High-level High School Math and College Success: Correlation or Causation?

I was reading the Common Core Math Appendix A (attached) and found an interesting paragraph. I highlighted in red the part that I found particularly interesting.

The number of students taking high school mathematics in eighth grade has increased steadily for years. Part of this trend is the result of a concerted effort to get more students to take Calculus and other college-level mathematics courses in high school. Enrollment in both AP Statistics and AP Calculus, for example, have essentially doubled over the last decade (College Board, 2009). There is also powerful research showing that among academic factors, the strongest predictor of whether a student will earn a bachelor’s degree is the highest level of mathematics taken in high school (Adelman, 1999). A recent study completed by The College Board confirms this. Using data from 65,000 students enrolled in 110 colleges, students’ high school coursework was evaluated to determine which courses were closely associated with students’ successful performance in college. The study confirmed the importance of a rigorous curriculum throughout a students’ high school career. Among other conclusions, the study found that students who took more advanced courses, such as Pre-Calculus in the 11th grade or Calculus in 12th grade, were more successful in college. Students who took AP Calculus at any time during their high school careers were most successful (Wyatt & Wiley, 2010). And even as more students are enrolled in more demanding courses, it does not necessarily follow that there must be a corresponding decrease in engagement and success (Cooney & Bottoms, 2009, p. 2).


I think this relationship might be due more to correlation than causation. That is, while taking hard math classes in high school might help a student be more successful in college, a student that can persevere through a hard math class in high school probably has learned the perseverance skills—hard work, self-advocacy, determination, stick-with-it-ness, etc—necessary to overcome the challenges and setbacks that one often experiences in college. The conclusion that I take, therefore, is that it’s important for us to teach our students—explicitly—how to work through adversity and challenge. We need to talk them through, and give them strategies for, dealing with stress, managing a high workload, how to study, how to communicate with teachers/professors, etc.

My Fourteenth First Year of Teaching

Even though most of my first year of teaching is a distant haze, there are some memories that remain very strong. They’re not so much memories of things that happened as they are feelings of what happened. I remember a feeling of excitement mixed with an overwhelming sense of dread, which of course makes no sense and therefore describes my first year perfectly.

One of the memories that is the most clear is that of a phone conversation with my mom, an amazing educator with nearly four decades of experience. The conversation occurred somewhere in the middle part of the school year, after I had survived the vanguard of the profession’s attack. It was right about the time when that second wave of frustration/failure kicks in after you’ve put in months of 70 hour weeks but still muster the motivation to craft a lesson that you think will be really engaging for the kids but then you give it to them and they’re like… “meh”. It was right about that time.

I remember feeling so broken and so spent and so… empty. Anybody that has been there knows how it feels to be exhausted physically, mentally, emotionally…

My mom, being the amazing mother that she is, was giving me a pep-talk to try and lift my spirits. I had so much lesson planning to do but was simultaneously so behind in grading that I just felt overwhelmed. I was at that place–mentally–where I was trying to talk myself into not going to school that night to do work (the ol’ bury-your-head-in-the-sand-and-pretend-it-will-all-go-away approach), even though I knew I needed to go in and work. I called Mom–probably because I knew that she would tell me what I needed to hear, but also because I needed someone to relate to that nothing-more-to-give feeling. What Mom said at the time gave me the strength I needed in that moment, but every year I continue in the profession, I am reminded of just how completely wrong she is.

She said, “I know it’s hard now, but it gets easier every year.”

She reminded me that the work I was doing in my first year would be work that I didn’t have to do my second year, that I’d be able to use lesson plans over, that I wouldn’t have to make out tests and quizzes and projects each year from scratch. At the time, that made sense.

However, she didn’t tell me that–sixteen years later–I would have two new preps, the fourteenth and fifteenth of my career (Math Tech 1, Math Tech 2, SAT Math, Geometry, Math Tech 3, Pre-Calculus, Algebra 1, Algebra 2/Trig, IMP I, IMP II, IMP III, IMP IV, Integrated 1, The Art of Mathematics, Integrated 3).

She didn’t tell me that every day I would leave work feeling like I still needed another hour and a half more of work in order to feel “ready” to leave.

