What is Math? What do Mathematicians do?

UPDATE – SPRING/SUMMER 2020 – During the start of the pandemic, I recorded a video for families titled “What is Math?”.  Too many adults and students I meet say “I can’t do math” or “I am not good at math” – When they say this, I think, ‘Are they basing this belief on the real definition of math – or on a misconception based on memorization and speed.”  If you prefer to get the idea of this post from a video instead of reading, click below.  Otherwise – keep reading for more details for the classroom.

I started teaching in 1991 when NSF funded reform curricula entered many schools around the nation.  My district started using IMP (someday I will write a blog post, “A love letter to IMP”) and for 13 years I taught math out of these texts.  Many times during my first years of teaching I would think “Oh, this is why math works”.  Here I was a math teacher and in some ways I was discovering math and what it means to be a mathematician for the first time.  I honestly was a bit pissed off about how I had been taught math throughout my entire education.  I received A’s in every math class I ever took, but I knew in my heart this was just because I was good at the system of school.  I was OK with not knowing why and just doing what they told me to do.  When my friend became frustrated when her questions of “Why does this work” were unsatisfactorily answered I would quietly think “she needs to let that go and just do what they say”.  My poor friend.  Math classes were not built for her then.  The sad thing is 30+ years later many still do not work for those like my friend.  I was lucky, my early years of teaching were full of best practices PD that I lovingly say brainwashed me into the way I would think about mathematics and what mathematicians do for the rest of my career.

For the last 20 years of my teaching career I’ve given the same homework assignment sometime during the first week of school.  One of my goals week one of school is to convince students that all of them, yes ALL, are mathematicians.  Many students enter my room not believing this.  In recent years there has been lots of people talking about this and many, many resources created to help students believe that they can get smarter in math.  Jo Boaler, Stanford and the YouCubed team have created some of the best resources with their week of Inspirational Mathematics.  We are lucky to live in the era of easy access to newly created resources we can immediately bring into our classroom.

Despite all the resources created recently,  I think there is something that most math teachers miss in their early messages to students about math.  In my experience many students – even those in HS – have a misconception of what mathematics is and what mathematicians do.  Once I clear up this misconception and give consistent messaging around our work in mathematics class, I’ve found that more students engage in the work of becoming mathematicians.  This messaging to students has ultimately changed me as a teacher and what I choose to put in front of them as they learn mathematics.

What is Math?  You need to do this week 1!

I have a couple of week 1 posts with a ton of views in the last year.  Tons of you like my group work 100# task for week 1 and have used my name tents to build relationships with students.  What follows though is maybe more important to me week 1.  This messaging follows through all school year.  It is how I set the tone for what the work of y classroom will look like all school year.  Here is how it plays out during the first week of school.  I’ll give you a script of sorts.

  • daily practiceOn day 1 or 2 I assign this homework (HW #1 What is Math I call them daily practice assignments vs homwork) to my students.  “Tonight I want you to complete this homework.  There are 2 rules.  You can not use your phones or computers to look up the answers.  I just want to know your real thoughts.  Also, you need to ask and record a question of at least 2 people that are not at South HS.  Ask your little sister, call your grandma or uncle, go across the street and ask your nosy neighbor.   Tomorrow I will ask you to report back what you heard.”
  • The next day I start class with these GIFS of babies eating lemons for the first time.  (btw, there are tons of these gifs out there – just google ‘babies eating lemon gif’.  who knew this was a thing.)

    After putting these GIFs up I say, “What do these 3 GIF’s have in common?  What do you think they have to do with mathematics?”  I just listen to their ideas and tell them I will answer this in a few minutes.

