Cre8tive Math


A few years ago, I completed a questionnaire to determine my personal operating style. I’m green. Creativity. At first, I questioned the validity of the assessment. I didn’t see myself as creative. I’m not… artsy.

But taking a closer look, the results made sense. I scored very highly in the four strategies that made up creative thinking in this system: brainstorm ideas, challenge assumptions, reframe problems into opportunities, and envision possibilities. To be clear, this was an assessment of preferences, not proficiencies. Also, there are trade-offs; to choose one thing is to reject another. For example, my 98 in reframe and 91 in envision meant zero — zero! — in tune-in to feelings and ten in empathize with others. These results did not suggest that I can’t tune-in and empathize; they did suggest that I don’t want to. Preferences, not proficiencies.

More important to this post, this assessment tool offered a different definition of creativity: “the generation of a wide variety of options, ideas, alternatives and fresh ways of approaching difficult situations and everyday challenges.” BC’s Ministry of Education defines creative thinking, one of the core competencies, as “the generation of new ideas and concepts that have value to the individual or others, and the development of these ideas and concepts from thought to reality.” There are similarities between these two definitions: both talk of the generation of novel ideas; neither talk of art.

Broccoli with Cheese Sauce

The MoE also has this to say: “Core competencies are evident in every area of learning; however, they manifest themselves uniquely in each discipline.

Over the last few years, I’ve sat through many presentations where examples of creative thinking across subject areas have been shared. The examples from mathematics almost always make me cringe. The math song is a common offender. (Usually the topic tips towards the procedural — BEDMAS, the quadratic formula, etc. — but that’s a different post.) Here, creative thinking manifests itself outside of mathematics. It happens in language/fine arts. (Maybe. Talk to a language/fine arts teacher.) You can substitute provincial capitals for divisibility rules and the nature of creative thinking within the task remains unchanged. Math is merely the context.

Worse, the message is that math is unappetizing in and of itself. Broccoli. The cheese sauce that is the math song (or poster, or skit, or diorama, or …) comes at a cost. Limited time means tension — time spent on products versus time spent solving interesting problems and having interesting conversations. Note: in my mind, the opportunity cost isn’t coverage of content; it is engaging students in the “doing” of mathematics.

Yesterday, I attended a meeting where the MoE repeated the message: “By doing the curricular competencies, students will be developing their core competencies.” The math song attempts to have students develop a core competency without doing the curricular competencies.

Et tu, Desmos?

The connection between creativity and art is strong:


I’d like to suggest a better title:


Creative Math is clearly evident. Just click on one of the staff picks and look to the left. Focus not on the equations themselves, but on the thinking behind them. Not on “front mathematics,” but on “mathematics in back.” (A lovely metaphor from Reuben Hersh that I first came across in Tracy Zager’s Becoming The Math Teacher You Wish You’d Had.)

To most math teachers, this title makes no difference. Just me nitpicking. But it matters where teaching includes designing curriculum/learning experiences. If teachers think of creativity in terms of art, they may look to Pinterest when planning; if they think of creativity in terms of ideas, they may dive deeper into Desmos.

Last year, one of my highlights was being invited into a classroom to observe Marbleslides: Lines in action.


I observed students experimenting with new ideas by changing the variables one at a time. They asked “what if…” questions. They made — and checked — predictions. “New ideas” here means new to the students themselves. These new ideas had value, evident in cheers and high fives. “Right here, right now” value, not “real-world,” career, or “when you take Calculus” value.

(The Desmos Teaching Faculty designed the activity with students in mind who were familiar with equations for lines in slope-intercept form and the idea of domain. In the classroom that I visited, the students were not. We worried that introducing restrictions on the domain at the same time as slope-intercept form would overcomplicate things. It didn’t.)

More Mathematical Manifestations

I don’t fault my fellow educators for associating creativity with art. It’s a natural thing to do. We in mathematics education need to articulate better what creative thinking looks like in mathematics. I’ve had some success in asking teachers to sort curricular competencies by core competency. (Here they are, in random order. Venn diagrams work nicely; I let that idea come from teachers themselves.)

There’s still the leap required to go from making these connections to designing curriculum/planning learning experiences with these connections in mind. Rather than listing activities that elicit creativity, like Marbleslides above, it may be helpful to think about the attributes of these tasks.

Marbleslides is immediately accessible and highly extendable (“low floor, high ceiling”). It invites a wide range of responses (multiple solutions). (The teacher can view novel solutions at a glance on the dashboard.) Open questions, like Which one doesn’t belong?, share these attributes, as does Quarter the Cross.

A rich task can have a single solution, but invite a wide range of approaches (multiple strategies). To me, this has less to do with the task/problem itself and more to do with pedagogy. A curriculum that values creative thinking has pedagogical implications. Consider a typical What’s the best deal? task. A step-by-step-worked-examples-now-you-try-one approach to teaching leaves little room for creativity. The strategy — calculate and compare unit prices — is predetermined. What if students were presented with the problem before the strategy? The class would generate several different ways to solve the same problem. They’d see and discuss a wide range of ideas. Note: this doesn’t preclude the teacher from later bringing a particular strategy (e.g., compare unit prices) to the conversation, if necessary. Ask “Why does this strategy make sense?” or “What’s the best strategy?” and students develop another core competency: critical thinking.