She didn’t tell me that the school/district-assigned textbooks would be so [redacted] worthless that I would apologize to my students whenever we had to use them because I didn’t have time to make/find a lesson good enough for them.

She didn’t tell me that–in addition to grading and lesson planning and assessment-making–I’d need to spend hundreds of hours a year responding to emails, writing letters of rec, and updating SchoolLoop pages.

To be fair…

She also didn’t tell me that classroom management would get exponentially easier (like… the single-easiest part of my job).

She didn’t tell me that there would be a community of like-minded math teachers who would share their wisdom, their lessons, their mistakes, their questions… and that they would share so much  I would feel overwhelmed at not being able to keep up with it all.

She didn’t tell me that seeing cascading lightbulbs go off in a class of nearly forty teenagers because I asked/planned the right series of questions to help them make some discovery for themselves would feel like driving a Ferrari at top speed on the cliffs of the PCH.

Lastly, she didn’t tell me how conflicting it would feel to teach students for three years and realize with both pride and shame–as the kids and I both do a new POW for the first time–that their way of solving complex problems is more efficiently, more elegant, more creative, and vastly more connected and communicated than my way.

I don’t know what the me from Fall 2000 would think of my current feelings about (math) teaching, but it’s equal parts exhausting and fulfilling.

NOTE: So that’s why I abandoned Blaugust2016 and haven’t posted since.  It’s my fourteenth year of teaching, and I’m still scrambling to keep my head above water for tomorrow. (By the way, what’s that definition of insanity everyone quotes all the time…?)

Mistake #670: Missing the Landing

[NOTE: This post was inspired by a post on Closure via Exit Tickets by @druinok, who has motivated me to join MTBoS Blaugust2016, a blogging challenge for those of us new to the MTBoS.]

At the beginning of every year, I make a list of goals for myself. For many years in a row, one of those goals has been improving my lesson closure. This is concerning because it belies a much bigger concern: that I’m more content-centered than student-centered. This larger fear stems from the middle years of my career (years 4-9ish), but closure is still an issue for me. First, though, let me explain how the mistakes eventually led to some closure structures that now feel successful.

The name I’ve given to my closure is the Big Idea of the Day (BIotD). At the end of each class, with 3-5 minutes remaining, I ask students to write on a notecard what they feel is the biggest idea from the day’s class. When I look at these BIotD cards, if students are unable to clearly articulate the big idea, it means that I haven’t given them enough opportunity to clarify their thinking on the subject. Maybe they need more time to discuss in their groups or they need a better-worded question to lead their discussion. If they think some ancillary topic was the BIotD, that means I’ve drifted too far (which happens a lot with my tangential way of facilitation) and I haven’t effectively helped them to understand why that day’s topic is more important than other ideas (for some lessons, though, the big idea might not be clear until we get to a later lesson–for example, with Sine of a Sum).

One of the first reasons I’ve struggled with closure is the expense of exit slips (as a teacher, when I use the word “expense” or when I say something is “expensive”, I am always referring to the expense of time–the precious resource I never have enough of). Exit slips are expensive to me because I don’t have time to read 30+ sheets of paper when the bell rings. I usually don’t have time to even read 10 slips of paper because, after every class, I am mobbed either by students from the previous class or the next class who have absent work to show me, a question to ask about an assignment, or something that takes all/most of the available time between classes (this is a problem for a later post). I don’t like saving all the slips until the end of the day because a) I don’t want to read a hundred exit slips when school gets out and, more importantly, b) if I made a mistake in Block 1, I’d like to be able to fix it in Block 2.

I solved both problems by drawing a card at the end of class and only taking those BIotD cards. That is, if I draw the 9 of spades, since spades = spies, all of the Spies–not just the Spy from table 9–turn in their BIotD cards.

I immediately liked this system for many reasons. Firstly, it doesn’t take long to read 7-10 notecards. Additionally, I found that having less cards allowed me to read each one more carefully than I was when I was rushing through to try and finish them all. The depth that I gained from each card allowed me to implement changes for–and share insight with–students in the very next block. It was cheap for the kids and it was cheap for me. [Side question: Maybe it’s not good, though, to want it to be cheap for the students… maybe they need to spend longer at the end of each class reflecting?]

