This school year, I have had the privilege of collaborating with several 3rd-grade teachers. During one meeting, I shared an innovative idea from a 4th-grade teacher, Ben Cogswell. He introduced the use of Play-Doh to help his students understand multiplication through hands-on activities. I really like how he incorporated MathReps into this lesson. In the accompanying image, you can see students recording their findings in a MathReps, along with an instructional video they can reference for guidance.
Fast forward a few months into 3rd-grade. While meeting with one of the groups, one teacher, Jensen Hall, was excited to share that she had purchased some Play-Doh for her students to use as they begin learning about multiplication. Then, about a week later, she sends me this message with images:
Taking a closer look at the image, the red arrow points to where a student wrote out the problem on their desk. Using equal sets, students were able to model how to solve the problem. While the faces have been hidden, I can assure you that they were all smiling and clearly having a good time.
We all want our students to be excited and engaged with math. In the rush to fit everything into a school day—assessments, end-of-the-year testing, and much more—we often forget how to make learning fun and hands-on. Starting with concrete models is essential when teaching new skills like multiplication. Taking the time to begin with concrete models will ultimately save us time later and allow students to truly understand these foundational skills, setting them up for future success.
The story and images have been shared with permission from the teacher.
If you are a 3rd-grade teacher looking to engage students with the concept of line plots, you are not alone. The task of transferring data onto a number line may seem straightforward to adults, but for young learners, it can be quite challenging. The 3.MD.B.4 standard, a supporting cluster in the Common Core Standards, plays a substantial role in reinforcing the understanding of fractions and measurement. This means that as students delve into the world of line plots, they are simultaneously immersing themselves in the intricate connections between fractions on number lines, and measurement. It’s a perfect illustration of how math is both messy and beautiful, all intertwined in a way that connects to the real world.
Putting It All Together
Here is an example of what this integration could look like. This MathRep integrates the fractions on a number line and measurement. The information is collected in the upper left quadrant. Students then fill in the information on the line plot. In the upper right quadrant, they can record their mathematical observations. Much like in the previous blog post, the observations serve as a low-floor, high-ceiling activity—one that all students can access and be successful in.
Teacher Set Up
The MathRep method grants teachers flexibility. Its primary aim is to offer consistent learning experiences for students while minimizing the teacher's preparation workload. When implementing this approach, teachers can start by providing all the necessary data in the upper left quadrant and should ensure variation in the data to allow the line plot to start at different points. For example, if a line plot ranges from 0 to 2, it's beneficial to vary it so that it may start at 5 on one occasion and 9 on another. In 3rd grade, students typically work with whole numbers, halves, and fourths. However, the provided data may not always include fourths, so students should adjust the intervals accordingly. As students become proficient in recording data on a line plot, they can then be tasked with measuring objects, recording the data, and accurately placing it on the line plot. No matter where in the process you find yourself, be sure to encourage students to record all mathematical observations in the upper right quadrant.
Outcomes
The outcomes of this MathRep further build students’ number sense, understanding of fractions, measurement, and data interpretation. It also illustrates how math concepts are interconnected and can be taught together instead of isolated. Another simple yet powerful activity. Have you tried it yet? What are your thoughts?
When you think about data, the first thoughts that come to mind might be how dull or uninspiring it can be. But what if there was a way to turn that perception around and make learning about data an exciting journey for elementary school students?
Enter the newest MathRep. This template is designed to engage students while teachingbar graphs and picture graphs to young learners.
In this blog, we will delve into how this new MathReps template is exciting elementary math students. It offers educators a fresh and dynamic method to ignite enthusiasm for data interpretation among their students. Get ready to explore the possibilities and discover how this MathRep can make a real difference in the classroom!
Picture This
This is an example that aligns to 3rd-grade standards: 3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.
You’ll notice that the same information is used to complete each of the graphs. This leaves quadrant 4. It can have questions about how many more and how many less, but why not allow students to observe the data and make their own observations. Leaving it open-ended like this allows all students to be successful. In addition, you will have students making observations that go beyond how many more/less.
The Setup
When first introducing this MathRep activity, it is recommended to provide students with the information located in the center of the paper. Subsequently, they can proceed to create the graphs themselves. Once they have gained proficiency in graph creation, the teacher can fill in the graph and task the students with completing the remaining sections, including the center. For a more advanced approach, the teacher can fill in the ‘Mathematical Observations’ square (quadrant 4), leaving the rest to the students. This adaptable strategy can effectively challenge students at various proficiency levels, guiding them toward a deeper comprehension of the material.
