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Four Best Practices for the Math Classroom

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Four Best Practices for the Math Classroom

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Strategies to Support Learners Who Are Below Grade Level

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Strategies to Support Learners Who Are Below Grade Level

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Recommendations for Targeted Math Support

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Recommendations for Targeted Math Support

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Illustrative Mathematics provides resources and support for giving their students an enduring understanding of mathematics.

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Responsible Use of Technology and Social Media Policy.

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Enter your name and email below for your free front row seat at the Build Math Minds Virtual Summit.

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K-5 Math Teaching Resources is a leading New York City based math consultancy firm. Our mission is to transform the teaching and learning of mathematics, improving outcomes for all students in elementary and middle school. Click on a tab below for information on our research-based professional development and customizable consulting services. Our results-driven elementary and middle school consultants are committed to working side-by-side with administrators, coaches and teachers to build exceptional quality math instruction and curriculum for schools and school districts, transforming students' math education and achievement.

On this website you will find mathematics to support standards-aligned math instruction. Use the grade level tabs at the top of this page to download sample resources for each grade level from Kindergarten through 5th Grade or visit our EBook Store to purchase a complete set.

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Research based resources and professional learning opportunities to transform math teaching and learning communities.

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Find the connections between the Common Core State Standards for Mathematics

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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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The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

The progressions can explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics. They would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks. Progressions documents also provide a transmission mechanism between mathematics education research and standards.

For more information on the project, please visit the project website.

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Progressions Documents for the Common Core State Standards for Mathematics

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The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

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University of Arizona Progression Documents

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University of Arizona Progression Documents

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Cathy Kessel wrote an article with Liping Ma that discuss "unit fraction" vs "fractional unit" and has added a remark to the appendix about this.

Here is the new version.

Hi everybody, Cathy Kessel has compiled all the suggestions that people have made here and other places over the years to produce a final version of the progressions. Here is the compiled document of all the progressions.

And here are two documents outlining the major changes from previous versions.

Note that the University of Arizona has deleted the website that used to host the progressions. I am working to have that url redirect here.

Hi everybody, in order to deal with the spam problem I have deleted all users regarded by WordPress as spam users. These are defined as users who have never posted. Unfortunately this seems to have caught some legitimate users as well. Their content is still on the blog, with the author marked as anonymous.

Cathy Kessel is working on editing the progressions documents into final versions, incorporating comments from this blog and elsewhere and filling in gaps. Here is the fractions progression. We will be rolling out the others over the next few months. Comments welcome!

With the high school writing work in full swing, I've been remiss in posting here lately. I do have some ideas saved up! Meanwhile, Kate Nowak and I have a piece about instructional routines over at the IM blog.

I've been meaning to let you know about the Illustrative Mathematics blog, which launched a few weeks ago. It has blog posts by members of the IM community about our grades 6-8 curriculum and about teaching practice, including a whole series on the 5 practices framework of Smith and Stein. Also, we will be cross posting any IM related posts I write here over there as well. I hope you find our new blog useful!

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AASD Coherence Map Continuum Planning Template

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AASD Coherence Map Continuum Planning Template

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The New York State Education Department discontinued support for the EngageNY.org website on July 7, 2022. The NYSED encourages educators to download any EngageNY content they wish to use in the future from our archive sites below.

All ELA and mathematics curriculum files will be available at the links below, and will remain free and licensed under the Creative Commons Attribution-NonCommercial-ShareAlike (CC BY-NC-SA) license.

Archived EngageNY Curriculum Files:

Grades 3-8 ELA and Mathematics Test Materials:
Questions regarding this change may be directed to the EngageNY Help Desk.

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The New York State Education Department discontinued support for the EngageNY.org website on July 7, 2022. The NYSED encourages educators to download any EngageNY content they wish to use in the future from our archive sites below.

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Designing Shifts-Aligned Interventions in the Math Classroom

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Designing Shifts-Aligned Interventions in the Math Classroom

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As teachers, we want to help all kids learn and when we're faced with the realization that some students just aren't "getting it," we work to figure out how to catch kids up. It's a prevalent struggle many educators grapple with and a frequent conversation in schools: How do we address unfinished learning in math and fill gaps in student understanding?

Over the course of my career I have seen, all too often, students behind grade level denied access to grade-level mathematics, in an effort to catch them up or "fill the holes." All students deserve access to content-rich instruction at their grade level, and when students have unfinished learning from previous grades, teachers need to take appropriate action-and that doesn't mean reverting solely to materials and content from previous grades.

So, how do we address unfinished learning and support students in developing understanding and mastery of content? Chrissy Allison, Director of Math Professional Learning at Achievement Network (ANet), and I were knee deep in conversations centered on unfinished learning when we first penned a "Do's and Don'ts Chart" for catching kids up. The chart identifies some missteps districts, schools, and teachers make when working with students who have gaps in their math knowledge. We've included some recommendations to consider to avoid these missteps. You can download the chart at the top of this blog post.
Recommendations for Targeted Math and Intervention
Evidence of the Shifts in Mathematics are found in the Recommendations for Targeted Math and Intervention chart. When teachers plan universal instruction with Focus, Coherence, and Rigor in mind, but ignore these Shifts when it comes to intervention and support time, there is an immediate disconnect in instruction. The Shifts must be top of mind for both whole-class instruction and extra support. Let's break a few of these down.