I started doing the BIotD about 7 or 8 years ago, but by some time in October or November each year the routine usually disappears. In some years, the students have helped me maintain the system by requesting the BIotD because they find the process helpful for studying. But in most years, the routine falls away as I feel the stress of falling even further behind in the curriculum.

This past year, however, I kept the BIotD going through the end of the year (with hiccups here and there). The reason for the change was twofold. Firstly, I had a student teacher again and at the beginning of the year I showed her my past goals and how I had failed with the BIotD for a number of years in the past. I wanted to model to her that veteran teachers have struggles but that making goals is a way to focus on solving those struggles. Secondly, to solve this problem, I incorporated the precise wording of the BIotD into my planning. I’m sure better teachers than me plan their closure every day, but I had not previously done so and on many days during the year, what I came up with off the top of my head was not optimally worded. Although it increased planning time, I felt like the students really benefited from having something more tangible to hold on to before starting the next lesson.

By no means is the problem solved, but I do hope to continue the lesson closure improvements I made this year during the 2016-17 school year.

Mistake #669: Maintaining Group Norms

NOTE: This is the seventh in a series of posts about Complex Instruction

Previously, I wrote that despite learning so much about Complex Instruction in Seattle, when I returned to the classroom, I made a lot of group-related mistakes in those first two years back. My first mistake was “Dumping all that information on students on the second day of school (when they’re still trying to scope out who’s cute and what’s up with Oscar’s hair) and expecting them to remember it later in the year.” I learned from this in 2007, and in 2008 I “fixed” the problem by splitting up the initial dump of group roles, norms, etc over the course of the first two weeks.

Thinking that things would be better, I applauded myself for the adjustment.

Naturally, my groups in 2008 struggled just about as much as they did in 2007, which is to be expected. Whether the initial explanation of Complex Instruction takes place on Day 2 or if it’s spread out over Days 2 through 4, the information is still going to fade by December, much less May. So in 2009 I finally came up with a real solution. I knew that I needed a way of regularly refreshing students of our norms for group work–the norms they so expertly came up with themselves. Also, in addition to not using huddles enough (shame!), I was too overwhelmed launching and teaching a new math program by myself to regularly make task cards with norms on them.

However, I was already doing a Quote of the Day, which the kids interacted with. So I started making a Group Norm of the Week.


It only takes 3 minutes on the first class of the week to discuss, so it’s very cheap to implement. Since then, my groups have maintained the norms of the class much better and the groups have been more effective. Since there are ~36 weeks in a school year, there are not enough weeks to do all the group norms (theirs and mine). But I think it works out fine, as some norms are more important than others. Not only that, but there are a couple norms that make multiple appearances on the GNotW. These are the ones that I seem to come back to the most often:

  • Answers aren’t as important as understanding.
  • Before insisting that you are right, listen—truly listen—to others’ ideas.
  • Question each other.
  • Disagree without being disagreeable.
  • Saying it louder does not make you right.
  • Listen before speaking.

Whenever I pick the last one, I always remind students about the talk-first status research (that the first person to talk during a task has more status), so that the talk-first kids learn to lay back in the cut a little and give others a chance to go first. As a student (and frankly, as an adult), I need this reminder when interacting with others, and the GNotW has given me a system for reminding students of class norms.

Other good techniques for maintaining group norms are include 1) using huddles and 2) listing norms relevant to a specific task on its task card. I hope to write more about those soon.

Group Norms (part 1: establishing them)

NOTE: This is the sixth in a series of posts about Complex Instruction

One of the best mistakes I’ve made came by accident in 2007. At that point in my career, I had learned not to teach Group Roles, Member Qualities, and Norms all on the second day of school. I made a slide show (I think it was my first ever) for Group Norms, but when I went to show it to my class, the projector bulb was so dim that the font didn’t show up against the background (rookie mistake). I decided it would just be easier to open the document in Microsoft Word and show the students that way.

While I was at the computer, trying to find the file, my students started getting restless (as students are wont to do). To buy myself some time while I clicked through my far-too-layered filing system, I asked the students to think about the prior experiences they’d had working in groups. I think I gave them a prompt along the lines of “spend 90 seconds brainstorming norms that you’d like for group work… try to come up with norms that prevent some of the problems that have occurred when you’ve worked in groups before this class”. After I found the file and got ready to take the reigns again, I walked around and listened/read the students’ ideas.