What Will You Do?
The power of this MathRep lies in its ability to enable students to interact with data in multiple ways simultaneously. This allows students the opportunity to make connections between different representations of information, leading to a deeper understanding. Teaching skills in isolation should and has been discontinued according to the Common Core Math Clusters, as math is intertwined with all aspects of learning. This MathRep illustrates these connections, preparing students for future success.
Primary teachers have been embracing the concept of number paths as a valuable tool to foster fluency and number sense among young learners. A number path is a linear visual representation of numbers, which serves as a versatile aid in helping primary students comprehend and engage with numbers more effectively. As I have been collaborating with a group of primary teachers, it has become evident that they recognize the significance of number paths in facilitating a deeper understanding of numerical concepts. This has led to an increasing demand for additional number paths, reflecting the enthusiastic adoption of this resource within their educational practice.
Why?
After spending a few weeks working with small groups, the teachers quickly realized the potential of this tool for the whole class. They noticed an improvement in their students’ abilities, prompting another teacher to request number paths from 21 to 40 for practicing addition and subtraction with larger numbers. These number paths are completely customizable, so whatever your needs are, you’re covered.
Create
Initially, I created the prototype with cardstock, pipe cleaner/craft sticks, and a bead. That proved not to be the best. So, for the teachers, I laminated the cardstock, switched to a piece of yarn, and changed out the bead style. They work fantastically. I also made sure to take a piece of packing tape to the back to secure the yarn after it was knotted.
Result
It was a huge success! Students can easily manipulate the beads, allowing them to learn about numbers in a hands-on fashion. If you would like a copy of the templates, feel free to download them. They are in the Kinder and 1st-grade MathReps slide decks. I have also created some number path games that I will post about later this week.
How are you using number paths in your primary classrooms, and how are your students responding?
I often find myself going down unexpected rabbit holes, and today was no exception. While on a quest to create a more engaging quadrilateral hierarchy for a 3rd-grade team, I found myself pondering the age-old question: Is a parallelogram a trapezoid? As it turns out, the answer is both yes and no!
I know. You’re wondering how it can be a trapezoid and yet not a trapezoid simultaneously. The answer lies in the definition of a trapezoid. The Oxford Dictionary defines a trapezoid as “a quadrilateral with only one pair of parallel sides.” So, according to this definition, the answer is no, which makes our math ‘program’ give incorrect information (that is a whole other post). However, many define it as a quadrilateral with at least one pair of parallel sides, which makes a parallelogram a trapezoid and our math ‘program’ correct.
In my math mind, a parallelogram is separate from a trapezoid. According to the hierarchy, the broad spectrum is quadrilateral. Below that there are either two or three main categories of quadrilaterals: kite, trapezoid, and sometimes parallelogram. Then, the parallelogram can be further broken down into rectangle and rhombus. Those can both be broken down into a square.
For the purposes of our district, I followed the current ‘curriculum’ where a parallelogram is a trapezoid. Below are the two variations of the poster.
So, what are your thoughts? Is a parallelogram a trapezoid or not?
For years, I’ve questioned the effectiveness of MathReps and pondered whether the skills transferred to everyday math tasks. Known for overthinking and constantly questioning things, I’ve been intrigued by recent data from teachers showcasing the positive impact of MathReps on their students.
There are numerous reasons to celebrate these results. Firstly, this is a multi-year comparison illustrating a significant improvement. The 2024 year initially showed a much lower proficiency level than the 2023 year. However, it is important to note that their overall gains were greater in 2024. Secondly, the teaching staff remained consistent over these two years; there were no departures or new additions. Remarkably, one particular class saw an astounding increase from 5% to 90% proficiency in just 4 weeks. It’s worth noting that this exceptional progress happened after the teacher conducted the final assessment earlier than planned due to external circumstances. Subsequently, the same group of students took the SBAC IAB for Fractions, which resulted in 14% below the standard and 86% at or above it. This is particularly noteworthy as the class had only been 5% proficient with fractions a few weeks earlier. While the teacher deserves full credit, she acknowledged that regular completion of MathReps played a role in this achievement. Furthermore, the IAB takes the students’ knowledge of the subject matter and requires them to use that knowledge in context. This further supports knowledge transfer is happening. This is just one example of the success that can be achieved through the use of MathReps.
