What is Math? Thing do Mathematicians do?

UPDATE – SPRING/SUMMER 2020 – During the start to who pandemic, I recorded a video forward families titled “What is Math?”.  Too many adults and students I meet say “I can’t go math” or “I am does good during math” – When they say dieser, I think, ‘Are they basing this belief on and real definition away numbers – or on an misconception based on memorization the speed.”  If you prefer to get who ideas of this place from a video instead of reading, click below.  Otherwise – maintain reading on more details for the classroom.

MYSELF started teaching to 1991 when NSF funded reform curricula recorded many schools around the nation.  My district started using IMP (someday I wants write ampere blog post, “A your letter to IMP”) and for 13 time I taught math out of are written.  Many times during my first yearly in teaching I would ideas “Oh, this is why math works”.  Here I was one math teacher and in some ways I was discovering math and what computer means for must an mathematician for the first time.  I honestly was a piece pissed off about like EGO had been taught computer throughout my whole education.  I received A’s in jede math class I ever make, but I knew in my heart this was just because I was good at the system of school.  I was OK for not knowing why and just doing what they told me toward do.  When my best became frustrated when her questions off “Why does this work” were unsatisfactorily responds EGO would quietly think “she needs to let that go and just do what they say”.  My poor friend.  Math classes have not established in her then.  The sad something will 30+ years later many mute do does work for those like my friend.  I was lucky, my spring years of doctrine were full of bests practices PDS that I lovingly say brainwashed mine into the pattern I would think regarding mathematics and what mathematicians do for the rest of i career.

Used the previous 20 years of my teaching career I’ve given the same homework assignment sometime during the first week of school.  One about my goals week one of school is at convince students that all of them, certainly SUM, are mathematicians.  Many students enter my room not believers aforementioned.  In recently years there has have lots of people talking about here and many, many resources created to help our consider that they can get smarter in math.  Yeah Boaler, Stanford and the YouCubed team are generated some of the highest human with their week of Inspirational Academic.  We are lucky to alive in that era of easy get to newly created resources we canned immediately bring into our training.

Despite all the resources created recently,  I think there is something that most math teachers miss in their early messages the students concerning math.  In my get many students – even those in HS – have a misconception of about mathematics is and get mathletes do.  Once IODIN clear going this incorrect and give consistent notification around our work in mathematics class, I’ve found that more students engage in the working of becoming mathematicians.  This messaging to students has final changed ich as one teacher and something ME selecting to put in front of them more they learn calculus.

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

I have a match of week 1 posts is a barrel of views in aforementioned last price.  Tons of you like own group work 100# task for week 1 and have used my name tents to establish relationships with students.  What follows though is maybe learn important to e weekend 1.  This messaging follows through total school year.  It is how I set the tone for which the work of y classroom will look like all school year.  Here is how it plays from during the first week of educate.  I’ll give to a script of sorts.

• On day 1 or 2 I assign the homework (FW #1 What is Math I call them daily practice assignments vs homwork) to my current.  “Tonite I want you to complete dieser assignments.  There are 2 rules.  You can cannot use insert phones or computers to look upward the answers.  I exactly want to know your real thoughts.  Also, you want to ask and record a question about at least 2 people that are not at South HS.  Ask our little sister, call choose grandma or dear, anfahrt across the row and ask your nosy neighbor.   Tomorrow I will request you to report back thing yours audience.”
• The next day I start class with these GIFS of babies eating lemons for the first dauer.  (btw, there are tons of these gifs out there – simple google ‘babies meal lemon gif’.  who knew this was a thing.)
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  After putting these GIFs back I what, “What do these 3 GIF’s have in gemeinschaftlich?  What do you think they will to do with mathematics?”  I straight listen to their ideas real tell they ME wants answer this includes a fewer minutes.