What I heard and saw was so amazing that I immediately knew I would give the same prompt to every class I taught in the future.

I am only including a small sampling of the norms that students have suggested over the years because the list now is almost 500 suggested norms deep. Please remember that I typed these suggestions verbatim. As any veteran teacher knows, whenever you type student suggestions verbatim, you are bound to get some clunkers and some catfish, but the overall haul makes me excited and optimistic about the last decade’s worth of teenagers.

  • 2007
    • 1st Block suggestions:
      • Help absent members catch up
      • Wait—don’t leave group members behind
      • Complaining about lunch won’t make it come any sooner.
      • Make sure everyone understands the instructions
    • 3rd Block suggestions:
      • So that everyone understands, don’t just explain your answer—explain why.
      • Don’t be hard-headed. Compromise.
      • Don’t bring lunch drama into class.
      • When someone’s like really smart, don’t like make slower people feel like really slow or whatever. So… yeah.
    • 4th Block suggestions:
      • Don’t insist you are right. Listen to other people’s ideas.
    • 5th Block suggestions:
      • Be accepting of people that aren’t visual artists. [Editor: I teach at an arts school made up of students from over a dozen conservatories.]
      • Keep “outside” conversations outside.
    • 6th Block suggestions:
      • Some people want to learn concepts more deeply and don’t want to be rushed ahead.
  • 2008
    • 1st Block suggestions:
      • Don’t discriminate based on grade  [Editor: Most of my classes contain students from 4 grades.]
    • 2nd Block suggestions:
      • Don’t let one person do all the work
      • Don’t be scared to ask questions
    • 5th block:
      • Share your epiphanies  [Editor: one of my favorite ever!]
      • Don’t say you get it when you don’t
  • 2009
    • 5th Block suggestions:
      • Do your HW so that you are prepared to discuss with your group.
      • Don’t stress out. Stress makes it harder to understand.
      • Don’t share your stress.
    • 6th Block suggestions:
      • Don’t pull others into your vortex of distraction.
      • Don’t be afraid to share your answers, because if you’re wrong, you can still learn something.  [Editor: Just to reiterate–STUDENTS said these things. Every teacher that uses groups should give their students this prompt.]
      • Attendance is important—but STAY HOME if you’re sick.
  • 2010
    • Block 5
      • Make sure your blood sugar levels are fine so you don’t get grumpy.
      • Get enough sleep.
  • 2011
    • Block 6
      • Don’t be bossy and don’t be bullied.
      • If you’re having a problem with your group and you’re trying to resolve it, be open to hearing other people’s suggestions.
      • Saying it louder does not make you right.  [Confession: I remember paraphrasing this one.]
  • 2012
    • Block 1
      • Don’t be that person who copies off everyone else on the group test. Contribute.
      • If you’re having a bad day—don’t spread it. Do something nice for someone else—it will make you feel better.
    • Block 2
      • Just because you’re like really good at math doesn’t mean that everyone else is stupid.
      • Don’t be that person that gloats about grades.
    • Block 3
      • Contribute without taking over.
      • Don’t nag group members (this means you, too, Mr. H).
      • If you think you’re going to fail, you might. Have a positive attitude.
      • Don’t be that guy.
      • The majority is not always in the right.
  • 2013
    • Block 1
      • Disagree without being disagreeable. [Confession: I remember paraphrasing this one.]
      • We all think differently.
      • Stress cuts years off your life.
      • Bring your own supplies. By like the fourth time your table loans you supplies, it gets annoying.
      • Don’t take off your shoes.
    • Block 2
      • No dictators (except Mr. H)—this is a democracy.
      • If you don’t do your homework or like whatever, then the whole group is affected.
      • Be appropriate with your clothing.
  • 2014
    • Block 1
      • Nothing is impossible. The word even says ‘I’m possible’.  [Editor: see *NOTE* below.]
      • Don’t be that person that gets an A- on a test and acts like they failed.
    • Block 2
      • Spoiler alert!!!—don’t spoil the answers for other groups/classes
      • Don’t bring dishonor upon your house


*NOTE*: Once I heard this one, I realized that many of the norms that the students were advocating for were norms that they’d heard from other teachers. I think what makes my accidental prompt so effective, however, is that all these amazing norms are coming out of the mouths of students. When teachers say things, we have a certain status that carries a weight with it. However, when trying to get students to buy into a set of norms for how they would like to work together, it is infinitely more valuable for these words to be spoken by students.

