The number line is a seemingly simple object: a straight line with two points marked 0 and 1. Those two points are the seeds of great complexity, however. Whole numbers are located at positions marked off by iterating the interval. Fractions are located at equal subdivisions of the spaces between whole numbers. Flip all those numbers to the other side of 0 and you have negative rational numbers. Then, although the line is completely dense with rational numbers, you find you can sneak between them with infinite decimal expansions to define a whole universe of irrational numbers. Given all of these layers of complexity, when exactly is the right moment to introduce this marvelous object to students?

Continue reading “When is a number line not a number line?”# Shifting Practices: Helping Everyone—from Students to Administration—Find their Voice in the Math Classroom

It was easy to say yes!

Last spring, I was approached by our Math Coordinator and asked about piloting a new math program. I knew my staff was ready for building-wide consistency and we were ready to try something new. I easily said yes!

My teachers were offered training over the summer and access to the resources to begin teaching this fall.

After just a few weeks of instruction, my staff began to voice concerns.

Continue reading “Shifting Practices: Helping Everyone—from Students to Administration—Find their Voice in the Math Classroom”# The Art of Reflection

“In times of stress, the best thing we can do for each other is to listen with our ears and our hearts and to be assured that our questions are just as important as our answers.” —Mr. (Fred) Rogers

We are never more “on” than when we are teaching a lesson. All of our senses are heightened and all of our energy is focused on understanding students and being understood by the students we are teaching. Often times, it is not until the lesson is over that we have the mental space to look back over the student work samples and anecdotal notes, or replay scenes from the lesson in our minds to gain insight. I was reminded of this recently when I was invited to test out new problem-solving structures from IM K–5 Math’s Grade 4 Unit 8 in my colleague’s classroom.

Continue reading “The Art of Reflection”# K–5 Curriculum Design Features that Support Equity and Inclusion

Before I joined the K–5 curriculum writing team at IM, I was a K–8 regional math content specialist for a public charter organization that largely consisted of Title I schools, or schools receiving federal funding to support a large concentration of students in poverty. Prior to that I had experienced the joys and challenges of serving communities like these as a teacher and math coach in South Central Los Angeles and Crown Heights Brooklyn.

Continue reading “K–5 Curriculum Design Features that Support Equity and Inclusion”# Ratio Tables are not Elementary

In grade 3, as students start to learn about multiplication, they think about products like 6 x 7 in terms of equal groups. 6 x 7 is the number of things when you have 6 groups with 7 things in each group. They might start out calculating that number by drawing a picture of the 6 groups and counting how many things they are. They might use a 6 x 7 array to organize the count. They might then see that the total number is 7 + 7 + 7 + 7 + 7 + 7 and do the additions 7 + 7 = 14, 14 + 7 = 21, etc. From there they might learn to simply write down the multiples, doing the additions mentally:

7, 14, 21, 28, 35, 42

Continue reading “Ratio Tables are not Elementary”# Could you—or someone you know—be our newest IM Certified Facilitator? The Critical Role of IM Certified Facilitators.

“What I find distinguishes IM is that IM Certified Facilitators are uniquely supported by the IM authoring team to ensure the integrity of the curriculum remains intact.”

I was always interested in mathematics as a student but I only began *enjoying* mathematics when I was in high school. Until then, I didn’t think it was something I could get excited about. Now, as an educator and as the Manager of IM Certified Facilitators at Illustrative Mathematics, I am seeing first-hand how teachers are enabling students to embrace and enjoy mathematics and be enthusiastic learners from their first interaction with mathematics. This is why I was drawn to IM—for their mission to create a world where learners know, use, and enjoy mathematics.

# Using Diagrams to Build and Extend Student Understanding

Take a moment to think about the value of each expression below.

What do you notice? How would you explain the things you notice?

If you are like us, or the students in Ms. Stark’s grade 5 classroom, you may have noticed many things. Things such as each expression has the same denominator, or the way in which the values increased as the problems progressed. When students notice these things, we often ask, ‘Why is that happening?” but it can be challenging to explain why beyond the procedure one followed.

