“What if students aren’t ready?”


We’ve all seen this before on Facebook and laughed internally. But one practice in our classrooms proves we aren’t so far beyond this as we’d like to think. It’s happened to all of us – You have a great idea for an engaging and authentic problem about similarity! Or about writing equations! Or…[fill in the blank]!

There’s just one problem. You know that your students have arrived in your classroom without all of the prior knowledge hoped for at their current grade level. A student gives you a blank look when you ask about scale factor or about how to solve a proportion. Or a student can’t articulate what a variable represents. Or…[fill in the blank]. Sound familiar? We’re left lamenting, “What if my students aren’t ready?.”

My admission:

We have all experienced this in some form or another. It is daunting to present students with a problem at grade level when you know that for many (or even most) students you can’t draw on the appropriate prior knowledge. Our natural instinct is to pause, review what should be prior knowledge, and then re-engage in the work at grade level.

My ask:

I ask that you present your great problem ideas to all of your students and allow them to identify and ask for the learning they need. Our most successful math teachers in the New Tech Network fight the natural instinct to front-load needed skills, and instead present the problem at grade level to all students first. As I’m sure you’ve guessed, I ask you also try to fight your natural urge as well.

The “why”:

Students who arrive lacking basic skills do not show up this way because they have never been taught those skills. Quite the contrary, they likely have been formally taught those skills at least twice and perhaps had some remediation on top of that. Even with potentially three passes at these skills, they didn’t ‘stick’. They didn’t make enough sense to students for them to be able to call upon that knowledge when necessary in your course. And in all reality, a fourth pass at practicing that skill in the same way likely won’t be successful either. You must strategically change the way you teach that skill, and provide context for why that skill matters, to change a student’s understanding of that skill. Change the teaching/learning method to change the learning/understanding result.

Students who arrive in your classroom also are likely lacking a mathematical mindset, or even the mindset that they can be successful in a math class. The importance of a student’s mindset about his/her ability to learn a skill overwhelmingly outweighs current knowledge of the actual skill itself. If a student doesn’t believe she can learn, you’re sunk before you’ve even begun. But a student who is willing to engage in the great problem you want to do, and who believes it is worthwhile and is smart to ask questions? Yes, foster that! If you present your students with challenging problems, and with the scaffolding to support them, you are sending the not-so-subtle message that you believe your kids are ready for real mathematics. Please let your students see the real and beautiful and connected math we so appreciate. If you let your students engage, coach them how to engage, and support them as they struggle, the results might be surprising to you. And they’ll certainly be rewarding.



Fostering True Math Learning

Some introductory brainstorming:

(One clarification first: When I use the words “our” or “we”, I mean “us” as math educators. Now the prompts!)

  1. What limitations from our own math learning are we bringing to our planning and the math experiences we provide our students?
  1. How can we model the skills and mindsets of a savvy problem solver for our students, and how can we model the learning of new skills and mindsets that we as adults don’t yet have ourselves?

Some context and elaboration:

In a recent visit to Cincinnati, Ohio to work with math teachers in the Winton Woods City School District, I pushed the room to take a step back. As a coach for the New Tech Network, it is common practice to push teachers, leaders, schools, and fellow coaches to articulate their purpose, to borrow from Simon Sinek and ask folks to state their “why”. In this case, we brainstormed how each of us might complete the following sentence:

“Math learning, true math learning, is essential because…”

One key phrase kept bubbling up in our sharing was that true math learning fosters and develops savvy problem solvers. So we pushed into what being a savvy problem solver really means and what indicators we might look for.

