What does it mean to be a “mathematician”?
What does it mean to be a “mathematician”? What does it mean to identify oneself as such?
During the last twenty plus years as a mathematics educator, I feel as if I have thought a lot about these two questions. However, I learned recently that I have not thought about them nearly as much about them as I might have supposed.
During a recent book discussion, I was asked to consider the question, “Do you consider yourself to be a mathematician?” Without much thought, I raised my hand in the “no” group – a response which might surprise some people reading this and which also surprised a few of my colleagues in the room.
In the immediate and later discussion of this question that followed, I found myself challenged to articulate why I do not tend to identify myself as a “mathematician”. One idea that came to mind fairly readily is that I know that I associate this word with people whom we might also call “research [or ‘career’ or ‘professional’] mathematicians” – that is, people who make their living exploring mathematical ideas in (often) a formal way. I actually made an intentional career choice toward the end of my undergraduate program not to continue with graduate study in mathematics because I was very passionate about mathematics education in K-12. And, to be honest, I found that even though I was doing well in my mathematics classes (“grade-wise”), I had no desire to continue in what I felt would be even more abstract, theoretical mathematics. I believed that I was capable (as did the mathematics department); I simply chose not to do so.
However, I am engaged in an active community of mathematics educators who believe, as I firmly do, that ALL students should have equal access to mathematics as a discipline. We must stop marginalizing students (which happens in the vast majority of classrooms in many different, often unintended ways), and we must intentionally include, engage, and empower every student. Students should leave mathematics classrooms with rich understandings as well as belief in the value of their own thinking and the extent of their own potential. Students should develop mathematical habits of mind that they can apply anywhere, anytime, throughout the rest of their lives.
So, one might ask – as I was asked – does the description above not constitute the definition of a “mathematician”? Don’t we want all students, all people, to consider themselves mathematicians?
I found myself struggling to answer this question – simply from my own perspective, let alone from the perspective of every other human being. I commented that I would never tell a group of students that I don’t consider myself to be a mathematician. Still, I have explained to students many times that I don’t engage in mathematics exploration as my primary career focus, but that there are many people from all backgrounds who do! Without question, I want all students to believe that they can be “career mathematicians” or that they can choose any careers that interest them. And I also want them to believe that they can think mathematically, apply mathematical principles, and enjoy mathematics in countless ways in their own lives. But does this mean that I personally should identify as a mathematician?
In thinking more about this, I considered the word “teacher”. Many professional educators feel a bit bothered when members of the general public who do not work in schools seem to imply that anyone can be a “teacher” if you find a way to teach people something. Of course, in this sense, anyone *can* truly be a “teacher”. But we do mean something different when we talk about professionals who have committed years to developing as teachers, both through experience and through continued formal learning. There seems to be a parallel here with the word “mathematician”; perhaps this is why I am not sure that I feel comfortable identifying myself as a mathematician.
Further, I recognized after the book discussion that I have never felt as if identifying as a “mathematician” was out of reach for me; I was always encouraged by teachers, professors, family, and friends to pursue my mathematical interests. That is, it is entirely of my own choosing that I don’t identify this way (caveat: see the next paragraph). In contrast, for many people who have felt left out of mathematics, being able to finally identify this way – and to help young people to do so – could feel incredibly empowering, and I would never want to deny anyone that opportunity. I don’t think I would ever say to someone, “You are not a mathematician.” If a person feels like a “mathematician” because they are thinking mathematically, then more power to them – truly!
Another thought – since the idea of a “mathematician” has traditionally been intimidating to many people, perhaps I have avoided identifying myself this way in order not to appear as if I am placing myself in a position of power or privilege. But if this is at all the case, it has been entirely subconscious!
So, what has this resolved for me? Well… let’s say that we do define “mathematician” as any person who thinks mathematically and applies mathematical principles for many purposes on a daily basis. If that is the case, then I can agree to identify myself that way; after all, proceeding from a definition is a very mathematical thing to do! But perhaps, alternately, we could consider the use of some adjectives to accompany the word – “everyday”, “career" – what are other possibilities? How can we include and celebrate everyone equitably in this discipline while allowing people to choose words that they feel best identify themselves? So, are YOU a mathematician? What are your thoughts?
How many of 13? (or, Which group do you want to be in?)
Your students will have about 13 math teachers from K-12. Will you be one who truly helps them develops growth mindsets?