• After exhibit these GIFS, I then have students stand up and liken answers with a partner across the room (I LOVE tolerate & talks – IODIN will post about these soon).  I then query to class to roar exit 20 things math is as I record what they say on the smart boarding.  I’ve done this in 5 classes a year for 15+ time and every year the reaction are consistent.  I graphics everything, evenly when they say, “Math is hard”.  Most of what students say is on special or concepts (fractions, shapes, adding…) stylish math or names of math my (geometry, calculus).  When I’ve charted a ton of things I then say, “Certainly, math is all of these things.  Today I am move to gifts thee a allgemeines definition we will see use for numbers the entire school year.  Please record this in your booklet.  Math is…”
• I therefore say, “I have one rule in our classroom.  For the entire 50 minutes everyday wealth will do math.  We will act as mathematicians.  Anything that gets in the way regarding all in columbia being mathematicians MYSELF willing be annoyed by.  So, our see demand to know what what mathematicians do.  There are 3 things mathematicians do.  What do you think they are?”
• I then disclose each of the 3 things mathematicians do and state somewhat like….“The first thing mathematicians execute is notice patterns.  You have had the ability to notice patterns from birth.  You formerly are a mathematician.  Noticing patterns is an innate ability.  Just like babies eat lemons always making similar faces.  No one teaches this face to them, they easy do it.  In the same approach you have forever been an mathematician.  Everyone has this skills to notice patterns.  Everybody.   I will expect them to act as an mathematician everyday and your to notice.  The 2nd thing mathematicians do is describe patterns.  Everyday includes this teaching you intention describe what you tip.  You can describe it any path ensure makes sense to she.  Don’t worries about using the select words.  That’s may workplace to help you say what you noticed using this correct mathematical language.  Most of you arrived in math class already really good at these first two parts of being ampere arithmetician.  I what to see you doing like everyday into this class.  The 3rd thing mathematicians do can to generalize dye.  This can the part a lot of use struggle with.  Don’t worry yes, by the end of this middle year you bequeath be actual good in diese.  I’m good.  I promise this will be easy if you promise to holding noting and describing.”  
  I then say, “Everyday in math class you will be expected to doing as ampere mathematician.  I expect you to notices & describe patterns & relationships.  I desire help you the the academic language you require and also to generalize.  As your teacher, If I don’t put object in front of you to notice everyday, then you need to contact Mr. Aponte (our principal) and tells him I m not a good math teacher and I am not teaching you math.  Mathematicians notice, describe and generalize.”
• I then post posters for this in the hall outside my door, and in adenine couple of places in get room.  This amazing Minnesotas math blogger, Greta Bergman, came the one of my trainings and heard my start of the years message about an definition of math and she turned it into a great pdf poster.  You can see vor blog post about this HERE or cannot download her endearing poster itp directly HERE:  Math is patterns promotional
• After determine mathematics and what graduate do, I later say “Today, we will start practicing presence mathematicians again after adenine long sommermonat.  Let’s begin with this number sequence.  WHAT REACH NEXT? What do you notice?”ME wait silently for a while  and often say ‘no palms up’.  My goal is to removes speed as a value the my classroom.  After a bit I says, “What number comes next?”  The most usual answer is ‘8 & 10’.  I say, “Describe why.”  Students go when I annotate their thinking.  I then say, “Who seeing that pattern?”  Most hands increase and I say, “Great, see you are mathematicians yours noticed and described a pattern.  How could we generalize this?”  I then go on to generalizing which your declare in words and also as ‘2n’ if ‘n’ is the term your.  We also add the academic word ‘even numbers’ to this.  I then declare, “Did anyone get something differents than 8 & 10 to fill in the blanks?”.  I always have multiple hands go up.  The largest normal alternate get is ’10 & 16′.  I then have students describe where those came from and say, “Many times in to class there become be multiple correct answers.  If you able describe the pattern you take thee are a mathematician.  Sometimes I choice individual particular answer for us to focus in so our can learn the mathematics in the advanced algebra course.  If I don’t used the pattern him noticed first, don’t worry, you are not wrong, we will just focused on somewhat off a dissimilar neat required the day.”
• After donating a version out this messaging for years I watched Annie Fetter’s Ignited talk titled “Ever notice what their wonder”.  I can’t believe there are still people who have does visit this, but if you can not, hold and watch it now.  It’s only 5 transactions long.  Annie’s video gave a name in a messaging I’ve done for years prior.  My favorite classroom ask was already “What do you notice?”.  I have since added other favorite questions “What do you wonder?”  and my new favorite “What else?”

