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Episode of Tammy McMorrow's Podcast Little Minds, Big Thinkers where Ann Elise shares about BTC and Learning Trajectories

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Episode of Tammy McMorrow's Podcast Little Minds, Big Thinkers where Ann Elise shares about BTC and Learning Trajectories

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‎Little Minds Big Thinkers: 20. Facilitating the Mathematical Journey with Ann Elise Record on Apple Podcasts

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‎Show Little Minds Big Thinkers, Ep 20. Facilitating the Mathematical Journey with Ann Elise Record - Mar 22, 2024

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NHASCD's podcast with me discussing Building Thinking Classrooms  October, 2023

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NHASCD's podcast with me discussing Building Thinking Classrooms  October, 2023

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Building Thinking Math Classrooms with Ann Elise Record by NHASCD Spotlight: A Podcast

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ABOUT
Ann Elise Record is a well-known New Hampshire math educator, writer, and consultant. She has been in the field for 22 years as a K-5 Math Coach, adjunct faculty member for Plymouth State University, Bureau of Education and Research presenter, and currently an independent elementary math consu…

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NCTM Fluency Position Statement April, 2023 

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NCTM Fluency Position Statement April, 2023 

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Procedural Fluency in Mathematics - National Council of Teachers of Mathematics

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Procedural fluency is a critical component mathematical proficiency and is more than memorizing facts and procedures.

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NCTM Fluency Position Statement April, 2023 

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NCTM Fluency Position Statement April, 2023 

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Procedural Fluency in Mathematics - National Council of Teachers of Mathematics

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Procedural fluency is a critical component mathematical proficiency and is more than memorizing facts and procedures.

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All of Ann Elise's favorite games for all operations

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All of Ann Elise's favorite games for all operations

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Teaching Like Ted Lasso!  This episode is my favorite quote - Be Curious, Not Judgmental! 

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Teaching Like Ted Lasso!  This episode is my favorite quote - Be Curious, Not Judgmental! 

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Description
Dr. Amanda Jansen joins John and Dave on the Teaching Like Ted Lasso Podcast to discuss what it means to BE CURIOUS: in the show and in the classroom. And Dave and John do a play-by-play commentary on an assessment interview featuring a first-grader.
Video: https://youtu.be/5TCTYxCQdaw

Show Notes for Episode 2 on Be Curious
IMDB "The Diamond Dogs" episode: https://www.imdb.com/title/tt11193418/?ref_=ttep_ep8
John's Problem Solving graphic: http://mathhombre.blogspot.com/2009/06/polyas-army.html
George Polya's How to solve it: https://math.hawaii.edu/home/pdf/putnam/PolyaHowToSolveIt.pdf
The Literacy Network's parent company - Richard C. Owen Publishers, Inc: https://rcowen.com
Dave's What's your paradigm? post: http://deltascape.blogspot.com/2011/03/whose-job-is-it-anyway.html
IMDB "Inverting the Pyramid of Success" episode: https://www.imdb.com/title/tt14968664/?ref_=ttep_ep12
Innovators' Compass website: https://innovatorscompass.org
John Hattie's Visible Learning for Teachers: https://www.google.com/books/edition/Visible_Learning_for_Teachers/xY-6MyDgks8C?hl=en&gbpv=0
Clements and Sarama (2000) Young Children's Ideas about Geometric Shapes: https://www.researchgate.net/publication/258933238_Young_Children%27s_Ideas_about_Geometric_Shapes
Target Measure from Illustrative Mathematics, Grade 2: https://curriculum.illustrativemathematics.org/k5/teachers/grade-2/unit-3/lesson-13/lesson.html
Amanda Jansen on Twitter: @MandyMathEd
Amanda Jansen's website: https://jansenamanda.wixsite.com/professional
IMDB Ratatouille: https://www.imdb.com/title/tt0382932/
IMDB Stranger than Fiction: https://www.imdb.com/title/tt0420223/
Mandy's article with Denise McCunney: https://democracyeducationjournal.org/cgi/viewcontent.cgi?article=1557&context=home
Association of Mathematics Teacher Educators website: https://amte.net
Frances Harper on Twitter: @WorldMathProblems
Three-read protocol: https://www.sfusdmath.org/3-read-protocol.html
Ewan McIntosh TED Talk: https://www.youtube.com/watch?v=JUnhyyw8_kY
Sara Van Der Werf blog on Name Tents: https://www.saravanderwerf.com/week-1-day-1-name-tents-with-feedback/
Jack Smith's Misconceptions Reconceived: https://faculty.weber.edu/eamsel/Classes/Practicum/TA%20Practicum/papers/Smith,%20DiSessa,%20&%20Roschelle%20(1993-4).pdf
Three-Act Math Tasks: https://www.sfusdmath.org/3-act-tasks.html
Joshua Davis Music: https://joshuadavismusic.com
Other Resources:
Dylan Wiliam on Eliciting evidence of learning: https://youtu.be/jPo1fFQshUY
John Spencer's blog post - Ask the Teachers: https://spencerauthor.com/ask-the-teachers/
Christopher Emdin's article on 7 Cs of Effective Teaching: https://www.ascd.org/el/articles/seven-cs-for-effective-teaching
Brian Cambourne's Conditions of Learning: http://sydneysclass.weebly.com/uploads/3/8/7/0/38708599/cambourne_relevant_theroy1.pdf
Dave's article: Are they wrong? Or did they just answer a different question? https://pubs.nctm.org/view/journals/tcm/12/4/article-p202.xml

Hosted on Acast. See acast.com/privacy for more information.

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‎Teaching Like Ted Lasso: Be Curious on Apple Podcasts

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Description Dr. Amanda Jansen joins John and Dave on the Teaching Like Ted Lasso Podcast to discuss what it means to BE CURIOUS: in the show and in the classroom. And Dave and John do a play-by-play commentary on an assessment interview featuring a first-grader. Video: https://youtu.be/5TCTYxCQdaw...

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Choral Counting photos of examples for all grade levels curated site by Berkeley Everett

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Choral Counting photos of examples for all grade levels curated site by Berkeley Everett

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This simple 10-15 minute routine builds number sense across many grade levels and content areas. Use organized images below (under Examples) for inspiration as you plan your Choral Counting. Check out this book for detailed support.

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Paint Splatter Arrays (.key)
Paint Splatter Arrays (.pdf)

In Steve Wyborney's Splat!, the total number of dots is given and the number of dots under each splat is unknown. In my Paint Splatter Arrays, the total number of dots is unknown. My paint splatters do cover some dots but how many is beside the point. Also, Steve's dots are scattered; mine are arranged in arrays. (More on that below.) Steve's splats splat. My splatters are there from the get-go. See? Not the same.

h/t Andrew Stadel

Here's why I created this activity...

T: "How many do you see?"

S: "Twenty-five."

T: "How do you see them?"

S: "Two, four, six, ..."

Every. Single. Time.

Not all students. Most students do see and use groups or arrays to figure out how many. Those strategies are described in this post. But some students don't seem to make sense of others' ideas. That's a greater challenge than I'll tackle here. (Recommended: Intentional Talk by Elham Kazemi and Allison Hintz.) Instead, I designed the activity above to (gently) shove students towards looking for and making use of arrays.

The first three are softballs. For example, the second:

Students can still see each dot and count all by ones or twos. But a more efficient strategy is to see 3 × 5 (3 rows, 5 columns).

The next several slides completely cover at least one dot, so students can't count all by counting what they can see. In each, at least one complete row and one complete column is visible. For example:

I had some fun with the last two. In the next-to-last one, the middle column is completely concealed.

In the last one, most of the dots are hidden. A bit of estimation. How many?

How confident are you?

What about now?

I test-drove these on my daughters. (Keira likes Booger Math! over Paint Splatter Arrays, by the way. It is catchier.) I'm looking forward to trying this out in Surrey classrooms. Feedback welcome!

And mine goes ding ding ding di di ding ding DING ding ding ding di di ding ding.

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This isn't Splat! Paint Splatter Arrays (.key) Paint Splatter Arrays (.pdf) In Steve Wyborney's Splat!, the total number of dots is given and the number of dots under each splat is unknown. In my Paint Splatter Arrays, the total number of dots is unknown. My paint splatters do cover some dots but how many is...

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Search Google

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Archive of all Tape Diagram Tuesday problems

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Introductory Video Discussing Exploring Multiplication Facts Through Strategies

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Introductory Video Discussing Exploring Multiplication Facts Through Strategies

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Dr Nicki and Ann Elise's Top Ten Ways to Get Students Problem Solving - Teacher as podcast discussion

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Dr Nicki and Ann Elise's Top Ten Ways to Get Students Problem Solving - Teacher as podcast discussion

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Episode 64: Top Ten Ways to Get Students Problem Solving with Dr. Nicki Newton and Ann Elise Record - The Teacher As

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Dr. Nicki Newton and Ann Elise Record are back! They each have their own lists of the best ways for students to practice problem solving and they collaborate to create one essential top ten list.

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5 Weeks of Daily Routines by Berkeley Everett

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5 Weeks of Daily Routines by Berkeley Everett

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Here are 5 weeks of number sense routines (5-10 minute math talks) tailored to each grade K-5. Every slide deck includes a "how to" and a week each of:

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Build Math Minds Podcast. Host Christina Tondevold interviews Cathy Palkovic about when her students determined how many cards were in 6 boxes of hundreds of decks of cards. Wow!

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Build Math Minds Podcast. Host Christina Tondevold interviews Cathy Palkovic about when her students determined how many cards were in 6 boxes of hundreds of decks of cards. Wow!

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Episode 60: Building Self-Reliance in Math

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Welcome, fellow recovering traditionalist, to episode 60. Today is an interview with third-grade teacher, Cathy Palkovic, and a theme of our conversation is about building self-reliance in math. Christina - In this interview with Cathy, we get into it all kinds of things, from what her distance learning looks like to Math Running Records, to how her third-graders solved 898 x 52 and they did it without any guidance from her.

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Kids Math Talk Podcast interview with Dr. Nicki Newton, Dr. Alison Mello, and Ann Elise Record

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Kids Math Talk Podcast interview with Dr. Nicki Newton, Dr. Alison Mello, and Ann Elise Record

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Exploring Basic Math Fact Fluency - An Interview with Dr. Nicki, Alison, and Ann Elise

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Desiree Harrison 0:00 Fluency is a concept we have talked a lot about on the podcast. We need to keep talking about this because there are so many deep-seated core beliefs about how to best help children with addition and subtraction. We need to move past our feelings and get into the facts.

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My conversation with Laura and Karina on the Learning Through Math Podcast  Released Oct 17, 2021

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My conversation with Laura and Karina on the Learning Through Math Podcast  Released Oct 17, 2021

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Learning Through Math with Laura and Karina

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Episode 55: Learning About Math Running Records - Learning Through Math with Laura and Karina

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In this episode, Laura and Karina discuss Math Running Records with Ann Elise Record. CHALLENGE: Give a Math Running Record to one of your students and let us know how it goes!

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Top Ten Manipulatives - Teacher as podcast interview with Dr. Nicki Newton and Ann Elise!

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Top Ten Manipulatives - Teacher as podcast interview with Dr. Nicki Newton and Ann Elise!

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Episode 44: Top Ten Math Manipulatives — The Teacher As

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In this second edition of Top Ten, Dr. Nicki Newton and Ann Elise Record return with their own lists of top ten math manipulatives and collaborate to create one essential top ten list. Dr. Nicki and Ann Elise are so knowledgeable and share their favorite manipulatives and how they like to use them w

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Learning Through Math with Laura and Karina

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Learning Through Math with Laura and Karina

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Learning Through Math with Laura and Karina

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Podcast - Learning Through Math with Laura and Karina

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CHALLENGE: Share books with us that are on your radar! Email or DM us if you want to share a book on our podcast! Order your copy of Principles to Actions - Ensuring Mathematical Success for All so that you are ready for our Summer Book Club!

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My Fractions Operations Webinar for Brainingcamp

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My Fractions Operations Webinar for Brainingcamp

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The most popular manipulatives at your fingertips. Powerful yet easy-to-use.

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My Fractions Conceptual Understanding for Brainingcamp

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My Fractions Conceptual Understanding for Brainingcamp

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The most popular manipulatives at your fingertips. Powerful yet easy-to-use.

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Keep current on topics in math education, and learn from the perspectives and expertise of Black women.

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Ann Elise: Well, first of all, we need to define fluency. We need to be all on the same page of what fluency means. So in my presentations I ask people, "What words come to mind when you think about being fluent with your math facts?" And, overwhelmingly, people will put words like 'speed.' Speed, fast, automaticity, all words about speed. And then occasionally, I'll get 'accuracy,' right, because kids can give the right answer but then be wrong. Right? So we want to have the accuracy as well as it being, I call it relatively speedy, because it's the least of the four things I want to talk about. But there are two more pieces of fluency, which is crosstalk.

Jon Orr: Today we speak with Ann Elise Record. Ann Elise is a seasoned educator with a career spanning over two decades of elementary classroom experience. These days, she's a K-5 math coach who works with districts on developing strategies to increase students' fact fluency as well as how to use manipulatives in the classroom.

Kyle Pearce: Stick around as we'll discuss effective strategies to help students build their fact fluency, how to help students with the progression from additive thinking to multiplicative thinking, and, how can we help students with concepts that hold them back from their learning?

Jon Orr: Let's do this, Kyle.

Kyle Pearce: Welcome to the Making Math Moments That Matter Podcast, I'm Kyle Pearce from tapintoteenminds.com.

Jon Orr: And I'm Jon Orr, from mrorrisageek.com. We are two math teachers who, together...

Kyle Pearce: With you, the community of Math Moment-Makers worldwide who want to build and deliver math lessons that spark curiosity...

Jon Orr: Fuel sense-making...

Kyle Pearce: And ignite your teacher action. Are you ready, Jon, to dive in to this awesome conversation with a dear, dear math friend?

Jon Orr: Mm-hmm (affirmative), mm-hmm (affirmative), of course, Kyle. We are honored to bring Ann Elise Record onto this show as we've been looking forward to this conversation.

Kyle Pearce: All right, but before we dive in here, Jon, we want to make sure that you folks at home have taken the time to explore some of our most recent full units of problem-based math lessons.

Jon Orr: Right, right. We've been busy adding full units to our Make Math Moments website. And last episode here, we shared some details about our six day unit called the Wooly Worm Race that was created to help educators address teaching fractions, decimals, and percentages conceptually.

Kyle Pearce: Well, we've also tossed in some more context, rich, problem-based math lesson fun with our unit called Snack Time, where students explore the tapintoteenminds.com classic three-act math task called Cheese and Crackers, to use partitive division to partition and distribute slices of cheese to a specified number of crackers.

Jon Orr: That's definitely a classic I've used with my class, and now I'm set up with a whole unit of connected math talks. Sense-making prompts, purposeful practice, and summative assessments, all scraped from my web browser by visiting our test page at makemathmoments.com/tasks.

Kyle Pearce: Yes, it's super exciting, Jon. We hope to see you over at makemathmoments.com/tasks, so that you can run one of these awesome problem-based units in your classroom today.

Jon Orr: All right, enough from us. Let's get on with our great, great, great, great conversation with Ann Elise.
Hey there, Ann Elise, welcome to the Making Math Moments That Matter podcast. As always, we are super excited to chat with someone we've met, someone we know... How are you doing today?

Ann Elise: I'm doing great, doing fabulous.

Jon Orr: Awesome stuff, awesome stuff. Well, we know you quite well, but maybe there's a few Math Moment-Makers out there who aren't as familiar with Ann Elise Record. Can you help us out? Let's learn a little more about yourself. What's your role in education and how did you get to that part in your learning journey?

Ann Elise: Sure. Currently I'm a consultant. So currently I work for myself out of my home and I was traveling around a lot and working in schools and districts, and also presenting around the country for the Bureau of Education and Research. But then once we had the shutdown, I've been working from home. But I had gone to get to, my undergraduate degree was in sociology. So I was not certified or able to teach. And so I got back to school, once our kids were becoming of school age, I went back to school and got my master's in order to teach. And then I taught grade five for 13 years. And then a math coach position opened up in my district, and so I became the math coach K five.
And that was really the start of my journey to truly understand the trajectories of the learning and progressions of kids from the early numeracy on up, and all those connections. But also at that time, I taught some pre-service college classes, the math methods, I helped people, tutored them to help pass the Praxis test, because I've experienced in this role, working with so many different populations of people. And I've seen the anxiety and the stress and the phobias, firsthand, of people.
And I didn't have that growing up. When I grew up, I could answer things quickly and accurately, so I felt good about myself. But to have seen the phobia up close for a variety of people, it was surprising to me how many of the college kids were terrified of math going into teach elementary. And it just became a passion of mine, the more that I learned, why I wanted to share with people. And my husband got a new job two hours south of where we had been living. And then I'm like, "Well, what am I going to do now?" There weren't any math coaching positions. I loved working with the teachers, and the kids, and everyone. And that's when my two close friends and mentors, Dr. Nicki Newton and Christina Tondevold, said, "Well, why don't you go off on your own?" And that's what I did. So my life has been completely different these last few years, doing this new part of my life.

Jon Orr: Yeah. And I think you've brought up a huge point about math anxiety being a huge thing, and I think still is for so many people, and I'm wondering, what suggestions can you give our listeners right now, and teachers, that can help reduce that anxiety? If you've been recognizing it's a problem in your past, in your career, now I'm wondering, I guess, two things. One is, what can we do now to help kids, and even teachers, reduce some of that math anxiety? But then I'm also wondering about, what did Ann Elise look like as a early teacher? Did you recognize that right from the get-go, that we've got this math anxiety, and it's something that you've been using in your career right from the start? Or is that a progression that built up over time? And so those are, I guess, two things to talk about.

Ann Elise: Yes, okay. So first of all, when I think back to when I first began teaching, so I think we all know what we know at the time and we don't know what we don't know. And so, in my master's program to teach, the only class that I had to take that involved math to become certified K eight was math and literature. So it was a literacy-based math class. So I came out certified K eight, really being no more qualified to understand early numeracy than I was to teach an eighth grade math class. I think the assumption was that you'll be given a textbook of whatever district hires you, and you follow the textbook. So that's what I did. I became the grade five teacher, we had a textbook, I was very procedural in my teaching. It's all that I knew when I grew up learning math. It was a very procedural, rote procedure. There was no open tasks. There was no talking about our thinking or having any kind of flexibility of thought.
And so I gave time test to my classroom, all 13 years I was in the classroom. And over the course of that time, I know I did concrete and pictorial things, but it was leading toward the algorithm. I didn't do flexibility with number sense, I was trying to show them why the algorithms worked. I didn't know anything else. So it definitely was not in play the first many years that I was teaching. But as I learned more, and we had a consultant come to our district that showed me how to teach math in a visual way, that was totally new to me. I had no idea that you could teach math visually, like pattern blocks, refractions. It was a mind-blowing thing to me. I had never seen it before.
So it's been, certainly, I call it a journey because I think we're all on journeys. We as educators, as well as the kiddos, right? As we get to find out where we are on that journey, and then you continue and when, Dr. Maya Angelou says, "When you know more, you do better." And that's exactly what I believe and I've been living, that, I know the more that I learn, the more I realize I have to learn. This is a never-ending process. And so, I think we need, I need to send the message to people, I want to let the people know that there is so much more to mathematics and connections and a beautiful journey to it, so far beyond rote memorization and procedures. Because I think that when we think about giving kids worksheets that are just 20 kill and drill things, that's not going to be encouraging a love of math, nor will it build the flexibility of thought that we need to build our number sense and to develop the confidence to try harder problems.
The idea of the growth mindset messages of Jo Boaler from Stanford, the fact that our brains have the way of growing new parts. There is no such thing as a math brain, but think what happens is people have gone on their journey, and I truly believe time tests have begun, well, research is out there that proves it, but, that it's the start of math anxiety. Kids think that because they're not fast at the math, and they're not good at math, and it begins disassociating themselves from the idea that they can be a mathematician. And then it gets harder over the course of the year, as if we're only doing procedures and formulas. That's the only way they've been exposed to thinking and algorithms... Algorithms take lots of steps. And I think, I'm like, a three digit by two digit multiplication problem, that's a lot of calculational work.
But until we explore these things in more flexible ways, it can make it much easier. And I find that my consulting work, the more that I share these ideas, and that they're seeing the math in a different way. Even the parents, I have my districts do math nights, because I want to talk to the parents. It's huge we get the parents involved. And by the end of that 30 minutes, the parents are like, "Oh my gosh, this is a whole new world to me. That's why my child was doing that this morning." Literally they've said to me, "We were arguing with my child this morning about eight plus six. We just told her, 'It's 14.' She's like, 'But no, mom, that's not the way I was supposed to do it.'" And the parents were like, "But now we know why, and we totally get it. We've seen the future impact of that work."
And so, the whole anxiety piece comes from, not the math, but the feeling of insecurity and embarrassment and humiliation people have felt growing up, that they were not a math person, as if it takes some extra kind of brain to be good at math. And it's not true. We all can grow those parts of our brains, and in fun and engaging ways. I want our students to have no bad stories of math growing up. In all my consulting, I would say probably 80% of the people that I meet tell me the most awful, traumatic math journeys of their lives. It's just always negative. They're always about 20% like, "Oh, it was good. It was fun. I loved it." It's like, "Well, you're like me then, because I got my answers correct, I was able to get them done fast, so I thought I was good." But it wasn't until I really began this deep journey that I realized I didn't have any understanding of mathematics, so I didn't see connections to topics within the math. But now the more that I learn, the more I'm like, "Oh my gosh, that connects to that. And throughout the grade levels, even, it's a beautiful thing, to finally see those connections.