problem solver

While this is a great ideal, we all know that putting this ideal into practice is difficult. On my trip home from Cincinnati, I re-stumbled across a 2014 New York Times Magazine article titled “Why Do Americans Stinks at Math?” in which Elizabeth Green investigates why math education in the United States historically struggles to achieve this desired true math learning despite repeated, national initiatives (think: “new math” of the 1960s and the challenges of CCSS implementation now to name a few). Green interviews professor Magdalene Lampert who challenges us to assess our role in this struggle. While many of us, me included, were highly successful math students in school – so successful in fact that we became math teachers – Lampert notes that our math learning is likely the product of a traditional math classroom. I admit this line of thought brings up an uncomfortable sense of vulnerability for me. I think to myself: “Is she really saying that I am somehow ‘less than’ in my understanding of mathematics. This can’t be. I am a math teacher and a math coach after all, right?” I could get defensive. I could get defiant. I could foster learning environments that are comfortable for me because they are familiar. Instead, I take her words as a caution and as a nudge. A caution because I must admit that I may not have personally experienced what I try to foster in math classrooms now. And a nudge to be honest about where I can grow and improve.

Some closing brainstorming:

As you head into another school year and now will some additional context, I’ll again pose two questions for further brainstorming:

  1. What limitations from our own math learning are we bringing to our planning and the math experiences we provide our students?
  1. How can we model the skills and mindsets of a savvy problem solver for our students, and how can we model the learning of new mindsets and skills that we as adults don’t yet have ourselves?

Go ahead! Teach to the Test!

Teaching to the test. The four words that seems to simultaneously bring a sense of security to teachers and make them shudder at the same time. It’s a tempting option: prepare students for a standardized test by having them practice each piece of content that could be tested in the format it will be tested, primarily multiple choice. This should prepare students. This should improve test scores. This should keep teacher jobs secure. Repeatedly exposing students to content and test format should work, right?

Sadly we all know given both personal experience and national-level data that this doesn’t work. I would posit however that students do poorly on tests not because they haven’t been exposed to all of the content (or test format), but because they have only just been exposed to the content. The content has been covered, not learned. For example, picture the following two scenarios occurring while you proctor a test come May.

Scenario 1: A student sees content she is unsure of on the test, she doesn’t know how to start investigating the problem, anxiety spikes, and the question is skipped.

Scenario 2: A student sees content she is unsure of, she pauses, re-reads the instructions, tries a few things she feels like might help lead her to the answer, and ultimately works backwards from the answer choices thinking about which might be the most reasonable option given the context.

In the first scenario, the student is likely armed with a vast, breadth of surface-level content knowledge. In the second, the same student is likely armed with admittedly a smaller subset of content knowledge, as well as persistence, initiative, estimation, bravery, attention to givens, confidence, willingness to restart, and ingenuity. The difference is that one classroom experience is focused on covering content, and one classroom experience is focused on how to approach content. One classroom fosters problem knowers while the other fosters problems solvers.

So go ahead and teach to the test! But first make one critical shift in your test preparation mantra. Heading into testing season, I encourage you to actively shift your mindset from, students will do well if they know the content –> students will do well if they know how to approach the content.

Caution: the Narrow-Minded Misstep

This week I am putting final touches on plans for an upcoming convening that is pulling together dedicated core content teachers to grapple with improving literacy in their classrooms. My preparatory work demanded that I dig into real examples of true disciplinary literacy in social studies, language arts, science, and math. The convening is separated into two groups: humanities one day and math/science the next, and this separation jogged my memory about an ‘incident’, a very telling conversation, I had last spring.

While talking to coaching colleagues in May, I casually made a sweeping statement about English/Language Arts (ELA) classrooms and, in listing key things that I thought to be taught in this course, grammar was near the top of my list. My idea of a focus on learning punctuation and essay structure was lovingly but literally scoffed at. The incredibly thoughtful and invested humanities’ minds at my table instead made it clear that they wanted students to write not to critique grammar, but to expose, discuss and critique ideas. Grammar and essay structure, they continued, are only important as they aid in better communication of these ideas.

In other words, I made the exact assumptions that I fight against every day. I literally cringed at myself.