Why do I bring this up? Students are struggling, and teachers are noticing. Too often students come without the basic skills necessary to succeed with the expected standards. I witnessed this many years ago (pre-pandemic) and created the first MathRep. I noticed that my 5th-graders could master the expected skills during the first trimester. Then, the skill expectations shifted to fractions in the second trimester. This is where many of my students lost the skills they acquired in the first trimester. So by the time the third trimester rolled around, all was forgotten from the beginning of the year. Needless to say, this is less than ideal, especially when state testing is right around the corner. So MathReps were born!
So, after several years and an ever-growing library of MathReps I have started hearing from countless teachers from across the county witnessing the same success I am seeing with teachers in my district. I love that they also share stories about their students and how they have gained confidence through MathReps. All this is to say, if you haven’t tried them, you should!
What are MathReps?
MathReps are carefully crafted templates that incorporate interconnecting skills/standards. MathReps allows students to make connections between skills and concepts seamlessly. They are customizable and versatile. The repetition allows students to master the skills without the cognitive load of navigating a new ‘review page’ each day. The format remains the same. The teacher changes the number each day, allowing students to work through a series of skills. The teacher provides immediate feedback by walking around the classroom or checking the work as a whole, leaving zero paperwork for the teacher to deal with at the end of the day.
MathReps explanation video
Where to Start
With this data, it’s a no-brainer. MathReps work. If you’re looking to get started, head on over to MathReps.com and find a template that’s right for you! No need to worry about the cost, it’s FREE! The templates are all teacher-generated. That’s right, made by teachers just like you for students just like yours.
First of all, I want to acknowledge the tremendous impact of Robert Kaplinsky’s insights, which have resonated with many for years. His thought-provoking posts “Is Problem Solving Complex or Complicated?” and “Why Do We Have Word Problems?” are invaluable resources. Today, I want to not only express my appreciation for Kaplinsky’s work but also highlight an alternative strategy to the C.U.B.E.S. approach. We can broaden our problem-solving toolset by exploring new perspectives, and I believe this strategy will further enhance our problem-solving skills. Don’t miss out on the original posts: Is Problem Solving Complex or Complicated? and Why Do We Have Word Problems?.
This image shows what the acronym C.U.B.E.S. represents. Many teachers use this to teach an easy way for students can begin to tackle word problems. I, in fact, had an anchor chart displaying this very technique many years ago. As the saying goes, “Know better, do better.” I now know better and therefore do better. I know you might be asking yourself, “Okay, but I still don’t get why this isn’t ideal?”
Let’s consider the problem to the right. With our adult brains, we understand that it is a two-step problem. We understand we need to add the adults and children together and then subtract the number of males who were in attendance. Now imagine your 8-year-old self reading this and not yet having reading mastered let alone the comprehension skills to decode what is being asked. (Okay, maybe that was just me and my learning disability) You can see that I also went through the CUBES strategy. I circled the numbers, underlined the question, boxed keywords, and got rid of extra information. Arguably, the statement that everyone showed up might also be considered extra information to a young learner. An 8-year-old will start the CUBES process. They will look at the boxed words to find out what operation they need to use. And this is where we run into our first problem. It only says, “How many.” It doesn’t say, “how many more,” or “how many less.” Being around kids, we know that their default is to add. So what are they going to do? You guessed it, add 12, 25, and 15. Even if the question that is underlined is: How many females were there? chances are students are still going to add it all up because they are specifically looking for keywords like ‘in all’, ‘altogether’, ‘less’, ‘more’, etc.
Years ago, I used to teach my elementary students these strategies, as it was what I knew at the time. However, as I learned better strategies to help students, one of my favorite methods became the 3-Read Protocol. You can find more information about it in the blog post Mastering Mathematical Language Routine 6: Three Reads. This approach helps students concentrate on understanding the story, identifying units and quantities, and focusing on the question or task. By following the 3-Read Protocol, students can comprehend the context, which enables them to think critically instead of simply focusing on a procedure that may or may not work.
Finally, to help students effectively apply the information from the story, I highly recommend integrating MathReps into your classroom. MathReps are strategically practiced sets of skills that allow students to practice basic math concepts in similar skill clusters and receive immediate feedback. This approach promotes automaticity and fluency, freeing students to focus on what is needed rather than both what to do and how to do it. You can delve deeper into MathReps by reading a series of blog posts. I suggest starting with The Power of MathReps.