  • After showing these GIFS, I then have students stand up and compare answers with math isa partner across the room (I LOVE stand & talks – I will post on these soon).  I then ask the class to shout out 20 things math is as I record what they say on the smart board.  I’ve done this in 5 classes a year for 15+ years and every year the responses are consistent.  I chart everything, even when they say, “Math is hard”.  Most of what students say is on skills or concepts (fractions, shapes, adding…) in math or names of math classes (geometry, calculus).  When I’ve charted a ton of things I then say,Yes, math is all of these things.  Today I am going to give you a common definition we will all use for math the entire school year.  Please record this in your notebook.  Math is…”math def
  • I then say, “I have one rule in my classroom.  For the entire 50 minutes everyday we will do math.  We will act as mathematicians.  Anything that gets in the way of all of us being mathematicians I will be annoyed by.  So, we also need to know what what mathematicians do.  There are 3 things mathematicians do.  What do you think they are?”math ques 3 things
  • I then reveal each of the 3 things mathematicians do and say something like….“The first thing mathematicians do is notice patterns.  You have had the ability to notice patterns since birth.  You already are a mathematician.  Noticing patterns is an innate ability. baby lemon 1 Just like babies eating lemons always making similar faces.  No one teaches this face to them, they just do it.  In the same way you have always been a mathematician.  Everyone has the ability to notice patterns.  Everybody.   I will expect you to act as a mathematician everyday and work to notice.  The 2nd thing mathematicians do is describe patterns.  Everyday in this class you will describe what you notice.  You can describe it any way that makes sense to you.  Don’t worry about using the right words.  That’s may job to help you say what you noticed using the correct mathematical language.  Most of you arrived in math class already really good at these first two parts of being a mathematician.  I expect to see you doing this everyday in this class.  The 3rd thing mathematicians do is to generalize patterns.  This is the part a lot of us struggle with.  Don’t worry though, by the end of this school year you will be really good at this.  I’m good.  I promise this will be easy if you promise to keep noticing and describing.”  math 3 things
    I then say, “Everyday in math class you will be expected to act as a mathematician.  I expect you to notice & describe patterns & relationships.  I will help you with the academic language you need and also to generalize.  As your teacher, If I don’t put something in front of you to notice everyday, then you need to contact Mr. Aponte (our principal) and tell him I am not a good math teacher and I am not teaching you math.  Mathematicians notice, describe and generalize.”
  • I then post posters of this in the hall outside my door, and in a couple of places in my room.  This amazing Minnesota math blogger, Greta Bergman, came to one of my trainings and heard my start of the year message about the definition of math and she turned it into a great pdf poster.  You can see her blog post about this HERE or can download her lovely poster it directly HERE:  Math is patterns posterposter
  • After defining mathematics and what mathematicians do, I then say “Today, we will start practicing being mathematicians again after a long summer.  Let’s begin with this number sequence.  WHAT COMES NEXT? What do you notice?”2 4 6I wait silently for a while  and often say ‘no hands up’.  My goal is to remove speed as a value in my classroom.  After a bit I say, “What number comes next?”  The most usual answer is ‘8 & 10’.  I say, “Describe why.”  Students do as I annotate their thinking.  I then say, “Who saw that pattern?”  Most hands raise and I say, “Great, see you are mathematicians you noticed and described a pattern.  How could we generalize this?”  I then go on to generalize what they say in words and also as ‘2n’ if ‘n’ is the term number.  We also add the academic word ‘even numbers’ to this.  I then say, “Did anyone get something different than 8 & 10 to fill in the blanks?”.  I always have multiple hands go up.  The most normal alternate answer is ’10 & 16′.  I then have students describe where those came from and say, “Many times in this class there will be multiple correct answers.  If you can describe the pattern you notice you are a mathematician.  Sometimes I select one particular answer for us to focus on so we can learn the mathematics in the advanced algebra course.  If I don’t use the pattern you noticed first, don’t worry, you are not wrong, we will just focus on something on a different one for the day.”
  • After giving a version of this messaging for years I watched Annie Fetter’s Ignite talk titled “Ever notice what they wonder”.  I can’t believe there are still people who have not see this, but if you have not, stop and watch it now.  It’s only 5 minutes long.  Annie’s video gave a name to a messaging I’ve done for years prior.  My favorite classroom question was already “What do you notice?”.  I have since added other favorite questions “What do you wonder?”  and my new favorite “What else?”