Below are the norms that I curated and created myself. As you can see, they have a different flavor than the students’ norms.

Group Norms

Getting work done

  • Pull your own weight—everyone participates equally.
  • Focus & stay on task.
  • Everybody does the work together, but each person writes it in their own notebook.
  • No one’s done until everyone’s done.
  • Do your HW so that you are prepared to discuss with your group.

Interacting with others

  • Listen before speaking.
  • Everyone takes turns and gets equal time to talk.
  • You must ask for help if you don’t understand.
  • You must help someone if they ask a question.
  • Don’t criticize or put someone down—even if you’re joking.
  • Help other group members without doing their work for them. (Don’t write on a classmate’s paper—you can show them your paper, but let them write for themselves!)
  • Before insisting that you are right, listen—truly listen—to others’ ideas.
  • No talking or interaction with students outside of your group.

 Learning the key math concept

  • Answers aren’t as important as understanding.
  • Question each other.
  • Explain your thinking.
  • There are many correct ways to solve a problem.
  • If you have a question, see if anyone else knows the answer. If not, have the spy call the teacher over.
  • Learning takes time (i.e. I don’t get it… YET!!!)

Being part of a group

  • Have one of the group roles make sure that the group norms are being followed.
  • Fulfill the duties of your role in the group.
  • Everybody cleans up.
  • Keep the desks clear of extraneous stuff to make it easier to see and focus during paper powwows.

Mistake #668: I Said It, So They Learned It

I know it’s a classic teacher mistake (one of Tom Sallee’s Two Lies of Teaching) to think that anything the teacher says will be learned by the students. I want to explain, though, what I learned from a specific example of making this mistake.

When I returned to the classroom after going back to grad school, I thought I knew everything. I had learned so much and I was excited to implement what I had learned. One of my biggest areas of growth came in my implementation of Complex Instruction in place of the basic groupwork I had done prior to grad school. So on the second day of school, I spent nearly the full class delineating the group roles we’d be using for the year, outlining the desired qualities I wanted in group member, and laying out expected group norms. It was the first time I had ever used a slideshow in the classroom and I spent way too long choosing the theme/font/colors for my 20+ slide onslaught.

In hindsight, it seems foolish to think students would learn and retain 20+ slides packed with information, but I think that underscores an even bigger mistake I made. I didn’t realize it until later in the year, but during that first year back in the classroom, the groupwork in my class was not nearly as effective as it is now–and it was my fault. Dumping all that information on students on the second day of school (when they’re still trying to scope out who’s cute and what’s up with Oscar’s hair) and expecting them to remember it later in the year is silly.

After reflecting about what went wrong, I made changes the next year and decided to spread out the setup of groups over the course of three classes instead of one:

Day 2: Group Roles
Day 3: Group Member Qualities (see below)
Day 4: Group Norms

Even though there were many more mistakes to come (spoiler alert: how can students on Day 86 remember what they learned on Day 4), I felt like the students had a much better understanding–at least initially–of how I wanted them to work in groups.

I can’t remember where I got the following group member qualities, but I’m pretty sure it was from some IMP teacher:

Group Member Qualities

A skillful group member…

  • Fulfills their group role & stays on task
  • Explains ideas
  • Puts ideas together
  • Requests or provides information
  • Asks if everyone is ready to decide what to do

An especially skillful group member…

  • Asks quiet group members what they think
  • Listens with interest to what other people say
  • Praises good ideas and suggestions
  • Is willing to compromise
  • Is concerned with understanding the problem, not just getting the answer
  • Challenges others in a respectful way when there is disagreement

A destructive group member…

  • Talks too much
  • Listens very little
  • Insists on having his or her ideas accepted
  • Fails to do something about the destructive behavior of others
  • Criticizes people rather than their ideas
  • Lets other people do all the work
  • Is impatient or sarcastic with questions that may seem too obvious

Delineating Group Roles

NOTE: This is the fifth in a series of posts about Complex Instruction

When teachers take their first forays into groupwork, the first organizational decision that most make is in delineating group roles. I was no different. Understanding the implications of status on group dynamics, however, is when I finally felt like I wasn’t causing more problems than I was solving by grouping students in my classroom. Having roles for groupwork, though, is still important.

