Growth and Fixed Mindsets in Everyday Life

Growth and Fixed Mindsets

My students last year kept bringing up mindsets and then at National Council of Supervisors of Mathematics (NCSM), Jo Boaler was a keynote presenter and spoke about mindsets (see more at

Mindsets in Mathematics

Jo Boaler spoke about the misconception that some people are born with a “math brain” and others are not. So some people are unable to math at all. She pointed out that there some people who really are physically unable to do math, but those represent fewer than 1% of the population. She argued that we need to support “growth mindsets” which basically means that we support the idea that anyone can learn anything if they work hard, have appropriate experiences, and give it time.

Mindsets in Everyday Life

Pretty frequently I tell myself I have a “___ brain” where lots of things can go into the blank: I am just not a social person, and nothing I can do will ever get me to the point where I can interact effectively with other people on an ongoing basis. My mind just doesn’t work that way. Or, I am an imaginative person but I am not a creative person. I can’t draw or paint because my mind just doesn’t work that way. Or, I’m not an organized person – I am scatter-brained and can’t keep up on notes or planning. 

So, I’ve started to try to talk back to myself and say: If I expect my students, who may feel the same way about math as I do about socializing, to change to a growth mindset, then I need to change to a growth mindset about all of these things too.

This realization was pretty amazing for me: All this time, I have thought I was born a certain person who just can’t do things like other people. And I realize that some of those things may never come easily for me (because I probably still will be an introvert who prefers a quiet reading chair in the evening as opposed to an exciting evening with friends) but I can work on these things and improve. I can reflect on my feelings and strategies, and I can talk to other people about their feelings and strategies, and get better at anything that I want to.  It feels freeing.


What is the learning goal anyway?

In my years at Michigan State University, I have learned that having a learning goal is one of the critical elements of lessons and assessments, research plans, presentations, and guiding your own learning. (I think this post is supported to some degree by All About Models.)

Learning Goals in Mathematics Lessons and Assessments

For teaching, learning goals are important in helping a teacher choose relevant and focused questions, tasks, and discussions. Sometimes learning goals can be too restricting, closing off the potential learning and interpretations. Deciding what the true take-away from a lesson should be, is incredibly difficult! A teacher needs to think about immediate mathematical needs of her students, as well as future needs – in future math classes, of course, but also for the future adults the students will become. Those adults need to be able to make sense of their world, think critically as consumers and citizens, and put their mathematics thinking to use in their future profession. Once a learning goal is chose, thinking critically about the assessment is also important and really questioning whether the kind of learning and thinking does the assessment test for is relevant to the learning goals. (Which comes first, the assessment or the learning goals?)

Learning Goals in Research Plans, aka “research questions”

In my time on research projects, especially Preparing to Teach Algebra, I have seen again and again how important it is to work for a while and then re-read the research questions. The research questions indicate what you hoped to get smarter about before the project began – they indicate your learning goals for the research. Dr. Sharon Senk told me over and over: “I think you need to go back to the research goals: Does your current analysis help or should you reorient and what we hoped to find out?”

Learning Goals for Presentations

A large part of my job as a research is “disseminating results of research.” In class or at conferences, it is tempting to – and almost impossible to avoid – sharing everything you have learned with the audience. Because it all is amazing and interesting – and, yes, the audience would be enthralled if they had several days of time to chat with you. I try to fight against that need because I have been the perpetrator and the victim of such presentations. I remember serving on a Math Education faculty search committee. A candidate visited and shared her dissertation – it seemed like the whole dissertation – in a 50 minute presentation. She had over 50 slides, and only managed to talk through a couple of them, because each slide had tiny graphs and figures. I think it was frustrating to her, but also to us – we wanted to understand something she shared, but she talked very fast and it was difficult to see any details on the slides. In my mind, I think of that example as my reason to focus on only a handful of take-aways from a presentation. (I fail sometimes.)

Learning Goals for Guiding your Own Learning (as a grad student … and beyond?)