fav questions

  • For the next 10 days of school I put something in front of students to notice.  Every day I repeat the definition of math and the 3 things mathematicians do.  Every day I ask, “What do you notice?” and give students space to notice.   I walk around the room and catch students noticing and I say “you are a mathematician!  You described what you noticed…”.  My goal is to catch every student acting as a mathematician and name it when I see it.  I want to change how they see themselves as mathematicians.
  • This messaging is also pretty much the sum total of my behavior management plan for my classroom.  Anything that gets in the way of all of us acting as mathematicians for the full 50 minutes is annoying to me.  Tardies therefore are annoying.  Being on social media is annoying.  Sleeping is annoying.  Any disengagement is annoying.  I tell students anything outside of being a mathematician for 50 minutes is not OK with me.
  • My messaging about the definition of math also transforms my planning as a teacher.  I must have multiple opportunities every class period for students to notice and wonder.  I often work with principals and others who observe teachers and I tell them – any time you observe a math lesson, if the teacher does not ask some form of “What do you notice?” of students, then the math lesson is not as good as it could be.  They are not empowering students to be mathematicians and own their own learning of math.  When you plan a lesson this school year, I encourage you to start with this question, ‘What mathematics will I ask students to notice today?’ and/or ‘How will I set up this math concept so my students have something to notice?’.  
  • A word of caution.  It is easy to ask students to always practice what I am talking about using number sequences.  Be careful of this.  If you do this than student’s powerful ability to notice patterns will tell them that math is only about number sequences.  For this reason, I mix things up the first few weeks.  I use all kinds of things.  Some are related to the content we are studying and some are silly things I find on twitter or Facebook.  Here are just a few examples.  Most use up 5-10 minutes of class maximum.  My goal is to convince students that they are mathematicians and they do have the ability to notice and wonder.
  1. First a visual I got from Christopher Danielson.  What do you notice?  How many avocados?  How did you see that number of avocados?avacados
  2. I found this sign in some tweet.  I ask, “what do you notice? ” coffee wordsI then doctor the photo and ask ” How much would ‘Hello, one small coffee’ cost?”  “Justify this cost”.
  3. I put up the first picture of the green shirt Kristen Fouss tweeted out and ask, “What do you notice?”  I follow up with Jim Pardun’s question, “If the pattern continues, how many tickets could you get for $40?” and have partners work on justifying their amount on white boards.
  4. I give student partners the following 3 graphs & equations in factored form and ask, “What do you notice?”  what do you noticeI ask students, “How could we find the coordinates of the vertex if all we have is an equation in factored form?”  It does not take long for students to notice that the vertex is in the middle of the 2 x-intercepts/roots.  In a 10-minute discussion students discover a visual method for finding the vertex without having to convert the form of the equation.  vertex
  5. power funcThis visual has I lot of patterns.  I made it from screen shots from this cool animated factorization site.  Can you see the patterns in the rows and columns?   I use this at the start of my unit on exponential and power functions.  Lots to notice and describe….and generalize.  Lots of chances to practice being a mathematician.
  6. c then cIf you are not following the hashtags #cthenc (contemplate then calculate) and #connecingreps (connecting representations) – 2 instructional routines from the free online New Visions math curriculum – on twitter, then you are missing out on tons of things to have students look at to notice, wonder and then calculate.
  7. which one 2Any ‘Which one doesn’t belong?’ is great to use to practice being mathematicians.
  8. asphalt 1I found this picture posted by a pet-loving relative on facebook.  I covered up some of the information.  Ask students to notice & wonder.  You can’t help but try and make some meaning out of this.  I then follow up with more information using one of theses photos at a time.

    When I reveal the 2nd photo (2 pieces of data), students want to make this linear and find a rate of change – even without me telling them too.  The third photo is the reveal later and students are annoyed (which I love)….but they are acting as mathematicians (and I name it for them) and thinking critically about making sense out of data.

  9. geo scenarioOne huge thing I’ve learned from Annie Fetter, Max Ray and all the other amazing people at The Math Forum is to turn tasks into scenarios (remove all question and some details) and ask what do you notice & what do you wonder.  geo originalHere is the original task.  In my experience with 5 minutes of noticing and wondering,  Students solve all these questions without me giving them first.  They also see more things than this.  They ask/say, “are these right triangles?”, “I wonder if we can use the Pythagorean theorem.”  I give information like, ‘these triangles are similar’ as students notice the details.  I give the question to solve after they’ve said it first.  Try it for yourself.  Create ‘scenarios’ instead of questions.  See how that changes your students ability to see themselves as mathematicians.
  10. visual patternAny pattern from Visual Patterns would be amazing.
Noticing Patterns Goes Wrong