Mathematics is creative. Is math class?

In her ShadowCon talk, Tracy Zager shared a word cloud generated from the language mathematicians use to describe their work. Creative sticks out. And invent, curiosity, play, imagination, wonder, etc. The image generated from the language society/teachers use to describe math… not so much.

But I know that there are places where school math is creative. In Surrey Schools (#sd36learn), in the “MathTwitterBlogoSphere” (#MTBoS), and beyond. When I wrote “We in mathematics education need to articulate better what creative thinking looks like in mathematics” above, I really meant “I need to articulate better…” So, I need your help. Did I get it right in this post? Artsy or not, what does mathematical creativity look like in your classroom?


[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.


[BC’s Curriculum] “Know-Do-Understand” Model

This year, BC teachers (K-9) implement a new curriculum. For the past two years, much of my focus has been on helping teachers–in all subjects–make sense of the framework of this “concept-based, competency-driven” curriculum. This will be the topic of these next few posts.

In this series on curriculum, I’ll do my best not to use curriculum. There is no agreed upon definition. I imagine that if any educator in the “MathTwitterBlogoSphere” (#MTBoS) followed the link above, she’d be shouting “Those are standards, not curriculum!” Similarly, when #MTBoS folks talk about adopting curriculum, I’m shouting “That’s a resource, not curriculum!”

My union makes the following distinction: “Pedagogy is how we teach. Curriculum is what we teach.” Curriculum as standards. For the most part, this jibes with how curriculum is used in conversations with colleagues and is echoed in this Ministry of Education document. But Dylan Wiliam doesn’t make this distinction: “Because the real curriculum – sometimes called the ‘enacted’ or ‘achieved’ curriculum – is the lived daily experience of young people in classrooms, curriculum is pedagogy.” Curriculum as experiences. Or pedagogy.

Rather than curriculum, I’ll try to stick with learning standards, learning resources, or learning experiences.

Three elements–Content, Curricular Competencies, and Big Ideas–make up the “what” in each subject and at each grade level. Last summer, the Ministry of Education simplified this as the “Know-Do-Understand” (“KDU”) model. The video below describes how content (what students will know), curricular competencies (what students will do), and big ideas (what students will understand) can be combined to direct the design of learning activities in the classroom.

I imagined planning a proportional reasoning unit in Mathematics 8 using the KDU model and shared my thinking throughout this process.

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Teachers can start with any of the three elements; I started by identifying content. (It’s a math teacher thing.) Then, I paired this content with a big idea. In English Language Arts and Social Studies, it makes sense to talk about you, as the teacher, making decisions about these combinations. In Mathematics and Science, this mapping is straightforward: algebra content pairs with a big idea in algebra, not statistics; biology content pairs with a big idea in biology, not Earth sciences. (BC math teachers may notice that the big idea above is different than the one currently posted on the Ministry of Education website. It may reflect a big idea from a previous draft. I can’t bring myself to make that change.)

Identifying curricular competencies to combine with content and big ideas is where it gets interesting. Here, my rationale for choosing these two curricular competencies was simple: problems involving ratios, rates, and percent lend themselves to multiple strategies… we should talk about them. The video makes the point that I could go in the opposite direction; if I had started with “use multiple strategies,” I likely would have landed at proportional reasoning. Of course, other curricular competencies will come into play, but they won’t be a focus of this unit. This raises questions about assessment. (More on assessment in an upcoming post.)

Note that “represent” is missing from my chosen curricular competencies. Why is that? My informed decision? Professional autonomy for the win? Or my blindspot? A teacher who sees proportional reasoning as “cross-multiply and divide,” who is unfamiliar with bar models, or double number lines, or ratio tables, or who sees graphs as belonging to a separate and disconnected linear relations chapter wouldn’t think of connecting this content to “represent.” Making connections between these representations is an important part of making sense of proportional reasoning. Will this build-a-standard approach mean missed learning opportunities for students? This speaks to the importance of collaboration, coaching, and curriculum, er, I mean quality learning resources.

In early talks, having these three elements fit on one page was seen as a crucial design feature. Imagine an elementary school teacher being able to view–all at once!–the standards for nine different subjects, spread out across her desk. As a consequence, the learning standards are brief. Some embraced the openness; others railed at the vagueness. In some circles, previous prescribed learning outcomes are described using the pejorative “checklist”; in others, there is a clamouring for “limiting examples.” (Math teachers, compare these content standards with similar Common Core content standards.)

I wonder if the KDU model oversimplifies things. If you believe that there is a difference between to know and to understand, then you probably want your students to understand ratios, rates, proportions, and percent. For a “concept-based” curriculum, it’s light on concepts. Under content, a (check)list of topics. To that end, I fleshed out each of the three elements (below). But I have the standards I have, not the standards I wish I had. (Free advice if you give this a try: don’t lose the that in that stem below.)



I wonder if the KDU model overcomplicates things. Again, U is for what students will understand. But “understanding” is one of the headers within the D, what students will do.