I personally found the standard Leader, Reader, Resource Monitor, etc roles to be a little dry for my taste, so I tried to spice up the titles a little and went with Emperor, Ambassador, Designer, and Spy. I used to call my Designer role ‘Scribe’, but I felt like ‘Scribe’ has connotations of labor whereas the Designer label feels more creative. A student suggested a couple of years ago that I switch the name to ‘Architect’, but I think more students can connect to the name ‘Designer’–especially at my school.

A couple of years ago, one of my colleagues had a CSI theme in her room and gave her roles names like Lead Detective, Forensics Expert, and Investigator. Another colleague who teaches middle schoolers is really into superheroes and gave superhero names to her roles. I like that these connected their classes to an interest of theirs. Their are many ideas for group roles on Pinterest, though I would caution teachers to realize that the names they use for their group roles are orders of magnitude less important than the other elements of Complex Instruction.

Here are the duties of each of my roles:

The ♣ Emperor ♣ is in charge of:

  • Making official decisions
  • Reading* materials to the rest of the group
  • Coordinating presentations
  • Filling in for absent group members

The Ambassador ♦ is in charge of:

  • Making sure that everyone in the group participates and understands the activity
  • Checking for understanding
  • Facilitating discussions

The ♥ Designer ♥ is in charge of:

  • Acquiring & returning materials
  • Making diagrams and drawings
  • Making sure that everyone at the group writes the work in their own notebook

The Spy ♠ is in charge of:

  • Making sure the group finishes the activity on time
  • Asking questions to the teacher (The Spy is only allowed to ask a question if no one at the group knows the answer)
  • Using HINT sheets

*When reading to the rest of the group, the Emperor has 4 options: they can read it aloud themselves, they can ask another group member to read it aloud (they may decline), they can popcorn read it, or they can ask everyone to read it silently by themselves.

My favorite role/duty I’ve come across is that of Devil’s Advocate, which the University of Waterloo has just nailed:

  • Remains on guard against “groupthink” scenarios (i.e., when the pressure to reach the group goal is so great that the individual members surrender their own opinions to avoid conflict and view issues solely from the group’s perspective).
  • Ensures that all arguments have been heard, and looks for holes in the group’s decision-making process, in case there is something overlooked.
  • Keeps his or her mind open to problems, possibilities, and opposing ideas.
  • Serves as a quality-control person who double-checks every detail to make sure errors have not been made and searches for aspects of the work that need more attention. Keeps an eye out for mistakes, especially those that may fall between the responsibilities of two group members.

Typical phrases:

  • “Let’s give Mike’s idea a chance.”
  • “OK, we’ve decided to go with plan C, but I noticed that we still haven’t dealt with the same problem that plan A didn’t address. What can we do to solve this?”


Assigning Competence: a weapon used to fight status

NOTE: This is the fourth in a series of posts about Complex Instruction

The first time I went into a classroom after reading an article about status, I felt like Neo beginning to realize his power in The Matrix. I saw things that had been right in front of me for years that I had completely missed. I started noticing how students perked up or tuned out depending on the status of the classmate who was speaking.

Once I learned about status, I never saw a classroom the same way again.

Students who self-assign low status within a group may sit back and play a very passive role even though their ideas may be valid.

Elizabeth Cohen, Designing Groupwork: Strategies for the Heterogeneous Classroom

Notice that Cohen writes “self-assign”. Peers may assign one status to a student, but students may self-assign a completely different one. In my experience, the default self-assigned status for math is as low as Samwell Tarly’s Season 1 self-confidence. In my class, however, I want student ideas to be judged on their merit–not the status of the person to which the idea belongs. I want to value the voices of all students. Part of the way I do this is by establishing a multidimensional classroom, so that students learn to recognize the many ways they are like an expert mathematician. But status is especially dangerous because it is the perception of students’ abilitiesnot the reality. Teachers need a weapon to counteract the inexorable attack of these perceptions and the damage they inflict (particularly  in math classes to females and underrepresented minorities).