Kyle Pearce: Yeah. Well, and we love asking about these things because I worry that when teachers and educators who are at that place in their own journey where they feel like maybe they don't feel comfortable with the content, or they maybe remember their own math anxiety that they had as a student, or they just think plain out that they're doing it wrong and they see someone like yourself, Ann Elise, or Jon and I, and we are so into thinking about how mathematics develops, and thinking about the conceptual connections, all of those pieces. But yet, most of us, most of us didn't have that same experience growing up. And many of us, as you just mentioned, and it's great to hear, that, many of us actually taught it the way we were taught. And it's okay if that's where someone is as they're listening. So I think that is such a huge message for people to see that it takes time.
And I immediately connected when you were describing how you were teaching very procedurally and you were trying to help kids to understand why the procedure works, which I think is part of that trajectory. If you learned procedurally, you tend to teach procedurally, and then hopefully over time you start to see that, "Oh, connections are important." But then what we do, it seems like for me, anyway, I started to still teach the procedures, but then I tried to explain them afterwards. Instead, now we're realizing, "Oh, we really need to start with kids making meaning, and making sense first, and then working towards and allowing those student-generated algorithms to start popping out and helping them to make their connections so that we can essentially get more efficient along the way, but not with the end goal just being an algorithm or a procedure, right?

Ann Elise: Absolutely, 100%. And in fact, the more learning that I'm doing, I've been trying to work on my own knowledge of middle school mathematics, in terms of the conceptual understanding. I can get right answers, I help people pass the Praxis test. That's not the issue of getting right answers. It's, I want to get the connections that I missed first time around. And so I've been doing a lot of work on that. And the more of the work that I'm doing, I am doing mental math or ratio tables and things of using chunks of numbers that are nothing to do with the algorithm. And I'm getting easy answers really quickly, I'm breaking apart numbers in ways I never had done before, and it's just a matter over time. I do a lot of Pam Harris' Math Strat Chat every week, and inaudible Academy. I took a class with Pam Harris on proportional reasoning, that is just like, I'm learning these ways of thinking about it. When I learned it the first time, I had no idea of those connections, I was just following procedures. And it's just a beautiful extension to what I already know as a foundation of understanding K five, and to extend that to that middle school level.

Kyle Pearce: I love that. And it's actually ironic that, nowadays you'll see, both in your messaging and so many others who come on our show and obviously Jon and I, who, we do not believe, like math fluency is not speed. Yet, when we actually start to help kids become more flexible in their thinking, and start to play with different strategies, and really just give them the freedom to think and use critical thinking in order to reason and solve through problems along a trajectory, along a progression, what I end up seeing is actually kids end up becoming more quick with their math facts than kids who are memorizing the procedure. It's actually mind-blowing when you're saying ,"No, no, we're..." It's not about speed, but then the end goal ends up oftentimes becoming speed anyway, which isn't the holy grail, it's just like a symptom of having flexibility and fluency with number. And I'm hearing that in your voice, I hear that in your message.
So before we dive deeper into fact fluency, because we have a lot of questions for you, we know your connections with Christina Tondevold, as well as Nicki Newton. We know you're big into that, and we'll be diving into those questions, but before we let it get out of hand here and we dive too deep, we've got to hear about a math moment from your past. So you've described a few things about how you began teaching, how you taught for a while, also, a bit about your experience learning in the classroom as a student, but is there a moment that pops out in your mind? It could be defining, maybe it's something that, like a celebratory moment, or it could be one that maybe you remember, and it's one of those memories that you sort of wish you didn't remember. What's a math moment that pops out in your mind that you're willing to share with our audience?

Ann Elise: First of all, I'm just so happy to be able to share you my math moment, because I've been listening to your podcast forever. And I walk my dog listening to it. Whenever you ask your guests, I'm like, "Oh, I would tell this story if I was ever inaudible them." When I became math coach, K five, before I began that part of my job, I took a one-week class with Dr. Mahesh Sharma, on, really, it was K eight type content. And it was the first time I saw the connection between grade levels of things I could do at a lower grade level that affected what came up. It was my eye-opening to that kind of procedure. Well, we did the area model of multiplication, and he loves Cuisenaire rods, they're my favorite math manipulative. I think they're the key to helping our kids move past the counting phase of reasoning and into strategic, additive thinking, multiplicative reasoning.
inaudible we had these Cuisenaire rods and we built area models of basic facts. So in the third grade content, right? Well, over the course of that week, we kept using the area model to do double digit by double digit, or to do decimals by decimals, or mixed numbers by, like the same area model, I was able to use with harder content. Well then he said, "We get to division, and we're using the area model to do division," which I had never, ever thought of before. And we were doing it concurrently with the Cuisenaire rods. And he put on the board 'X squared plus five X plus four.' That's the exact problem that he gave. And he wanted us to divide it by X plus one. Now, I remember my brain saying, "I'm out." I took calculus in high school and again in college, it wasn't like, I really thought I did well with math. But, it had been decades since I had done that before.
So I'm like, "Okay, well I'm out. I'm not even going to attempt this." But then he said, "Try doing the same thinking we've been doing with the division for the area model. Can you make a rectangle that is X plus one by something?" And within 20 seconds, I built the area model, I knew the answer was X plus four, and I could explain to you why. And that was my moment that I went, "Oh my gosh, a model that I can show the thinking with the kids in third grade, can follow them through their entire math journey, and the math journey will make sense." It's not just getting a right answer. And I was sold on the whole idea of, "Wow, this is just a learning progression." And the more that I see that, the more that I know, it's all about that journey.
My husband has been working from home since the shutdown in March, and he says, and after doing a lot of trainings online, he's like, "Hon, you say the word 'journey' all of the time." I'm like, "I know, but it is true." We're all on a journey, right?

Jon Orr: Yeah, yeah, for sure. And I'm glad you brought up the area model because we love using it, we love talking about it, and we think it's...

Kyle Pearce: Didn't we just talk about it last night, Jon?

Jon Orr: Yeah. I was just going to bring that up, Kyle. Kyle and I talked about it last night, but I just had a recent discussion with a friend who couldn't see this new math, and why did their son or daughter have to learn this diagram or this way of multiplying? And when I can show her how to multiply, and get done in 30 seconds, and he referenced that Facebook post that we've all seen crosstalk...

Ann Elise: Oh my gosh, I know...

Jon Orr: Somebody being done in 30 seconds, and then twiddling their thumbs for the rest of the time and...

Ann Elise: Yup, exactly.

Jon Orr: Versus somebody who's drawing the area model. And we had this long conversation about that, and that was exactly what I referenced, Ann Elise, is that this model can carry forward, whereas this algorithm works in this one instance, and... We use it in high school and, listeners of this show know I'm a high school math teacher, and I just helped model my students this morning. And yesterday with completing the square, using the area model, and completing the square for a grade 10 student can be tough. And some teachers think it's one of those procedures you just got to show kids. "They just got to memorize this procedure." And it's hard to explain why it works unless you use a square to model it. And the area model works perfectly. And I love how it carries forward from there. And I really like how you referenced, both Kyle and you referenced that, by using these models, kids can get that fluency.

Kyle Pearce: I was sold on the area model from high school and then downwards. Whereas you went from an area model from lower elementary upwards, because I was sold downwards, because after using the area model to help kids complete the square in high school, they're completing the square in their heads. They didn't have to draw anything, they didn't have to write anything, because they could actually imagine the square and what the dimensions of the square would be. And so that they could write the vertex form of a parabola or quadratic expression, so it is a great tool, and I know that we'll talk about many other things.

Jon Orr: And I'm being a high school teacher and still trying to learn more about elementary math. I know Kyle's been on a journey for that for the last few years, but I don't get a lot of experience into the elementary math unless I do that on my own, or I talk to fantastic teachers like yourself. So I'm really hoping for this episode and the remainder of this episode to be a master class in fluency, and a master class that I can learn from, and all our teachers can learn from. So if we just think about diving into early mathematics in particular, and if we could help educators and parents learn effective strategies to help their children and students with fact fluency, let's think about some practical suggestions, some tips that we can help our teachers with on fact fluency at an early age.

Ann Elise: Well, first of all, we need to define fluency. We need to be all on the same page of what fluency means. So in my presentations I ask people, "What words come to mind when you think about being fluent with your math facts?" And, overwhelmingly, people will put words like 'speed.' Speed, fast, automaticity, all words about speed. And then occasionally, I'll get 'accuracy,' right, because kids can give the right answer but then be wrong. Right? So we want to have the accuracy as well as it being, I call it relatively speedy, because it's the least of the four things I want to talk about. But there are two more pieces of fluency, which is flexibility and efficiency. Those are the four pillars of math fact fluency. And I want to honor the fact, also, that there may be some of our students that are not ever going to fast, but they are using gray strategies and number sense.
I have to mention to you right off the bat, that, a lot of my consulting work has been focusing around what's called a math running record. It was a protocol designed by Dr. Nicki Newton. She did 10 years of research on math affluency, and created this interview protocol for us to talk to our kids about their thinking. I believe it is the key to transform schools and districts and the mathematical journey for our kids, because it's looking for that flexibility of thought. And if we explore flexibility with our different operations with the kiddos, it'll set a foundation for thinking that will then apply to the actual grade level content we're trying to teach. Without having to go to algorithms, they'll become much more efficient problem-solvers, and with their fluency, with the calculations, because they're using strategies and number relationships, which will build their number sense.
It's the start of everything. I've found it's been my key to educate people on early numeracy, because myself, having gotten, honestly, a master's degree, certified K eight, I had no idea the world of early numeracy, until I took a course from Dr. Sharma, that whole beginning of that journey. But then I also took the Number Sense 101 course with Christina Tondevold. And that was like, "Oh my gosh, what is this world?" And then Dr. Nicki as well, and it filled in more information. And then I just continued to learn about the early numeracy. So I want us to think, first of all, about that, there are developmental journeys our kids go through. Typically what our kids do for math inaudible journey is they begin counting on their fingers, right? Fingers are very important for the brain, very important for kids to do that.
But there's also something called finger agnosia, which is the discrimination of the fingers, but not finger counting. So the idea that the kids are going to start by counting, they usually count all, our youngest kiddos. If you ask them three plus two, in a context, I like putting all the fluency work in context, but you ask them three plus two, they put out three fingers and two fingers, and they would count them all, starting at one. Then kids developed to count on. So they may have all five fingers up, but they know this hand has three, and it's four or five. That progresses to head counting on. So the three isn't even out anymore, they might finger count four or five, or then, in their head, go, "four or five."
Then we have a whole section of derived facts. And that's typically the phase people skip. They tend to go, "Okay, you're counting on." So kids are counting on their fingers when they do their teen sums, because how else do you count on eight without keeping track on your fingers? And then we tell them, "Memorize them. I want you to play games with all the math facts, I want you to memorize these." As the kids are waiting for numbers to pop in their brain, when you ask them the questions, so that they don't know what it is. If they don't know what the answer is, that nothing pops in their head, what do they do? They count on their fingers, which is their comfort level, to find out what the sum is. And then when you think about a double digit addition problem, where we have raised to put the algorithm in front of the kids so soon in their journey, they're usually grade one, I find, a lot. Well, within the algorithm, it's just single digit basic facts. And in each of those basic facts, they're counting on their fingers. So they're in a counting phase of reasoning, within the context of addition problem.
I love this work of Pam Harris on the development of mathematical reasoning. That image of her graphic that she has, to go from the counting phase into additive thinking, and then by grade three, the multiplicative reasoning, before the proportional, and then the functional, the whole K-12 journey. So when I try to think about the math the kids are doing, what phase of reasoning is it? And the algorithms, adding and subtracting, are usually, kids are counting, they're counting on their fingers. So how can we get that? Well, we can work on the flexibility of thought. The idea that I could ask kids what's eight plus five, and that the kids could break apart that five and give two to the eight to make a 10, and then I can rename eight plus five as 10 plus three. That's a lot easier for my brain.
So the heart of that addition of breaking apart numbers and putting them back together in flexible ways will be the key, because I can then forward that to 48 plus five, or 98 plus five, or 98 plus 25. I can take two from it to make a 100, so 100 plus 23. It progresses all the way through their journey, including to fractions. Adding four fifths and three-fifths... Rather than adding numerators, keep denominators the same, uh oh, it feels weird, this large number on top, right? Break apart the three fifths and give one fifth to the four fifths and rename it as a whole plus two-fifths. It just will affect everything down their journey, starting with the basic facts.
So the Math Running Record allows us, when we're interviewing our kids, to be aware of that learning journey. Because there are codes for that. This is all based on Carpenter and Moser's research, inaudible research of the developmental trajectories, that the kiddos, we ask them sets of facts as a benchmark inaudible question for each of the set of facts. So you find which set of facts are causing a super slowdown, inaccuracies, or inefficient strategies, like just counting on. And then within that zone, where are they on that trajectory of the having no strategy to even try it, the counting all, the counting on, the head counting, the derived strategies, and the automaticity. Because we can't take kids from counting all to automaticity in one step, that's not how kids learn and develop. So it's a protocol that allows us to learn about the expertise needed for early numeracy. It's a brilliant protocol, and it's been a lot of my consulting work, because Dr. Nicki is just brilliant and she's so prolific, and she just talks about all different areas of mathematics. But I've latched on to this Math Running Record of the fluency piece, because I really feel it's my gateway to get people to understand, teachers and the parents, to understand the foundations of the operations, and to build the fluency through flexibility. And it alters everything, because it applies to every other content that you're doing.

Kyle Pearce: I love how you reference, even, how, we jump the gun on the memorization of those derived facts, right? They're not really memorizing derived facts or memorizing facts, when we skip all of these parts, and composing and decomposing number is so important. And I'm hearing that in what you're saying here, and this idea of equivalence pops out. And as you're working with young kids and, my son, who's an SK and my daughter who's in grade two, I get to have fun with these things, by playing with numbers and using strategies and seeing the strategies that they're using. And it really is pretty magical when you see them thinking.
And I just had this, I guess, a bit of a sad moment there, reflecting, as you had mentioned, like a student who's sitting there trying to bring up a fact that they've tried to memorize, but now it's like they're recalling facts they haven't derived themselves, so they don't have a connection. They also don't have a path to help them get out of a jam. And we talk about that a lot on this podcast and it continues to cause more and more problems down the road, because if students don't have a understanding of how they got to where they are, which I think many of us are like, as adults, many of us are like that, right? Those adults, like Jon, you had mentioned that friend who was thinking of the algorithm like, "I can multiply two digit by two digit numbers, no problem." But could they do it any other way? Or if they forgot one step of that algorithm, well, how are they going to get themself out of that jam?
And this is happening to so many young children, because I think for so long, we've made them take that leap, and the students who had the memory, the ability to memorize a little more easily or fluently than some other students, they were rewarded in the schooling system, but not necessarily rewarded with any sort of extra reasoning skills in mathematics. They just got the big gold star, and that was about it. Jon always references his puffy stickers that he used to get as a child. So I love that you're referencing this trajectory. And when this episode comes out, it'll be a few weeks after we had Doug Clements on, and his work.

Ann Elise: Oh, my goodness.

Kyle Pearce: Yeah, how fantastic.

Ann Elise: Clements and Sarama, they're just like the king and queen of early numeracy.

Kyle Pearce: Exactly. She was stuck out of town and we wanted to have both on the show.

Ann Elise: Oh, my goodness.

Kyle Pearce: I'm telling you right now, hearing you and your passion about this trajectory, it's so, so critical. And again, I know Nicki Newton thinks so highly, as well, of Doug Clements and Julie Sarama, and...

Ann Elise: Absolutely.

Kyle Pearce: It really all does come back full circle. So whether you're sitting at home right now, and you're thinking to yourself, "Maybe this episode isn't for me, I'm a middle school teacher, or I'm a high school teacher..." I hope that people are seeing the connections, and you brought up Pam Harris and the development of mathematical reasoning that, way down the road in high school, some of our students that are struggling the most, their struggles began way down here in this area where we didn't help them make those connections and really start to use reasoning to get themself through number fluency. And that's really hindered them.
And it makes me think, "What do we do?" Do you have any thoughts, if someone's sitting there thinking, and maybe I have a vision of a specific student in their class, and they're like, "Oh, this student is really struggling." What might be a get started tip for someone who, maybe this is a new idea for them, what are you going to say to them? Where might they begin? Might it be PD of some type, or might it be just a website to get them going? What are your thoughts on that?

Ann Elise: Well, I typically go right back to the Math Running Record, honestly, because it's going to help you understand where each child is on their journeys. You can develop instructional response derived for the kid, and then to get a sense of where all of your kiddos are. Because typically what I find, I'm training districts to implement this and then to do the next steps of how we explore fluency with our kids, in concrete, pictorial, and abstract ways. That's what our book is about. Dr. Nicki, Dr. Alison Mello, and I collaborated on a book about fluency doesn't just happen with addition and subtraction. So the idea that, now I know where I need to begin working with each kid, what can I do then to explore these concretely pictorial and abstract, and develop this fluency program for my kiddos?
So the Math Running Record, I have a Math Running Record Facebook group that I began close to three years ago now, because Dr. Nicki, she's doing so much all the time. I said, "Yo, do you mind if I make a group, of a support group for those of us that are really going in depth of implementing these?" She's like, "Oh, go right ahead." So there are videos in there of she and I talking for an hour about Running Records, and I have a whole Padlet of resources for the group that has all kinds of information, great resources on fluency as well. So it's a wealth of knowledge in one place that you can learn all about this. And I'm always right there. A lot of my time is online, providing advice to people within Facebook, because it's a great platform to be able to share things like that. And so a lot of my life is being in there, and providing advice, and guiding people along.
inaudible beginning a book study starting next Monday, but I know this will not be out in time for that, but all the information in the book study is going to be within the Facebook group. So no matter when anyone joins the group, it'll be there for you. So you can see the posting, and the questions, and the comments, and all. Because until we understand that trajectory of learning, and, then you can't really be looking for that, right? We've got to know, "Oh, that child is counting on. They're counting all." Now, what I want to make sure I'm clear about a derived strategy. So, for example, one really important fact for me is when I get to the progression, the beginning of the progression, say to add zero, and add one to numbers, and then counting on within five, and then within 10... But then I'm going to get you teen numbers, you have the doubles, and then doubles plus one.
So when I ask kids five plus six, most kids tell me that they know it's an 11, because five and five is 10, and one more is an 11. So it's a derived fact. The using a fact that they know to figure out one that they don't know. But the very next question in part two gives them more problems in that zone. The very next question is six plus seven. And that's where I can tell the kids who have begun using strategies, and breaking apart numbers in flexible ways, versus the counting on, because the kids will count on six from the seven. So the very same student that tells me five and five is 10, one more makes an 11, so they know five plus six makes an 11, will count on their fingers, six, higher than the seven. So I can tell that there's no strategy thought there.
So for that kiddo, I'd want them to go on to explore what doubles plus one is. And I would show them like a rekenrek. So a rekenrek is a tool that, one version of the tool, you can have two rows of 10, and the beads are color-coded in each five. So I can slide six beads over and seven down in the bottom row, so kids can see that amount of the 13. And I'll ask them, "What do you see? How do you know?" And some kids will count all the beads. So I know they're in a counting all phase, some count on the six from the sevens, they're in counting on, but other kids will say, "Well, I'm seeing five and five red, and that's 10. And then I see one white and two white, that's three. So I see ten plus three, which is a 13." But other kids say, "Well, I see six and six is 12, and one more makes 13." Other kids say, "I see seven and seven is 14, I take away one is 13." And still other kids say, "Take three from the six and make a 10 with the seven. So it's 10 plus three."
So I'm aware of all these different look fors to then develop my own instructional response with them. Because the kiddo that tells me that six plus seven is 13, because they could break apart that six to be a 10 and a three? They go to the end, they do all their math facts, and do whatever strategies they want with any of the addition math facts. But the kiddos that are counting on their fingers at that point? They go to just having visuals, to see the quantities, and describe what flexible ways can we get to that total, and make it easier for our brain. So that's where I find it's a great place to begin the journey.

