Small Change, Big Difference part 1. Why you should eliminate ‘POINT’ from your vocabulary.

Hello my friends. we are moments away from the start of 2019. When you return to your classrooms this January it is difficult to make big changes to your classroom culture (these work best at the start of the school year), but NOW is the perfect time to implement small changes. My favorite changes are small things, easy to implement, that result in big differences in what you are doing in your classroom.

I asked math twitter (#MTBoS) to tell me about small changes they’ve made that resulted in big differences in their mathematics classrooms. I HIGHLY recommend you read the tons of responses I received (click the link below to read the entire thread). I promise something in this thread will inspire you.

Reading through all the responses, one thing that surprised me was how many responses had nothing specific to do with math, but had everything to do with building relationships with students. You can’t teach math until students trust you and believe that you believe in them. If you have a small change you love for the math classroom, please comment below.

My answer to this question!

I could write a book on small changes in the math classroom that have had a huge impact on student engagement and learning, but I thought I would start with something that will enhance your students conceptual understanding. I selected this small change as the first one to share with you based on observations in classrooms this fall and working with math teachers from all over the country. It is clear to me that some things I took for granted as normal in my own classroom and district are not well known by many secondary math teachers in other places. Small changes I and others in my distract made years ago are new to many teachers I’ve worked with. So what is this small change that can lead to enhanced conceptual understanding?

Let me begin with a question for you.

What is the goal of K-5 mathematics?

I’ve asked this question to lots of people. Most secondary math teachers shrug and are unsure. Many elementary math teachers and leaders say ‘number sense‘. While I agree number sense is the goal, the definition of the phrase is vastly different depending on who you ask. When many of us hear ‘number sense’ we think, ‘yes! I am doing this’. Unfortunately, some of us hear ‘number sense’ & view this phrase through a faulty lens of what we believe elementary math is, specifically to be good at elementary math is to be fast at arithmetic calculations – this is not what my elementary math leader friends mean when they say ‘number sense’.

So what is a good definition of ‘number sense’? From Wikipedia (they have all the answers, don’t they?) Gersten and Chard say number sense “refers to a child’s fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparisons. Here is another definition from Christina Tondevold’s website ‘Build Math Minds‘. (check her site out – she is passionate about not just knowing what number sense is, but how you actually build number sense in your students)

How I talk about the goal of K-5 mathematics…

I don’t say the goal of K-5 mathematics is ‘number sense’ since many have faulty definitions of this phrase. I’ve been blessed in my state to be surrounded by amazing elementary math thinkers and somewhere in my own learning on the topic in the last 10 years I was told the goal of elementary mathematics to build a flexible understanding of the base-10 number system. This is the phrase I now say and also informs my own practice.  The key words in this statement are ‘flexible‘ and ‘base-10‘. These words have informed my work in the last 10 years more than pretty much anything else. What does flexibility in the base-10 number system mean? Let me give you a few examples (this is literally just the tip of the tip of the iceberg that is K-5 math)…..

A student who is flexible within the base-10 number system would decompose the number 437.

437 = 400 + 30 + 7 = 4x100 + 3x10 + 7x1

Students may say ‘437 is 4 hundreds, 3 tens and 7 ones‘ using place value language.
Note: many of us secondary people are comfortable with this example – but not the next…students are flexible if they also see 437 in other ways…for example:

437 = 430 + 7 = 43x10 + 7

Flexible students might also say ‘437 is 43 tens and 7 ones‘. Students would represent this understanding using a variety of models.

See what ‘a flexible understanding of the base-ten number system’ also looks like in a few photos below:

So K-5 mathematics is all about building a flexible understanding of the base-10 numbers system. My question now is for every secondary and above mathematics teacher….What are YOU DOING in your own 6-12th grade classroom to build a flexible understanding of the base-ten number system?  Unfortunately for many of us we are doing very little. It is time to change this. You can begin this change with a small change that will give you immediate results!

Small Change #1:
Outlaw the word ‘point’ in your classroom.