Assigning competence is that weapon.

Assigning competence is a form of praise where teachers catch students being smart.

Lani Horn, Strength in Numbers: Collaborative Learning in Secondary Math

To assign competence, Cohen explains that the praise to a student must be:

  1. public
  2. intellectually meaningful
  3. specific to the task

Because status is all about perception, the first of these seems obvious to me. However, I constantly battle with myself to make sure I’m doing the last two components. How many times have I told a student “That’s amazing!” or “Good job”? I’m embarrassed at how often such meaningless compliments come out of my mouth. My friend Sam Hilkey has really helped me to be more aware of this through our intense discussion of the implications of Carol Dweck’s Mindset, but I still have a long way to go.

Another mistake I’ve made is not posting the “An Expert Mathematician,…” list publicly in my room. I used to do it, and I need to do it again this year because it makes it so much easier to be intellectually meaningful when I can use the exact wording referenced on the document.

Lastly, many teachers incorrectly believe that assigning competence is a remedy for low-status students. However, I assign competence more for its effects on the group than the individual because I’m trying to change the group’s perceptions.

An Expert Mathematician…

In 2010, Avery Pickford posted an entry on his edublog called “Habits of Mind”,  which inspired me to think about the skills I think are important for secondary math students. Based on the lists of Avery and other edubloggers (it’s been 6 years, so I can’t remember the others), I compiled a list of my own, organizing skills by their location in the process of working on a problem. I call this list “An Expert Mathematician…” because I want students to internalize the commonalities they share with an expert of mathematics.

To have a multidimensional classroom, there needs to be many ways for students to be smart. “An Expert Mathematician…” is my list of ways to be smart.


“An Expert Mathematician, …”

 …when starting/playing with a problem, ..

  • … can restate a problem in her own words
  • … states given information, eliminates unimportant information, and/or makes simplifying assumptions
  • … can begin to work on a problem independently
  • … explores with curiosity (and is thoughtful and purposeful about which case(s) to explore)
  • … estimates efficiently
  • … determines lower and upper bounds
  • … makes the problem smaller
  • … looks for patterns and/or shortcuts
  • … is able to look at problem from completely different angle
  • … makes conjectures

…when working towards a solution, …

  • … organizes a plan of attack
  • … finds patterns
  • … is creative
  • … extrapolates results from smaller problems
  • … uses multiple representations to visualize relationships
  • … creates models/diagrams/pictures or uses manipulatives
  • … invents notation and language when helpful
  • … organizes information
  • … applies old skills and concepts to new problems
  • … can work backwards to solve a problem
  • … computes/solves efficiently
  • … visualizes processes (e.g. graphic organizers)
  • … can explain her thinking to others
  • … reflects on the viability of current  plan-of-attack and considers alternate plans/approaches

…when solving, …

  • … generalizes patterns
  • … creates algorithms
  • … tests models
  • … refines models
  • … is thorough and exhaustive, covering all possibilities
  • … is accurate with detail
  • … checks reasonableness of answers
  • … looks for additional or easier solution paths
  • … finds multiple solutions
  • … makes the problem bigger
  • … discovers new problems (i.e. extensions)

…when presenting a solution, …

  • … explains how
  • … justifies why
  • … displays results/diagrams/graphs/tables so they are clear, easy-to-understand (i.e. labeled, color-coded, etc)
  • … focuses communication towards audience understanding (and draws on previous knowledge)
  • … communicates clearly, concisely, and convincingly

…when thinking/communicating mathematically, …

  • … formulates quality questions
  • … asks questions that clarify misunderstandings
  • … asks creative, outside-the-box questions
  • … works on one problem for greater and greater lengths of time
  • … spends more and more time stuck without giving up
  • … finds beauty in mathematics
  • … looks for mistakes or holes in proofs
  • … connects different skills and concepts together




Establishing a Multidimensional Classroom

NOTE: This is the third in a series of posts about Complex Instruction

In the last post, I wrote about how status affects all interactions between humans–none more so that those between teenagers.

