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…

I want to do PrBL, but where do I start?!

The Common Core State Standards (CCSS) for Math are a blend of content standards, the content of our courses, and practice standards, the manner with which students tackle that content. In an ideal world, our students would bring the mindset of these practice standards, their math disposition if you will, to class each day. But these, too, have to be developed, facilitated, and practiced, which admittedly adds to a teacher’s load. Many teachers, and I would be bold enough to say rightly so, feel that a problem-based learning experience (mini-projects essentially) is a great way to allow students to participate in experiences that will ask for a blend of these two types of standards. NCTM research, New Tech Network work with math facilitators, and my own experience in the classroom back this up. But that doesn’t make it any less daunting to take this on – to go against the way you yourself was taught (and were likely successful at) math and probably the way you were taught to teach math as well. So where do you start? I have an idea!

The CCSS math practice standards come with some guidance stating, “Expectations that begin with the word “understand” are often especially good opportunities to connect the practice to the content” (CCSS for Math, page 8). If you are looking for spots to help develop your students’ ability to engage with the content, I suggest starting here. As an example, below is the Geometry Overview taken from CCSS for Math with the appropriate “understand” expectations highlighted. If you view the full list of standards, there are a few smaller strands that also begin with the word “understand”, but I think the big picture perspective of the Overview is just the ticket if you as you think about curriculum planning for next year.

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Geometry Overview

Congruence

• Experiment with transformations in the plane
• Understand congruence in terms of rigid motions
• Prove geometric theorems
• Make geometric constructions

Similarity, Right Triangles, and Trigonometry

• Understand similarity in terms of similarity transformations
• Prove theorems involving similarity
• Define trigonometric ratios and solve problems involving right triangles
• Apply trigonometry to general triangles

Circles

• Understand and apply theorems about circles
• Find arc lengths and areas of sectors of circles

Expressing Geometric Properties with Equations

• Translate between the geometric description and the equation for a conic section
• Use coordinates to prove simple geometric theorems algebraically

Geometric Measurement and Dimension

• Explain volume formulas and use them to solve problems
• Visualize relationships between two-dimensional and three-dimensional objects

Modeling with Geometry

• Apply geometric concepts in modeling situations

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There are three critical places where you have key opportunities to dive into PrBL, places that the authors of the standards highlight as excellent crosswalks between the math content and the math practices. I do not mean to understate the importance of the others expectations that ask students to use, apply, describe, or explain; these are important to gain a well-rounded and thorough view of Geometry. But if you are looking for a spot to start your PrBL journey, here it is! The standards themselves narrow down the scope of expectations and hopefully guide you to standards/units where you can throw some energy into really creating PrBL experiences for your students. So grab a copy of your standards, bust out a highlighter, and dig in!

P.S. Once you’re done highlighting and know where to kick off your PrBL planning, I recommend stopping by emergentmath.com and checking out all of problem ideas Geoff Krall has gathered together under the “Curriculum Maps” tab.

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 🙂

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.

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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.