As a former math teacher and a current math coach, I pretty consistently experience the moment of telling a hair stylist or rental car agent or family member that I’m a math educator only to receive grimaces relating their own poor math experiences.These facial expressions are inevitably followed up by an explanation of the very narrow view of mathematics being referenced. It becomes clear that the math experienced in these remembered classrooms is not the math I try to expose to students. Like ELA classrooms where structure in and organization of writing is important, similar math structures are not the primary driver of a math classroom either. Whether talking about grammar, essay structure, procedural fluency, or mathematical notation, students develop this type of skill set only in service of being able to better communicate the complex ideas of the content. There is no doubt that these are skills that we must help our students develop, but we must continually remind ourselves, our students, and our communities that they are not the end game. These skills are simply an aid to grapple with and express ideas about deep content connections.

So often, as I’m sure I will next week during the convening, I hear a reference to being either a ‘math person’ or an ‘English person’, but perhaps we’re more similar than we let on…

Summer Reading for Grownups

Summer is admittedly the time when we as educators get to unwind, reflect, and relax. Invariably, it can also often be the time where our minds and hearts are actually free to really digest new learning and big ideas. To assist in this digestion and growth, I’m borrowing and sharing a set of Inquiry Math Badges. Each badge represents an aspect of being a math practitioner: the hope is that in trying to attain each badge, they might help you experiment and grow. For each badge, you start with level one, and steadily work your way to level four. There is no particular order for the list of badges, so let your personal interest and inspiration lead the way!


For summertime growth, I’ll zoom in on the badge entitled Higher Ed: Continued Growth and Learning. Below the badge image, I’m listing a few suggested starter resources for each level to get you going. The list will be woefully under representative of the wealth of information out there, but my hope is these this starter set will beget more resource links which will beget more resources links which…well, you get the idea 🙂

A push for continued growth and learning

Higher Ed badge


Level 1

Blog: You’re already reading this one!

Blog: emergentmath.com

Blog: teachingmathculture.wordpress.com

Article: Why is Problem Solving Important to Student Learning?

Level 2

The two listed above, or this one if you haven’t followed already!

Level 3

Strength in Numbers by Ilana Horn

What’s Math Got to Do with It? by Jo Boaler

Designing Groupwork by Elizabeth Cohen

Other suggestions via Edutopia

Level 4

Develop a plan for how to digest, reflect, and discuss your learning with peers. A few possibilities:

  • Is there 15 minutes that could be spared in every other department meeting to talk about current learning?
  • Could one lunch a week be designated to be ‘growth and learning’ discussion?
  • Or my personal favorite, is there a weekly happy hour crew that could convene to learn and grow together, beers in hand.

Whatever you decide, the time doesn’t necessarily have to be great to be effective and meaningful. The real need is for consistent and accountable time to talk with peers about your influential learning.

I wish every teacher a fun and rejuvenating summer, and (hopefully) one that includes a bit of learning and growth! Enjoy!


No One of Us Alone is as Smart as All of Us Together

“No one of us alone is as smart as all of us together.” This is a quote to live by in any collaborative environment; but it admittedly isn’t easy to implement and embody in a classroom, department meeting, or even a circle of friends. For this post, I’d like to focus on how this mantra could be woven into student collaboration. In essence, how do we go from the ‘divide and conquer’ mentality to a place where student groups genuinely work together to create shared knowledge? There are of course a few important things at play, but one striking current that can work against this ideal is the issue of status. Status, high or low, can stem from gender, race, socioeconomic status, social status, strength in other classes, prior math experiences, etc. And whether it is voiced or not, this sense of status affects daily work and interactions in a huge and meaningful way. By this I mean that when presented with a task or problem, students with high status are expected to do well and so, they often do. The opposite happens for low status students. And this is not just self-perception. The truly tragic part is that this status is propagated by self, other students, and occasionally (even if inadvertently) the teacher.