Most of us have heard about research (granted – it is based on standardized test scores) that says that if a student has ineffective teachers in mathematics for three years in a row, it is nearly impossible for the student ever to “catch up”. I have been thinking about this same idea – not focused on test scores – but in the context of students’ mindsets in mathematics class (though I would argue that test scores are affected by mindsets). What subtle or direct actions and words encourage all students equally, rather than favoring some and “rescuing” others?* How many of 13 teachers does it take to convince a student that he is not capable of achieving much in mathematics? How many of 13 does it take to convince a student otherwise? Which group do you want to be in?
*Here are 10 “easy” teacher moves — that any of us can use at any time — that are very powerful in terms of helping students to believe in their own mathematical thinking (and not rely on the teacher as the ultimate and only authority) — but we have to be consistent about these for them to be effective!
Choose students randomly to share their thinking — every day, as often as possible (e.g., names on popsicle sticks, index cards, or an app); remind students that you are doing this because everyone’s thinking is equally valuable and because we learn from each other
If a student isn’t sure what to share at a given moment, ask the student to call on a classmate with a hand raised for a thought or an idea (I prefer not to say “for help”), and THEN come BACK to the first student and ask this student to explain what was said — because it’s important that the first student learns in this moment
When a student shares an idea, DON’T show whether YOU agree or disagree — ask the class to show thumbs up/down/sideways to indicate their agreement/disagreement/uncertainty with the idea; then ask the student to call on someone who agrees/disagrees to ask why (and allow them to continue the discussion as appropriate)
When you ask a question that requires some extra thinking, ask students to think on their own for a moment (or longer) and then to turn and talk with a partner for an appropriate amount of time before you invite whole group discussion
Ask questions like: “What do you notice?” “What questions could we ask about this?” “What do you see?” “How are you thinking about this?”
Don’t allow just a single word or number as an an answer from a student, and if you start a sentence for a student to finish, have the student restate the entire idea afterward (e.g., rather than saying, “The slope of this line is…?”, say, “Tell me what you know about the slope of this line” — and require more of an answer than “It’s 6”… rather: “The slope of the line is 6 because when the x-value increases by 1, the y-value increases by 6”)
Expect students to use visual/physical representations whenever possible, especially when new ideas are developing (err on the side of having students create them even after you think they "shouldn’t need" them) — these representations are a key part of mathematics, and they never go away!
Emphasize that our brains expand and develop when we make mistakes and when we are thinking deeply
Create, and have students help to create, written/visual records of students’ thinking and learning (both in individual notebooks/digital files and in classroom references for all)
Use the Standards for Mathematical Practice as a key basis for our planning for ALL students, every day
This list could go on… The more we can consistently utilize the teacher moves above, the more likely it will be that ALL students develop autonomy and depth of understanding. How many of 13? Which group do you want to be in?
Math in the Movies: Why does Lady Bird throw out her math teacher’s grade book, and what can we do about it?
Having seen the wonderful movie Lady Bird this week (not at all connected with First Lady “Lady Bird” Johnson), I have been thinking about the portrayal of the main character’s high school math class and how it parallels the feelings of so many people about school mathematics. In the scenes involving the class (set in the year 2002), the very pleasant teacher is seen lecturing from an overhead, with students sitting in rows taking notes. There is no real engagement for students in what is happening, and they look incredibly bored. Some students are identified by the well-meaning teacher as being good at math, while others (including Lady Bird) are essentially told the opposite. And Lady Bird questions why she is not good at math even though her father and brother are, while her friend suggests that perhaps she takes after her mom in this regard. The feelings of inefficacy and frustration are palpable. When Lady Bird has a chance, she takes the teacher’s grade book and throws it away in an outside trash can.
There is a reason that this portrayal is a stereotype for mathematics classes. We as educators have a responsibility to ensure not only that we are “providing” opportunity to all students but also (more importantly) that we are actively and intentionally including ALL students in questions, discussion, and mathematical activity and showing them that we believe they are not only capable but vital in our classroom learning communities. It is very easy to leave too many students behind because we say they are choosing to withdraw or that they just don’t come to us with enough knowledge. It is easy to continue the cycle of telling these students how to get through every step of a problem and to not ask them what and how they are thinking. When we do this on a regular basis, we are furthering the inequity that already exists in so many ways in our society. Let us make active choices to create equity in our classrooms instead – we have the power to make change for students and for the future.
Reflection: What it means to be thankful for teachers
I am thankful every day for teachers who think critically and reflectively about students’ learning.