Kyle Pearce: Right? And, forgive me, I'm not an elementary school teacher, haven't had experience working, other than with my own kids, talking about numeracy at that level... I'm wondering, if I'm a teacher who wants to have that interaction, that lesson with my students, and, what do I do beforehand? How do I prepare for that? We often talk about, that preparation is so important. Another quote that we've used from Tom Schimmer is that you want to plan with precision, so that you can proceed with great flexibility. In that preparation, we can't underestimate that. So I'm wondering, what does a teacher do to help prepare so that when they get in a conversation like that, they know to look for those strategies, and they know, "If I have that strategy, I'm going to push them this way." What would you suggest the teacher can do to help get ready for something like that?

Ann Elise: Well, I think it's a matter of us building our own knowledge about that. Dr. Nicki has a class, an online class, on her site. I've also, on my Padlet, have created videos for each of the strategies, and how we can explore them with math tools. And there's no cost to any of this. The protocols to interview the kids are free, there's no cost to the Running Records. But really it's a matter of us learning what we're looking for, right? And so how do we do that? Well, her Math Running Record book describes all the four operations, and the thought behind each of the operations, and the strategies.
And then our book we've written so far, the addition/subtraction, we are currently working on a multiplication and division book to go along with it. But that's a matter of learning about it, the matter of just building our own... Christina Tondevold, building our math minds, so we can then build the math minds of the kids, right? Because if you don't know what you're looking for... I ask people all the time, when kids are counting. You put eight objects in front of a student and ask them to count - what are you looking for? And very basically I always hear about one-to-one correspondence, and saying the numbers in the correct order, and that's pretty much it.
Well, Clements and Sarama has 20 levels of counting, right? So their site, learningtrajectories.org, I send all my participants to that site, because it is the most wealth of information about, from birth to grade three, of early numeracy and mathematics. But until you know there are 20 things you could be looking for with counting, you're going to look for one-to-one correspondence and if they say it in the right order, right? So it's a journey.

Kyle Pearce: I know. And I love that response because I think that's where the content knowledge piece is so important. And I think in mathematics, I mean, in education in general, oftentimes teachers, we have this drive to find new pedagogical moves. And while they are important, and they are a huge part of how you're going to organize and run your lessons, the content knowledge is key because if you don't have that, then it doesn't matter how hard you try to prepare, you're underprepared, because that look for is not there. Being able to notice and name, and then being able to, essentially, once you've noticed and named what a student has done, to know what comes next. And Jon and I are constantly talking about that with our igniting the next moves piece of the framework, which is this idea of really understanding that trajectory of math learning that students travel down. And that's something that we continue to try to do, and, clearly, you are also an advocate for that, continuing to try to grow your own math mind as well.
And we're just thrilled that we've got some great ideas and some great links to share. We're going to definitely get that Facebook group up there in the resource page. We'll throw in learningtrajectories.org, which is a great resource. And we'll also grab some of those Running Records links, as well as to the book, so... I'm wondering, before we wrap up though, is there anywhere that the Math Moment-Makers can learn more about you specifically in your work, some of which that you've already mentioned here, but where can they find more about you and get connected?

Ann Elise: Sure. I have a webpage, anneliserecord.com. You can check that out, and check out all the different offerings and things I have for you there. And honestly, I have found that connection in the Facebook community because I was living in Northern New Hampshire for over 20 years, and I found this amazing collection of people online. So they're just brilliant and generous and kind. And so, I began in Twitter a little bit, and then in Facebook, creating communities of people willing to learn. So I run a Facebook group of my own called Elementary Math with Ann Elise, and that's any math topic, K five is on the table, anything you want to talk about to give advice and all.
And then I facilitate the Math Running Records one. We have over 13,000 people in there right now. And so that group is focused on fluency. So that's the fluency piece. So the Elementary Math with Ann Elise is that, anything, like fractions or any other topic on mathematics, but the Math Running Records is that fluency one. So joining in those groups or following me on Facebook... I have a growth mindset tree as my picture, because that's my math profile on Facebook. So I only share math things there. So if you follow me there on Facebook, you could continue learning about me.
And I also have a newsletter that I do because I know and respect the fact that people may not want to be on Facebook or social media, so they can subscribe to my newsletter, and I send it out but once every week or two, with all the links and the great things that I come along.

Kyle Pearce: Awesome stuff, thanks for sharing that. We'll make sure that all of that is in the show notes page for this particular episode. Ann Elise, we want to thank you for taking time out of your day to share your expertise with us and our listeners. And we hope you enjoy the rest of your day. And we really appreciate talking to you and hopefully we can catch you at a face-to-face conference sometime in the future.

Ann Elise: Let's hope, absolutely. Thank you so much for having me, I really appreciate it.

Kyle Pearce: Thanks so much, we'll talk to you soon.

Ann Elise: Okay, take care.

Kyle Pearce: It was awesome to finally welcome Ann Elise onto the show to chat with us about her amazing efforts in conceptualizing mathematics.

Jon Orr: Hopefully this conversation has inspired you to dive into your next math class using a problem-based approach.

Kyle Pearce: Right? So be sure to check out the Wooly Worm Race, the Snack Time unit, or any of the other problem-based units that are available for you to dive into right now over at makemathmoments.com/tasks.

Jon Orr: Absolutely, Kyle. Get on over to makemathmoments.com/tasks to run one of these awesome problem-based units in your class today.

Kyle Pearce: In order to ensure you don't miss out on new episodes, make sure that you smash that subscribe button on Apple Podcasts or on your favorite podcast platform. And, did you know that these episodes are also on our YouTube channel? And you know what, Jon? We've been doing a lot of these YouTube lives lately, so be sure...

Jon Orr: Yeah, we're getting ready...

Kyle Pearce: To subscribe over on YouTube, because we got some pretty cool little tricks up our sleeve that we're hoping to do each and every week.

Jon Orr: Yeah, we've got some new stuff coming. Also, if you're liking what you're hearing, please share the podcast with a colleague, help us reach our wider audience by leaving a review on Apple Podcasts, and tweet us over at Make Math Moments on Twitter, or Instagram, or on Facebook.

Kyle Pearce: Show notes and links to resources, including full transcripts that can be downloaded or read right from the web for this episode, and all the other episodes can be found over at makemathmoments.com/episode113. Again, that's makemathmoments.com/episode113. Well, my friends, until next time, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us...and a big high five for you.

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Episode #113: Fluency Doesn't Just Happen - An Interview with Ann Elise Record - Make Math Moments

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Today we speak with Ann Elise Record. Ann Elise is a seasoned educator with a career spanning over two decades of elementary classroom experience. These days she's a K-5 math coach who works with districts on developing strategies to increase students' fact fluency as well as how to use manipulatives in the classroom.

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Dr icki Newton and I joined podcast host Melissa Milner to create a List of 10 favorite math daily routines. Such fun!

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The Teacher As...Top Ten episodes are a new addition to the podcast. Dr. Nicki Newton and Ann Elise Record collaborate to create one top ten list of math routines. The collaboration gets a bit intense when the two disagree as they get closer to the top five picks. These math educators have so much to share in this episode.

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030: The Teacher As...Top Ten Math Routines with Dr. Nicki Newton and Ann Elise Record - The Teacher As...

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The Teacher As...Top Ten episodes are a new addition to the podcast. Dr. Nicki Newton and Ann Elise Record collaborate to create one top ten list of math routines. The collaboration gets a bit intense when the two disagree as they get closer to th...

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My Brainingcamp Webinar on the Progression of  Addition K-5

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My Brainingcamp Webinar on the Progression of  Addition K-5

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by Dan Finkle

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These beautiful flash cards use spaced repetition to teach multiplication facts. Achieve fluency with just five minutes of practice per day!

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I was interviewed by Melissa Milner on her podcast. We talked everything elementary math!

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How to reach Ann Elise Record:

Website and newsletter signup: https://www.anneliserecord.com

Facebook: https://www.facebook.com/annelise.record.94

Twitter: @AnnEliseRecord

Instagram: ann.e.record

Email: aer@anneliserecord.com

Transcript:

(transcribed by kayla.r.fainer@gmail.com)

Melissa Milner 00:09

Hi, this is Melissa Milner. Welcome to The Teacher As... podcast. The goal of this weekly podcast is to help you explore your passions and learn from others in education and beyond to better your teaching. The Teacher As... will highlight uncommon parallels to teaching, as well as share practical ideas for the classroom.

In this episode, I interview Ann Elise Record. She is a New Hampshire certified K-8 educator, as well as a New Hampshire elementary math specialist. She's been an educator for 20 years in the roles of classroom teacher, K-5 math coach, adjunct faculty member for Plymouth State University, Bureau of Education and Research presenter, and currently an independent elementary math consultant providing training for both in person and virtually. She's a contributing author, too. Together with Dr. Alison Mello and Dr. Nicki Newton's book, Fluency Doesn't Just Happen with Addition and Subtraction.

Ann Elise works with school districts and presents at educational conferences. Her passion is working with educators to help them implement best practices within the three basic pillars of classroom math instruction that encourage growth mindset messages, math fact fluency, word problem structures, and understanding progressions of the standards. I hope you enjoy my interview with Ann Elise Record.

Welcome to The Teacher As..., Ann Elise. What would you like The Teacher As... listeners to know about you?

Ann Elise Record 01:37

Well, I am passionate about elementary mathematics and helping to create students that have a foundation of mathematical understanding that will follow them their whole journey and that they understand that everyone can learn math. It's not just certain math brains. That's a myth, and everyone can learn and grow.

Melissa Milner 01:55

Absolutely. You have done a lot of work with Dr. Nicki Newton with Running Records. I was part of the book club for Choral Collections & Counting Collections. You talk about the three pillars of math instruction. There's so much to talk about. Where do you want to start?

Ann Elise Record 02:15

Well, honestly, I think the start is Math Running Records, because I find that to be the key to help transform the math journey for not only the students, but the teachers as well. Because my undergraduate degree is in sociology. And so in order to teach, I needed to get my Master's degree. And I had taken some mathematics courses in college.

So when they look over my transcripts, they wanted to have the mathematics covered. I had one class in my grad program, and it was Math in Literature. So I love using picture books to explore math and all. But I really was not taught about the intricacies of building number sense and early numeracy. And so I came out certified K-8 to teach. But I really wasn't feeling any more qualified to teach kindergarten than I would have an eighth grade science class or something. I think the expectation was you would be getting a textbook and following whatever the lessons are in the textbook, and so indeed, I did.

I taught grade 5 for 13 years, and we had a textbook for most of those years. But all I knew was how I taught, which was rote procedure, and memorization, and valuing getting the right answer. And over the course of my career now of understanding mathematics and having kids talk about their thinking and build flexibility of thought and that the algorithm may not be the destination, all that work is embedded within the Math Running Record. Because it's getting us to be able to interview kids about their thinking and knowing for the research--

So Dr. Nicki spent 10 years researching math fact fluency, and then created this protocol of how we can interview students. She created a progression of strategies for each of the four operations. So we can find out where the students are, what set of facts are causing them a super slow down or inaccuracies or inefficient strategies, and then find out where they are on the trajectory of Baroody's research of counting all and counting on and then counting it all in their heads, and then derive strategies until finally developing a mastery with understanding. So it becomes a blueprint and a layout for the instructional response for the kids.

But in the process, when I'm training districts on how to do this, I'm really teaching them about early numeracy. And so everything else benefits. And then if we apply those same strategies to larger numbers, and eventually decimals and fractions, I mean, it unlocks the math journey for so many more of our students that typically had been excluded, because they weren't able to memorize or they weren't able to follow the 10 step procedures of long division. And so it just transforms the idea of mathematics. And I find a lot that teachers that really did not like teaching math, it becomes a favorite part of their day. And they had been literacy describedpeople. But they're like, this is so amazing.

Melissa Milner 04:53

It is.

Ann Elise Record 04:53

I know!

Melissa Milner 04:54

It's so diagnostic. It is time consuming, but it is so worth it. And it actually saves you time in the end. Because you realize how your kids think and what strategies they do use. And you know, I started doing them this year. I found I was underestimating what these kids could do.

Ann Elise Record 05:14

And the opposite, too.

Melissa Milner 05:16

Yeah. And these were kids that were historically struggling math students. And they were doing the most sophisticated mental math and taking these steps. And I'm like, oh, I get it now. I can play on those strategies that work for them and make them more efficient. And you know, it was amazing. It was amazing.

Ann Elise Record 05:39

I think it works the other way too, though. There are a lot of teachers that tell me that the kids they thought that would ace the Running Record did not.

Melissa Milner 05:47

Nope.

Ann Elise Record 05:47

Because they've gone down the memorization path, and they've got no strategy sense and no number sense at all. They're just-- they've memorized the answers. And they won't even try harder ones, which is why after I give the multiplication running record, that tends to be something that kids have traditionally memorized. I don't find many that have done it successfully.

But there are many kids who have successfully memorized the math facts. So I asked them when it's all done, when I'm assured that most of the higher facts are accurate and relatively speedy, I asked them 4 times 17. And that separates out the kids who have some kind of number sense from kids that don't.

The kids who are going down a path of memorizing, they literally will tell me, I haven't learned that one yet. They think it is a discrete thing. The kids who are going down a path of exploring their fluency, they've developed automaticity. They've developed their mastery of the facts. When I asked them 4 times 17, they're like, well, I don't know it offhand. But I could double double!

Melissa Milner 06:41

4 times 10. 4 times 7. Right, yeah.

Ann Elise Record 06:46

Exactly. They have some strategy of number sense to attack it, and they understand that they might make a mistake, but that's okay. And it just unlocks that math journey, and it can be much more accessible for many more of our students.

Melissa Milner 06:59

It's so exciting. And I mean, I was like you. I graduated in '91.

Ann Elise Record 07:04

Oh, I was '90.

Melissa Milner 07:06

Yeah. And it was absolutely like, here's the textbook, and here you go. And just the Build Math Minds site, being a member on that has changed my whole world. And I have to give my district credit. The other thing that pushed me into really looking into Build Math Minds and basic number sense, and Van de Walle, and all of it was that my district decided not only are we doing reading workshop and writing workshop, but we're going to be doing math workshop.

And I'm like, okay. I've got to think a different way to make math workshop work, you know? I can't be sitting in that book all day long in a math workshop. So that's what pushed me into it. And then I mean, Dr. Nicki, you know, we made the beaded number lines, and it's so fun, and it's so creative. I had no idea math was creative.

Ann Elise Record 08:00

Yes, exactly. The two of them have changed my life honestly. I had gone to a Bureau of Education and Research conference. It was a two day conference back when the core standards had just come out. So there were literacy sessions and math, you could choose which ones you wanted to go to. And one of the first ones I went to was Dr. Nicki. And she was showing me things I had never seen before with math, open number lines - it was the first time I'd ever seen an open number line - and double number lines. I'm like, what world is this?

And I stayed with her the entire two days. And then when I became a math coach in my district, we had some money to be able to hire her to come to our district for a week and do some PD with all the teachers. And she doesn't like to drive, and so I became a taxi driver. Because in Berlin, New Hampshire, in the north, there are no taxis. At least there weren't at that point. And so we became fast, close friends, and we put the entire week together. And she's one of my treasured friends, and so that began our collaboration together.

And then during that same year was when Christina Tondevold did her first number sense one on one course. And I knew becoming a coach K-5 that I had to learn about the early numeracy. I did not know how kids learned about early numbers. And so I took that course, and that changed my trajectory, as well. And so then that continued my journey with Christina and working with Build Math Minds, and then creating content for her site. I still do that to this day.

Melissa Milner 09:17

Yeah, it's fantastic.

Ann Elise Record 09:18

So those two have been my mentors, they have changed my life.

Melissa Milner 09:22

I was about to teach fluency for multiplication. So I went on, and I looked at the number sense, there's the four pieces, the subitizing, the--

Ann Elise Record 09:34

Is it the verbal counting, the object counting, the decomposition, subitizing. Yes.

Melissa Milner 09:39

As part of my instruction for multiplication, I was really checking and making sure that the kids could do those things. And then I was getting ready for fractions. And she was like, same thing for fractions. I'm like, oh yeah, benchmarks. Okay.

Ann Elise Record 09:57

It all fits together. That's the beauty of it.

Melissa Milner 10:01

Amazing.

Ann Elise Record 10:02

It really is. The biggest one for me, when I'm doing parent nights with districts, I do like to have a parent night in the districts that I work with. Because we have to get the parents on board, they have to understand why we're doing what we're doing. And so I share with them some foundational understanding of fluency. But then I show some future impact. And I say 4/5 plus 3/5.

And a lot of the people have trauma with fractions, because that tends to be a gateway where some people have very negative experiences. Multiple I meet have negative experiences with math, but fractions tended to be one place where they're like, oh, I can't do it fractions. They have anxiety about it.

But then I talked to them about the same thing. I talked to them about basic fact fluency to break apart numbers to make a 10. And I'm like, well, what if you broke apart that 3/5 into 1/5 and 1/5, and just rename it as 1 and 2/5? I mean, it's so much more accessible than just add numerators, keep denominators the same. And it feels weird, because there's a large number on top, that whole thing just goes away, and we have conceptual understanding. And the parents are like, if I had learned it this way when I was younger, I wouldn't feel the way that I do now.

Melissa Milner 11:05

It's so true. It's so true.

Ann Elise Record 11:07

It's so true!

Melissa Milner 11:08

Yeah, it's really-- I'm like, I'm a mathematician. That is a shocker, because I really struggled in math. And I worked and worked and worked to just get by. And if it had been taught this way, I would have gotten it. I would have been zooming along, and maybe I would have decided to become a veterinarian instead. The math and science scared me.

Ann Elise Record 11:33

But I think that's also true of a lot of our students. A lot of our kiddos get put in interventions, because they can't follow the procedures. Because what we're expecting of them is to get a right answer and to follow algorithms. And they're not able to follow that with all those different steps when it has no meaning attached. Whereas if we taught it in these conceptual ways, we wouldn't have as many kids needing an intervention.

Melissa Milner 11:52

Yes. You mentioned the consulting. What do you usually ask to come and do in schools? Or what is your goal when you walk into a school or a district to make their math curriculum and teaching and pedagogy better?

Ann Elise Record 12:07

Well, first of all, I focus on the eight math practices. So these are gonna be in place for kindergarten classrooms all the way to high school. Things like having kids problem solving and persevering at that, modeling with mathematics, using math tools, talking about their thinking, the underlying structure, not only our number system, but also of word problems structures, as well.

So I imagine they're like three pillars to our math instruction. We want to give instructional time, too. There's the fluency piece, which needs to be given some instructional time and attention. There's the word problem and structures of those, so it's not just solving word problems and certainly not keywords. Those lead the kids astray.

That develops kids that are number pluckers, as Graham Fletcher calls them, where they pluck the numbers out and do a random operation of trying to find one word that tells them what operation to use to get the right answer. But if we explore the structures of thetypes based on the research of Thomas Carpenter and his colleagues in Cognitively Guided Instruction, then we can show what's happening with the situation and then give the flexibility to the kids to numerically solve it however they want.

But if we give a typical problem that people think of as being division, which really, there are three different types. And if one of the factors is unknown, then it could use division to solve it. But I could use addition, subtraction, multiplication or division to solve it. I don't have to use the division operation for it. So we need to get away from talking about problem types as being defined by the operation that we want kids to use to solve it, and instead talk about the structure of a situation.

Melissa Milner 13:34

Yeah, take that poster down, take it down.

Ann Elise Record 13:38

Yeah, oh my gosh, take down those posters. So when I go into a school, I'm certainly looking on the walls. Because Dr. Nicki says that the walls reflect what's important to you. And generally, what I see is 75% of the walls are covered in literacy posters. There usually is a calendar corner or a hundreds chart that I'll see.

But beyond that, I'm not seeing much else for mathematics in many places. And so I'm looking for that. And I'm looking for those keyword posters to have a conversation about why we don't want to have those on the wall. So typically, I've been brought in-- this is my second school year being a consultant. And I've also been presenting for the Bureau of Education and Research, which is a full circle for me to be a presenter at the same conference I met Dr. Nicki at.

Melissa Milner 14:20

Wow.

Ann Elise Record 14:21

Fabulous for me to have that. But so I work with districts. Very often the Math Running Records is the one that gets me in the door, because the fluency piece is just so important. And that is, I believe, the gateway to help fix everything else. But then once you've gone beyond exploring the strategies with the basic facts and how they apply to larger ones, then I tend to go into the problem solving.

And so the idea that there are 15 different problem types and involve adding and subtracting, and there are 9 that involve multiplication, division, single step, teachers are like, what? So if we don't know that stuff ourselves, then certainly we're not facilitating that learning with our students. So that tends to go into the next time, next phases, and then we go into daily routines, things we can do every day that can kind of get that fun and engaging way to get the mathematics out in front of the kids and having them talk about their thinking and building the flexibility and respecting each other's viewpoints. So there's not just one way to get an answer.