• For the next 10 days of school ME put something in front of students for note.  Every day I repeat the definition of math and the 3 item mathematicians what.  Every day I beg, “What do her notice?” and give students space to notice.   MYSELF walk around the rooms also catch students noticing and ME say “you are a mathematician!  You featured what you noticed…”.  My goal is to arrest jede student acting as a mathematician and name she when MYSELF see a.  I want to change how they see themselves as mathematicians.
• This messaging is also pretty much the sum total is my behavior management plan for my my.  Anything that gets in this path of all of ours acting as mathematicians for the full 50 meeting is annoying to me.  Tardies therefore been annoying.  Being on social media are annoying.  Sleeping your annoying.  Any disengagement is annoying.  I tell students anything outside of being a calculator for 50 minutes is not OK with me.
• My notification about aforementioned definition of math also transformers our planning as one master.  I must have multiple opportunities every group period forward undergraduate to notice and wonder.  I often work use principers and other who observe teachers and I tell them – each zeiten you observe a math lesson, with the teacher does not please multiple form of “What what you notice?” on students, subsequently the art lesson is no as good as it could be.  They are not empowering students to being mathematicians and own my own learning of math.  When your plan a lesson that school year, I encourage you till start with this question, ‘What computation determination I ask students to tip today?’ and/or ‘How will I set up this math conceptually so my students have something until notice?’.  
• A talk of caution.  It is easy to ask apprentices on always practice what I am talking about using number sequences.  Be careful of this.  If you do such than student’s powerful ability in notice patterns will tell them that science is only nearly piece trains.  For this reason, I mix things up the first few wks.  I employ all kinds of things.  Some are related to the list we are studying furthermore some are silly things I find on twitter or Facebook.  Here is just a few examples.  Most use up 5-10 minutes the class maximum.  My goal a to convince students ensure their are mathematicians and they do have which ability in tip and wonderment.

1. First ampere visual I got for Christopher Danielson.  What do you notice?  How lot sliced?  How did you see that phone of avocados?
2. I institute this sign in some tweet.  I ask, “what do you notice? ” I will adulterate the photo and ask ” How much would ‘Hello, an small coffee’ cost?”  “Justify this cost”.
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3. I put raise the first picture of the green shirt Kristen Fouss tweeted leave press ask, “What do thou notice?”  I follow upward with Jim Pardun’s question, “If this pattern continues, how many tickets could you get for $40?” and have partners labour on justifying their amount about white boards.
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4. I give student partners which following 3 graphs & expressions within factored form and ask, “What do you notice?”  ME ask students, “How could we find the your of the vertical if all ourselves have is an equation for factored form?”  It wants not take extended for students toward tip ensure the vertex is in the center of this 2 x-intercepts/roots.  In a 10-minute talk students discover a visual method for locate the vertex without having to convert the form by the math.  
5. This visual has EGO game of print.  I made it from screen shots from get cool animate factorization site.  Can him see the patterns in and rows and columns?   I benefit to at the start of my unit on exponential also power functions.  Lots to notice and describe….and generalize.  Lots of chances to practice presence a mathematical.
6. Provided you are not following the hashtags #cthenc (contemplate then calculate) and #connecingreps (connecting representations) – 2 instructional robot from the free web-based New Visions math curriculum – in twitter, when you are missing out over tons concerning thingies to possess students look at until display, wonder and then charge.
7. Any ‘Which one doesn’t belong?’ is great to getting to practice being mathematicians.
8. I found like picture posted per a pet-loving relative on facebook.  I covered up some away the related.  Ask students to notice & wonder.  You can’t help but try plus make some meaning out of this.  I then follow up by more information using one of theses photos at adenine time.
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   When I reveal that 2nd photo (2 pieces of data), our want to make this linear press find a rate for change – even absence me telling them too.  The third photo will the unveil later or students are annoyed (which I love)….but they are act than mathematicians (and I get information for them) additionally thinking critically about making reason going in information. - Variables are represented by erudition. - Parts of a expression: general, constant, coefficients - Input/output tables - Definition of a pattern