As a fifth-year graduate student, I have spent a great deal of time in the past four years completely overwhelmed by a program requirement or course assignment. It is easy to lose sight of the meaning of these learning opportunities, and I believe that many faculty are not focused on the learning goals and many faculty would disagree on the learning goals if they talked about them. I have tried to choose a learning goal that each assignment could support, and have successfully navigated each (so far). I would strongly recommend to other graduate students to think deeply about potential learning goals – what will you get from each project or assignment? Then re-evaluate you work regularly to see if the project is getting too big or if you are steering off-course.

That said, it is not always possible to reconcile your chosen learning goals with the learning goals of your committee, other faculty, or other graduate students. Try to argue for you perspective, but be open to other interpretations. And remember that at the end of your program, you will have to defend the choices you make. If you had no choice but to let others make choices for you, it could be difficult to explain – so make sure to talk to others to make sense of those “other” choices.

Navigating Grad School: Power and Communication

Over the past few years, I’ve spent a lot of time confront my own misconceptions about how to communicate with people who I feel hold power over me. Power can be real or imagined, but in grad school when you typically have about half a dozen “bosses” at any given time. I’m going to define “my boss” as someone who can tell me what to do and reasonably expect that I will do it or face unpleasant consequences like withholding money or dropping a grade or giving a poor evaluation. And then my boss is also someone I report to, who tells me I’ve done good or bad / right or wrong, and who – if they disagree with my choices or strategies – can cut me off in mid-stride and send me a different direction.

So, generally over the past four years I have had at least one boss per assistantship each semester, sometimes two. I have often continued on a project after my assistantship has ended, receiving hourly pay or volunteering my time. I consider the instructors of my courses as “bosses.” I also consider my advisor as a “boss” as well as committee members (to some extent, although advisor is “main boss”).

One of the difficulties with bosses is that most bosses in grad school don’t see the power they have for good or for bad and generally do not wield their power in thoughtful ways. On the other hand, it’s easy for me, as a grad student, to be overwhelmed with a sense of my bosses’ power. Partly because sometimes a boss is in a bad mood for a completely disconnected reason and uses their power to cause pain and distress, and to threaten dire consequences. And then the next day the same boss is happy and has forgotten all about what they said the previous day. So, as a grad student, it sometimes feels like walking in a mine field, trying to learn the rules when there aren’t actually “rules.”

It shows up in communication.

For example, emails. I can’t tell you how many days and nights I have stressed because a faculty member should have responded to an email and did not, and I don’t know if send a reminder is disrespectful or how they will react to it. Sometimes, if I send a reminder email the faculty member is thankful and responds quickly. Other times, I have sent reminders over months before getting an answer. And the question becomes: Whose responsibility is the communication? Ultimately, I pay the price when communication lines go down, so I am responsible. (I do recognize that faculty are so busy and have no way to accomplish all their tasks, much less respond as quickly as I want them to every time.)

Another example is communication in meetings. Faculty are not always sensitive to the power that their questions, jokes, or statements have on grad students. I remember in my first year talking to a 4th year student who was almost in tears because her advisor had made a joke that might mean she would fail comps. It’s easy for faculty to forget that we are often in situations that are high stakes, where we have little control, and where we don’t know the parameters because we’re doing it for the first time – that is, because we don’t know the parameters and because we haven’t seen successful and unsuccessful strategies, we don’t know whether our strategy is even close to reasonable.

A final example is communication about interaction. As a graduate student, it is difficult for me to tell the faculty around me when things are going well or poorly. Partly, I find it difficult because I am afraid and the fear is not always justified. Partly, I find it difficult because sometimes saying something has drastic repercussions. So, even though sometimes talking helps a situation, other times it makes it far worse, and there seems to be little consistency.

I try to think about what I can do when I am a faculty member and when I am an advisor. How can I create a situation where my students feel comfortable with helping me become a better mentor?