So all that kind of work tends to be what I've been doing so far in my consulting work. And then whatever districts want, I'll do. I've done workshops on fractions and place value, those kind of things. But generally, it's the Math Running Records, because it's really teaching everyone about early numeracy and then the problem types. Because I find those two pillars-- because we have the fluency, the problem solving, and then we have the regular content standards. How do I teach kids about subtracting two digit numbers in grade two? And that takes a lot of sophisticated thought.

It's not the algorithm. The algorithm should be in grade four. There's a reason why it's in grade four. There are other ways you can do it. But very often, when I ask people, I usually do 52 minus 38. And I ask them to think about how many strategies they can think of to figure out the answer. And I usually get two, three or four. But there are a lot more than that. But we just haven't exposed our brain to it yet.

So it's just a matter of exposing our brain and practicing it and just building those new neural pathways that allow us to think more flexibly about it. But I could apply that same thinking to mixed numbers and subtraction. So I no longer do that regrouping algorithm when you give me two mixed numbers. I have many other ways to think about it that are much more efficient for my brain.

Melissa Milner 16:31

Do you have any suggestions about doing this work that you've just talked all about, Running Records, and that deep word problem, delving into those different word problems and so on? Do you have any suggestions about how to do that with a remote model or a hybrid model when we're not in person with the kids all the time?

Ann Elise Record 16:57

Well, I've been doing a lot of virtual Math Running Records, actually. Because since the shutdown in March, all the districts I was planning on working in couldn't have me because the schools were closed. So my consulting and being on the road all the time came to a screeching halt. And so I had special time on my hands.

And I thought, you know, I should ask for volunteers to have kids talk to me about their thinking. And so I did a lot of virtual running records. And it actually worked out really awesome, because I could easily record it. I did it in Zoom. I would have my phone signed in as another device, and I just pointed it down. I put it on some Harry Potter books, because they're really thick, and put my phone facing down onto the desk and showed the benchmark expressions for the kids for the part one.

And then I talked about their thinking, and I could easily just record it and have it to go back to look at later. And then I could figure out from the interview where I want them to begin working on their fluency. And then I could suggest games and activities to the parents to help work on that with the kids. Because I want to engage parents in this learning. And I think basic facts and playing games is a really great entry level to it. They can feel comfortable with that.

And so it can help us fix that problem by encouraging the parents and encouraging positive interactions between the kids and the parents and not having it be this fearful, anxiety prone-- The parents don't understand what we're trying to do right now with math, and so they freak out, and they call their friend. Did you do tonight's homework. Or they go on Facebook, and they bash it. They need to understand that there's a larger thing at work here beyond what they see, what they don't understand. It's scary to them, you know?

So if we can have them play games with their kids and encourage flexibility of thought, yeah, it's a win win. And so certainly with the fluency piece, I like interviewing the kids. And a lot of teachers were meeting with their kids one on one anyway to kind of build a relationship. So while you have them with you, you take 10 minutes and ask them about their math thinking. And then that can help guide your decision on what you want them to do moving forward.

Melissa Milner 17:14

Yeah, that sounds doable. But my other question, I think it's still going on, your Cuisenaire course?

Ann Elise Record 18:54

Oh, yes.

Melissa Milner 18:55

Do you have remote ideas for manipulatives?

Ann Elise Record 19:01

Yeah, in fact, with my course, you get a four month subscription for a classroom to Brainingcamp. And Brainingcamp is a web page that has virtual manipulatives that are fabulous. And in fact, research has shown that virtual manipulatives have shown to improve engagement with mathematics, as well.

So doing it virtually, I mean, I totally believe that doing hands on things and getting concrete objects in the hands of the kids is really important. But virtual is not a bad second place right now when we're dealing with a pandemic and don't want to have to wash off any hands on things with germs and whatnot, right? So having it virtually is not a bad second choice.

Melissa Milner 19:38

Even if you're in, even if you're in school.

Ann Elise Record 19:40

Yes, exactly. And Brainingcamp actually is affiliated with Hand2Mind, which is the company that has the trademark on Cuisenaire Rods.

Melissa Milner 19:47

Oh, yeah.

Ann Elise Record 19:48

And so they've offered to give a four month subscription for a classroom to their app. And they have a Cuisenaire Rod app that is amazing. It's the best that I've seen. So yeah, that is a part of the course. I mean, in the course, I do go through the three pillars. I go through fluency, problem solving, as well as the content standards for each grade level. I have a K-2 version out now, and then also, I just released my 3-5 version.

Melissa Milner 20:11

And is it too late to get into the K-2 if people are listening and interested?

Ann Elise Record 20:16

I don't have plans to close it. I want it to be available for people when they want to learn about it. So no, I don't have any plans to close it. It is asynchronous. So there are videos and some resources within it that people can do it at their own pace.

Melissa Milner 20:29

Gotcha.

Ann Elise Record 20:30

So yeah, you've got the K-2 version and the 3-5, and then I go through each grade level of the content standards and how Cuisenaire Rods, they're my favorite manipulative. Because I think they have the key to help us move our kids from the counting phase of reasoning into strategies into additive thinking, and then from additive thinking into multiplicative reasoning. So it helps us move them all on the journey. Pam Harris has a wonderful graphic, that she developed on the development of mathematical reasoning.

And so we think of our kids entering kindergarten, they're in the counting phase of reasoning. When they go into addition, and particularly when we use the algorithm too soon, kids are doing the algorithm, which of course, is just single digit facts. And if they haven't developed the fluency yet with the math facts, they end up counting on within the basic facts of let's say a two or three digit addition problem. So they're doing a counting phase of reasoning in the context of a multi-digit addition.

So instead, I want to help move them into additive thinking, which has place value attached to it, as well. And then, very often, kids - beginning in multiplication - will start skip counting. And then they continue doing the skip counting for the basic facts. So in multiplication, we want multiplicative reasoning. They're doing a multi-digit multiplication problem, which is, again, just basic facts. They're ending up skip counting on the basic facts.

And so these are the things we're aware of in the Math Running Record to see where they are on that phase of that journey. And then we're able to explore those things with games, activities and daily routines to help move our kids forward on that journey. And the Cuisenaire Rods, the numbers exist as a group. So when I use Base Ten blocks, which I did in my fifth grade classroom to do subtraction for my kiddos who still couldn't do the algorithm up, I wanted them to understand why the algorithm worked.

So we traded in a 10 rod for 10 cubes, and we already had 8 cubes sitting there. And you can't subitize 18 cubes. The kids, then, would take away the ones that were being removed. And then they would count the ones that were left. And I realized, all they're doing is counting. They're not doing any strategies. So that, to me, the Cuisenaire Rods are the key to help moving that forward. And so that's why I just love them so much.

Melissa Milner 22:29

That sounds pretty amazing. And you said you're doing a grade 3-5, as well?

Ann Elise Record 22:33

It's already open. I opened it a few days ago.

Melissa Milner 22:36

Oh, great.

Ann Elise Record 22:37

And you can get a bundle, as well. You can get both classes for even less money.

Melissa Milner 22:41

You'll give us that information at the end.

Ann Elise Record 22:43

Yes, absolutely.

Melissa Milner 22:44

All right. So what are you zooming in on right now with your work?

[ZOOMING IN SOUNDBITE]

Ann Elise Record 22:51

So certainly, being in person and doing trainings is not feasible right now. But it's been actually kind of nice. I've been doing virtual trainings from my house. So I've had lots of sessions with Nebraska. I had Texas last week. I did New Jersey this morning. So I'm able to still meet with people in this virtual space. I do it through Zoom. It allows me to have breakout groups, and people can still be having those small group conversations.

There are also some ways virtually to get people active, like doing a Google Drive, like a sheet, a Google Sheets file. So people can access that file and collaboratively work together. So I'm still finding some ways to be able to do the trainings. But I'm able to have dinner at night with my husband and be home with our puppy.

Melissa Milner 23:30

That's amazing.

Ann Elise Record 23:32

And so I also still work. I do make content for Christina Tondevold's Build Math Minds for pay, professional development site. And facilitating the Facebook groups takes me a lot of time. I do the Build Math Minds for Christina. That's got over 56,000 members. And then my Math Running Records one, all working on fluency with Dr. Nicki's work, that's got over 13,000 people. I think for me, creating these online courses that teachers can access at a reasonable price for teachers and to be that support to people online is really the heart of what I'm doing right now.

Melissa Milner 24:08

That's great.

Ann Elise Record 24:09

I'll get back out in person as soon as there's a vaccine for it, but I'm not in any hurry to get out there because this is scary stuff. And Dr. Nicki had COVID.

Melissa Milner 24:19

I know.

Ann Elise Record 24:20

And she was in a coma for two months, but she is getting stronger every day, which we talked yesterday. She's looking great. But you know, it's a long journey to recover. So, you know, this is no joke. So I'm not in any hurry to get out there traveling or flying around the country like I was doing.

Melissa Milner 24:36

Absolutely. It's not worth it. Can you tell us about the book that you worked on with Dr. Nicki?

Ann Elise Record 24:42

Sure. Dr. Nicki and also Dr. Alison Mello and I collaborated together on a book called Fluency Doesn't Just Happen with Addition and Subtraction. So it's all about this fluency work and exploring strategies and number relationships with kids. But what we're bringing to the conversation is also about using concrete pictorial and abstract ways to explore the strategies.

So in the book, there's guides about creating fluency plans and including parents on the journey and the foundation of the flexibility of thought. But also, for each of the strategies, you get ideas for videos you could show the students, and picture books you could use, and all kinds of ideas for your workstations that are concrete, pictorial and abstract, word problems you can use, and even a quiz to help measure at the end of working on a strategy whether a student's ready to move on to the next strategy.

So we have not yet begun to write the next one on multiplication and division. But that's certainly our goal, to get that one done, as well. But yeah, it's a wonderful, wonderful book. It's really the next steps to the Math Running Records. The Math Running Records by Dr. Nicki Newton, that book is about how we talk to the kids about their thinking with all four operations and then the codes involved with that, because the research informed that went behind that. And then this book is the next steps.

Now that I have the Math Running Record done and I know what strategy I want each student to be working on and what level they are with that progression of counting all, counting on, doing that also within their head, and then derive strategies to mastery, how can I help them move forward? These are getting all kinds of ideas at your fingertips that you can do with the kids.

Melissa Milner 26:13

That makes me so happy. Because it is! It's like, great, I have all this great information about each of my students. And I think I know what to do next. So that's really, really helpful. And I know everything you guys do is based in research and work with real students.

Ann Elise Record 26:30

Absolutely. And if anyone's interested in that, I do run the Facebook group called Math Running Records. And with that, I've created what's called a Padlet, which is a web page. So it's not affiliated to Facebook or social media. So if you're not doing social media, totally fine. It's a web page that has lots of resources on there for you. And there are also some videos of me giving running records that you can see. And Dr. Nicki has her own site called mathrunningrecords.com. And that's where you can get the downloadable sheets.

It's all free. There's no cost to them. And so you can check out her site, and there's some videos of her giving them, as well as the downloadable recording sheets. And then on my Padlet, I have lots more information for you. It's a rabbit hole of time, but you can get access to that, as well. So there's lots of resources for no cost out there.

Melissa Milner 27:16

Very cool. Are you on Facebook and Twitter and stuff like that, too?

Ann Elise Record 27:21

I sure am.

Melissa Milner 27:21

Do you want to share that? You can share that with us, yeah.

Ann Elise Record 27:24

Sure. So on Facebook, you can befriend me at Ann Elise Record. But look for the logo that is a growth mindset tree. If you see a picture of me and my adorable Marty McFly puppy, then that's my personal one, and you'll see pictures of my kids and my three year old niece and the dog. But if the growth mindset message tree, that's all my math postings. I only post in Facebook on that profile math things. And so I have that.

And then within the groups, the Math Running Records Facebook group, as well as the Build Math Minds Facebook group. And I have my own one called Elementary Math with Ann Elise. And that's for any topic of elementary mathematics, fractions, anything else you want to talk about. And then on Twitter, I'm just @AnnEliseRecord, just my name. And in Instagram, I'm ann.e.record.

So I'm active in all three areas, mostly Facebook. I find Facebook is my home. That's where I live. Because I'm able to share something so easily within the Facebook, videos, and links to webpages, and then the Padlet that has links to everything.

So the Padlet has everything that I offer. I have the free version for the Math Running Records. But I also have a version for my consulting work, because of course, that's my livelihood. So I will not give that link away, but I will give the free one away.

Melissa Milner 28:42

No, that's awesome. Have you ever thought of doing a Twitter chat?

Ann Elise Record 28:47

I had actually led several within the Elementary Math Chat. So it's a #elemmathchat. So I've led several of those. I've got one that was coming up at the end of September. I'm not sure whether it will correlate to the date that this releases. So I do participate in those. I haven't yet led my own special hashtag one, no. I have not done that.

Melissa Milner 29:11

I think you should.

Ann Elise Record 29:13

An interesting idea. I've never thought of that.

Melissa Milner 29:17

Yeah, it's time consuming. But I think a lot of people would love to, once a week, be able to interact with you and ask questions or however you want to structure the chat. I think it would be great.

Ann Elise Record 29:30

Well, what I've been doing, actually, I have held some Zoom calls with my Elementary Math with Ann Elise group.

Melissa Milner 29:38

Oh, cool.

Ann Elise Record 29:39

So if you're a member of that Facebook group, then I share a Zoom link, and then we get together and just chat.

Melissa Milner 29:44

That's great!

Ann Elise Record 29:45

Oh, yeah. I find Twitter to be a little constraining with the characters.

Melissa Milner 29:50

Yes, I agree. All right. So my last question is very-- I don't know how to say it. A lot of people have found it to be difficult. What is your favorite movie, and why?

Ann Elise Record 30:01

One movie that I will watch over and over and over again is While You Were Sleeping.

Melissa Milner 30:06

Oh, I love that movie!

Ann Elise Record 30:07

Have you seen that one with Sandra Bullock? I just love that whole storyline of the family taking her in and falling in love with the other brother. And yeah, that's the one that if it's on television, I will just sit and watch it. I just love it.

Melissa Milner 30:21

That's definitely a guilty pleasure movie.

Ann Elise Record 30:25

Yep.

Melissa Milner 30:26

Bill Pullman is great in that.

Ann Elise Record 30:29

Oh, he's so good.

Melissa Milner 30:30

He's so lovable.

Ann Elise Record 30:31

I know it. And then the end when he puts the ring in the-- Oh my gosh, I just-- And then the whole family is right there, too. It's like, oh!

Melissa Milner 30:40

That is such a funny movie. And she's so lovable. She's such a great actress.

Ann Elise Record 30:43

Oh, she totally is. Yeah.

Melissa Milner 30:45

Thank you, Ann Elise. This was absolutely wonderful to talk to you. And I appreciate you taking the time out to do it.

Ann Elise Record 30:51

Oh, thanks so much for having me. It's been a treat.

Melissa Milner 30:56

If you enjoyed this episode, and have not done so already, please hit the subscribe button for The Teacher As... podcast so you can get future episodes. I would love for you to leave a review and a rating, as well, if you have time. For my blog, transcripts of this episode and links to any resources mentioned, visit my website at www.theteacheras.com. You can reach me on Twitter and Instagram @melissabmilner. And I hope you check out The Teacher As... Facebook page for episode updates.

I am sending a special thanks to Linda and Lester Fleishman, my mom and dad, for being so supportive. They are the voices you hear in the Zooming In soundbite. And my dad composed and performed the background music you are listening to right now. My intro music was "Upbeat Party" by Scott Holmes.

So what are you zooming in on? I would love to hear from you. My hope is that we all share what we are doing in the classroom in order to teach, remind, affirm and inspire each other. Thanks for listening. And that's a wrap!

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The Teacher As Mathematician with Ann Elise Record - The Teacher As

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How to reach Ann Elise Record: Website and newsletter signup: https://www.anneliserecord.com Facebook: https://www.facebook.com/annelise.record.94 Twitter: @AnnEliseRecord Instagram: ann.e.record Email: aer@anneliserecord.com Transcript: (transcribed by kayla.r.fainer@gmail.com) Melissa Milner 00:09 Hi, this is Melissa Milner. Welcome to The Teacher As... podcast. The goal of this weekly podcast is to help you explore your passions and learn from others in education and beyond to better your teaching.

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Document I wrote for Getting Started with Cuisenaire Rods

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Document I wrote for Getting Started with Cuisenaire Rods

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Cuisenaire Rods Getting Started (copy 01)

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Steve Wyborney's Free Multiplication Course on Youtube

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Steve Wyborney's Free Multiplication Course on Youtube

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math "steve wyborney" "number sense" multiplication teaching elementary

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Share your videos with friends, family, and the world

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10 Anti-Racism Professional Development Books for Teachers

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A reading list for thinking about racial bias in schools.

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Each teacher and student brings many identities to the classroom. What is their impact on the student's learning and the teacher's teaching of mathematics?

This book invites K-8 teachers to reflect on their own and their students' multiple identities. Rich possibilities for learning result when teachers draw on these identities to offer high-quality, equity-based teaching to all students. Reflecting on identity and re-envisioning learning and teaching through this lens especially benefits students who have been marginalized by race, class, ethnicity, or gender. The authors encourage teachers to reframe instruction by using five equity-based mathematics teaching practices:

Going deep with mathematics

Leveraging multiple mathematical competencies

Affirming mathematics learners' identities

Challenging spaces of marginality

Drawing on multiple resources of knowledge

Special features of the book:

Classroom vignettes, lessons, and assessments showing equity-based practices

Tools for teachers' self-reflection and professional development, including a mathematics learning autobiography and teacher identity activity at nctm.org/more4u

Suggestions for partnering with parents and community organizations

End-of-chapter discussion questions

About the Authors:
Julia Maria Aguirre is assistant professor of education at the University of Washington-Tacoma. Her work focuses on mathematics teaching and learning, teacher education, and culturally responsive mathematics pedagogy, with a primary goal of strengthening access and advancement in mathematics education for historically marginalized youth. The National Science Foundation, Carnegie Foundation, and Spencer Foundation have funded her work.

Karen Mayfield-Ingram is associate director for the EQUALS programs at the Lawrence Hall of Science at the University of California, Berkeley. Her work focuses on mathematics and equity professional development, teacher leadership, and parent involvement. She is the author of The Journey--Through Middle School Math(EQUALS, 2005) and co-editor of the EQUALS Middle School Investigations series.

Danny Bernard Martin is professor of education and mathematics at the University of Illinois at Chicago. His research focuses on understanding the salience of race and identity in black children's mathematical experiences. He is the author of Mathematics Success and Failure among African Youth(Erlbaum, 2000), editor of Mathematics Teaching, Learning, and Liberation in the Lives of Black Children(Routledge, 2009), and co-editor of The Brilliance of Black Children in Mathematics: Beyond the Numbers and Toward New Discourse(Information Age, 2013).

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The Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices - National Council of Teachers of Mathematics

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Each teacher and student brings many identities to the classroom. What is their impact on the student's learning and the teacher's teaching of mathematics? This book invites K-8 teachers to reflect on their own and their students' multiple identities. Rich possibilities for learning result when teachers draw on these identities to offer high-quality, equity-based teaching to all students.

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Robin DiAngelo author of White Fragility

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Robin DiAngelo author of White Fragility

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* Ryan Honeyman from Next Economy, interviews Robin DiAngelo, and examines why it's difficult for white people to discuss issues of race, racism, and white fragility. * How can we start to evaluate and move past our white fragility around speaking about racism? DiAngelo says that racism is a system and is embedded in language. Why is it important to understand race and racism as a system?
* Read about white fragility in teaching and education.

Robin DiAngelo is a former associate professor of multicultural education at Westfield State University and currently affiliate faculty at the University of Washington. DiAngelo's scholarship is in critical discourse analysis and whiteness studies. In addition to her academic work, she has been a consultant, mediator and workplace racial equity trainer for over 20 years.

DiAngelo has written numerous publications, including her latest book, White Fragility: Why It's So Hard For White People To Talk About Racism, which has been on The New York Times bestseller list since it debuted in June 2018.

Next Economy's Ryan Honeyman interviewed DiAngelo, and here are some highlights from that conversation:

How Dr. DiAngelo first got into this work as a "classic white progressive" who was "clueless about racism."

Why good, open-minded, liberal progressives (who marched in the '60s) still have a fundamentally racist worldview.

How having one or more historically marginalized identities (e.g., being a woman, low-income, LGBTQ, Jewish) does not mean that one understands the experience of racism.

Why naming, disrupting and dismantling white supremacy shifts the problem to white people, where it belongs.

How the unexamined values of individualism, meritocracy, objectivity and conflict avoidance are part of the dominant culture and lead to problematic outcomes for people of color.