When you return to your classrooms in January, post a few signs around the room that look like this:

Perhaps you don’t say anything about them for a few days. Put one on your classroom door and between classes while in the hallway greeting students (did you read all the ideas shared in the tweet above? you need to be in the hallway between classes) shrug your shoulders and say ‘I don’t like the word point, it is not allowed in this classroom.’ when they ask about the sign. Have fun with it. If you have fun with it, your students will too.

Note: Here is a sign you can download and use in your classroom:

Why should you outlaw the world ‘point’? If you don’t know why yet, then I recommend putting up the following problem for your students on the board:

Say to your class, “Quick, no calculators, add this as fast as you can, go!” Watch as tons of students sit there having no idea what to do and others furiously working to get common denominators. After 30 seconds, say to the class, “Stop. Look up here. <insert student name> read this out loud to the class.” As the student reads the expression you write 5.307 and listen to the chorus of ‘Ohs!’ ring out in the classroom. Your secondary students are very weak at connecting fractions to decimals.

You should eliminate the word ‘point’ in your classroom, because it is the easiest and fastest way to build students understanding of the base-10 number system past the decimal point in your classroom. Here is how I introduce this word being outlawed to my students.

Step 1. Write the following number on the board: 5.45. Invite your class to say the number out loud. When they say ‘Five point four five‘, say “No, that is not how you say it, who can say it in another way?” It will take little time for someone in the class to say ‘Five and forty-five hundredths‘. When they do, say “YES! From this time forward in this class we will never use the word ‘point’ in this classroom. All decimals will be read using place value language.”

Step 2: Write up 3-6 more decimal numbers and as a classroom practice this together. Read these chorally.

 3.2 ‘Three and two tenths’
7.009 ‘Seven and nine thousandths’
16.182 ‘Sixteen and one-hundred eighty two thousandths’ …and so on….

Step 3: After you say each number, write the appropriate fraction.

TIPS: I am not a fan of much on my classroom walls that is not student work, but one poster I think every secondary classroom should have is is a place value poster. As you say these numbers point at the poster. I recommend limiting most examples to the thousandths place. Also, no need to go to billions in your examples either. The goal here is to get your students using place value language.

When you introduce saying decimal numbers without using the word ‘point’ – especially if you do it in a fun and playful way – you will be surprised how easily your students will adapt to this. They start doing this immediately. They catch each other and hold each other accountable. Honestly, more than likely, the person who in your classroom who will have the hardest time with this will be YOU, the teacher. I have been doing this for years and I still find myself saying ‘point’ at times.

Not only will this be an easy transition for your students, you will notice your students understanding of place value improve without you having to do much else. I introduced this idea to middle school teachers at a training I was doing recently. We were training a unit on teaching calculations with decimals. One model in the unit was using the area model to visualize partial products in the expression 2.4 x 1.3.

Area Model

Teachers were worried in students first experience with decimals that students would not know what do do when they multiplied three-tenths and four-tenths. As we said these decimals aloud (without saying ‘point three’ and ‘point four’) teachers had an A-HA moment and wrote these as fractions. They believed their students could connect multiplying fractions to the decimal work. Teachers asked me, ‘does this always work?

Our teacher PD, where I had outlawed the word ‘point’ for teachers led to teachers themselves making connections between fractions and decimals and making themselves more flexible within the base-ten number system.

Once again, my challenge to you, is to make a small change in your classroom that will lead to a big difference in students conceptual understanding of the base-10 numbers system. Outlaw the word ‘point’ from your classroom.  One small note, I was religious about this change in my middle school classrooms. When I returned to teach HS three years ago, I got a little lax with this and noticed a difference immediately. This is for MS and HS. We all need to work on our flexibility within the base-10 number system.

In future posts I will reveal 8 additional small language changes I’ve made in my classroom that have resulted in big conceptual understanding differences in my students. For now, start 2019 by not saying ‘point’ and using place value language to name all decimals.

9 language changes that build conceptual understanding.

If you want a preview to my 9 changes to mathematical language to build conceptual understanding, enter your email below and I will connect you with a one pager you can download of 9 things to say in your classroom (with 9 things to stop saying).