All About Models

This semester, I’m taking a course (CEP 902, for those of you in PRIME) called “Psychology of Learning School Subjects.”  I don’t think the following was a goal of the course, but I have come to think of this course as “All About Models,” in an interesting way. Let’s see if I can communicate why!

On the one hand, it seems (reasonably) that we read about models constructed by researchers to explain student learning in each broad topic area: writing, reading, mathematics, physics, science, and engineering. On the other hand, it seems as though in at least some of these areas that students create their own models as part of their learning.

What is a model?

Merriam-Webster says, “a usually small copy of something.” Well, that’s kind of close. But I’m really meaning to say a “scientific model” perhaps. American Heritage Science Dictionary says, “A systematic description of an object or phenomenon that shares important characteristics with the object or phenomenon. Scientific models can be material, visual, mathematical, or computational and are often used in the construction of scientific theories. See also hypothesis, theory.

Good. The latter definition is what I was looking for:

  • systematic
  • description (or representation? or simplification?)
  • “shares important characteristics with the object or phenomenon.”

I think that my model of definition is: a representation of a complex object or phenomenon that reduces the complexity in a way that focuses users’ attention on chosen aspects or features.

We use these types of models in everyday life. For example, maps are models. Depending on your goals, the “important characteristics” may change. Map designers make decisions about what features are important. Here are a few different examples of models / maps / representations of Washington, D.C.:

Can you think of the creator’s goals for each map? Which characteristics are important in each map? How does the focus help the viewer? Are the maps above valid “as maps”? Or should a map of Washington, D.C. always look like this:MapDC-05My point is: there is no “best” or “more correct” model for all time and in every circumstance – a model is always a simplification, which means the creator must always be thoughtful about their goals and how some characteristics support their goals.

Models of Student Learning

A question that keeps coming up in this course is: “Does this model describe every person’s learning? If not, how many models would we need to describe every person’s learning?” I’m not sure I understand the question fully. In my mind, no model can completely describe a person’s learning – because then it would not be a model but would be the thing itself. My perspective on creating models of students’ learning is not accurately describe every student, but as a useful way to think about learning as support for creating better learning opportunities.

Models and Scaffolds in Student Learning

In different subject areas, we have talked about ways students can simplify their task. For example, creating a template to structure your writing or creating an algorithm to structure your problem-solving. These scaffolds can support students in arriving at the finished essay or answer, but can also proceduralize the work. That is, a scaffold can support students but can also block their learning. At what point can the student begin to remove a scaffold? Or is it okay for them to continue using it forever?

Another way of simplifying a task (or reducing complexity) that might be scaffolding but might not be – I’m not sure – is use of models. I’m going to stretch the definition of “model” a little bit, but say that in most fields students can create models to support their learning. In math or science, we talk explicitly about modeling situations. In writing, I mean “model” as an outline for notes or for writing – a small, lightweight  representation of their ideas that can help organize their thoughts around the important characteristics, whatever those may be.

In any case, talking explicitly about how to think critically about those important characteristics should be important. Supporting students in understanding that it isn’t possible to create a model that accounts for every variable or idea can help students develop into more critical consumers or citizens who will ask: “This advertisement or article makes a particular claim and backs it up with data. What choices did they make in the characteristics to keep and the characteristics to lose?”


model. (n.d.). The American Heritage® Science Dictionary. Retrieved September 22, 2015, from website:

Dumbest-Kid-in-the-Room Syndrome / Imposter Syndrome

Dumbest-Kid-in-the-Room Syndrome

This syndrome is basically when you sit in a class, amazed by the questions others ask or the insights they have, and think “I could never be like them. I’m the dumbest one here.”

I first saw evidence of this misconception in a Differential Geometry course in my last year of undergrad. All of my math courses before then had been mostly silent, and this one was no different. One day, I turned and asked someone a question before class. Soon we all began chatting. We quickly realized that we all assumed each other was brilliant, and we each assumed we were the dunce of the course. Because each of us understood different pieces. If I understand piece A but not piece B, then I assume anyone could understand A – if I understand it, and I’m a dunce, then everyone must understand A. But my classmate understands B and not A. I assume she understands both – she assumes I understand both.