One of my main messaging day one, week one, month one is that ‘We are all mathematicians.  We all have the power to notice, describe and generalize patterns. You have all had this ability since birth.’  One way to be convinced of this is based on how powerful students misconceptions in math are.  I would argue most if not all misconceptions come from acting as a mathematician and noticing patterns given to them by their teachers.  Here is an example of what I mean.  equal signTwenty years ago my friend, Terry Wyberg ( U of MN prof), told me that 30% of my HS students would get this problem wrong.  I did not believe him and tried it out.  He was wrong (only about the % – not the underlying problem), only about 20% of my HS students got this wrong, but OMG, 20% got it wrong.  20%.  20%.  No wonder my students struggled to solve math equations.  The most common answer was 19.  Students saw the equal sign as meaning ‘do whatever is on the left side of the equal sign’.

In the last 20 years there has been a ton written about improving students understanding of the concept of equality, but I find many teachers don’t know our part in creating this misconception.  Students often created their misconception of the equal sign from noticing the patterns in the work we gave them.  If most of our work for students look like these…

..than no wonder students created a misconception about the equal sign.  They simply had noticed that to solve math problems I just do what is on the left.

The good news about our students errors is they know how to notice patterns.  We need to see ours students assets, not their deficits.  The challenge then for us math teachers is to put things in front of our students to notice that don’t create misconceptions when our students notice patterns.  Since realizing this many years ago I am always writing equations in a variety of ways.  For example y=2x + 3 vs. 3 + 2x = y so that they don’t see linear equations as having to start with y=.

Here are a few other classic ways we – the teachers – create math misconceptions.

Google ‘area of triangle worksheet’ and take a look at the vast majority of problems given students.  No wonder students think ‘base’ means ‘bottom’ of the triangle.

area triangle

Be careful not to give examples that can give correct answers by doing the problem incorrectly.  Here are 2 classics….simplifying experessions

The teachers responsibility.

If you subscribe to the idea that ‘math is the study of patterns’ and that ‘mathematicians notice, describe and generalize patterns’ then this will change how you teach.  If we believe this than everyday we must plan lessons that allow students to act as mathematicians.  We must put something in front of our students to notice.  We must put something in front of our students to describe, to generalize.  How do we expect our students to become mathematicians if we do the noticing and describing for them.  My experience is that students feel empowered in their ability to do math when we let them act as mathematicians every single day in our classrooms.  This process of noticing, describing and generalizing becomes easier and easier for them.

A Challenge for the new school year.  
This school year I challenge you to start the school year by defining math and the role of of a mathematician.  Give your students the big picture and don’t start with the details.  Us math teachers tend to treat the definition of math as a set of details and don’t honor our students who need to see the big picture of what mathematics is to be successful.Kevin-parr-1
Additional Resources:
  • I’ve been reading Tracy Johnston Zager’s “Becoming the math teacher you wish you’d Had” book this summer and it is amazing.  You need to order it and read it today.  The first chapter of the book touches on the definition of math starting with her own mother’s definition.  There is more great stuff throughout this book to support what I’ve talked about above in much further detail.IMG_0489
  • Math as the science of patterns from the Mathematical Association of America (MAA)
  • Keith Devlin’s Mathematics as the Science of Patterns book. “Mathematicians now see their work as the study of patterns—real or imagined, visual or mental, arising from the natural world or from within the human mind.”   Using this basic definition as his central theme, Devlin explores the patterns of counting, measuring, reasoning, motion, shape, position, and prediction, revealing the powerful influence mathematics has over our perception of reality. Interweaving historical highlights and current developments, and using a minimum of formulas, Devlin celebrates the precision, purity, and elegance of mathematics.

math pattern quote

Follow-up:  My friend Laura Wagenman posted this great ‘Math is’ video on Facebook today.  Love when others find resources connected to my passions.  Give it a view.

Sara VanDerWerf

I am Sara Van Der Werf, a 24-year mathematics teacher in Minneapolis Public Schools. I have taught math in grades 7-12 as well as spent several years leading mathematics at the district office. I currently teach Advanced Algebra at South High School and I'm also the current President of the Minnesota Council of Teachers of Mathematics (MCTM). I am passionate about encouraging and connecting with mathematics teachers. I'd love to connect via twitter.  Join the community.  Tweet me @saravdwerf.

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