BONUS MATERIAL for Secondary Math teachers. Calling all 6-12+ math teachers. it is time for us to improve our understanding of how to teach elementary mathematics (number sense). Many of us struggle knowing how to teach a student to subtract, who can’t subtract. If that is you, below are some easy ways you can improve your understanding of elementary education from the comfort of your classroom our couch. At minimum commit 34 minutes of your life to doing #1 below. Seriously. If you don’t have 34 minutes to work on your own practice then don’t you dare give your students another homework or project again. We need to learn and practice just like our students.

What can you do to learn about K-5 Mathematics?

  1. Graham Fletcher has five great videos (slightly more than 34 minutes total – don’t tell me you don’t have time to get smarter!) for the progression of learning mathematics in Elementary.  Watch all of these.  Ideally, watch one video with a teaching peer (How great would it be if you had a group of 2-5 teachers at a coffee shop from various grades to watch each video have this conversation). I recommend watching each video eat least twice. Listen for Graham saying things like ‘model’, ‘efficiency’, ‘flexible’, ‘context’, ‘let’s not rush them’, ‘let students explain’…..
    • The Progression of Early Number and Counting. (7 min 34 sec) Here are some terms to ‘google’ to learn more after watching the video.
      • Subatizing
      • Rote Counting
      • 1-1 correspondence
      • Cardinality
    • The Progression of Addition and Subtraction (7 min 20 sec) Here are some terms to ‘google’ to learn more after watching the video.
      • 5 Frames
      • Unitizing
      • 10 Frames
      • decompose numbers
      • partial sums
    • The Progression of Multiplication (5 min 56 sec) Here are some terms to ‘google’ to learn more after watching the video.
      • partition
      • array
      • relational thinking
      • area model
      • base 10 blocks (10 rods…)
      • partial products
    • The Progression of Division (7 min 49 sec) Here are some terms to ‘google’ to learn more after watching the video.
      • partitioning
      • base-10 blocks
      • area model
      • decompose numbers
      • concrete model to representation to written expression
      • fair share model
      • partial quotients
    • The Progression of Fractions. (7 min 26 sec) Here are some terms to ‘google’ to learn more after watching the video.
      • partition
      • length model
      • set model
      • area model
      • number line
      • counting circles
      • unit fractions
      • common denominator and common numerator
      • Benchmark fractions
  2. Print out and read the Common Core progressions HERE, particularly these two:
  3. Check out Christina Tondevold’s website ‘Build Math Minds‘. She has great resources and offers virtual PD on elementary math. Join her Facebook group too.
  4. Learn about ‘CGI’. CGI stands for ‘Cognitively Guided Instruction’ and comes from research out of the University of Wisconsin Madison. I highly recommend reading ‘Children’s Mathematics’ by Thomas Carpenter to begin your journey in understanding more.
  5. Check out Pam Weber Harris’ website ‘Math is Figureoutable’. She is the perfect transition resource linking K-5 math to 6-12 math. I love her levels of Algebraic Reasoning. If you don’t know the difference between additive and multiplicative thinking, you have no business teaching proportional reasoning (just sayin).

6.  In my home state of Minnesota the godfather of Elementary Mathematics – he has trained almost every great Elementary Leader I know in my state – is James Brickwedde. His PD is a master class for people who want to go deep in their knowledge of how students develop numeracy. Google his name and read things he has written or check out his website, The Project for Elementary Math (note: Jim is the organizer behind this summer’s CGI conference in Minneapolis)

7.  The rest….there are lots of other K-5 words you can google and learn mor about….: Fact Families, Additive Thinking , Open Number Lines, Tape Diagrams (bar model) , Rekenrek, Counting On, Direct Modeler, Compensate, Multiplicative Thinking, Relational thinking…..also get familiar with all types of K-5 models.  Here are a few places to start.

Lastly, what do you do to improve your K-5 math understanding.  Let me know in the comments below.  I highly recommend looking for good K-5 PD in your area this summer and attending it as a 6-12 person.  Not sure where the PD is, contact your local NCTM affiliate and ask them.  They will know.

That is all for now – I know, I know – my usual ridiculously long post. Let me know what you think by commenting below or by tweeting me at @saravdwerf. I love hearing from you all.

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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