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    • This post is part of a week of math education-themed blog posts hosted by @StephenL (@apm_writer on Twitter). Next week is APM's annual weekend-long mathematics panel discussion.

      Jenna usually blogs at jennalaib.wordpress.com. She is currently a K-8 math specialist for Brookline Public Schools in Brookline, MA, USA.

      _____

      "I mean, you know... I'm not a math teacher," said Jessica, who teaches Kindergarten.

      She didn't whisper this confession in an empty classroom, or sneak it into a long paragraph in an e-mail. She said it loudly. Plaintively. In the faculty room. Because she believes it's true.

      I wrote about some experiences collaborating with Jessica (and her kindergarten mathematicians) in the blog posts about kindergarten geometry (Same or Different & Shape Hunt). Look at the way that Jessica helped her students build understandings, and make their thinking visible. Jessica is decidedly a math teacher.

      In preparation for the return to school in the new year, I reread the beginning of Kassia Wedekind's Math Exchanges (Stenhouse, 2010). (Kassia can be found on twitter at @kassiaowedekind.) One thing I love about Kassia's book is that it is profoundly respectful of mathematical learners -- of all ages.

      In the first chapter, "Creating Space for Math Workshop," Kassia writes about classroom culture that breeds opportunities for learning.

      "In math workshop, learning occurs when children are actively engaged in their environment and create a system of meaning and deep understanding. Within this framework of learning and teaching, teachers are not the guardians of knowledge whose job it is to pass down information held sacred and untouchable under the rules that govern math. Quite the contrary.

      Kassia Omohundro Wedekind, Math Exchanges, page 9

      Kassia describes the role of the teachers and students in math workshop -- no matter what format or definition of workshop is being used (page 7).

      Teachers should identify themselves as mathematicians who are continually growing and learning.

      Students should see teachers not as a source of mathematical knowledge, but as fellow mathematicians who are continuing to learn both inside and outside the workshop.

      Kassia Omohundro Wedekind, Math Exchanges, page 7

      I had to read that part twice! It continues:

      Teachers should believe that all students are powerful mathematicians and treat them as such.

      Students should identify themselves as mathematicians who have valuable ideas to contribute to the field of mathematics.

      Kassia Omohundro Wedekind, Math Exchanges, page 7

      What does that mean for a coach?

      I work with plenty of teachers like Jessica: strong classroom leaders, who, for some reason or another, do not consider themselves to be a real "math teacher." It's my job to illuminate the work these teachers do that makes them math teachers.

      "When I first began teaching math I fully believed that if I only explained a concept clearly enough and provided enough engaging experiences to reinforce the math skills I was teaching, all of my students would learn. It took me a while to realize that this approach simply wasn't working for all of my learners and did not support the development of strong mathematicians."

      Kassia Omohundro Wedekind, Math Exchanges p. 9

      I see all of these beautiful strengths in Jessica's instruction. She listens to students. She makes sense of their thinking, and helps them make it visible, often for the whole class to digest. Jessica supports making connections across representations, and across mathematical ideas. 

      Jessica might not know the same for a 12-sided polygon, but what qualities in a teacher are high leverage when it comes to student learning? It's hard to learn how to listen. It's easy to google "name of a 12-sided polygon."

      Kassia wrote about the use of "mathematician statements" in Math Exchanges. I wrote about the idea in a blog post, too, in the context of the presentation I gave with Heidi @heidifessenden at NCTM Hartford 2018. 

      Mathematicians take notice of patterns and relationships between numbers.

      Mathematicians make connections to problems they've seen before.

      These are also true of math teachers. More additions:

      Math teachers listen to student thinking.

      Math teachers support students in making their thinking visible.

      Math teachers pose purposeful questions.

      Math teachers facilitate meaningful mathematics discourse.

      The last two came directly from the 8 Effective Teaching Practices in NCTM's Principles to Actions (2014). They are statements I wholeheartedly agree with, and also ones that may feel less accessible to teachers like Jessica. They're more formal. What if we started with things like:

      Math teachers ask questions that get students to pause.