Read the full article about white fragility by Andrew Baskin at B the Change.

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White Fragility and Racial Equity: What's Stopping the Conversations [Audio]

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Ryan Honeyman from Next Economy, interviews Robin DiAngelo, and examines why it's difficult for white people to discuss issues of race, racism, and white fragility. * How can we start to evaluate and move past our white fragility around speaking about racism? DiAngelo says that racism is a system and is embedded in language.

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If you previously purchased a course from this school, you can login if you wish to contact the owner or view your transaction history.

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Leading Equity delivers an eye-opening and actionable discussion of how to transform a classroom or school into a more equitable place. Through explorations of ten concrete steps that you can take right now, Dr. Sheldon L. Eakins offers you the skills, resources, and concepts you'll need to address common equity deficiencies in education.

BUY THE BOOK

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Leading Equity delivers an eye-opening and actionable discussion of how to transform a classroom or school into a more equitable place. Through explorations of ten concrete steps that you can take right now, Dr. Sheldon L. Eakins offers you the skills, resources, and concepts you'll need to address common equity deficiencies in education.

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Facebook group that focuses on Math and also Equity and Access Issues

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Facebook group that focuses on Math and also Equity and Access Issues

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Log into Facebook to start sharing and connecting with your friends, family, and people you know.

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Comprehensive list of printable manipulatives

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Comprehensive list of printable manipulatives

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Ultimate List of Printable Math Manipulatives & Games

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The best websites for finding free printable math manipulatives.

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NHTM/DOE Webinar on engaging digital learning activities

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NHTM/DOE Webinar on engaging digital learning activities

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Virtual Tools for elementary math zoom_0.mp4

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NHTM/DOE Webinar slidedeck on Engaging Learning both in-person and virtual

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NHTM/DOE Webinar slidedeck on Engaging Learning both in-person and virtual

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Illustrative Math's blog on planning for the fall

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Illustrative Math's blog on planning for the fall

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By Kristin Gray, Director K-5 Curriculum and Professional Learning
and Kevin Liner, IM K-5 Professional Learning Lead

In our previous post, we highlighted important considerations in planning to support students in the fall. While we need to first explore these ideas conceptually, we must also consider what this looks like in practice. In this post, we explore the unit adaptation materials we are creating under these considerations. We explore this structure through the IM K-5 Math beta curriculum materials being piloted this fall, but we believe this is a generalizable process and structure that could be applied to other materials.

In deciding which content to emphasize and ways in which to support students in accessing grade-level content, we worked under beliefs in which the instructional resources would:

Incorporate prior grade-level knowledge and skills, when necessary, to support access to current grade-level content.

Use assessments meaningfully yet sparingly to make strategic instructional decisions.

Integrate ongoing practice and review into the teaching of current grade-level content, instead of disrupting it.

Be manageable and useful for teachers.

So, what does a resource designed around these overarching principles look like?

For each grade-level unit we are creating a compilation of recommended additions and supports, called a "Unit Adaptation Pack." These packs will include: rationale for recommended adaptations, prior grade-level lessons, assessment to inform instructional decisions, ongoing practice to address unfinished learning, and extensions and explorations for students who may have previously learned the content. Because we know teachers will not have added days for added lessons, we also will give guidance on lessons to combine or skip within each unit and ways in which to prioritize the major work of the grade. Here is our current thinking on how these four ideas might support students and teachers next school year in order to start the community conversation.
1. Incorporate prior grade-level knowledge and skills when necessary to support access to current grade-level content.
Relying heavily on the coherence in the standards, we first determined where to add lessons in a coherent and "just in time" manner across the year. Then, to balance the number of allocated school days, we must analyze each unit to identify lessons that can be adjusted and combined or skipped.

We are taking two approaches to the placement of the add-in lessons, depending on the content of the unit. The first approach is highlighted in this example from Grade 4, Unit 1, which focuses on factors and multiples within the context of area. Since this unit relies on grade 3 understanding of area and multiplication fluency within 100, we are adding key lessons on those ideas, before the start of the unit. Below are sample activities that represent the math progression the unit takes, including the add-in lessons.

Grade 4 Unit 1: Factors and Multiples

The second approach is demonstrated in this example from Grade 3, Unit 1 which introduces multiplication through students' work with scaled picture and bar graphs. Because of this, Section A is dependent on grade 2 measurement and data standards while Sections B and C rely heavily on students' understanding of arrays and skip counting by twos, fives, and tens. In this case, teaching all of the add-in lessons upfront would feel disconnected, so we recommend teaching the add-in lessons where they most coherently fit in the mathematical story of the unit, between the sections.

Grade 3 Unit 1: Multiplication

2. Use assessments meaningfully, yet sparingly, to make strategic instructional decisions.
We know that if you spend too much time administering formal assessments, you not only lose valuable instructional time, but can also be left with more questions than answers about what to do next. For this reason we embedded a formative mini-assessment to be used after each set of add-in lessons. This mini-assessment is 1-2 problems that can be given at the end of the last add-in lesson as the cool-down.

Take the grade 3 example above. You will notice that the add-in lessons before Section B come from Grade 2, Unit 7, which introduces arrays and uses what students know about addition and skip counting to lay the foundation for equal-groups thinking. After these lessons are taught, we want teachers to consider the following question: Can students engage in the invitational beginning lessons of Section B? That is the purpose of the mini assessment. We are not focusing on mastery or completion, but rather checking in and supporting students' ongoing learning towards the grade-level standards.

Here is an example of what this process may look like for the Grade 3 example for Section B:

3. Integrate ongoing practice into the teaching of current grade-level content, instead of disrupting it.
Integrating ongoing practice is easier when a curriculum develops concepts and representations coherently along a mathematical progression, like telling a story. Because of the intentional choices that were made in the IM K-5 beta materials, both with concepts and representations, we can integrate practice students may need from prior units and grade levels directly into the lessons that are being taught at their current grade level.

This comes in handy because, as we know, after the mini assessment there may be students that need extra practice with the knowledge and skills that were covered in the add-in lessons. We also know that stopping to reteach all of the prior grade-level content will never get students through their current grade-level material. To keep the learning going, we leverage the coherence of the standards and our curriculum to align ongoing practice, in the form of centers and routines that can be integrated into the current grade-level unit.

In the grade 3 example above, a teacher may learn that students need additional support with the array structure or skip-counting. For these students, we are suggesting the use of a center from Grade 2 Unit 7 that asks students to match groups of objects, first in varied group structures and then in an array pattern, with an expression that represents those groups. In addition, we pulled a collection of choral counts from grade 2 to give students more experience counting by twos, fives, and tens, as needed.

Below is how this process may look for a teacher between Sections A and B.

4. Be manageable and useful for teachers.
While so much is still unknown, one thing we are certain of is that nothing will be seamless or easy about school in the fall. Our hope is that we can work together to best support students in feeling safe and successful after they have experienced so much recent change and uncertainty. To this end, we want to take special care to ensure students are building on their current understandings as they progress through grade-level content in a way that is inviting and safe and prioritizes the community building that happens particularly in the fall.

And while we have spent a lot of time discussing the unfinished learning most students will have, we wanted to make sure we also added in materials for teachers to support students who are feeling more confident in the content as well. In addition to the add-in lessons, mini assessments, and ongoing practice, we are providing explorations and extensions for students who may have had exposure to the prior grade-level content.

We share this sample Grade 3, Unit 1 Adaptation Pack overview to start a conversation. While identifying the content to support unfinished learning is a first step in planning for next year, there are many other considerations we need to think about moving forward. We look forward to hosting community chats with our K-5 beta teachers and anyone else who is interested in learning and brainstorming with us. Thank you all for everything you do. We appreciate you.

Next Steps

You can share your thinking via twitter #LearnWithIM or commenting on this blog post. Look for more information on social media ( Twitter, Facebook, LinkedIn) to learn more about upcoming community chats and brainstorming sessions.

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Looking to the Fall, Part 2: Creating a Supportive Resource for K-5 Teachers - IM CERTIFIED® BLOG

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By Kristin Gray, Director K-5 Curriculum and Professional Learningand Kevin Liner, IM K-5 Professional Learning Lead In our previous post, we highlighted important considerations in planning to support students in the fall. While we need to first explore these ideas conceptually, we must also consider what this looks like in practice.

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Moving Forward: NCSM/NCTM Joint Statement

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Moving Forward: NCSM/NCTM Joint Statement

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Moving Forward - NCSM and NCTM Joint Statement - National Council of Teachers of Mathematics

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"We live in uncertain times. Public health is at the forefront of our minds, and our schools have been disrupted in ways we have never seen. Although no one can predict how education might

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The post below is an excerpt from our Learning Acceleration Guide, which we wrote to help districts and educators prepare for and address to learning loss due to COVID-19 school closures. Visit our COVID-19 School Response Toolkit t o see more resources. I f you have questions or are interested in learning more, please .
Many students will need to fit even more than a year of learning into a single school year to make up for the learning they've lost. We recommend identifying the content knowledge and skills your students might struggle with in their current grade level, and filling those potential gaps "just in time," when the material occurs in the school year.

This may feel like a radically different approach for some of your teachers and leaders. And in many ways, it is. Schools typically address gaps in content knowledge or proficiency by engaging in some form of remediation (or reteaching) around skills from previous years. While well intentioned, this approach often means that students who fell behind never encounter the work of their grade, instead remaining trapped in a cycle of low-quality or below-grade-level work. Instead of sending students backwards to fill in all the potential gaps in their learning, leaders and teachers should focus on filling in only the most critical gaps-and not in isolation, but at the moment they're needed.

Addressing unfinished learning starts with two steps:

Understanding the demands of grade-level materials and content. To begin this process, study the standards alongside year-long scopes and sequences to understand what standards and topics will be covered when, how students will apply their knowledge of the standards, and how they will show that knowledge. This means looking closely at the standards, the topics, and the tasks within a unit and across the year.

Identifying the most critical prerequisite skills and content knowledge students will need to access that grade level content when they encounter it throughout the year.

For example, a ninth-grade English Language Arts teacher might look across the year and note that students will be reading Lorraine Hansberry's A Raisin in the Sun in the fall. The teacher could give students an easier text to read and remove this book from the year-long sequence, out of fear that students are too far behind to tackle such a complex text at the beginning of the year.

Instead of removing the text and opting for something of lesser value, the teacher could note that tackling this text requires ready-made scaffolds for building students' content knowledge about the topics and time period covered in Raisin. Additionally, because Hansberry obscures concepts like the Great Migration and redlining in rich figurative language, the teacher might identify understanding the purpose and meaning of figurative language as a critical prerequisite skill for tackling this complex text. Here we are thinking about the prerequisite knowledge that students need by considering the techniques the author uses to convey key ideas, along with the intentional topical knowledge that the curriculum seeks to build.

While engaging in complex texts is crucial, it is not the only component of strong literacy instruction. Foundational skills instruction (K-2) and writing are also key elements. As always, we want to make sure students are writing about what they are reading. We'll tackle an approach to foundational skills in the section below, as it requires systematic assessments of student proficiency.

Now let's imagine that you're a math teacher or someone who supports mathematics in a system. The natural inclination here might be to make room for reteaching last year's missed units by wholly eliminating any on-grade level units focused on "additional standards" and then, once the "reteach" portion of the year is complete, turning your attention to units focused on the "major work of the grade. But we know this approach did not lead to student growth in New Orleans after Hurricane Katrina.

Instead, you should identify the most important skills and concepts of each unit of study or standard. For example, if you're a seventh-grade teacher who knows your sixth-graders missed their Statistics and Probability Unit, look to the seventh-grade Statistics and Probability Unit as a mechanism for recovering missed learning. You'd see that the most important sixth-grade skill for students to recover is their understanding of "measures of center such as mean" because, in seventh grade, students "use measures of center to draw inferences about a random sample." Therefore, doing a multi-week sixth-grade Statistics Unit isn't necessary. The teacher (and students) may only need one or two lessons on this sixth-grade concept to be ready to tackle the work of their grade.

We've just highlighted an example from seventh grade that covers an "additional standard" to show how those units can still be done, but even "major work of the grade" requires this thoughtful planning. For example, third-grade students might have missed their unit on fractions because these concepts often come at the end of the year. Understanding fractions is "major work" and a critical concept for students, but repeating the entire third-grade unit isn't necessary. Teachers could focus more narrowly on the third-grade fraction concepts related to understanding 4 Hill, P. learning-losses. 9 equivalence of fractions (3.NF.A.3) because that is really what is carried forward in fourth grade. Again, a fourth grade teacher could ground a unit in one to two lessons on understanding fractions as a quantity and understanding equivalence with common denominators or common numerators (third grade content) before moving to the major work of fourth grade.

Identifying prerequisite skills can begin now and continue throughout the summer. This kind of work is best done collaboratively, ensuring that your curriculum and instruction team works with your district-level teams who focus on serving English Language Learners and students with IEPs. This can reduce potential blind spots by highlighting unique needs of special populations.

For example, multi-lingual learners or students receiving special education services may need an additional layer of support. Teams should reserve dedicated time to think through additional prerequisite content knowledge or skills for these populations as well. In addition to working collaboratively, there are tools that can help teachers with this process. In math for example, tools like standards' focus by grade level guidance and the coherence map can help educators better understand the connection between standards and concepts.

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To Accelerate Learning, Focus on the Most Important Skills and Knowledge

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The post below is an excerpt from our Learning Acceleration Guide, which we wrote to help districts and educators prepare for and address to learning loss due to COVID-19 school closures. Visit our COVID-19 School Response Toolkit t o see more resources. I f you have questions or are interested in learning more, please .

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Achieve the Core Guidance for new School Year 2020-21

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Achieve the Core Guidance for new School Year 2020-21

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Human-Centered Learning

Create positive, affirming learning spaces that center student identities and experiences.

Teaching For Equity: A Guide to Integrating Academics, Well-Being & Anti-Racism in Student Experiences | Create supportive learning environments using this teacher-designed framework from Leading Educators. The framework includes five strands that facilitate structured reflection of classroom practice, school and community relationships, and your own identity as an educator. Working through the framework will help you translate beliefs into action.

EdResearch for Recovery Project School Climate Evidence Briefs | Identify research-based strategies to improve school climate and foster safe, inclusive learning environments. This project from Brown University asked researchers from across the country to develop evidence briefs that point to research-based recommendations for supporting students. The school climate briefs cover topics such as trauma-responsive educational practices, community engagement, and how to rebuild a positive school climate after disruptions to in-person learning.

Disrupting Inequity: Having Brave Conversations About Bias | Invite colleagues to join in a brave, honest discussion about bias in schools and its effect on students. The toolkit from UnboundEd contains individual PowerPoint presentations and materials containing detailed notes, resources, and activities designed to facilitate challenging discussions in a supportive professional learning space. Topics include microaggressions, historical implications of race, and actionable steps you can take to identify bias and improve school culture.

| Encourage students to be changemakers by creating a classroom environment where critical conversations about racial injustice, social inequality, and discrimination are addressed openly and often. This detailed guidance from Learning for Justice will help you develop the skills to facilitate honest communication around difficult topics through the implementation of classroom-ready strategies.Let's Talk: Facilitating Critical Conversations With Students

Social, Emotional, and Academic Learning

Addressing students' social and emotional needs and development is a critical component of human-centered learning environments.

Academic Acceleration

| These documents name instructional content priorities in mathematics (K-8, high school) and ELA/literacy (K-12). They support planning focused on keeping all students working on grade-level content. These were developed for the 2020-21 academic year in response to the disruption of the global pandemic, and they remain helpful for those setting academic priorities for the 2021-22 school year. Plan instruction that builds on the foundation of CCR-aligned standards to prioritize content.
Companion guidance is also available for families.
Learning acceleration is different than remediation - and more effective. Acceleration requires focusing on grade-level work, ensuring students access to that work, and offering support and connections to prior-grade learning, as necessary. Learn more in Accelerate, Don't Remediate, a report by TNTP and Zearn.

2020-21 Priority Instructional Content in English Language Arts/Literacy and Mathematics

Reading and Fractions

Customize content decisions in these critical areas based on research and the specific needs of students.

Reading science emphasizes two critical pieces in systematic early reading instruction: securing foundational skillsand building knowledge and vocabulary These "Early Reading Accelerators" are essential content that all readers need access to in order to become proficient. While each student and their experiences are unique, we can accelerate the development of reading proficiency for all students by implementing these practices strategically through classroom instruction and materials. To learn more and examine the research base for literacy accelerators and the intersections of equitable personalization for students in ALL grades, explore Reading as Liberation. Invite students and their communities to understand how we learn to read and be partners in that learning with How We Read: A Graphic Guide to Literacy.
A robust body of research tells us that flexibility and accuracy with fractions are among the most critical elements for student success in later mathematics. The resources and support in this fractions collection are designed to develop educator understanding of fractions as well as facilitate the teaching and learning of fractions with students. For support on approaching unfinished learning in math on other topics visit Instruction Partners' Supporting Unfinished Learning in Math.

Integrated Scaffolds and Support

Support access to grade-level work for all students.

rning by creating learningAchieve the Core Learning Supports Collection | Address unfinished le a Prepare to meet all students where they are with these proven strategies and practices. environments that focus on scaffolding and just-in-time instruction, rather than remediation. Every student is capable of grade-level work that keeps them on track for success after high school. Nearly all students contend with some degree of unfinished learning each year.

The Rethinking Intervention series from Instruction Partners features videos, PLC guides, and blog posts to help you rethink the concept of intervention to focus on student-teacher relationships and access to grade-level content.

Perspectives on Scaffolding Student Work | Discover teacher-tested scaffolding strategies in this collection of blog posts.

English Learner Success Forum Resource | Deepen and accelerate English learners' content area learning by implementing the strategies included in this collection of teacher-facing guidance. The activities and scaffolds in this collection from the English Learner Success Forum can be built into existing lessons and units to help English learners engage fully with the curriculum.

| Expand access to grade-level content with these guidance documents from the New York State Department of Education offering grade-specific scaffolding guidance for instruction in grades 3-8. Examples are drawn from the open-source curriculum EngageNY, but can be applied to any set of materials.Supporting all Students: Resource Guides for Scaffolding Instruction of English Langauge Arts and Mathematics

Culturally Relevant and Responsive Pedagogy

Meet student needs by honoring all aspects of their identities and by helping them to make connections to the world around them.

Whether you are practicing culturally relevant pedagogy (Dr. Gloria Ladson-Billings) or culturally
responsive pedagogy (Zaretta Hammond), teaching requires thoughtful planning and a commitment from educators to get to know each and every one of their students each year. This work not only helps all students excel academically, but also helps expand perspectives, reaffirm identities, and prepare students to create a better world, using the skills and knowledge they gain in the classroom to demand and design change.

For a quick introduction to culturally relevant pedagogy, read this three-part blog series. Educators preparing to design culturally responsive learning environments can find tools, guidance, research, case studies, and more in NYU Metro Center's Culturally Responsive Education Hub. Make sure to download Transforming Our Public Schools: A Guide to Culturally Responsive-Sustaining Education and the Culturally Responsive Curriculum Scorecard Toolkit, to jumpstart your work.

Looking to take the first step toward translating these concepts into action in your classroom? Begin by learning about the students you serve and their unique identities. Creating lessons and learning environments that are culturally relevant and/or culturally responsive will look different from classroom to classroom--because the students are different!

Understanding Student Progress

Rethink how to understand student progress and integrate ongoing practices to understand where unfinished learning exists and support students with grade-level content.
Humanizing Assessment Blog Series | Gain practical, research-based advice on expanding the ways educators can gain insight on student progress that go beyond traditional assessment. These student-centered monitoring methods will transform the way you think about assessment.

| Reenvision how you collect and use data on student progress. This collection by Instruction Partners includes a short video, professional learning materials and resources, and an overview blog post.Rethinking Intervention Blog Series

What the Data Can't Tell Us | Learn about the limitations of summative data collected at the end of the 2020-21 school year. This blog post outlines cautions and concerns about using summative assessment data and presents alternatives for monitoring student progress.

Are there additional tools and resources that should be a part of this collection? Share them here.

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Priorities for Equitable Instruction: 2021 & Beyond

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Create positive, affirming learning spaces that center student identities and experiences. Teaching For Equity: A Guide to Integrating Academics, Well-Being & Anti-Racism in Student Experiences | Create supportive learning environments using this teacher-designed framework from Leading Educators. The framework includes five strands that facilitate structured reflection of classroom practice, school and community relationships, and your own identity as an educator.

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I was interviewed by Dr D. on this podcast on fluency and Math Running Records.

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Get to know Spotify for Podcasters, the free, all-in-one podcast platform for every creator.

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Synchronous Online Teaching Strategies and all material on this website by Theresa Wills is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.Based on a work at theresawills.com.Permissions beyond the scope of this license may be available at theresawills.com/permissions.

