[BC’s Curriculum] “How do we assess it?” (Part 1)

Sometime last year, this question, or some variation on the theme, leapfrogged “Where can we find good problems?” as the most frequently asked FAQ asked of me. Below, my answers, as of today.


“You clarify and share learning intentions and success criteria. You implement rich tasks that elicit evidence of student thinking. You pose questions that cause thinking.”

I presented teachers with four sample student responses to the following question:

A store sells a box of nine 200 g bags of chips for $12. How much should the store charge for twenty-four 200 g bags?


I asked teachers to consider (1) Where is the learner going? (2) Where is the learner right now? and (3) How can the learner get to where she needs to go?

This sparked some interesting conversations. The students in the top left and top right know that a unit price is an equivalent rate where one term–number of bags in TL, dollars in TR–is one. The student in the bottom left also knows that proportion problems can be solved by looking for a scale factor–albeit an inaccurate one–between ratios. What’s going on with the student in the bottom right? What’s the learning goal in terms of content? What’s the learning goal in terms of curricular competency? This activity was preceded by a conversation about the KDU model, so teachers were thinking “use multiple strategies” and “communicate mathematical thinking.” Is it fair to consider “use multiple strategies” using this–or any single–task as evidence? (A good time to bring up triangulation–products, observations, conversations with students.) What does “good” communication look like in mathematics? Do the bottom two responses need words? Would a ratio table help answer what’s going on in the bottom right?

While this was a worthwhile exercise, this answer was “not yet meeting expectations.” One reason for this is that assess is often a euphemism for evaluate. Or grade. Or report. As a student teacher, my school associate once asked me how I planned to assess. I began to tell him about upcoming quizzes. “That’s all well and good, but that’s evaluation. Minute-by-minute, day-by-day, how will you know they know?” This has been helpful for me as I’ve navigated through assessment by preposition (assessment of, for, or as learning) and “Is this formative or summative?”

“Assess what?”

Answering with another question is probably unsatisfactory, but, to me, what is a much more important consideration than how.

The Ministry of Education released the following in the summer:

At the end of the school year or semester, Boards must provide a written summative report to parents that address the student’s progress in relation to the learning standards of the curriculum in all areas of learning as set out in the Required Areas of Study Ministerial Order.

(Emphasis added.)

Learning standards in BC’s curriculum are made up of curricular competencies (“what students are expected to do“) and content (“what students are expected to know“). (#MTBoS, think practice and content CCSS-M standards.) As late as June, some teachers were still wondering if there would be a requirement to assess–or evaluate? or report on?–the curricular competencies. To me, the MoE’s choice of “learning standards” makes this clear.

At the same time, there’s another message out there: learning standards and curricular competencies are synonymous.  The gist of this idea is that content is interchangeable. And maybe that’s more true in other areas of learning. (I still take issue with “If you enjoy teaching ancient Egypt and ancient Egypt has moved, then you can still teach ancient Egypt” but social studies isn’t the hill I’ll die on.) And I’m all in favour of a greater emphasis on students doing mathematics. Helping teachers make this happen is my work–it’s what I (try to) do. Still, I’m baffled.

Of course, nobody argues that process and content exist without one another other. In the classroom, “I can use multiple strategies to solve problems involving ratios and rates” or “I can communicate my thinking when solving proportional problems” work as learning intentions. I can design learning experiences around these. My question is about evaluating: together or separately? Consider the student in the bottom right above. If she “fully meets expectations,” or is “proficient,” or is a “Jedi Knight,” it’s easy–the learning intentions above still work. But if she, as most agreed, isn’t, then why is that? My take is that she is proficient with respect to content (proportional reasoning)–or, at least, here’s one piece of supporting evidence–but not quite there yet with respect to competency (communicate thinking). What are some implications surrounding reassessment? And is it possible to fully meet with respect to competency without also possessing a deep level of content knowledge?

I’m beginning to enter the Land of the Gradebook, which, nine times out of ten, is at the heart of teachers’ “How do we assess it?” Standards-based grading, depth of knowledge, learning maps, rubrics, portfolios, etc. will be part of part two.


Always, Sometimes, Never

This week should have been my first official – third unofficial – week back. Instead, I’m starting this school year as I ended the last – walking the picket line. I haven’t been up to blogging since this started.  Below is a draft from June. I never got around to finishing it. The ending has a “pack up your personal belongings” feel. I left it as-is; seems fitting that this post should come up short… I mean, 10% of my pay – and my colleague’s – was being deducted at the time. 

Recently, I invited myself to a colleague’s Math 8 class to try out Always, Sometimes, Never. In this formative assessment lesson – originally by Swan & Ridgway, I think – students classify statements as always, sometimes, or never true and explain their reasoning.