Continue reading “Using Diagrams to Build and Extend Student Understanding”# The 5 Practices: Looking at Differentiation Through a New Lens

Our district had seen a downhill trend in standardized test scores in mathematics. This forced us, as educators, to take an intentional look at our teaching practices.

The past few years have been an exciting time in math instruction. Research on brain plasticity and mindset have caused a shift in the idea of what it means to know and do mathematics.

Continue reading “The 5 Practices: Looking at Differentiation Through a New Lens”# Learning through Teaching

I was in New Orleans a couple of weeks ago visiting a school using IM 6–8 Math and was inspired by the efforts the school was making to implement problem-based instruction. I saw teachers at different stages on a learning curve with the instructional routines in the curriculum and realized how important it was to *have* a learning curve, and not a learning cliff, for teachers to grow into this way of teaching. We have tried to achieve this in many ways in our curriculum.

# Building a Math Community with IM K–5 Math

“I’m not sure this is working. Only five of my students are participating and commenting each day. The rest sit there and look at me.”

This was my conversation with our math coordinator after my first few days of teaching IM K–5 Math^{TM} with my third graders. Those five students were having great conversations. However, my other students just sat there wide-eyed, silent, and staring blankly at their papers. I felt lost. Was this the best for my students? Could we survive a whole year of math like this? I wanted my students to love math and have a deeper understanding of mathematical concepts. How would this get them there?

# Which Vertex is the Center of a Triangle?

I am sometimes asked what is the secret to the success of our curriculum, what is the special property that sets it apart from other curricula. That question is like the one in the title of this blog post, “Which vertex is the center of a triangle?” It doesn’t have an answer. None of the vertices is the center of a triangle; all three are equally necessary for it to exist. Similarly, all three vertices of the instructional triangle—students, teachers, and content—need equal attention in the work of teaching mathematics. And not only the vertices but the arrows between them, which “represent the dynamic process of interpretation and mutual adjustment that shapes student learning [and] instructional practice.”^{1} If there is something special about what we do in writing curriculum it is to pay equal attention to all parts of the instructional triangle.

# Updates to Supports for Students with Disabilities and English Language Learners in IM 6–8 Math

Continue reading “Updates to Supports for Students with Disabilities and English Language Learners in IM 6–8 Math”At Illustrative Mathematics we are committed to creating a world where learners know, use, and enjoy mathematics. We believe that every student can learn grade-level mathematics with the right opportunities and support. Our approach is to remove unnecessary barriers and provide teachers with options for additional support so that every student can engage in rigorous mathematical content. We’ve been busy this year working on some exciting enhancements to the teacher tools and supports to empower teachers to deliver instruction that meets the specialized needs of English learners and students with disabilities.

# First Impressions: The First Units in IM K–5 Math

“I’ve learned that people will forget what you said, people will forget what you did, but people will never forget how you made them feel.”

― Maya Angelou

When I think back to my 8th grade math class, I cannot recall the exact problems I struggled with or exact things the teacher said or did, but I can distinctly remember how I felt each day walking into that classroom: anxious. From the very first day of school, I struggled, and my feelings of failure and self-doubt only compounded as the year progressed. I just could not keep up. While many, *many* years have passed, and details have faded from my memory, I have never forgotten how badly I felt about myself as a learner of mathematics each day.

# Realizing the promise of open resources, part II

In my first post on the topic of realizing the promise of open educational resources, I described the IM Certified program. Our partners offer multiple versions, including a free online version and enhanced versions with different options for users. This is IM’s way of reaching teachers and students from a wide variety of districts who may be looking for those different options, while assuring that, as these versions evolve, they will stay true to the original design. However, by the terms of the CC BY license, anybody can use the curriculum with or without certification. This freedom further supports our mission to get these carefully crafted materials into the hands of as many students and teachers as possible.