Despite this, I have found the KDU model to be helpful. In particular, it’s been helpful when discussing what it means to do mathematics. The math verbs that we’re talking about are visualize, model, justify, problem-solve, etc., not factor, graph, simplify, or solveforx. Similar discussions take place around doing science (scientific inquiry) and social studies (historical thinking).

More broadly, the model has been helpful in making sense of the framework of our new curriculum, or standards. It’s a useful exercise to have to think about specific combinations–far more useful than:

Q: “Which competencies did we engage in?”
A: “All of ’em!”

We’re still some distance from “the lived daily experience of young people in classrooms” but it isn’t difficult to imagine learning experiences in which this specific combination of the three elements come together.


Kids to learn to do math the old way, just like previous generations didn’t

Last week, Nancy Allan, Manitoba’s Education Minister, announced the province’s revised back to basics math curriculum. The move was applauded in a Winnipeg Free Press article.

So, why the changes? From the Winnipeg Free Press: “I heard from parents that their kids were lacking basic arithmetic skills,” Allan said. “It was during the (2011) election, and I picked this up on the doorstep.”

What we know about how children learn mathematics is no match against a politician with a political football. Allan spent “two years of serious work with [the province’s] education partners. We have met with all the math professors, the superintendents have been part of this.” Notably absent from the list: math educators, faculties of education.

The bottom half of the internet gives us a glimpse of Allan’s doorstep. I could have saved her those two serious years. Rather than a revision, her “kids these days” constituents could have been placated with the addition of one prescribed learning outcome, one PLO to rule them all:

Students will be able to make change.


Where an appeal to logic has failed the reform movement, an army of pimply-faced cashiers able to count back change to John Q. Public – even after he’s thrown down an extra nickel after the fact – might just succeed. Of course, this skill involves applying the mental math strategy of thinking addition for subtraction, not the standard subtraction algorithm. But let’s not tell them that.

The Winnipeg Free Press article featured a poll asking readers “Which of these everyday math tasks could you tackle without a calculator?”

Two of my favourites:

  • Determine how much you should leave for a 20% tip at a restaurant.
    14% (6192 of 45357 votes)
  • Halve a recipe that calls for 2/3 of a cup of an ingredient.
    12% (5577 of 45357 votes)

The results are interesting in light of the following: “The minister said the revised curriculum makes Manitoba the first province in Western Canada to go back to placing an emphasis on basic skills previous generations had.” At best, it looks like previous generations have misplaced those basic skills. At worst, they never had ’em. Psst, hey kids… remember this the next time your uncle quizzes you with “What’s 7 times 8?” He may be asking because he doesn’t know.


So, what happened? Previous generations learned to do math, by definition, “the old way.” They were taught the standard pencil and paper algorithms. In the two tasks above, the standard algorithms are far from efficient. In the first task, a more efficient strategy is to divide by ten and double. In the second task, it’s two thirds of a cup… two thirds… twoOOOoooOOOooo thirds.

Buried in the announcement: “Allan said the province will provide parents with a website to help them understand what their kids are learning.” Admirable. But its necessity is an indictment of the parent’s, not the child’s, mathematics education. Not many politicians are going to pick that up on the doorstep.

Recommended: Dr. Keith Devlin’s response to a recent New York Times article

Update: Frank beat me to it.

December 10, 2013: From a Grade 4 WNCP approved textbook, no less:


A Deconstructed Learning Outcome: Sum of Its Parts

Maybe I’ve seen one too many deconstructed Caesar salad or peanut butter and jam sandwich on TV. Or maybe I’ve heard “This workbook covers the curriculum” one too many times¹.
Whatever my reason, I wanted to take a closer look at a learning outcome from the WNCP Math 8 curriculum document:
It is expected that students will demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially, and symbolically [C, CN, ME, PS]
“It is expected that students will”
It’s about students’ learning. Worked examples on the whiteboard or in a textbook may be evidence of the teacher’s or publisher’s learning.
“demonstrate an understanding of”
Not will be able to. Students need to make sense of mathematics. Justifications and explanations are required for answers and methods.
“multiplying and dividing positive fractions and mixed numbers”
This is a topic. Curriculum is more than a collection of these.
“concretely, pictorially, and symbolically”
No longer just suggested, the use of concrete materials (i.e., manipulatives) is prescribed² as is having students draw to represent their thinking (diagrams not decorations).
[C, CN, ME, PS]
From K to 12, seven processes are to be integrated within the learning of mathematics. The ‘C’, for example, means that students should be provided with opportunities to communicate their learning– to write about and discuss mathematical ideas.
¹ To my US reader(s)– in my province, curriculum is different than recommended learning resource (i.e., the textbook). In theory, the textbook is not the course. In practice…
² For many teachers, this is probably the biggest change to the curriculum. Earlier this year, I created the posters below. My intent was to generate conversations among teachers, not to teach the concept. Plus, I got to be artsy-fartsy. Enjoy.

CPS Poster Algebra Tiles
CPS Poster Counters
CPS Poster Pattern Blocks
CPS Poster Toothpicks