Because it’s June, we created a set of statements that spanned topics students encountered throughout the course. Mostly, this involved rephrasing questions from a textbook, Math Makes Sense 8, as well as from Marian Small’s More Good Questions, as statements. That, and stealing from Fawn Nguyen.

.doc .pdf

To introduce this activity, I displayed the following statement: When you add three consecutive numbers, your answer is a multiple of three.

Pairs of students began crunching numbers. “It works!”

“You’ve shown me it’s true for a few values. Is there a counterexample? What about negative numbers? Does it always work? How do you know? Convince me.”

Some students noticed that their calculators kept spitting out the middle number, e.g, (17 + 18 + 19)/3 = 18. This observation lead to a proof: take one away from the largest number, which is one more than the middle number, and give it to the smallest number, which is one less than the middle number; each number is now the same as the middle number; there are three of them. For example, 17 + 18 + 19 = (17 + 1) + 18 + (19 – 1) = 18 + 18 + 18 = 3(18).

I avoided explaining my proof: x + (x + 1) + (x + 2) = 3x + 3 = 3(x + 1). This may have been a missed opportunity to connect the two methods, but I didn’t want to send the message that my algebraic reasoning trumped their approach. “Convince me,” I said. And they did.

To encourage students to consider different types of examples, I displayed a ‘sometimes’ statement: When you divide a whole number by a fraction, the quotient is greater than the whole number. Students were quick to pick up on proper vs. improper fractions.

Next, students were given eight mathematical statements. We discussed some of the statements as a whole-class. Some highlights:

A whole number has an odd number of factors. It is a perfect square.

I called on a student who categorized the statement as always true because “not all of the factors are doubled.” We challenged doubled before she landed on square roots being their own factor pair. For example, 1 & 36, 2 & 18, 3 & 12, 4 & 9 are each counted as factors of 36, but 6 in 6 × 6 is counted only once.

The price of an item is decreased by 25%. After a couple of weeks, it is increased by 25%. The final price is the same as the original price.

Like the three consecutive numbers statement above, students began playing with numbers – an original price of $100 being the most popular choice. I anticipated this as well as the conceptual explanation that followed: “The percent of the increase is the same, but it’s of a smaller amount.” I love having students futz around with numbers; it’s so much more empowering than having them “complete the table.”

The number 25 was chosen carefully in hopes that some students might think fractions: (1 + 1/4)(1 − 1/4) = (5/4)(3/4) = 15/16. None did. There’s a connection to algebra here, too: (1 − x)(1 + x) = 1 − x². Again, I didn’t bring these up. Same reason as above.

One side of a right triangle is 5 cm and another side is 12 cm. The third side is 13 cm.

All but one pair of students classified this as always true. That somewhat surprised us. More surprising was how this one pair of students came to realize

BTW, the blog-less Tracy Zager has a crowd-sourced a set of elementary Always, Sometimes, Never statements.

Update: I stand corrected.

It’s pronounced ‘soobitizing’

Subitizing – two years ago, I had no idea what it was.

In September 2010, I was asked to do my first demo lesson as Numeracy Helping Teacher. In a Kindergarten classroom. I taught Math 8 to 12. I was terrified a little nervous. Thankfully, Sandra Ball was there to hold my hand provide moral support. In these last two years, I have become much more comfortable in primary classrooms. And I can pronounce subitizing and tell you what it is – it’s recognizing, without counting, one to five objects (“1, 2, 3, What do you see?”).

That’s me. The one on the left.

This year, it has been very rewarding to support Surrey Kindergarten/Grade 1 teachers with an assessment package developed by Carole Fullerton and Sandra Ball. “What Do They Know?” focuses on three areas: subitizing, partitioning/decomposing, and patterning. In addition to fall and spring assessment tools (instructions, blackline masters, materials, rubrics), an instructional resource with suggestions for subitizing, partitioning/decomposing, and patterning lessons is also included. Carole and Sandra wrote about WDTK in a special elementary mathematics issue of BCAMT’s journal, Vector. Please read the article here.

With all-day-K in effect this year, the timing is perfect. There is time (It is time?) to focus on early numeracy. The number of Kindergarten teachers in Surrey has almost doubled this year, many of them teaching Kindergarten for the first time.

WDTK provided me with opportunities this year to work with K/1 teachers. When teachers invited me into their classrooms, I asked them to choose three kids. Then, I modeled each of the three assessment tasks with these three students. After, the teachers and I discussed the results. It was common for these teachers to be surprised by their students. Often students who were identified as struggling demonstrated capacity in at least one of the three areas. Sometimes these students even outperformed their high-achieving classmates. Later, these teachers were able to complete the assessment tasks on their own with the remaining kids.

I look forward to spending more time in Kindergarten classrooms – every secondary math teacher should get the opportunity at some point in his or her career.