      Math teachers encourage students to talk about their ideas with one another.

      ...and then connected these back to the 8 Effective Teaching Practices.

      I wonder how, as a coach, I could build these math teacher statements strategically, to support all teachers in developing a math teacher/learner identity, while also celebrating that there are ways to grow.

      Because Jessica is definitely, unequivocally, without-a-shadow-of-a-doubt a math teacher.
      _______

      Jenna Laib (@jennalaib)

      5 recommended posts from my blog to read next:

      1. Presuming Competence: Using Clinical Interviews to Support Classroom Instruction

      2. Talking about Geometry in Kindergarten, Part 2: The Shape Hunt

      3. Leading Act 3: Using the 5 Practices & Intentional Talk to Deepen Discourse

      4. No More Mathematical Matchmaking: The Return of the Inaba Place Value Puzzles

      5. Making Connections During Number Talks

    • One of the problems is that Jessica does not grasp the incremental nature of good math teaching. Math teaching should focus on today's building block and not on next year's tactical studies.

      When I was small, students were expected in second grade to know what a minuend and a subtrahend are in second grade. That kind of approach fails to grasp how baby steps yield growth.

      I had a problem with small motor skills (still do but keyboards make it easier) and handwriting became painful the longer that I had to employ it. So imagine what it was like for me when I got to a grade where one was required to write (by hand) Two hundreds and three tens and five ones plus Three hundreds and two tens and zero ones equals Five hundreds and Five tens and five ones. Homework became an ordeal for me and rather than learning three digit arithmetic I was learning to hate math homework.

      Children needs to be taught with patience, with understanding, and with an awareness of the short term goal which when added to the sum total of twelve years (and possibly college) will gradually produce the results desired. And if a child happens to be grasping the concepts and procedures quickly, they should not be made to suffer boredom for the convenience of the slower students.

    • Great guest blog post, @jlaib , and welcome to Cake!  (Anyone reading this who wants to join in the conversation, sign up here.)

      Are you familiar with the work of Liping Ma?  Her book “Knowing and teaching elementary mathematics” is one of my favorite texts.

      Ma shares that the typical elementary school teacher in China has a dozen math books at her disposal for lesson planning while the typical elementary school teacher in the United States has minimal pre-service training in math.

      Learning numbers in Chinese is also infinitely easier for young minds to learn.  In Chinese, the numbers from 10-29 are read as

      ten, ten-one, ten-two, ten-three, ten-four, ten-five, ten-six, ten-seven, ten-eight, ten-nine, twenty, twenty one, twenty two, twenty three, twenty four, twenty five, twenty six, twenty seven, twenty eight, twenty nine

      I’ve often felt that our math language overloads the limits of young minds with abstract numbers (eleven, twelve, thirteen, fifteen) and that elementary teachers are asked to teach that other language without the benefit of being “fluent speakers.”

      In your role as a coach, when working with elementary teachers who hated math, where do you start in order to increase their fluency and growth mindset?

    • I guess I should have written 13 years instead of twelve since tthe whole point of the Jessica illustration is that Jessica is teaching math before first grade. As Maxwell Smart would say, "Sorry about that, Chief."

    • It is a long road for sure, and mathematics is so incremental that it’s critical to meet students where they currently are with understanding. That must be a challenge in those younger grades where everyone has such a different starting level. I’ve had a kindergartner intuitively grasp fractions, decimals, and negative numbers and another kindergartner who really struggled with counting objects consistently despite knowing the names of numbers.

      I admire the approach that seeks to build confidence in those early teachers of math. Too often we label ourselves as either mathy or not, and kids pick up on that and sort themselves into the same categories, charting a course that is confirmed daily by their experiences in learning math. Many count themselves out early on. My kiddo who struggled to count is now thriving two years later, but only because he stayed in the game, recognizing that mathematical understanding yields to diligence and an open mind.