If a teacher assigns a group of students a task of typical academic work, the student who is perceived as getting the best grades in that subject is likely to dominate the group, regardless of the merit of their ideas in the specific task.

Elizabeth Cohen, Designing Groupwork: Strategies for the Heterogeneous Classroom

Some teachers want that and intentionally pre-select groups so that each group has at least one “strong student” to lead the group. I made that mistake until Lani taught me a solution to that and much more when I went back to get my Masters in Teaching. In a later post, I hope to explain the specifics of why this is a bad idea–despite its good intentions. Ideally, though, we should want groups of students to make decisions based on the merit, logic, and reasoning of the ideas presented and not the status of the group members.

However, our students are slaves to status. They will make decisions based on status unless we do something to counteract it. The entire system of Complex Instruction is built on fighting status and creating equity in the classroom, but the first step towards a solution is establishing a multidimensional classroom (those thinking about spacetime and fifth dimensions should settle down). A multidimensional classroom is just one in which there are many ways to be smart. Despite this sounding very simple, many of the classrooms I’ve seen–especially math–are very one-dimensional.

In fact, when you ask most Americans what it means to be good at math, there are three common answers. Being good at math, the fallacy goes, means that a person…

  1. can do math quickly
  2. can do math in one’s head
  3. doesn’t make mistakes

All three of these, of course, are completely false.

The 2014 winner of the Fields Medal (“the Nobel Prize of Mathematics”), Maryam Mirzakhani, is a self-described “slow thinker“. Students that think too quickly often think quite shallowly and miss mathematical connections (as a former student who embarrassingly cared a lot about finishing first, I speak from experience). There are many opportunities for creativity in deliberate thinking. With regards to the fallacy of needing to do math in one’s head to be good at math, research on the neuroscience of math learning shows that thinking mathematically involves using distributed networks of the brain that involve visual processing regions. Someone that does math only in their head will show less neurological activity–less connections being made–than someone who uses visuals (paper, their hand, technology, whatever). The third fallacy, that good mathematicians don’t make mistakes, is like saying that good athletes don’t ever have a bad game and is so absurd that I don’t know whether to refute it with a link to an article or an entire book, Mathematical Mindsets, by Jo Boaler.

There are many ways to be good at math just like there are many ways to be good at cooking. Making a delicious dessert or a six-minute stir-fry or a health-conscious casserole all require vastly different skills but nonetheless display culinary acumen. Being good at math or dancing or writing or Australian Rules Football–or anything–is no different. It should be accepted that there are many ways to be successful at any discipline.

Every teacher should think deeply about what it means to be good in their discipline. The important step, however, is communicating these skills to our students. After all, if you think identifying connections between Shakespeare and Sorkin makes a student a strong writer but she thinks she sucks at writing because she can’t spell, then you’re fighting an uphill battle. I think every teacher, at the beginning of the year, should ask their students what it means to be good at their discipline. When they hear their students’ responses, I think they’ll understand why there are so many status issues in schools and why it’s so important to create a dialogue in the differences between what’s valued by teachers and what’s valued by their students.

One of the mistakes I’ve made in the past is not communicating these skills effectively enough to my students. So about seven years ago, I compiled/created a list that I call “An Expert Mathematician…” and I share it with students.  In it, students see 48 skills, like:

An Expert Mathematician…
… can restate a problem in their own words
… organizes a plan of attack
… spends more and more time stuck without giving up
… creates models/diagrams/pictures or uses manipulatives
… discovers new problems (i.e. extensions)
… asks creative, outside-the-box questions
… communicates clearly, concisely, and convincingly

In the second assignment of the year, students must identify one of the skills at which they are already good. Even though perhaps a majority of students enter my class claiming that they’re not good at math, a quick look at the list enables everyone to find at least one strength. Students then must artistically render their chosen skill on a piece of computer paper. If I’m good, I’ll have squared away an entire wall of my classroom to display these skills so that–throughout the school year–any time a student displays one of the skills, I can point it out to them and the class. But I’ll write more about that in the next installment about Assigning Competence.