I am thankful for teachers who listen to students without just hearing.
I am thankful for teachers who make professional decisions based on what they know, see, and hear from students.
I am thankful for teachers who learn from each other and from their students.
I am thankful for teachers who see and respond to students’ emotions and experiences.
I am thankful for teachers who communicate with students’ families.
I am thankful for teachers who model respect, compassion, and justice.
I am thankful for teachers who are human beings – because no computer and no set of data could do what is listed above.
Thank you, teachers! You are beyond value! And thank you to all who support teachers!
A challenge: Are you ready to break out of the “test scores” box?
Are you tired of feeling beaten down by test scores? Are you tired of always having to think about “fitting in” a whole laundry list of standards during the year? If you answered “yes” to either of these questions, then I offer a challenge to you – both teachers and administrators. There is real, growing (and in some cases long-term) evidence* that intentional instruction that focuses on the big ideas of the grade level or course, and that focuses on habits of mind like those in the Standards for Mathematical Practice, can deepen students’ learning and understanding while producing laudable test data as well. Of course, “intentional” instruction includes ongoing reflection on the part of the teacher and a purposeful inclusion of ALL students in EVERY element of these classroom practices. As teachers, it means that our primary task in class is to really LISTEN to students and to show them that we honestly believe that their thinking is valuable and important – no matter how sophisticated at that given moment.
Yes, this is a risk – but perhaps you are ready to take that risk! Possible stumbling blocks can include not actually getting to all of the big ideas of the course (but you have months!) and not truly including all students in the same way (but we really just have to work at this!). Teachers and administrators need to start talking more about making this leap together, because, oh, the rewards can be terrific in so many respects. I am glad to talk more about this with anyone! And I would love to hear your stories and to start to create our own NE Ohio body of evidence!
*Some examples of research evidence:
https://www2.ed.gov/programs/nclbbrs/math.pdf
https://www.youcubed.org/evidence/research-articles/
Promoting equity in mathematics classrooms
A reflection on what we can do every day to build a better, more equitable classroom and world
As this school year begins, I feel a more profound sense of responsibility and humility as an educator than I have felt in many years. We begin this year facing, in some sense, a different world than we faced at the start of last year – but perhaps the world is not as different as our new awareness of its complexity may be. Perhaps our collective awareness is what has been most changed during the last several months, because teachers deal with extremely complex issues every day of the year.
More than ever, we must acknowledge our opportunity – and obligation – as educators to provide a safe, respect-filled, thoughtful learning environment for EVERY student. We must demonstrate unequivocally that we equally value the thinking and potential of every student, through our words, actions, practices, and policies.
How can we do so in a mathematics classroom? Are you committed to equity, and are you ready to show it?
- We can stop – right now – using the language “high” and “low” to describe students. (Achievement on a given assessment may be high or low, but these terms should not apply to students. What do we really mean when we say “high” or “low”??)
- We can promote all students’ equal participation in class by using random name drawing on a frequent and consistent basis.
- We can promote collaboration and respect among students by asking them to listen to each other and to respond to (or re-explain) what others have said.
- We can listen – really listen – to students’ thinking and questions, not just listen for right answers and then move on.
- We can stop encouraging fast thinking and fast answers – slow down, and allow all students to think – really think – about a given question or idea.
- We can celebrate mistakes, because brain science actually shows that our brains grow when we make mistakes – more so than if we do not make mistakes.
- We can ask questions rather than telling – see the article “Never say anything a kid can say”.
- We can stop using the word “math” when we mean “computation” and acknowledge that mathematics is what the Standards for Mathematical Practice describe – a dynamic, multi-faceted, open discipline in which every learner has equal opportunity to learn and grow.
Resources to help you initiate equity in the classroom:
- Standards for Mathematical Practice signs, gr. 2-12 (student-friendly, easy to read across the room)
- Standards for Mathematical Practice signs, gr. K-1 (student-friendly, easy to read across the room)
- Jo Boaler’s 7 Positive Norms to Encourage in Math Class
Suggestions for new standards in grades 1-6
I invite you to view the following linked files to explore my suggestions for new standards across grades 1-6 that I think would go a long way in helping students to develop deeper understanding…
Part/part/total as the basis for addition/subtraction in grades 1-3
Decomposition of whole numbers in various ways in grades 1-3
Fraction/decimal standards (restructured) across grades 3-6
The way NOT to get better at basketball — or math
Imagine… a basketball coach (let’s say Coach K at Duke) and his team are in the gym. Coach K starts practice by demonstrating a new skill for his team to learn. He asks the two most experienced players on the team to demonstrate it again while the others watch. Then he tells all of the players to take the floor and try the skill. When Coach sees one of the players doing something wrong, he takes the ball from the player, demonstrates the skill again, and then hands the ball back and walks away. After about 15 minutes, he sends most of the team off the court and has his two most experienced players practice the skill for about 10 more minutes, with everyone else watching. Coach K repeats this process with other skills for the rest of practice.