    • meet students where they currently are

      That is exactly right. Instead of expecting the learner to adapt to the teacher or to match up with age standard expectations good teaching (regardless of topic) meets the learner where they are—not where the teacher might wish that they were.

      (For any who are interested, the examples of individual teaching found in Matthew-Acts use this particular principle. If a woman is interested in water, the teacher started with water. If a man was reading from Isaiah, the teacher went through Isaiah 53 and following explaining what Isaiah was prophesying. If a city was uncertain about God's identity, the teacher discussed who God is and even quoted two local poets.)

    • Thank you, Stephen!

      I love that Liping Ma book! I haven't read it in years, so I should revisit it.

      As 'teen' numbers are currently the bane of my existence when working with K students, I have been thinking a lot about what Liping Ma wrote about how the numbers are named in Mandarin. The K teachers at my school identified a number of students who 'can't count to 20' accurately. However, many of these same students are quite comfortable counting from 20 to 29. They have latched onto the pattern of our digits. How nonsensical is it that the ones digit is more prominently voiced in the 'teen' numbers? SIXteen... SEVENteen... even knowing that the 'teen' stands for 'ten' doesn't help, because the tens are name first in every other two digit number.

      "In your role as a coach, when working with elementary teachers who hated math, where do you start in order to increase their fluency and growth mindset?”

      I often start with images. This removes numbers as a barrier to experimenting with mathematical ideas. We do dot talks and use images from Pierre Tranchemontagne’s website (http://ntimages.weebly.com). I support them linking their mathematical ideas to expressions and equations. Numbers are symbolic of the ideas that they have.

      Also: for teachers that feel less comfortable with their own math background, I like to try to do the math together under the framework of anticipating student thinking. (Always focus on the students!) What do we think the students will do with this task? What student work might we want to share? (a lá Five Practices for Orchestrating Productive Discussion) Where do we want to nudge/push students next? The focus on the student thinking allows us to spotlight thinking, without making the teachers as self-conscious.

    • Ugh, I have a hard time understanding why anyone would think it necessary to teach 7 year olds the words "minuend" and "subtrahend." I can appreciate the development of precise vocabulary, but... (I think I may have been taught these words at a young age, too, although I promptly forgot them again until my preservice teacher training. Ha.)

      I hate that you dealt with so many unnecessary barriers as a child! Writing out numbers in the word form (two hundreds and three tens and five ones plus three hundreds and two tens, etc." is not only painful, but it's truly not symbolic of any of the learning. There's a reason we use standard form and not word form in most cases. There is sometimes a fine balance between reducing barriers and lowing expectations, but it is important work. I like to draw teachers back to the heart of the lesson. What's important here? How can we show it? Where will we push the learning next?

    • You're right that young, young children pick up on how we identify -- as 'mathy' or not! I think it's human nature to categorize, and that this extends to how we self-identify. I know there's a movement now to tell people that they are a "math person." I have mixed feelings on it. To me, a math person is someone that seeks to self-identify as someone who enjoys math. I don't even care if everyone enjoys math, but anyone who teaches it to children should be able to see themselves in the role! It's a professional classification more than a hobby.

      I do think that the kindergarten classroom is a truly diverse learning environment, mathematically speaking. How do you engage and encourage the K student who figured out negative numbers while also giving students learning how to count access to instruction around those early math concepts? It's really challenging -- and critically important. Every student deserves to learn, and the primary years are so important to building a positive identity as a math learning! Honestly, I'm stressed out just thinking about it. Primary teachers have such important jobs!

      I'm so glad to hear that your kid is thriving now, @amacbean16 :) There are so many factors in developing mathematical proficiency! (Developing understanding of concepts and procedures is surprisingly non-linear.)

      I admire the approach that seeks to build confidence in those early teachers of math. Too often we label ourselves as either mathy or not, and kids pick up on that and sort themselves into the same categories, charting a course that is confirmed daily by their experiences in learning math. Many count themselves out early on. My kiddo who struggled to count is now thriving two years later, but only because he stayed in the game, recognizing that mathematical understanding yields to diligence and an open mind.