students when we are making mathematical connections.
Connecting a diagram to a situation, connecting symbols to a
diagram, connecting a new skill with one previously learned,
observing things about the mathematics like a pattern that becomes evident, or a generality that can be
applied (extended) outside the problem. Our rubric is worded in a way that makes us think that writing a
new problem is an “expert” mathematical connection. This is one type of connection, but it can be
inauthentic if it isn’t a reasonable way to extend the mathematical content of a problem. An authentic
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extension would be where a student solves a problem and then recognizes a generality that could be
applied to any situation.
Math connections may be math to self, math to life, or math to math. For younger grades, math to self
and life connections are very valid: “This reminds me of when my uncle took me fishing”. These kinds
of connections are still valuable for older grades, but we would hope to see other connections as well,
involving math to math. For example “I can express the number of fish caught as 6 fish for every boat, so
6b. I can substitute any number of boats for b, so if there were one hundred boats, that is 6(100) or 600
fish”.
Other math to math connections may be finding patterns, finding alternative strategies, alternative
representations, showing algorithms, making representations (yes, we find the areas of the rubric
overlap. Making representations not only is scored as “representation” but is also a form of
mathematical “communication” and a mathematical “connection”).
The rubrics applied to the student work here will help you understand mathematical connections.
http://www.exemplars.com/ (Click download samples, click summative assessment tasks with anchor
papers)
Some of our templates list connections separately at the bottom. This may be confusing to students and
markers alike. For instance, last year we had some students make very rich mathematical connections
throughout the problem. Then the template prompted the student by asking “Can you make a
mathematical connection” and the student said “no”.
Perhaps we can have students look back through their work or their partner’s work, recognize and
highlight mathematical connections. Again, alternative strategies, representations, extending solutions,
recognizing patterns, verifying solutions, and comparing math concepts to other related math concepts
are all rich, valid, math- to –math connections. The “expert” mathematician not only makes connections
but recognizes them.
Read more about connections here: https://drive.google.com/file/d/0BwKqA1qYaPddXF2bFpCeENoMHc/edit?usp=sharing
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(The above problem is gr 3 to 5 level)
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Q: The rubric is difficult to understand and apply
We use an analytic rubric (breaks the task into 5 areas to be assessed, as opposed to a holistic rubric,
which just gave an overall letter grade). Note that on the Exemplars © site (link above) they are now
also using and showing an analytic rubric.
The criteria in the rubric seem vague at first. If you put yourself or your class through the process of
trying to create a rubric for these 5 processes, you come to realize this one is not so bad!
There is a teacher version, which we use at benchmarking (see online version)
And a student version
You could create your own class-friendly rubric, like Candice Gale and Kelsey Shields did
and there are many other rubrics for
various grade levels available online.
The most authentic way to immerse
your students in the rubric is to have
them evaluate some anchor papers. You
can use previous class work, online
anchors, or use the ones I saved from
last year. Work on one process at a time
(communication, representation, etc).
Have your students decide what makes one paper more clearly communicated than another? What are
some criteria we use to judge communication? Show them several levels of achievement. Have them
categorize and scale their criteria to create 4 levels of success, from Novice to Expert. Steer them to the
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criteria in the rubric or similar criteria to the rubric. Then have them create their own rubric or use the
exemplar rubric to self-assess some of their own work, or peer assess (this takes some preparation,
training and norms). Provide specific, descriptive feedback so students can improve their work.

Consider focusing on one category of the rubric (one process) at a time. Susan Muir (our former
math coach) had a great idea: she cut the rubric into strips and put them on rings so kids could
flip to one process at a time.
 Remember that you don’t have to do a whole exemplar problem at once to practice these
processes. Use your rubric to assess problems from the text book, journal entries, collaborative
writings, and other math work.
 Keep assessment part of your ongoing conversation in the classroom.
 Revisit worked exemplars over and over. Even if it’s one you did early in the year, bring it back
out and work on the writing, connecting, and representing some more. Self and peer assess,
using descriptive feedback.
Some people feel the rubric is not specific enough. How much math language do students need? What
types of representations?
The rubric must be fitted to each problem, but like a barbecue cover, it only fits loosely! If we create
criteria that is too specific, we get trapped in our own criteria. For
example, if we say we need a certain number/type of vocabulary
words, or a certain type of representation, then we see a brilliant
paper that is different from what we expected but very valid, and our
own criteria will prevent us from giving that student credit for great
thinking and reasoning. Therefore, the rubric has to stay a little
“loose fitting” to allow our professional discretion.
Douglas Reeves talks about the “perils of specificity” when it comes to rubrics. He states clearly that
rubrics are subjective, and that we need to have appropriate modesty about our grading practices.
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Q: What about templates?