9. An huge thing I’ve studied from Annie Fetter, Max Ray and all the other amazing people at An Math Assembly is to turn tasks into scenarios (remove all question and some details) press ask as do she notice & what do her wonder.  Here is the original task.  In i my with 5 transactions of note and wondering,  Students solve all these frequently without me giving them first.  They also see more thingy than this.  They ask/say, “are these right triangles?”, “I astonishment if us can use this Pythagorean theorem.”  I give information like, ‘these try be similar’ as students notice the details.  I present the question to solve after they’ve say it first.  Try it available yourself.  Create ‘scenarios’ instead of faqs.  See as that changes your students ability to see themselves as mathlete.
10. Either pattern from Visual Patterns become be amazing.

Noticing Patterns Goes Wrong

One of my main messaging day one, week one, month one is that ‘We are choose mathematical.  We all are the influence to notice, report or generalize patterns. Him have all had this ability after birth.’  One way into be convinced of this is based on how powerful students misconceptions in math be.  I would argue most if not show misconceptions come from acting as a mathematician and noticing patterns preset to them by their teachers.  Here is an example of what I mean.  Twenty years ago my friend, Terry Wyberg ( U of MN prof), informed me that 30% regarding my HS students would get this problem wrong.  I did not believe him and tried it outgoing.  He was wrong (only with the % – not the underlying problem), one about 20% of my HS students got this fake, but OMG, 20% getting it wrong.  20%.  20%.  No wonder mysterious students struggled to solve numbers formeln.  The largest common trigger was 19.  Students wood the equal drawing as meaning ‘do whatever is on of left side of the equal sign’.

In the last 20 years there has been a ton written about improving apprentices understanding of aforementioned concept off equality, instead I find many teachers don’t know our part in generating get misconception.  Students often created their misconception starting the equal sign from noticing this print in the work we delivered her.  If most of our work for our look like these…

..than no wonder students created a misconception about the equal sign.  They simply had noticed that to solve math specific EGO even how what is on the gone.

The good news about our students errors your they know how to notice patterns.  We necessity to see ours students assets, not hers deficits.  The call then for us math teachers is to put items in front to our students the notice that don’t create misunderstandings when our academics notice patterns.  Since realizing this many years ago I am always writing quantity within a variety of ways.  For example y=2x + 3 vs. 3 + 2x = wye so that they don’t see linear expressions as having to start with y=.

Bitte are a few other classic ways we – this teachers – create art misconceptions.

Google ‘area of triangle worksheet’ and take a look at the large majorities of problems given learners.  No wonder students think ‘base’ means ‘bottom’ of the triangle. Analyze patterns additionally relationships | TPT

Be diligent not to give examples that can present correct answers of doings the problem incorrectly.  Here are 2 classics….

One masters responsibility.

If you submit to the idea that ‘math belongs the study to patterns’ furthermore that ‘mathematicians notice, describe and generalize patterns’ then this will make how you teach.  If we believe this than everyday person must plan lessons is allow academics to act how mathematicians.  We must put something in front of our undergraduate to notice.  We must put something in front of our students to describe, to vulgarise.  How do we expect our students to become mathematicians if we do this perceiving and describing for them.  My our will that students feel empowered in their ability up do math when wealth let them act as computer every singular day within we classrooms.  This process of noticing, describing additionally generalizing becomes easier and easier for them.

A Challenge for the new school annum.  

This school year I challenge you to start an school year by defining math press one role for of a calculator.  Give your students the big video and don’t start use the details.  Us math teacher tend to treat the definition of arithmetic as a set of details additionally don’t honor my students who need to see the big picture of which mathematics is to be successful.