Interactive slides for any content area

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Templates - Where THERE'S A WILLS, there's a way

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Daily Routine Resources

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Looney Math

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Same But Different is a powerful routine for use in math classrooms. The images can be used for teaching a way of thinking: grayscale thinking, categorical thinking; helping students to build a network of ideas and an approach for learning and thinking about mathematics.

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Solvemoji - Free teaching resources - Emoji math puzzle, great as a primary math starter, or to give your brain an emoji game workout - great for home schooling

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NHTM DOE Webinar on Steve Wyborney's Resources

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NHTM DOE Webinar on Steve Wyborney's Resources

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NHTM and DOE Workshop Steve Wyborney's Resources PDF

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NHTM and DOE Workshop Steve Wyborney's Resources PDF

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Multiplication Math Fact Journey Video Explanation

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Multiplication Math Fact Journey Video Explanation

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Steve Wyborney\\\'s Blog: I\\\'m on a Learning Mission.

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For my educator friends who have access to Google Slides, but do not have access to PowerPoint, this post is for you. Since Google Slides does not have the animation that makes the Splats become transparent, the Splat! reveal does not work in Slides. Now - with these materials - it will. I've rewritten the first 20 Splat! lessons and the first 20 Fraction Splat! lessons with a different method so they will now include the reveal when using Google Slides.

The content of the lessons is the same as the original lessons.

Visit these links to see the original videos about how to use the and the Fraction Splat! slides.

Quick Note: The file sharing power of Google Drive is impressive and highly useful. However, for the purposes of sharing these resources, I need to use a method that I can personally maintain, update and support - and I've learned that the best way to provide resources from my blog is through the download links and the sign-up forms.

I hope you enjoy these Splats! formatted to work in Google Slides!

Splat Through 10
How many dots are under the Splat?
SPLAT 1.1 - numbers through 10 (Google Slides compatible version)
SPLAT 2.1 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.2 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.3 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.4 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.5 - with numbers 12-20 (Google Slides compatible version) SPLAT 1.2 - numbers through 10 (Google Slides compatible version)SPLAT 3.1 - multiple splats (Google Slides compatible version) SPLAT 3.2 - multiple splats (Google Slides compatible version) SPLAT 3.3 - multiple splats (Google Slides compatible version) SPLAT 3.4 - multiple splats (Google Slides compatible version) SPLAT 3.5 - multiple splats (Google Slides compatible version)
SPLAT 4.1 - instant multiple splats (Google Slides compatible version) SPLAT 4.2 - instant multiple splats (Google Slides compatible version) SPLAT 4.3 - instant multiple splats (Google Slides compatible version) SPLAT 4.4 - instant multiple splats (Google Slides compatible version) SPLAT 4.5 - instant multiple splats (Google Slides compatible version) Note: These lessons use a single splat along with a mixture of unit fractions and whole numbers. In Level 11, you control the Splat! SPLAT 11.1 - fraction splats (Google Slides compatible version) SPLAT 11.2 - fraction splats (Google Slides compatible version) SPLAT 11.3 - fraction splats (Google Slides compatible version) SPLAT 11.4 - fraction splats (Google Slides compatible version) SPLAT 11.5 - fraction splats (Google Slides compatible version) SPLAT 1.3 - numbers through 10 (Google Slides compatible version)Note: These lessons use a single Splat! which appears at the same moment as the whole numbers and fractions. Each question instantly presents missing information. SPLAT 12.1 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.2 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.3 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.4 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.5 - Instant Fraction Splats (Google Slides compatible version) Multiple Splats! with Fractions Note: These lessons include multiple Splats! and you control the timing of the Splats! The entire screen can be seen, the total can be determined, and then Splat! Remember: Each Splat! that is the same color must be covering the same number. SPLAT 13.1 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.2 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.3 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.4 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.5 - Multiple Fraction Splats (Google Slides compatible version)
Note: These lessons use multiple Splats! which appear at the same moment as the whole numbers and fractions. Each question instantly presents missing information. Remember: Each Splat! that is the same color must be covering the same number. SPLAT 14.1 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.2 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.3 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.4 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.5 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 1.4 - numbers through 10 (Google Slides compatible version)
SPLAT 1.5 - numbers through 10 (Google Slides compatible version)

Splat Through 20
How many dots are under the Splat?

Multiple Splats
Note: Splats that are the same color must cover the same number.

Instant Multiple Splat
Note: An additional feature of this level is the opportunity to ask, "What could the total be?"

Instant Fraction Splat!

Multiple Instant Splats! with Fractions

After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.

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Splat! for Google Slides - 40 Lessons - Steve Wyborney's Blog: I'm on a Learning Mission.

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For my educator friends who have access to Google Slides, but do not have access to PowerPoint, this post is for you. Since Google Slides does not have the

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Steve Wyborney\\\'s Blog: I\\\'m on a Learning Mission.

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Steve Wyborney's Blog: I'm on a Learning Mission

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Steve Wyborney is on a learning mission to spread math joy to classrooms everywhere! On this blog you'll find math resources for teachers and for kids of all ages.

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Teacher use of Esti mystery  Mar 27, 2020 3 36 PM

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Worksheets: 5 tweaks for deeper understanding

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Worksheets: 5 tweaks for deeper understanding

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There is nothing elementary about teaching elementary math!

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Mathematics K-8 Remote Learning Activities

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Achieve the Core Grade Level Focus Areas

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Achieve the Core Grade Level Focus Areas

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Achievethecore.org :: Instructional Content Nav - Mathematics: Focus by Grade Level

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The Common Core and other college- and career-ready (CCR) standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, CCR standards require us to significantly narrow and deepen the way time and energy are spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom. The following documents illustrate these concepts using the Common Core State Standards, but Focus, Coherence, and Rigor are integral to all college- and career-ready standards.

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Pam Harris Development of Mathematical Reasoning

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Pam Harris Development of Mathematical Reasoning

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Algebraic puzzles

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Mashup Math activities make your math lessons fun and engaging for students

K-8 math activities designed to be highly engaging for students.

We inspire kids to learn math in a way that isentertaining and fun.

Our unique approach boosts academic achievement and participation.

Over 100,000 math teachers rely on Mashup Math resources to supplement their lessons.

Standards-aligned math resources for grades K-8+

Gain On-Demand Access to All of our K-8 Math Puzzles, Worksheets and Activities

Inspire Your Students to Learn Math in a Fun and Engaging Way

Join our mailing list to get free resources in your inbox every week

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Math Education Teacher Resources Common Core Cool Video Lesson Free Worksheets Elementary Middle High School Homeschool Algebra Geometry Lessons Help Tutor Homework

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Great story backgrounds

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Number Balance, counters, etc

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Hands-On Math Goes Digital In and Beyond the Classroom!

Explore Our Collection of FREE Virtual Manipulatives

From Number Lines to Base Ten, Pattern Blocks to Place Value, Disks our collection of free Virtual Manipulatives gives educators and families access to visual and hands-on digital resources that lead to understanding. Students can use these resources in class and at home to practice math concepts, reinforce their growing math understanding, technology they love in the process.

Free Resources & Instructions for Teachers and Families

We've developed these resources for easy access and easy use:

How-To Videos

There's a short video for each of the Virtual Manipulatives in our collection so teachers can use these digital materials easily and confidently with students.
Watch Now

Activities

Activities from the Didax books that teachers use most for easy download. Engaging challenges address core math operations, concepts and skills, from early numeracy all the way to introductory geometry and algebra.
Access Now

FAQ

Download this document of FAQ's to answer your Virtual Manipulative questions.
Download Now

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Demonstrate key math concepts in your classroom using our new virtual manipulatives!

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University of Washington classroom videos of counting collections, choral counting, and more!

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University of Washington classroom videos of counting collections, choral counting, and more!

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University of Washington Unique Video Links

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Graham Fletcher blog and video

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Question: If you memorizeyour basic math facts have you learned them?

The term from memory is used twice in the K-5 math standards, and they continue to be the most misunderstood words I encounter each day. The term "from memory" does not suggest that students mindlessly their basic facts the way we were taught.

Here's my first stab at an Ignite Talk explaining the difference between from memory and memorization that I recently shared with some teachers and administrators.

Here's the link to the Prezi

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From Memory and Memorization: There IS a Difference

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Question: If you memorize your basic math facts have you learned them? The term from memory is used twice in the K-5 math standards, and they continue to be the most misunderstood words I encounter...

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Children are listening and learning from everything we say. The goal of the Kids Math Talk Podcast is to give parents and educators practical tips and insights that will deepen mathematical understanding while also encouraging the conversation about math to remain active and positive.

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Making Math Moments that Matter interview with Christina Tondevold

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Making Math Moments that Matter interview with Christina Tondevold

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Christina: I think our end goal is that we want kids to be proficient in mathematics. The difference starts to happen in how we get there. How do we help children develop that proficiency? And to me, the foundation of that proficiency is number sense. That, if you don't do the things to make the foundation solid, the rest of it will start to be affected by-

Jon Orr: You're listening to the Recovering Traditionalist, Christina Tondevold. Christina previously taught mathematics traditionally, much like we did. And now, she gets to work with teachers in developing students' mathematical understandings in ways that are not limited by the traditional ways of doing and teaching mathematics, through her site, buildmathminds.com.

Kyle Pearce: Stick with us as we learn what is number sense, and why is it so important? What are the four early numeracy concepts that all teachers should know and understand how to teach? What are the four number relationships? And finally, what steps you can take to ensure you learn fundamental mathematical progressions that underlie students' understandings of number sense.

Jon Orr: All right, Kyle. Let's dive into this conversation.

Kyle Pearce: Hit it!

Kyle Pearce: Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pearce, from tapintoteenminds.com.

Jon Orr: And I'm Jon Orr from mrorr-isageek.com. We are two math teachers who, together-

Kyle Pearce: With you, the community of educators worldwide, who want to build and deliver math lessons that spark engagement-

Jon Orr: Fuel learning-

Kyle Pearce: And ignite teacher action. You are listening to episode number 44, Building Math Minds, an interview with Christina Tondevold. Are you ready to get started, Jon?

Jon Orr: Of course. Of course, Kyle. As always, we are super pumped to bring you this resource-rich episode. But before we get into this episode with Christina, we want to let you know that Christina herself is running a free webinar training called Math is Not a Worksheet. Registration is open now until October 12th, 2019. You can register by visiting makemathmoments.com/buildmathminds. Again, that's makemathmoments.com/buildmathminds. If you're listening to this after October 12th, 2019, then that link will either bring you to the webinar replay or a wait list for the next webinar.

Kyle Pearce: We also want to take a minute here to give a quick shout out to a math moment maker who left us amazing five-star review on iTunes. Passagenpiaffe says, "Truly paradigm-shifting. I always struggled with math, and I am now teaching third through sixth grade. I found this podcast to be truly life-changing and so helpful for me to fall in love with math. It has done wonders to shift my own personal mindset regarding math, as well. Thank you for all you do to bring revolutionary ideas and resources to the table."

Jon Orr: Thanks to Passagenpiaffe for the amazing five-star rating and review. You don't know how excited we get when we see that amazing math moment makers like yourself take a quick moment out of their day to show some love on Apple Podcasts, Google Podcasts, Spotify, or whatever platform you're listening on.

Kyle Pearce: Well, that's enough of that, Jon. Let's get into our chat with Christina to build our math minds.

Kyle Pearce: Hey there, Christina. Welcome to the Making Math Moments That Matter podcast. We are extremely excited to have you on the show today. How are things going in your neck of the woods?

Christina: Well, what I like to say is better than I deserve. Right?

Jon Orr: Wow.

Christina: Any Dave Ramsey fans out there? Right? Things are so much better.

Jon Orr: Oh, awesome. Tell us more about that. Where abouts are you, and what is going well on that side?

Christina: Well, I live in Idaho, in a very rural town. I have four kids and life is just always busy. And I have to remind myself sometimes that it is better than I deserve, because being a working momma with four kids, you tend to feel like, "Ah! I'm just so stressed out," and, "How am I ever going to do all the things that I need to do?" And the reality is, it never happens, but I just try to do the best that we can and say, "You know what? If the laundry doesn't get done, that's okay." Or whatever it might be. That's the hard part is, I do a lot of stuff. People see my face, I do a lot of videos, I do podcast stuff, too. And they see the front side, so everything always looks so great. But then what you don't see is the piles of laundry and piles of dishes and all of that fun stuff of life that we don't always see about people that we interact with online.

Kyle Pearce: Yeah, for sure.

Christina: So I am better than I deserve, but life is always full of trials and good things. And all four of my kids are in school, sports, a very active family, always out doing stuff and just love to be together. And luckily here in Idaho, we have beautiful country to be able to do all that stuff in. So I always feel blessed, that's for sure.

Kyle Pearce: Amazing, amazing. I think we can mimic that. We've all got some struggles on a day-to-day basis, but I think for me, also, Christina, I can't complain about many things. So let's dive in a little bit more about who Christina is. We've met a couple of times, but Christina can help our listeners a little more about yourself, and maybe more specifically now that you've let us in on kind of a little bit of your family life and where you are, some background on how you got into math education and then also what is your role in math education right now, for our listeners who don't know?

Christina: So I got into education by default, I say. I actually went to school to become an interior designer. And my first year of college, I had to take art classes that I failed at miserably. And I did not have a growth mindset at all. And I said, "Failed at miserably," I got a C in. Which is not failing, but it felt like a failure to me. I was always that kid who did well and never really struggled in school. And when I did struggle, I didn't know what to do. I look back at that point in my life, and I thought, "Oh my gosh. You really ran from that because you got a C?" But I am so thankful I did because I turned to education. And I say it was by default because my mother was a junior high teacher. And I didn't know what else to go into, because I thought I was going to be an interior designer.

Christina: So I went into education. And during my student teaching, it was a year-long student teaching, and I wanted to junior high, because that's what my mom was. And I thought I was going to be an English teacher like my mom, and they had us do a rotation where we did a quarter teaching every subject; the four main subjects. So I did English then science, I think was next; then the math, and then social studies. And when I was done, I thought to myself, "I want to teach math." Because math was the easiest thing to teach. There's one answer. It's easy to correct. When I did my student teaching, it was still ... The kids would bring home the homework, I would say the answers out loud ... They would switch papers, you would grade each other's papers, switch the papers. So I would say the answers, then when you're done, they would tell you their percentage out loud, and you would put it into the computer. Right? You remember those days?

Kyle Pearce: Right. Yeah, oh yeah.

Christina: [crosstalk 00:07:53] did that.

Kyle Pearce: Quick and efficient, right?

Christina: [crosstalk 00:07:56]

Jon Orr: Christina, I did that.

Christina: Yeah. I think back and think, "Oh my gosh, what did I do to my ... " That's just one of the things that you think back of and think, "Oh my goodness, I was such a horrible teacher." But in my mind, that was the easiest thing to teach. And I thought English was the worst because there was no back and white answer. I felt like teaching English and writing was subjective, and I didn't really get it. Math had rules that I could follow, and I was very much a rule follower. So I just felt comfortable in that mathematics space. And so I applied for teaching positions that were just mathematics in junior high setting. And I taught sixth, seventh, eighth grade in Nevada, Oregon, Idaho, just kind of all around this area. And it wasn't until I went back and got my master's degree that things opened up for me. And that was when I was introduced to cognitively-guided instruction and was introduced to this whole idea of a different way to view mathematics and to teach mathematics.

Christina: That led me into doing professional development. While I was doing my master's degree, my supervising professor had a grant to do professional development with teachers. And after I had that whole experience of doing professional development while getting my master's degree, I went back into the classroom, had my own class again, and got pulled back into professional development world. So it was kind of a long journey, but I've been doing lots of professional development with people all over the United States. And traveling around while also having four children, four young children.

Christina: And now I have the fortune to be able to do a lot of the trainings online. My focus is all on elementary education and laying a foundation for kids and what that looks like; what are the big pieces that our kids are missing? Because I saw it in junior high, where they had no sense of numbers. I was trying to teach them these rules and procedures, but it just wasn't making sense because they didn't understand how numbers worked. I could teach and have them memorize these rules and procedures, but it just didn't sit right. That's how I learned, and when I got my eyes opened to the different way of seeing mathematics, I didn't want my other students to be going through what I went through. I made it through, but I look back now and think, "Man, it could have been so much different." And that's really what I want for all of our elementary kids out there, is to have a different experience than what we had with mathematics.

Jon Orr: That's some big goals, some honorable goals, too. And you know, we always love asking our guests to think back to their experiences and what helped them, what molded them to get into what we're doing now. And there's a lot of similarities. And I know Kyle's got similarities to what you've done. What you've just outlined is almost kind of my path, also. My dad was a math teacher, not an English teacher. But I remember thinking I wanted to get into teaching. I wasn't going to be a teacher when I went to university, but when I was in university, I was like, "You know what? My dad's life was pretty good." You know? My dad was the ultimate high school traditional math teacher from the '80s and into the '90s. And I was like, "Yeah, he's got life pretty good. All summer he's sitting around reading books and swimming." And I'm like, "Maybe I can do something like that." And I kind of wanted to be a gym teacher, because I was like, "That's even better." And I was good at math, and I kind of fell in there, too. And I was that traditional math teacher, much like you said. And it's not until you realize that you were that rule follower and when you meet kids for the first time who are not that, you know that something has to be different.

Jon Orr: I didn't see those kids. You probably didn't see those kids, too, in your classes. We were in different classes. You know? We didn't hang out, maybe, with some of those kids that were just ... I know I didn't. All my friends were like me, you know? The math was pretty easy for us because we were all good rule followers and memorizers. And it wasn't until that I saw students that were not like me that it's like, "Something's got to change. I can't do it like this anymore." So I'm glad you pointed it out.

Jon Orr: But it's always amazing to me that many of our stories sound very similar and lead us to where we are now, helping this change happen. Because we all came from kind of that same place, but realized it has to change.

Christina: Yeah. It's definitely ... I don't tell the story too much of really why I chose math, but it was totally a selfish reason. Right? I thought it was going to be the easiest thing. You wanted to go into education because you're like, "Eh, it seems like a pretty [crosstalk 00:12:36] job."

Jon Orr: Right.

Christina: And then you get in there and you realize, "Oh my goodness. This is not what I thought it was going to be."

Jon Orr: Right.

Christina: And you know, I could have continued to have taught mathematics in that traditional way. But there was definitely points along the way that made me realize it opened your eyes. Right? Like you said, you have those moments that your eyes get opened and there's no turning back from that point on. Once you know that there's a different way out there, it's like, "I can't go back."

Jon Orr: Part of it for me, too, was is that I'm sure ... It's only when I reflect back on it that I'm sure no matter what I decided to do, I wanted to do a really good job. And I felt like seeing what I saw in my classroom, I'm not doing a really good job if I just keep teaching the way my dad taught.

Christina: Yeah.

Jon Orr: So my dad looks at what I'm doing in class now and he's awestruck because it's different than what he would have done, and we have many conversations about what mathematics education should be. And sometimes they get a little bit heated, but ...

Kyle Pearce: They no longer speak anymore.

Jon Orr: Yeah. No, it's ...

Christina: So at the dinner table, it's not just ... You can't talk about politics and religion, you can't talk about math education anymore at your house?

Jon Orr: Right. Partly. Partly, for sure.

Kyle Pearce: Well, you know what? Something that was interesting to me, just kind of backing up. You had named it really early on in the conversation, about having a fixed mindset. And I think it's really, really hard for us as adults when we're learning about this idea of growth mindset and so many people have seen Jo Boaler's work or Carol Dweck's work, and we read about it, but oftentimes it's easy for us to not actually reflect on our own experiences. And I know I was one of those students, as well. Luckily, it didn't affect me because I got through. But when things did get tough, it was like I played the blame game. I would blame the teacher. I would blame anything and everything but my own work ethic or my own perseverance, or any of those pieces. And it's like when you have this stumbling block, and for you, it was going through when you were intending to go through as an interior decorator, or was it ... Sorry.

Christina: Designer.

Kyle Pearce: Designer.

Christina: Same.

Kyle Pearce: Interior designer.

Christina: Similar, yeah.

Kyle Pearce: Apologies. As you were going through and you got that C, again, you're so used to kind of "getting it." It's almost like sometimes it's that perfectionist in us that sometimes that traditional math approach can sort of instill some of those perfectionist behaviors when we're only looking for right answers and we don't want to ever fail, we really struggle to dust ourselves off and pick ourselves back up. So I'm happy that we are making such a shift in math education right now.

Kyle Pearce: So what I'd love to do is, I want to back up and get you thinking a little bit more about your memorable math moment. When you think of your experience in math class, you had already said that you kind of were a rule follower. So I'm sort of guessing that you might have been like Jon and I where maybe math class wasn't extremely difficult for you, or at least at the way it was being taught and what you had to do; the hoops you had to jump through. But go ahead. Is there a memorable moment that you can share with us for folks to think about?

