Rethinking Instruction for Lasting Understanding: An Example

By Kate Nowak

How do we help our students build mathematical understandings that endure past the unit test? If we want students to construct strong, reliable bases of mathematical knowledge, our instruction needs to do more than present explicit procedures—even when that’s done well. Providing lots of opportunities for students to reason can help. So can understanding and leveraging the progression of learning across grade levels. But what does that look like in practice? Let’s examine a single topic in grade 7: solving inequalities. 

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When is a number line not a number line?

By William McCallum

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?

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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

By Kaneka Turner

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.

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Ratio Tables are not Elementary

By William McCallum

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

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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.”

By Kiana Porter-Isom

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. 

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Using Diagrams to Build and Extend Student Understanding

By Jenna Laib and Kristin Gray

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

\frac{1}{4}\times \frac{1}{3}

\frac{1}{4}\times \frac{2}{3}

\frac{2}{4}\times \frac{2}{3}

\frac{3}{4}\times \frac{2}{3}

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. 

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The 5 Practices: Looking at Differentiation Through a New Lens

By Catherine Castillo

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. 

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Learning through Teaching

By William McCallum

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.

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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.”

By Tabitha Eutsler

This was my conversation with our math coordinator after my first few days of teaching IM K–5 MathTM 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? 

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