 Which templates you use is up to you. They are available on the GSSD website
 You should let your students choose which template they like. Have several choices available to
them.
 Consider using a template that allows students a chance to “prewrite” or brainstorm ideas and
images before they start solving the problem. This may help them make connections, and is a
very non-threatening way to enter the problem. Consider adding a page like this to any template
you choose to use.
 Remember to have your students go back through practice problems (ones you practice in class
before the one you hand in. The one we benchmark must be done without prompting and
guidance). Did they recognize when they were making connections? Can they highlight those? It
doesn’t have to be at the “make a connection” prompt in the template.
 You may choose no template, just a checklist. This would be a tool students would choose to
use, and as such should be stapled on to the problem when they hand it in.
 Practicing communicating, reasoning, representing and connecting using problems students
have already solved, or working out solutions collaboratively first, or using problems a bit below
grade level can be a non-threatening way to get kids comfortable with the mathematical
processes involved. (Of course we still want to continue to challenge our students with
authentic problems at grade level.)
Q: My students have a hard time choosing a strategy and using it, or they choose one strategy and use
another.
Have you ever worked on a problem, trying several approaches until finally you determined a route to a
solution? Or suddenly seen the way to the solution and solved it, without even thinking about how you
got there? This is sometimes true in authentic problem solving. Some problems readily lend themselves
to a specific strategy, say “work backwards” or “guess and check”, and students, through practice,
recognize that as a feasible approach. But other times students will work through a problem without
knowing what strategy they will use. If this happens, have them reflect on their work, how they got
their answer, and then try to choose the problem solving strategy that they did use. In other words,
they may choose or reselect the appropriate strategy after working through the problem. That’s ok. Of
course our preference would be an insightful mathematician who chose an efficient strategy from the
start, but a second best would be a reflective mathematician who recognized the form her/his approach
took and learned something from the experience.
Q: Should I attend the marking day for my grade?
Please do! These days can be great collaborative days. Collaborative scoring of student work is one of
the most powerful professional development opportunities we can have (D. Reeves, 2010). As we
work to improve mathematics in our division, it’s important that we come to a collective understanding
about what success looks like, what level our own students should be at, what good instruction can look
like, and whether we are improving. Benchmark data really only reveals trends, it is imperfect (but we
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are trying to make it better), but it is still valuable. Most valuable is the way it impacts our practice,
keeps us focussed on teaching math through mathematical processes (as our curriculum demands) and
the coming together at marking time to engage in professional dialogue around achievement an
assessment. We are working together to come to an understanding of mathematics instruction and
to improve math proficiency for our students! Please be a proactive part of this process 
Attending the marking day will give you much more insight when you discuss your assessed papers with
your students.
Q: Are there any other ideas/strategies that teachers are using to try to help students with
benchmarking?
For sure!
 Remember that you can scribe for Kindergarten and grade 1 students
 Students may use manipulatives as one strategy. You can photograph their manipulative model
and attach it to their work.
 When doing practice problems in class, some teachers let students practice solving problems
collaboratively, then use a gallery walk to have a discussion about assessment. You could use
anchor papers as a gallery walk.
 Model appropriate math talk. “In math class we think and communicate like mathematicians”.
Provide opportunities for dialogue, and listen for appropriate vocabulary and highlight wellexpressed reasoning. Make mathematical conversation deliberate. Once we say it we can write
it.
 Consider using a checklist to guide students as they go back through their problems. Did you use
appropriate vocabulary? Have you highlighted any mathematical connections you’ve made? Did
you use appropriate representation? Check/verify/prove your answer? Extend your solution?
Show more than one strategy? Label things clearly for the reader? A checklist is like template
prompts except used for reflection and polished work. Remember that students must choose to
look through a checklist, just like any other tool. If it is posted in your classroom as an aid to all
problem solving activities, students will know it is an available tool.
 Involve students in dialogue about assessment, and explain what the benchmarking process is.
Remind them their papers will be read by people that don’t know them and can’t guess at what
they mean. They need to explain their reasoning very clearly!
 Problem solving is an ongoing part of instruction. We introduce new content concepts through
rich problem solving tasks. Writing the benchmarking Exemplar© problem will just be another
example of problem solving in your class. Make explaining reasoning a routine part of your
class. Making thinking and reasoning visible is an important math skill!
 Some people use exemplar work as one of their pod stations. Students could work on just one
part of problem, like the representation or the communication. You can bring back worked
problems over and over again to let students continue to polish them up (well, not the one you
hand in!)
 Have your math coach out to talk to you and/or your class about why we need to communicate
our reasoning, why the process is sometimes as important as the right answer. If you want a
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
little powerpoint with some images to guide that discussion, send me an email and I’ll send you
what I have.
I hope to see you at our benchmarking days!
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