The good news:

There is something you can do about this inequity when working to have students create shared knowledge while working on a task. Task design is of course crucial; tasks must be complex enough to genuinely require all students and provide equitably entry points. Group work norms are also a vital piece; do all students have a role and equitable access to the work at hand? But to grapple with, tackle, and reduce status issues, this is where smartnesses come in! Yes, I said smartnesses, or if you prefer, competencies. In her book Designing Groupwork, Elizabeth Cohen describes how addressing competencies can address status, “The strength of the treatment lies in the way that it attacks expectations for competence held by the low status student for himself as well as those held by the high status student for the low status student’s performance.” We must challenge current beliefs to show that all students have individual math strengths to share and contribute to the newly found, shared knowledge. Critically, students must genuinely believe and internalize this, too.


Making this practical:

Assigning competence in the classroom is something that takes prep work and practice to be sure. To assign competence, you are pointing out to a student (and his/her group) why and how he/she is smart in math and how this is useful to the group. Assigning competence has three requirements to be successful. The competence assigned…

  • Must be public – privately telling a student why she is smart may help her begin to change her sense of self, but will do nothing to help change the way other students see her and engage with her in the work
  • Must be math-related and specific – we want to focus on the skills/abilities that make mathematicians great at what they do
  • Must be relevant – we want to state why this skill/ability is useful to the group, raising this student’s status in the group


When designing your task, think about the ways a student could be smart while investigating the problem. This is a task that takes some practice and would be great to with peers when co-planning. These do not have to be, and in fact probably shouldn’t all be, traditional math skills like computation. Cohen gives the example of teaching that I think is a great framing thought here. A shallow view of smartnesses of a teacher would be that to be a good teacher, you need only to have good content knowledge, but we all know this is a drastically minimized view. In reality, “teaching requires great interpersonal intelligence, organizational ability, conventional academic ability, verbal ability, as well as creative ability.” These are abilities that may not be in the forefront when many think about teachers, but man are they vital. So as you plan your problems/tasks and as you listen and watch your students work, try to formalize a few traits and abilities that are essential but not always highlighted. Here are three examples of things that I have said to students as I circulate in the classroom as starting points:


“That color-coding that you did to show the point on the graph and in the table is a really smart way to show that    connection to others.”

“It is such a smart idea to do what I just saw you do; rotating your paper can help get a different perspective on a diagram so you can all see what you’ve been given.”

“I’m so glad to hear you say, ‘Well, let’s try it again.’ That perseverance and re-starting that we talked about is such an important piece to help your group get to that final solution.”


In essence here I’m highlighting connection-making, perspective-taking, and perseverance, but in ways that are clear to students. These are not things I necessarily ever thought to be vital math abilities when I was in school, but they are so powerful as students investigate and grapple with tough problems in their groups. If we as teachers can all begin to build a vocabulary of smartnesses, I believe we could dramatically shrink the deficit-based thinking of students and teachers alike. Many days, I got to walk around my classroom all day telling students how they were authentically smart. To me, that’s a pretty great way to spend a day.


Some extensions for your consideration:

  • Multiple Ability Orientation (MAO): instead of only sharing the smartnesses as you see them, identify them ahead of time and state them to students. Here is an example of a MAO I used before starting a new problem on inverse trigonometry. I posted the list and clearly stated that not one students will have all of this knowledge nor all of these skills, but that as a group, all abilities were present. At times, I’ve even had students identify which ones they know they are good at (and share them with their group) and which ones they wanted to work on improving during the problem. In this way, I hope students gain awareness and accountability.
  • Smartness List for Underachieving Students: print off your list of underachieving students (either by grade earned or by some rubric you set personally) and write at least one smartness next to the name of each student on the list. Carry this list with you throughout the next week trying to identify these smartnesses and share them in the moment with students.


Note: In writing this post, I must thank the math department at Cleveland High School in Seattle, Washington as well as our former coaches, Lisa Jilk and Karen O’Connell, from the University of Washington. It was with the guidance of these coaches and the collaborative work with these teachers that I have gained an understanding of status issues, the importance of equity, and the tenants of Complex Instruction.