With Focus, intervention time should be based on the content that matters most for success in later mathematics. For instance, if a third grade student's interim data illustrates weaknesses in two clusters: 3.G.A Reason with shapes and their attributes and 3. NF.A.1 Develop understanding of fractions as numbers, prioritize the fraction work since this reflects the Major Work of the Grade. Maintaining attention to Major Work of the Grade will ensure that additional time and attention is spent on foundational content necessary for students to be successful with future math work. When planning for Coherence, use the Coherence Map to trace content across the grades and design learning activities that support grade-level mathematics. Make connections explicit to students to support their developing thinking and understanding. Remember that the standards themselves call for particular aspects of Rigor. Teaching tricks that proceduralize concepts and make ideas "quick and easy" for students undermines learning. Stay true to the aspect of Rigor in the standard and plan as such. For example, if intervening on 6.NS.B.2, Fluently divide multi-digit numbers using the standard algorithm, teach the standard algorithm and practice, practice, practice! The suggestions in the chart likely can't all be tackled at one time. Determine where to start, what changes can be made, and how to move forward. It is important to consider interventions within the "big picture" of math education. This means taking a step back and applying the same principles you apply to your planning of whole-class instruction. At first, this may seem counterintuitive. It can be tempting to give in to gut reactions: "I need to address every gap, right now before I move on;" "I need to focus just on the unfinished learning-connecting grade-level content would just be confusing." The reality is, however, that if we give in to these inclinations, we're denying students access to grade-level content and Shifts-aligned instruction. And that isn't fair to our students.

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Designing Shifts-Aligned Interventions in the Math Classroom - Achieve the Core Aligned Materials

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As teachers, we want to help all kids learn and when we're faced with the realization that some students just aren't "getting it," we work to figure out how to catch kids up. It's a prevalent struggle many educators grapple with and a frequent conversation in schools: How do we address unfinished learning in math ...

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Addressing Unfinished Learning in the Context of Grade-Level Work

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Addressing Unfinished Learning in the Context of Grade-Level Work

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Addressing unfinished learning in the context of grade-level work - Peers and Pedagogy

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As a teacher, I had an understanding of the grade-level math content I was supposed to teach and the belief that students’ new learning had to build from their prior understanding. But the harsh reality was that most of the students in my class were several years below grade level, and I only had on…

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"How far back is too far?"

When discussing how to support mathematical learning for struggling students, this is a question I hear frequently. Of course, as with most things in education, there isn't a simple and straightforward answer to " How far should interventions flow back in order to access grade-level content?" In a recent blog post, I addressed unfinished learning in the mathematics classroom and included this Do's and Don'ts Chart showing some common missteps and recommendations for how to avoid them.

Since then, questions have surfaced about what it actually takes in the day-to-day bustle of instruction to enact these recommendations. Simply put, how do the recommendations come to life in the classroom? And maybe more importantly, how do they really help our most struggling learners?

First, as educators, we all know that each student is different; effective intervention is created and instructional decisions are made in the quickest flash of a day as we determine the changing needs of each student. All that to say, the chart is meant to be a beginning guide-a place to begin a discussion. So let's break down one of the missteps and its recommendation now: In her Aligned post, Chrissy Allison shared an approach to address unfinished learning that can be applied when reaching back in progressions to address unfinished learning. Of course, the trick is going back to the "right" content.

Let's try walking through an example of how we might design an intervention that goes back the appropriate amount in the learning progression. We're going to start with the "Understand" box. In planning for an upcoming unit on fraction multiplication, a fourth grade teacher digs into the standards. Some students struggled with the fraction work that came earlier in the year around equivalence and ordering. Others struggled with the operations of adding and subtracting. He anticipates that these same students, and maybe more, may have a hard time with multiplying a fraction by a whole number (4.NF.B.4). Where to start? What to do?

Spend some time digging into student work and digging into the standards.

1. In the Coherence Map, pull up the standard in question. (Make sure you can see grade 2 standards boxes.) It's likely that there are silent shrieks echoing across American classrooms when you see that a standard maps to so much pre-requisite work. Likely this fourth grade teacher is saying, "Oh boy, do I really have to re-teach grade 2 standards to my fourth grade class so they're successful with the fraction work?"

2. Now it's time to zero-in on the real source of the lack of understanding: the"Diagnose" step in the process pictured above. While it's always a good decision to build in opportunities to strengthen understanding during universal instruction and reinforce the content in other units throughout the year, this may not be enough for all students. For these struggling students, find out what they know about the concept, (in this case, fractions) first. Start with the closest connection in the Coherence Map: the same domain and then consider branching farther to find the missing link. When adults say they were "bad" at math, or they just "didn't get it," the root causes almost always stem from a weak understanding of underlying mathematical concepts. That's why, in this example, it's important to assess whether the students understand fractions as numbers by first going to the conceptual underpinning of current work, or the connection within the same domain.