A group of my classmates and I formed in informal support group for a class. Directly after class each day we would go somewhere for lunch and talk about how confusing the course was, and how brilliant everyone else was. I started to realize that “If every one of us is the stupidest kid in the class, then we must all be about as stupid and about as brilliant.”

In a way, this syndrome may go hand-in-hand with the “if I’m not clearly the most brilliant student in the room, then I must be the dumbest.”

Imposter Syndrome

This syndrome is similar. My advisor talked to me and classmates (he was also my instructor for both semesters my first year) about how normal it is for a graduate student to come into grad school thinking, “Why am I here? I don’t feel like I belong. If they knew who I really was — all my weakness and dumb thoughts — then they would kick me out.” I still have this syndrome and I’m almost graduating!

I have found my best strategy for both syndromes is to talk to classmates and faculty when I’m feeling dense. They can either reassure me (which is addicting – so try not to over-use this strategy!) or share their own struggles and strategies.

When I am feeling overwhelmed, it’s easy to quickly get sucked down and it’s easy to suck other people down with me in a spiral of negativity. So it’s important to approach conversations with a view that this struggle is temporary, you can be brilliant just like everyone else (remember: growth mindset, not fixed mindset!), and that there are strategies to practice.

“For some definition of …”

I think the best thing I’ve learned in prosem this semester (our “introduction to the math ed world” course) is the importance of definitions.

Definitions in Mathematics

In math, definitions are important. Definitions can build on axioms or other definitions, but basic defined terms rely on “primitives” that are assumptions about the “basic facts.”

Note: In order to be perfectly precise, we would need to define everything. But definitions rely on words.  One word cannot be defined without relying on other words.  For those other words to be defined, more words need to be used. This is why we use primitive terms even though they are not at all precise. This means we have to be very careful about assumptions we make regarding primitive terms – sometimes we will assign properties to the primitives but unless that happens we can’t assume properties.

It’s how mathematicians set the stage to do math. In mathematics, being precise is everything. Some choices when creating definitions are based on conventions and others are based on strategy. A mathematician could arbitrarily assign any meaning to a symbol or term, but they need to communicate to others so paying attention to conventional use is important. (Although some of the most interesting mathematics results from mathematicians choosing to ignore certain assumptions or to change elements of definitions.) Mathematicians also make strategic choices to make later theorems weaker or stronger and (often) easier or harder, respectively, to prove. So, definitions are important in mathematics to communicate and to set up later arguments.

Definitions in Mathematics Education

Choices about definitions of terms in Mathematics Education are similar to those in Mathematics. Still, a math ed researcher needs to make decisions based on convention and strategy, to communicate and support arguments. A difference is that it is more difficult to be precise in language, when that language has to describe the complexities of life, teaching, and human beings. It is still important to reflect on definitions and attempt to be precise, however. It is difficult to look at prior research that has “taken-as-shared” meanings of terms or processes.

For example, in my reading, I see many studies that attempt to measure the effects of technology in teaching. As one example, many studies measure the effects of “use of graphing calculators to support student learning of functions.” But those studies do not always describe how the graphing calculators were used: Did students use them only to calculate values or graph functions? Did students use graphing calculators to explore the parameters of functions, making and testing conjectures? Did students gather data with calculator-based rangers (CBRs) and the explain the effect of changes in the movement on the function? These are important differences.

Arguing without Defining

I have often found myself in heated “discussions” (and by “discussion,” I mean “argument”) that ultimately end when we realize we’re both fighting for the same idea, but using words in different ways. Taking the time to define terms is important in research and in real life!

Communication is so difficult in everyday life because we all come from different communities (even communities of one) where different norms are accepted as the only norms and where meanings remain unexamined. But, by examining our meanings, we can be more aware of others’ different meanings. This awareness can make communication easier and more satisfying, and can open up potential communities. (Well, I think so.)

Notice that I did not define terms in this blog post. Oops.