Of course, we know that Coach K would never actually conduct practice like this. Coach K knows well that every player, regardless of experience level, has to be on the court for the whole practice, working on every skill and every part of the game in order to become a better basketball player. He knows that the players with less experience aren’t going to get better by mostly watching.
As mathematics teachers, we have to guard against the first scenario above. How often do we as teachers actually do/talk about more mathematics in class than any of our students? How often do we allow many of the students to sit and watch while a few confident ones do most of the thinking? It is our job to be sure that every student is as engaged in doing and talking about mathematics as possible – and it is our job to be continually checking their understanding as they work. I think Coach K would be proud of this kind of math class.
“I want THAT for my child…” — Creating positive, deep learning experiences in mathematics
Positive, powerful learning experiences for students should be a primary goal of education. When we see students enjoying learning without significant anxiety — when we see students demonstrating deep learning — we get excited. “This is what we want for our students.” “I want that for my child.” What a joy it is to see the potential in young people as they shape their futures in school.
On the other hand, it is easy to be disheartened when we see students consistently feeling frustrated or bored by school experiences. “I don’t want that.” “We can’t let that happen for our children.”
I want everyone to be able to experience the joy that I have seen hundreds of times in the faces of students who have recognized their own deep learning in mathematics. By “deep”, I don’t mean just getting a series of right answers — I mean feeling that their own understanding has been changed substantively — feeling that they have confidence in solving new problems that are not just like 20 recent exercises. That is, we need to lessen mathematics anxiety while deepening learning.
We can create these deep experiences for every learner. It is absolutely possible. But we as adults also need to recognize what this looks and feels like so that we can work together to provide these opportunities for students. I want to work to create these kinds of positive learning experiences in mathematics for educators and the public at large. I urge others to join me.
When we have powerful learning experiences ourselves, then we can collaborate to provide the intensive, ongoing support and structures for ALL teachers so that they can foster these same opportunities for ALL students.
Once we see what is possible, there is no turning back!
Common Core State Standards and instructional strategies: Not the same things
I have been thinking today about the idea that much of the confusion that parents and community members have when they cite seemingly strange examples of problems related to the Common Core State Standards in Mathematics (CCSS-M) may have to do with the difference between end goals for student learning and the instructional strategies that we may use to reach these goals.
I should add, before proceeding, that there are two types of goals for student learning in the CCCS-M. We have the Standards for Mathematical Content (the “math”, so to speak) and the Standards for Mathematical Practice (the “habits of mind” that support mathematical thinking and problem-solving throughout our lives). Often, effective instructional strategies can relate very closely to the goals in the Standards for Mathematical Practice, so we as teachers need to try to be very clear about what the goals of any given assignment are (and, perhaps, what they are not). This alone may be helpful in easing some frustration on the part of parents.
One type of example that I often see online is a problem where the student is asked to show or explain a way to get to a given answer. Parents are often frustrated when they feel that their children are required to provide long, complicated explanations or diagrams when there are “quick and easy” ways to find the answers to such problems. It is important to recognize, however, that often these explanations and diagrams do not need to be overly complicated to demonstrate a child’s understanding — precision and conciseness are still important in mathematics! Also, that explanation or diagram is often simply an extension of an instructional strategy that has been employed in class to help scaffold a child’s understanding of a concept or a child’s fluency with a skill.
Again, it is important to point out that the Standards for Mathematical Practice include goals such as representing a problem or real situation in different ways for different purposes, justifying one’s thinking, understanding another’s thinking, recognizing and using particular patterns or structures within problems, and, in general, making sense of mathematics. So, sometimes assessing students’ movement toward these goals will require teachers to assign tasks with more complex responses. Still, parents and community members need to realize that these goals also help students to move toward fluency and efficiency that are built on understanding and not just memorization — we want the general population to feel they understand, rather than shy away from, mathematics.