3. Lastly, you need to actually plan the targeted intervention once you've narrowed the cause of the knowledge gap. Unfinished learning around the 4.NF.B.4 standard is a tough nut to crack since the conceptual foundation could pull from grade 3 fractions, 3.NF.A.1, or the meaning of multiplication in 3.OA.A.1. It is always hard to hear something like "it could be this, or it could be that,"but flexibility and listening and observing the students are important factors here since the teacher should be prepared to take an alternate path if needed. The teacher first draws on the 3.NF connection and if the understanding is secure with fractions, moves to the 3.OA connection to take action on unfinished learning on the meaning of multiplication. The Coherence Map has sample tasks and assessment items to help you get started with your intervention.

Our students' time, and our own time as educators, is incredibly valuable. Our job as math instructors is to help our struggling students plug the gaps, as quickly as possible, that are precluding them from accessing grade-level content. The longer they are not accessing grade-level content, the farther behind they will fall, leaving it in question whether they can stay on pace for the next year's learning/content and to eventually leave high school ready for college and their careers. While it may be tempting to start from square one with our interventions, our students don't have time to re-take 2 nd grade math in 4 th grade-and they don't need to. Strategic intervention planning helps us, and our students, tackle gaps more quickly. When students see themselves as incapable of doing the grade-level work their peers are doing, the more likely they are to convince themselves they "aren't math people," so our efficient interventions are critical now.

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How to Select Math Intervention Content - Peers and Pedagogy

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"How far back is too far?" When discussing how to support mathematical learning for struggling students, this is a question I hear frequently. Of course, as with most things in education, there isn't a simple and straightforward answer to "How far should interventions flow back in order to access grade-level content?"

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The Progression of Early Number & Counting 

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The Progression of Early Number & Counting 

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The Progression of Early Number and Counting

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If you’re not a pre-k, kindergarten, or 1st-grade teacher, you need to find one and give them a hug after watching this video. They do the work of an army and many times their work goes unno…

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The Progression of Addition & Subtraction

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The Progression of Addition & Subtraction

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Last month I posted The Progression of Multiplication hoping that a couple of friends and parents would find it helpful. Well, here's my stab at the progression of division. I understand that there's lots of ways to model division and this is only ONE of them.

Understanding the vertical progression of mathematics is really important in the conceptual development of everyone's understanding.

Hope it helps.

Stay thirsty my friends!

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Last month I posted The Progression of Multiplication hoping that a couple of friends and parents would find it helpful. Well, here's my stab at the progression of division. I understand th...

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I'm excited to share the 4th installment of the Making Sense Series which explores meaning, equivalence, and comparison of fractions. Fractions are the gatekeeper of algebraic thinking and pr...

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Progression Documents & Videos
Achieve the Core Progression Documents
Achieve the Core Progression Documents
 Achievethecore.org :: Progressions Documents for the Common Core State Standards for Mathematics
Achievethecore.org :: Progressions Documents for the Common Core State Standards for Mathematics
The progressions can explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics. They would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks.
achievethecore
University of Arizona Progression Documents
University of Arizona Progression Documents
 The University of Arizona - Institute for Mathematics & Education
The University of Arizona - Institute for Mathematics & Education
The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics. These documents were spliced together and then sliced into grade level standards.
arizona
The Progression of Early Number & Counting
The Progression of Early Number & Counting 
 The Progression of Early Number and Counting
The Progression of Early Number and Counting
If you're not a pre-k, kindergarten, or 1st-grade teacher, you need to find one and give them a hug after watching this video. They do the work of an army and many times their work goes unnoticed. There's so much happening in the early years of school, that without this progression of early number and counting,...
questioning my metacognition
The Progression of Addition & Subtraction
The Progression of Addition & Subtraction
 The Progression of Addition and Subtraction
The Progression of Addition and Subtraction
This is "The Progression of Addition and Subtraction" by Graham Fletcher on Vimeo, the home for high quality videos and the people who love them.
vimeo
The Progression of Multiplication
The Progression of Multiplication
 The Progression of Multiplication
The Progression of Multiplication
This video is about the progression of multiplication and the conceptual development of the standard algorithm.
vimeo
The Progression of Division
The Progression of Division
 The Progression of Division
The Progression of Division
Last month I posted The Progression of Multiplication hoping that a couple of friends and parents would find it helpful. Well, here's my stab at the progression of division. I understand that there's lots of ways to model division and this is only ONE of them.
questioning my metacognition
The Progression of Fractions
The Progression of Fractions
 The Progression of Fractions
The Progression of Fractions
I'm excited to share the 4th installment of the Making Sense Series which explores meaning, equivalence, and comparison of fractions. Fractions are the gatekeeper of algebraic thinking and probably a big reason why we suffer from arithmophobia as a society. I'm hoping this progression helps provide some relief and courage moving forward.
questioning my metacognition
 
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