Christina: Yeah. I definitely feel like math, when I was going through school, wasn't that memorable. I wasn't one of those kids who struggled, was traumatized by the timed tests. I definitely remember all of those things, but it wasn't a bad experience for me, so it didn't pop out. I also didn't have any wonderful experiences. You know? Our moments that stick in our minds are usually at the ends of the spectrum. They're either really negative experiences or really positive experiences. And through my education as a student, I had none of those. It wasn't like math was so awesome and I had these great activities that we were doing and things that we explored, but it also wasn't negative. So I don't have those big moments through being a student.

Christina: However, when I went back and got my master's degree, that's when things started opening up for me. And the moment that I remember distinctly was through my master's degree, my supervising professor was very much about cognitively guided instruction. So I got introduced to CGI, and we were watching these videos of kids solving problems with no instruction, and they had these amazing strategies. And to me, that was the math world opening up to me. And at the time ...A little bit of a backstory here. My boyfriend at the time, who is now my husband, we dated for like 10 years, but we had known each other our whole lives. We had gone through the same school. We live in a small town, we had some of the same teachers, same education, same upbringing. And I started telling him about these strategies that I saw kids using. And it was things as simple as 9 + 8. And I would say, "And this kid took one from the eight and gave it to the nine and made it a ten and had 10 + 7." And I was like, "Did you know you could do that stuff?" And he looks at me like, "Duh."

Kyle Pearce: I just did it.

Christina: Doesn't everybody know how to do that?

Kyle Pearce: Yeah.

Christina: Yeah. He'd say, "That's natural to me." And I was like, "What? Are you kidding me?" I never saw things like that. Even on things like something that we see as basic, 9 + 8. To me, that was just memorized. It was something that I just knew.

Kyle Pearce: Right.

Christina: It wasn't something that I could manipulate and make sense of. And my husband, boyfriend at the time, was like, "That's how I've always seen it." And it was just like the world opening up to me of, there are people who see things differently and who see mathematics as making sense, not as rules to be followed. I look back now and can totally see that, because he is such the rule breaker. He gets told, "This is the way you do it," and he's like, "Nah, I see a better way."

Christina: So there's all of these kids out there that are doing that, even if we aren't open to it; they're doing it. But then on their paper, they're writing down the way that we want it. And I could look back now at a lot of my students and think, "Oh my goodness. I did such a disservice to them." Because they were trying to tell me that they saw it differently, but I wasn't open to hearing it. It wasn't open to allowing them to share their strategies. I would just be like, "Eh, I don't get it. Do it this way." And then when they couldn't do it that way, I was just repeating it; saying it louder, slower for them.

Kyle Pearce: Say it louder, yeah. Yeah. Exactly. This is how you do it again.

Christina: I was totally that teacher. Oh my goodness. But that time when I was talking to my husband and saying like, "Did you know you could do this?" And he was like, "Yeah." I was like, "Why wasn't I allowed to see that?" My mind was not opened to this until I was 30 years old, 20 something. I don't want to date myself, how long ago that was and how old I was.

Kyle Pearce: Yeah. It's such an interesting idea, to think about. Not necessarily just in math, too, about rule followers and rule breakers. I, myself, growing up was always enforced to become a rule follower. I had so many rules. It was like, "We have to do this, and then you do this, and then you do this." You know? It was also like, "You're going to graduate high school and then you're going to go to university, and then you're going to get a job." And think about our parenting techniques. It's like, are we creating rule followers? Is that what we want to create? Or do we want to create kids who are thinkers and will break rules? And it's a double-edged sword sometimes, because you're seeing some of these kids who sometimes don't have the boundaries that you think that they should have. And then you're like, "Am I creating a rule follower, or am I creating the kid who is going to break the rules and think out of the box?" Because you want both, kind of. And it's like I don't know where the line is.

Kyle Pearce: As a parent, it's hard to think about that. And as a math teacher-

Christina: I struggle with it.

Kyle Pearce: Yeah. And as a math teacher, it's so clear to us now, in math class. But then to open the doors to just life in general, is scary.

Christina: Totally. I have one of my kids who is totally the questioner of everything. "Why do I need to do that? Why do I ... " And in school, that's kind of, in the past has been ... We stamp that out. Right? There's been research done of how many times kids will ask the question why at certain grade levels. And the higher up they go in grade levels, the less they ask that question. Because they learned that that's not something you ask in school. You just listen and learn and do what you're told to do. And I struggle with my one child, because he is such a, "Why do I need to do that? That doesn't make sense," kind of a kid, and we want to encourage that, because later in his life, that is going to be such a powerful trait; to be a questioner and thinker and go outside that box and don't follow the rules. But man, sometimes in school, they need to follow those rules.

Jon Orr: Yeah, exactly. It's like everything in life, and I feel like on this show if you were to go and do Wordle of everything we've ever said, I feel like some words come up quite often, and balance is one of them. We always say ... Depends, we say a lot. Like, "It depends on the scenario."

Christina: Yeah.

Jon Orr: And this is one of those cases, where it's really hard. You had already referenced ... You said you have children at home. I think you said four children at home. Am I right with that?

Kyle Pearce: Mm-hmm (affirmative), yeah.

Christina: Mm-hmm (affirmative). Yep.

Jon Orr: Four children at home.

Christina: Yeah.

Jon Orr: So you're trying to engage in your work life and you're trying to engage with your family. And sometimes you get tired. And usually when we show our true selves, how we're truly feeling if we're run down or if we're tired, it's usually at home. And our poor children sometimes, when they are inquisitive and they're asking us these questions, sometimes it's not even what we say, it's just how we react.

Christina: Yeah.

Jon Orr: Even if it's a look or the tone of our voice.

Christina: The tone, yes.

Jon Orr: Yeah! It's like it's sending that message to them that, "You know what? Stop doing this. You're annoying people." And we don't want to send that message, because we want them thinking outside the box. So I think this is a great way to segue into some of the questions that we had for you, because we're really curious. We know you are very passionate about working with some of our youngest learning in K through five especially.

Christina: Mm-hmm (affirmative).

Jon Orr: And as you mentioned, number sense, it kind of blew your mind when you started focusing on cognitively guided instruction and seeing these strategies and why we need them. What we'd really like to do is kind of dive into it. So if I'm a K through five teacher, why do we need to invest time in learning, deeply, how cognitively guided instruction can help us with number sense, or even compared to patterning or measurement, or any other topic? But in particular, why is it important that we focus in on cognitively guided instruction and really do our own homework since many of us didn't have the chance to learn this way? Why should we do that work for our students?

Christina: Okay so I like to start off a lot with ... You know, our end goal is really all the same. We want kids to become fluent and flexible thinkers of mathematics. Now, if you talk to anybody, even parents who are talking about how we shouldn't be doing this new math or anything like that, when they're kind of combative about how we're teaching mathematics, I think our end goal is that we want kids to be proficient in mathematics. The difference starts to happen in how we get there. How do we help children develop that proficiency? And to me, the foundation of that proficiency is number sense. I equate it to ... My husband is a contractor. He builds homes, does remodels, all that fun stuff. So he has seen lots of times where things are not done right the first time. Right? You can come back and fix it later. And oftentimes, we try to jump ahead; we will skip steps. Sometimes he sees that a lot in building; that if you don't do the things to make the foundation solid, the rest of it will start to be affected by that later on. It might not be right away. Right?

Christina: And so in education, sometimes we try to skip the foundational piece of building kids' sense of numbers and we jump straight into the operation of numbers. We want kids to add and subtract, multiply and divide as they move up in grade level, but without a foundation of a sense of what numbers are, how they work, how they relate to each other, then those operations that we're trying to teach them become just rules and procedures. If we can lay a foundation of numbers sense, we see kids who are flexible thinkers and can manipulate numbers. I mean, just this morning, we were headed to school this morning. I drop off my children every day, because it's just a good time to chat, also. You know? I like spending time with my kids. And we're talking, and my youngest son, he's really young for his age. He just turned six in August, but he's in first grade. So he's a young first grader.

Christina: And we're headed down and we're talking about math, and some of my kids don't like math. They've said before, "I don't like math." And my first grader, he goes, "I like math. I'm a master of math." And I go, "Oh, you're a master of it?" He goes, "Yeah. I've mastered addition and subtraction." And actually, he said, "I've mastered pluses and minuses." That's what he calls it. And I go, "Oh, really?" And my daughter goes, "Okay. If you think you've mastered it, what's 20 + 20?" And he instantly says, "40." And she was about to say something else. And I go, "Hold on. I got one for him. Okay. What's 20 + 19?" And he thinks for about three seconds and he goes, "39." I'm like, "What? How did you do that?"

Christina: "Because it's just one less." Right? He sees those relationships. He's not lining it up, doing the traditional algorithm like I used to do. If I would've been asked that question 20 some years ago, it would have been lining it up. Seriously. I could not do mental math. I had to do everything as ... We line it up and we add our ones, add our tens kind of a thing. And he just sees relationships. He doesn't have to be taught strategies. He sees how it works.

Christina: And I know that all the kids who have this foundation of number sense ... And yes, my children come from a very unique home life because they have me as their mother and we have done stuff that builds number sense, but I've never taught a strategy to them. We talk about numbers, we play with numbers. We play lots of games. You know? We don't know number talks, like teachers call number talks, but we definitely talk about numbers. We play lots of games where kids will see things, and they earn money for stuff.

Christina: Even just earning money for chores nowadays, kids don't have those experiences to think about how much they have, how much more they need to get something that they want. And thinking about, "I want this thing on Fortnite, the pack." My kids are all into Fortnite and they want to buy the V-Bucks or whatever it is, and how much money they need to earn in order to do that and how far away they are and how many chores they need that will get them there; all of those experiences. It's all built upon experiences. And a lot of our students don't have those experiences before they come to school. And in school, our standards jump us straight into thinking that our kids have to be able to add and subtract in kindergarten and first grade, so we focus on that end product of getting our kids to add and subtract, and we don't think about all the foundational pieces.

Christina: It's like jumping straight to putting in the cabinetry and stuff in your house, picking out the light fixtures and the doorknobs. Right? That's the end product that everybody sees, but what's the foundation of how we get that house to look nice? And that's what we got to think about in mathematics is, yeah, our standards may say we need to be able to get kids to do this certain thing, but what's the foundation of their understanding of numbers that get them there? That is such a huge piece.

Jon Orr: Yeah, and that's a huge analogy. That's really eye-opening. You know? I think it's a great analogy to start with, when you were talking about why we should be doing this. Because everyone would agree with you. We would never start with the finishing touches on a house, we'd start with the foundation. And you've made it very clear that number sense can be a great foundation.

Jon Orr: And one of the other things that related to both Kyle and I right now is that when you had talked about ... So many of us are rushing to that algorithm, and that's one of the phrases that we use quite regularly, when we go right to formulas. We go right to, "Let's just stack these numbers and let me show you how to add them," like the finishing touches. Sometimes we just say we rush way too much just to that algorithm and the abstract representation of numbers, or abstract representations of math concepts in general. We, too often, especially in middle school and high school, too, being from that area myself, jump right to just saying, "This is the formula. Let's apply it, let's move on." And we don't do enough foundational building there.

Jon Orr: And you know what? I think there's some teachers probably leaning in right now and thinking like, "This is awesome, and you're right, but where do I start?" What are some early numeracy concepts that all teachers should know, and then how to teach those?

Christina: Yeah. So I want to say a little bit to what you just talked about, about rushing to algorithms too quickly. Because I would say that I believe teachers even rush to strategies too quickly. Because the knew swing has been, "Okay, we're not going to teach the kind of traditional "algorithm," quote, unquote. You can't see me, there ... of stacking [inaudible 00:30:26] and doing that traditional algorithm. So instead, we're going to work on these strategies, like the one that my son came up with, that if it's 20 + 19, they think in their head, "20 + 20, and then I just subtract one."

Christina: So our textbooks, even, will have those strategies as something we need to directly teach kids. And I feel like we jump to those strategies even way too soon; that we try to directly teach these strategies and the kids ... If a kid doesn't understand how 19 relates to 20, that strategy makes no sense to them whatsoever. Right? So we need them to be able to have that foundation of number sense.

Christina: And none of this is my own. I learn from others, I compile it together. I'm curator of knowledge, basically. So I've taken what I've read. I love to research things. I'm just a researcher at heart, so none of this is my own research. I'm pulling from other people. But the first one that I ever discovered was from John Van de Walle's book. And we all call it John Van de Walle, but there's so many other authors who worked to compile that information together. So the book's Teaching Student-Centered Mathematics. And those books are ... I call them my math Bible. So there's one for pre-K to second grade, and there's one for third through fifth, and one for six through eight.

Christina: And the thing that I discovered in the pre-K to two, even though I was teaching middle school, I found it in the pre-K to two book. And it was the four number relationships that they talked about. And it's a little tiny part of that book. The book is really big. It talks about all kinds of content, but it was one little piece that I have just grabbed onto because I see the power of it. And I see the power of it not just for our pre-K to two kids, but all the way through. And these four number relationships were ... In the first version of Van de Walle's book, they talked about it as needed these for the numbers up to 12, but I feel like you need it for all numbers.

Christina: And one of the first one is spatial relationships. We need to have a visual of a quantity so that we can see the other relationships that we're about to talk about. All right? If I ask you to close your eyes and picture eight, most of us ... If you just asked a random stranger, right? And I know this is a different crowd, so you may have a different visual of eight. But if you picked a random stranger walking down a street, and said, "Close your eyes and picture eight," most everybody sees the digit eight. They see the abstract version of eight. And that abstract version of eight does nothing to tell me how eight relates to any other numbers. How does eight relate to five? How does eight relate to seven? How does eight relate to ten? You don't see that when the only thing you see is the numeral eight.

Christina: So the first thing is, we need to have some kind of visual of eight. And the hard part is, though, that it's hard to visualize anything past five, even for adults. You have to have some kind of grouping piece of it. So like your fingers, right? You may see finger pattern of eight. You may be able to see it in a 10-frame. Right? Lots of teachers are familiar with 10-frames. You may see tally marks. You are grouping it in some way that helps you be able to visualize eight things. But most of the time, you're seeing some kind of a group of five and then three more. Or other people may see a group of four and four. Right? But having some kind of visual of that eight then helps us with all the other relationships.

Christina: So the next relationship is knowing what is one and two more and less. When you see eight, you should also instantly know that it is one less than nine; it's two less than ten; it's one more than seven, two more than six. Right? All of those should just be natural things that we understand about the number eight.

Christina: Another one is how it relates to the benchmarks. So one of the number relationships in that book, they talked about how does a number relate to a benchmark number? Well, for our small numbers, the benchmarks are a five and a ten. We need to instantly know how eight relates to five and how it relates to ten. Because those help with all those strategies, right? We talked about how, if I was adding 8 + 9, how my husband goes, "Well, you just pluck one off and give it to the nine to make it 10." If a kid doesn't instantly know how those numbers relate to each other, how much to make the next ten, what happens when I pluck one off of the eight, all of that stuff goes into those strategies that we're trying to teach kids.

Christina: And the final one is that knowing ... Well, it's called part-part-whole, and that's the idea of knowing that if I take a whole amount and break it into parts, I still have that whole amount. For our young learners, that is a weird thing. If I count out eight things, but then I move three of them off to the side just a little bit, and I say, "How many do I have now?" They have to back and recount it. They don't understand that you can break apart the whole into different chunks, and it still stays the same. That's a weird thing for kids. And that's essential for kids to be able to do those strategies where they're breaking apart numbers and making new numbers. Because to them, it's making something new. But yet, we need them to understand it stayed the same.

Christina: So those four number relationships became a huge piece of my work with kids in helping them understand those strategies that we were trying to get them to do. And the work of cognitively guided instruction is you lay those foundation of number sense and kids will intuitively use these strategies when they're solving problems; that we don't have to directly teach strategies if they have this foundation of number sense, and we give them problems that are in a context that they care about. Right? If you just give them 8 + 9, there's nothing there for them to really latch onto, but if you turn it into a story that they care about; that you create a story problem out of it, that kids can solve those kinds of problems without any direct instruction, they have this intuitive knowledge about the situation that you've given them, and their information about numbers, and then they go from there.

Christina: And where kids will normally start is they'll count everything. So then that took me onto a journey of, "Okay, we get these kids who are just counting everything." Right? And everybody's like, "How do I get my kids to get past counting?" But counting is such a foundational piece. There are a time and a place where kids need to count, but there's definitely a time where we want kids to start using their grouping pieces. And I like to relate this to the idea of ... When we teach kids to read, we teach them to sound out certain letters. You sound it out letter by letter. It's B-A-T. We want them to sound it out ... But when they start getting into bigger words, we don't want them sounding it out. Right? So if it's something like baseball. My kid's still playing baseball right now. That's top of mind to me. I don't want kids at that stage to be reading it out sound by sound. We want them to see groups.

Christina: And so when kids are adding and they're adding 2 + 3, sure they're going to count. That's fine. But if they're doing 8 + 9 and they're still counting one by one on their fingers, that becomes an issue. Instead, we want them to use their number sense, the relationships of numbers and see groupings that they can use to make that problem friendlier, just like we want them to see groupings when they're reading; to see those chunks. Right? Chunking is so huge.

Christina: So there's a lot of work by Clements and Sarama of whole learning trajectories there, of getting kids to do more with subitizing. Counting is super important, but also being able to see groupings where you don't have to count; that is another foundational piece that helps kids be able to work in that number sense atmosphere instead of having to just count one by one by one all the time.

Kyle Pearce: Well, that was a huge value bomb for everyone who's listening. And over here, my math heart is full because when I hear you talking about Clements and you're talking about Van de Walle, for me, a book that's very similar to what you had been focusing on, but more or less kind of stretching a little further into the middle school mathematics is Elementary and Middle School Mathematics: Teaching Developmentally by John Van de Walle and friends, as you mentioned. Not just John Van de Walle, there was group in there; Karen Karp and Jennifer Bay-Williams and Lynn McGarvey.

Kyle Pearce: So there is so much goodness out there. And the part I really love about this, because Jon and I, while we work with K through 12 educators, we tend to kind of play a lot in that middle school land. And the work that you're doing to build that foundation in the K through five years is so important. When we go back and kind of recap here, talking about spatial relationships, we talk about spatial reasoning so often and the importance of it. And I'm hearing this idea, the visual [inaudible 00:39:32], that conceptual subitizing is popping out there. And when you're thinking of these ideas of one or two more or less, I'm hearing additive thinking, which is so important as we take that journey on our way to proportional reasoning and relationships, and your benchmark numbers as well as that part-part-whole relationship, which is so important as we get into fractions and we start thinking proportionally.

Kyle Pearce: So folks who are listening to this episode who are in our academy and taking our proportional reasoning mini course, they're going to touch on some of these ideas. But if they are teaching in some of the younger grades, they might want to take an even deeper dive into some of these ideas. So I'm really excited to continue this conversation.

Kyle Pearce: So these are four big ideas. You've referenced an amazing book which again, we will link up in the show notes. I'm wondering, do you have any sort of go-to ways to help teachers as they start learning about these number relationships, for example? How do they go about avoiding that, I love it, the rush to the strategy, we'll call it. We call it the rush to the algorithm, but you're even kind of going before that and saying, "We don't want kids to just memorize these strategies," like straight up memorize and repeat. "We want them to build automaticity with these strategies."

Kyle Pearce: So what are some ways teachers might be able to sort of embed this learning in their classroom?

Christina: Yeah, I think one of the biggest ones is having lots of visuals. Mathematics for so long has been very abstract. It's just a bunch of symbols on a piece of paper. And I think in the early grades, we tend to do that more. We bring out the manipulatives, we have lots of visuals. And as we go up in grade level, we start to kind of not use that as much. But it is so important. Any time that we are learning something new, even as adults, we have to go back to that concrete phase. So having great manipulatives, wonderful visuals; we need to be including those in everything that we do. Because no matter what grade level you teach, you have kids who are in need of seeing those visuals, having hands-on manipulatives to work with as well as attaching that to the symbols. You have some kids who are rushing to those symbols because that's their comfort zone. But a lot of those kids sometimes still need the work with having a visual and using the manipulatives to really make sense of what they're doing. There are some kids who are still great memorizers, great rule followers.

Christina: I mean, when we hear a parent say things like, "Yeah, my kid can do multiplication," and they're in first grade. Sure, they've memorized this, they can do the symbolic version of mathematics, but we need to have lots and lots of visuals and lots of manipulatives. So I'm a big proponent of the subitizing cards. I mean, if you just Google subitizing cards, tons are going to come up. I'm a big fan of the Rekenrek. Other people will call it a math rack, arithmetic rack. I even love that even for doing multiplication because you can get ones that will go up to 12 rows of 10 and even 10 rows of 10, to help kids create arrays for multiplication and see those groupings. I think that those are wonderful tools.

Christina: I mean, even 10-frames as you get up into the working with multi-digit stuff; it still helps to see those groupings. Kids need to see groupings of things. So I say it's not just about using manipulatives, it's about using structured manipulatives and being very intentional about giving kids groupings and helping them see those groupings. Not all the kids will see the groupings. Even if you hand the kid a Rekenrek. If you've used a Rekenrek, it is colored with those red and white beads, and we assume that kids will see these groupings that we adults see, but it's not. You have to bring that out for them sometimes and ask the kids, "Well, what do you see? When seven beads are you pushed over, what do you see?" Some of our kids still see it one by one by one. But letting the other kids talk about, "Well, I see three that are left over. I see five red ones and two white ones." Right? We want them to have those conversations.

Christina: So lots of visuals, lots of manipulatives and getting out of the way. It has been forever that we have been seen as the bearer of all the knowledge. I'm a firm believer of putting a problem out there and just seeing what your kids do with it. I've been on a kick lately about talking about how to teach math without a textbook. Because I feel like our textbooks make us feel like we have to provide all of this information; we have to give them the step-by-step stuff, where are kids actually come to us with a lot of intuitive math knowledge. And if we could just take a step back and feel like we don't have to be the bearer of knowledge, our roles is more of a guide and we need to listen to what our students have right now, and then build upon that. And the only way we do that is by posing a problem. Give them a problem. I love cognitively guided instruction. And it's as simple as creating a story problem and using numbers that are appropriate for your students. Right?

Christina: So if you work in the early grades, it might be something where you are ... I'm going to use Fortnite, because that's what my kids are all about, but you replace this whatever you want to. But my kids do it naturally. I hear them talk about it. And they will say things like ... I know it's all about guns and there's some violence and all of that kind of stuff. We talk about all that stuff with our children. So but they collect guns, and they're talking about, "Well, I have three of this kind of gun, and ... " I know you can't use this in your classroom, but I'm totally just doing it because we talk about it in our personal home. But I know you could not have this as a story problem you would use in your classroom.

Christina: But they collect this many of the ARs, whatever it might be. Whatever fits for your kids. That's the other piece, is you could leave them open. I'm getting totally sidetracked, so reign me back in, here. But the whole idea of story problems is you use a context that matters to your kids, but you don't have to dictate the numbers.

Christina: So I also love Brian Bushart's stuff about the numberless word problems, where you can leave it empty. You don't put the numbers in there, and then you can give the kids the number afterwards. And you may give some kids a different set of numbers. Some kids may still be working on something like 19 + 20, where some of your other students in your classroom are ready for something like 28 + 39. Right? And you can decide what numbers to put in there, and you let them solve it any way they can. It's the exact same story problem, it's just different sets of numbers, and then you watch how are they approaching it. You have them discuss and talk about the strategy that they used, and then you can look to see how many of your students are recognizing that they could make a 10, make one of those numbers friendly. How many of your kids are grouping their tens first and their ones first, and then putting it together? How many of your kids are still drawing the pictures? Right? They have to draw out 19 things and 20 things and then count them all, because we still see that, even when kids are working with multi-digit numbers.

Christina: But we only see that if we first give them a problem, and then to say, "Go for it." And if we first say, "Okay, here is how you solve a multi-digit addition problem. First, you add your ones together. Next, you add your tens together. Then we add those together, and that's our answer." Every kid in that classroom then, on every problem you give them, thinks that's the way they need to solve it, even if that doesn't make sense to them at that time. If the problem was 29 + 38, a ton of your kids want to take one from the 38 and give it to the 29 and say that's 30 and 37. But if we first dictate, "Our book says that we need to solve these problems in this way, and you're going to solve the next 20 problems in that way," they solve it that way thinking that's the way they need to do it, but it doesn't make sense to them to solve it that way.

Christina: So lots of visuals, give them manipulatives, but start with just giving them a problem and listen and see what they're doing.

Jon Orr: You know, you wanted us to reign you in a little bit. We just wanted you to keep talking, because you keep telling us great strategies to do with our students. And you know what I really love hearing you talk, and I know that we're talking right now about early numeracy number sense skills. But it's amazing what you're saying about give our students problems, see what they can do with that problem, and then that shapes your lesson. That's exactly what we're doing in trying to talk about in middle school and high school lessons, too. It's all about listening to our students. So it's great to hear that you're trying to make that happen in lower grades. We're trying to make all of that happen in higher grades. All the same stuff. It's all about what you just said. It's listening to your students, giving them a chance to showcase what they have, and then you can build on that using the strategies you've talked about for number sense, and then many other strategies for any other topic or grade level.

Jon Orr: I think it's awesome that that connection is right there. So if we think about this, and think about all these strategies, Christina, how can teachers become more comfortable with learning the strategies you're talking about and specifically, teaching with number sense?

Christina: Yeah. Okay I have two suggestions, here. Number one is the way I started out, which was ... I had no clue what I was doing, so I just tried it. And it failed miserably. Right? I put a problem out there, and I said, "Okay, just try it. Solve it. Let's see what you can do." And they all looked back at me like deer in headlights; big-eyed, just frozen. I don't know what to do. I don't know where to start. And I started to reflect because all of the other instruction I was doing up to that point had been, "Here's what we're going to do. Here's what I need you to do. Here's the steps I want you to take. Now, you go do it." It was the I do, you do kind of scenario where I do [inaudible 00:49:38] you do scenario. And so when I first opened it up and said, "Hey, go for it," they're like, "What? What do you want us to do? This is so foreign."

Christina: So I just had to kind of feel my way and try to figure it out. And it depends upon your kids. I feel like ... When I think back, I'm going to date myself, but I mean that was 16 years ago, after I had graduated with my master's degree and I'm trying to put this stuff in place and I think, "Things have come a long way, now." I think it's a lot easier now because more and more people are starting to do this, to where kids are getting more used to it. When I first started doing this, it was still very much we taught the traditional way. And kids weren't used to any other way than that. And so I feel like it's a little bit more easy to get started.

Christina: And so my advice to get started, too, is just when you open it up, just play dumb. It is very hard to do because as a teacher, we went into this to help our students. And I still will get sweaty sometimes when I open it up and the kids will just sit there like, "Tell me what to do." And I'm like, "Well, what do you think? What do you think this is telling us to think about?" All the questioning, and I start getting nervous about it because I feel like, "I just want to tell you. I just want to help you. I just want to tell you what to do, because I want to help." But sometimes giving that help is not actually helping. You have to think about that.

Christina: I love an analogy that I heard from Robert Kaplinsky once, was that you're like the spotter of a weightlifter. If you try to lift all of that weight for the person who's lifting the weights, they will never get stronger. You have to just be there to help support, you can't do it for them. So you've got to open it up and just play like you don't know. And it's so hard because we do know, and we are sitting there thinking, "Man, just do this one little thing, and it'll make it so much easier." But instead of telling, we have to question. And we have to dig deep and try to figure out what do they know about these numbers and how can I lead them based upon what they know?

Christina: So that brings me to the other piece that I didn't start out with, was really understanding the progression of how kids build their knowledge. And so for me, my focus has been on the progression of number sense. How do kids build this number sense? And it's very helpful when you understand that progression, because you can see like, "Oh, they're not getting this strategy because they don't understand that 49 is like making a 10." There are so many of our kids who can make a 10 when it's 9 + 8, but then magically, somehow when it's 49 + 38, they don't see how to make a 10. What's the issue? Do they understand that they can break apart numbers? Are they not able to visualize what benchmark number is close to 49? You have to understand and know what those number sense components are to figure out what the gap is.

Christina: And so oftentimes, we just see they aren't able to do the strategy, but really, what's at the root of that. Why can't they do that strategy? What is their misunderstanding or lack of understanding about the numbers that's causing them to not be able to understand that strategy that we're trying to get them to see?

Christina: So it takes a little bit of playing dumb but also being very knowledgeable about the progression of how kids develop their understanding so that you know how to question when you are playing dumb.

Kyle Pearce: Yeah. And that's fantastic. I think those two pieces are so important. And the second piece, I really want to talk about, and this idea of the progression; the trajectory. Jon and I are constantly talking about ... We have to understand where are students are coming from. Not only where they're coming from in their own experiences in math class, their home life and all of those pieces; those are very important, as well. But also our mathematical understanding of where they're coming from and what they're bringing with them. So the mathematics that we're hoping that they're coming to us with ... And we all know that a lot of students don't have that, even though our curriculum standards or expectations ask of certain things, often times not all students are at the same place.

Kyle Pearce: And if we don't understand the actual progression or that trajectory, then it's really difficult for us to be able to actually help our students. So I have to know where students are coming from. I have to know my own understanding of the current content that I have to teach. And then I also have to know where it is that they're heading to so that I can get them prepared.

Kyle Pearce: And you referenced some books earlier. I've referenced a book, as well. There's also some websites, as well. The Clements website, learningtrajectories.org is great. But I'm telling you, there is a lot of content for us to learn. So I'm wondering, someone sitting at home, and maybe they might even be getting a little bit antsy or anxious thinking about all of the work that we as math educators really have, because it's an ongoing process. I'm wondering if someone's looking for some way to get sort of a little bit of extra help, a little bit of guidance, what else could they do in order to kind of get a kickstart on doing some of that early math learning?

Christina: Yeah, so I definitely feel like we can spend our time digging into research and that's what I got to do. That was my job for a long time, was digging into that research and learning it as a math coach so that I could help the teachers that I work with. Because let's be honest, especially as elementary teachers, we got a lot. You have to do not just math. You've got all subjects. So I think it's a hard job right now as elementary teachers, because we are expecting teachers to understand all of this for mathematics plus all of this for reading and writing and all the other subjects. And it's a lot. And I really am a proponent of having coaches at schools. I think that's one of the biggest things that can make a difference longterm, is to advocate for coaches in the schools. Because they can focus on those particular subject areas and be able to help provide professional development.

Christina: The biggest piece of ... The only way we change is by learning. Right? We grow when we learn something different. So you can go and do it on your own, which is difficult because you have a lot of responsibilities as a teacher. We can hopefully start bringing in more math coaches into our schools so that we can have them consume that information and then find the pieces that are important to bring out to our teachers that they work with, their teachers they work with. And I feel like that was my role for a really long time. And luckily, I've been able to put what I've done into an online course.

Christina: So I'm also going to just kind of plug the course here a little bit. Because I know it's hard work to find the time to read all of that stuff that we've talked about. So I put it all together. That was my job, was to go out and do professional development in schools. But the hard part was that not everybody has access to that. It is very difficult to get professional development people to your schools. It costs a lot of money. It's hard to have time off for everybody to be there to come listen to the professional development speaker. Have a person there for one day and then have to go back and implement it.

Christina: So I put all of my stuff that I would do in an in-person training into an online course, which I know you guys have done the same thing with Making Math Moments That Matter. It's just a great way to be able to learn from somebody who has gone through this, who has sifted through all of the content that's out there. And I'll be honest; all of this stuff is in books. You can Google all this stuff and you can spend time doing it, but the power of having someone who has done that before you and can narrow it down to the biggest pieces that you need to know and understand, it's a powerful thing to do. And to be able to learn from that person who's taken the information and compiled it for you.

Christina: So I'm also going to give a little plug for, come take my number sense courses. Because I have one that is for what those number sense concepts look like in pre-K to two, but also three through five. Those same relationships hold true and help kids even as they get into fractions. Think about if you're adding three and a half plus five and three-fourths. We can have kids just memorize the procedure for doing that, but if we have built number sense, these strategies that kids have for things like 9 + 8 still apply when they're doing things like three and a half plus five and three-fourths. Kids can chunk numbers off, make numbers friendly. They can manipulate numbers and become really flexible if we lay the foundation of number sense. And it makes the operations so much easier on all kinds of numbers; decimals, fractions, multiplication, division. It is all so much easier if kids have this number sense.

Christina: But if we try to jump straight to the algorithms, to the strategies without the foundation of number sense, it really does ... It's like beating your head against the wall. You're like, "Why aren't these kids getting it?" And it's because we don't have the foundation set for them, first.

Jon Orr: You know, in building a course for any teacher anywhere to access from home, from school, on their own time, is an amazing thing. And you've built, and that's exactly why we've built our workshop, too. It's we can provide the PD to teachers wherever they want, and you can help change the students in those classrooms, because you're helping those teachers do those same things. And the moment that this is going live, if you're listening to this right now on the day it went live, that's when we've started a third cohort of our workshop.

Jon Orr: So I'm wondering, is your workshop similar to ours? Can you tell us a little bit about how it looks, and structured? How it's structured online? We're pretty curious about that, about yours, and what would ... Say a teacher enrolled in your workshop or in your course. What would that look like?

Christina: Yeah, so I want to back up really quickly. Because I hate only providing things that have to be paid for. So when this goes live, there's also ... I think we're going to have a button on the show notes page ...

Kyle Pearce: Yeah, yeah. We got lots of your information to put it in the show notes.

Christina: For the free webinar. Yeah. So I'm going to do a webinar all about that math is not a worksheet, and it's going to go way more in detail about how do we actually do this, and what are these number sense concepts. And so how do we take those number sense concepts and do it without relying upon worksheets, our textbook, that kind of thing. How do we provide kids these experiences? So there's definitely the free workshop that you can come and attend. You don't have to take the full course. But if you want to go deeper, there is the full course.

Christina: So in the course, we not only talk about those four number relationships from Van de Walle and friends, but we talk about the four early numeracy concepts that Clements and Sarama talk about.

Christina: And so each week when the course starts, I do what's called dripping. So you don't get the full course at once, because it's like drinking from a fire hose if you try to have it all at once. So the first week, you learn about the first early numeracy concept. And the first one we talk about is subitizing. Whether you take the pre-K to second grade course or you do the third through fifth grade course, because even kids in third through fifth, if they aren't able to subitize, we need to build that. So we talk about what it is, why it's important, and I provide you with a way to assess students. And my assessments are not like, "Sit down and let's do this worksheet." It's through observations. How can you observe this through the natural progression of the day? Right? What should you be watching for to see if kids can subitize, and how do we do that?

Christina: So we talk about how to look for that early numeracy concept. And then I give you a lot of my favorite experiences. I don't say activities. I call them experiences. Kids need to experience numbers and their relationships. So we talk about experiences, and then I'm big proponent of games. So every week, you get a download of my favorite games. And also, I'm a big proponent of including your parents and getting parents informed about what we're doing and why we're doing it. Because I think back to ... And I know you guys have talked about this before. I call myself the recovering traditionalist. And I think about if I didn't know this stuff, I would've been that parent who's like, "Why aren't you just drilling and killing them," basically. Like, "Let's give them flashcards. They just need to memorize this stuff." So at the end of each module for the week, you get letters that you can send home to the parents about what you're doing, why it's important, and some quick little things that they can do at home to help build that relationships or that early numeracy concept.

Christina: So each week, they get that for all of the eight number sense concepts. And so what it is, why it's important, how to assess it, activities or experiences and games and a resource to give your parents. So they go through it week-by-week, and then you have three months that you can access it, which is so amazing. Because that's what always bugged me about going out to do professional development, is I'm there for a day and then I'm gone. So I love the power of this online learning, because you get it continuously for three months. As you're trying it out, you can come in there and you can add comments underneath each video and talk about what you're trying, get advice from me and the others in the group. It's just a fun way to be able to find people who are working on the same thing. Because sometimes you might be the only person out there at your school who's wanting to try this, and so it's fun to be able to be online and learn with others.

Christina: And you don't have to be online at the same time. I used to call it PD in your PJs, but I was told somebody else uses that phrase, so I can't actually use that phrase. But you can do PD wherever you want in whatever ...

Jon Orr: In your most comfy clothes you have.

Kyle Pearce: In your [crosstalk 01:03:56]

Christina: ... clothes you want. Clothing of your choice and the drink of your choice. Right?

Jon Orr: Right.

Kyle Pearce: Oh.

Jon Orr: Right.

Christina: how often do you get professional development like that?

Kyle Pearce: That's up my alley.

Jon Orr: Yeah, for sure.

Kyle Pearce: Fantastic. Well, Christina, this has been such a great conversation. I know that some, especially our K through five educators who are listening, but again, the big idea is the same. It doesn't matter what grade level you're at. It's all about foundations. Whether it's a foundational skill for a grad nine course or a 12 course, it's all about the building blocks and making sure that we do it and we do it right. And I really liked the metaphor of your husband and going and building homes and looking at jobs that aren't done right the first time and the problem that that causes. You can still fix it, but it's going to take more time and effort and probably a lot more planning to try to figure out, "Okay, where is the problem? How do we help address this and help move on from here?"

Kyle Pearce: So this conversation has been huge. I know, because Jon and I, having met you and having seen your content online for so long, you provide all kinds of great value bombs with the vlogs you do. You also have a Build Math Minds podcast that's out, so we will definitely link that up. Are there any other website links that you'd like to share with folks before we say goodbye? We will be sure to post them in the show notes page.

Christina: I would definitely be looking up for those progressions. I know you guys are going to link my stuff, but for those who want to go learn on their own, looking up Clements and Sarama's learning trajectories, Cathy Fosnot, she calls them landscapes for learning; understanding those progression of how kids develop their understanding is super huge. So even just going and Googling math progression, or a specific content; fraction progression, multiplication progression, you're going to find lots of info out there to help you build your understanding of where kids were, where they hopefully were before they came to you, where they are now, and where we need to get them.

Christina: So that's just a big piece. And I don't care if you learn that from me or somewhere else. It's an important thing that all of our students need.

Kyle Pearce: All right. And I want to make sure that folks check out that webinar that Christina was talking about. It is wide open for everyone to participate in, so be sure to share with your colleagues. Makemathmoments.com/buildmathminds will bring you straight to that page. Again, makemathmoments.com/buildmathminds will bring you to the registration page, but we'll also link it up in the show notes.

Kyle Pearce: Christina, we are so happy to have had the time to have a chat with you tonight here on the Making Math Moments That Matter podcast. We're hoping that you have a fantastic evening, and we will be certain to check you out in that webinar coming up real soon.

Christina: Thank you Kyle and Jon. It was a pleasure to be on here, and thank you for letting me spread this message even further.

Jon Orr: Awesome. Thanks so much, Christina.

Kyle Pearce: We want to thank Christina again for spending some time with us to share her insights with us and you, the Math Moment Maker community.

Jon Orr: As always, how will you reflect on what you've heard from this episode? Have you written ideas down, drawn a sketch note, sent out a Tweet, called a colleague? Be sure to engage in some form of reflection to ensure that the learning sticks.

Kyle Pearce: Like you heard in the episode, Christina is offering a free webinar training called Math Is Not A Worksheet. Registration is open now, as this episode is released until October 12th, 2019. You can register by visiting makemathmoments.com/buildmathminds. Again, that's makemathmoments.com/buildmathminds. If you're listening to this after October 12th, 2019, don't worry. That link will either bring you to the webinar replay or a wait list for her next free webinar.

Jon Orr: In order to ensure you don't miss out on our new episodes as they come out each week, be sure to subscribe on iTunes or your favorite podcast platform.

Kyle Pearce: Also if you're liking what you're hearing, please share the podcast with a college and help us reach an even wider audience by leaving us a review on iTunes, Google Play or Spotify, and Tweet us your biggest takeaway by tagging @makemathmometns on Twitter and Instagram.

Jon Orr: Show notes and links to resources from this episode can be found at makemathmometns.com/44. Again, that's makemathmoments.com/44.

Kyle Pearce: Well, you know what that means, Jon. Until next time, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us. And high fives for you.

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Episode #44 - Building Math Minds: An Interview with Christina Tondevold - Make Math Moments

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Today we speak with the recovering traditionalist Christina Tondevold. Christina previously taught mathematics traditionally much like we did and Now she gets to work with teachers in developing students' mathematical understandings in ways that are not limited by the traditional ways of doing and teaching mathematics through her site BuildMathMinds.com where you can get PD in Your PJs.

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Build Math Mind's Podcast - Interview with Ann Elise is Episode 4

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Build Math Mind's Podcast - Interview with Ann Elise is Episode 4

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If you don't know whether or not you are a Recovering Traditionalist, here's how I define us. We are math educators who used to teach math the traditional way.

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Interview with Dr. Nicki on Learning Trajectories

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Interview with Dr. Nicki on Learning Trajectories

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Episode #62: Learning Trajectories & Running Records: An Interview with Dr. Nicki Newton - Make Math Moments

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We have a treat for you today. We're speaking with the great and obviously passionate Dr. Nicki Newton. By the end of this episode you'll wish Nicki could just keep on going. She shares with us today her passion about learning trajectories and why we all need to learn them for various mathematical ideas.

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Build Math Minds with The Recovering Traditionalist

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Working to help teachers think outside the traditional way of doing, and teaching, mathematics.