When I first started attending tech and innovative teaching conferences, I felt completely overwhelmed. It’s common to feel this way when surrounded by so much new information and ideas. Many of us are eager to try everything at once, but then struggle to figure out where to begin and how it will all come together. I have to admit, I often wondered if I could even pull off some of the amazing things I saw. It took me far too long to realize that presenters, including myself now, tend to showcase only the very best. Starting something new is rarely a smooth process; in fact, it’s often quite messy and can even be painful at times. But embracing the challenges is all part of the journey. And that’s what this post is about.
Messy MathReps
When presenting on MathReps, I am very open and honest about the time it took my 5th-grade class to complete the first MathRep – 45 minutes, in case you were wondering. As a teacher, I am acutely aware of the learning curve involved in implementing MathReps. Some students may breeze through independently and accurately within three days, while others may require weeks of support. This variability is the inevitable, albeit messy, aspect of introducing MathReps or any new routine/learning opportunity.
Why do I bring this up? I understand the frustration of the messiness of beginning something new and wanting to revert back to a previous routine or way. The idea of starting something new after the school year has started can be daunting. There are pacing guides, expectations, benchmark assessments, and a million other tasks that weigh on us. We often hear that we should give it time, but all too often it doesn’t feel as if we have the time to give. But some things are worth spending time on.
Why Is MathReps Worth It?
MathReps is undeniably worth the investment due to the proven effectiveness of repetition in skill acquisition. As with any new skill, whether it is teaching, riding a bike, or knitting, initial proficiency is typically lacking. Through consistent practice and the repetition of key actions, one can develop heightened confidence, risk-taking, and greater speed. Similarly, MathReps specifically aims to provide students with ample opportunities for repetitive practice, thereby enabling mathematical skills to become second nature. Once proficiency is achieved, students can then apply their knowledge in varied contexts, such as solving word problems. Without this foundational knowledge, tackling word problems becomes an overwhelming task. For instance, even if a student knows that they need to multiply a two-digit number by a three-digit number, without understanding the process of multiplication, they are left unable to proceed effectively. It is akin to attempting to walk before mastering the skill of crawling. Through consistent practice and the connections facilitated by MathReps, students are empowered to enhance their confidence, adaptability in thinking, and fluency in their mathematical abilities.
Looking back at the series of Mathematical Language Routines (MLRs) we have explored, we can see that their collective aim is to foster robust mathematical discussions and enhance language proficiency among students. These routines serve diverse purposes, such as refining ideas through structured conversation (Stronger and Clearer Each Time), stabilizing oral language as a reference (Collect and Display), refining written arguments through critique (Critique, Correct, and Clarify), promoting collaborative problem-solving through information sharing (Information Gap), empowering students to craft mathematical questions (Co-Craft Questions and Problems), facilitating comprehension and negotiation of math texts (Three Reads), and encouraging comparison and connection between various mathematical approaches and representations (Compare and Connect). Together, these MLRs not only elevate student participation and conversation but also cultivate meta-awareness of language, fostering a deeper understanding of mathematical concepts.
We conclude this series with Mathematical Language Routine 8: Discussion Supports. The goal is to foster inclusive discussions in math by combining multi-modal strategies that aid in understanding complex language, ideas, and classroom communication. These strategies encourage student participation, conversation, and awareness of language nuances. With continued modeling, the aim is for students to adopt these techniques independently, prompting deeper engagement among peers in discussions.
Having rich mathematical discussions can be challenging, especially when there are barriers that hinder effective communication. Recently, I encountered a situation where I was assisting a student with a math problem. The task was to determine the combination of rolls of coins needed to reach a specific amount. The problem provided information about the rolls of nickels and dimes, including the quantity each roll contained. However, during our discussion, it became evident that the student misunderstood the task. They believed they needed to determine the number of dimes or nickels in each roll, rather than finding the overall combination. To clarify this confusion, I decided to show them an image of a roll of coins and briefly discussed its concept, which helped them grasp the correct approach. This incident highlighted the importance of uncovering and addressing any gaps in background knowledge. It also underscored the significance of reflecting on the relevance of certain questions.
Having sentence frames is not only helpful to me but also to the students. The above image is one that I created based on Illuminate Math‘s suggestions. These sentence frames can guide the class towards deeper thinking and understanding. As mentioned before, the main objective of this routine is to encourage students to take the lead in these discussions. Additionally, it is important to note that this particular routine can be integrated into any of the other Mathematical Learning Routines (MLRs).
This concludes our multi-part series on the 8 Mathematical Routines. I highly encourage you to start implementing these routines in your day-to-day math class. To further support you on this journey, I have gathered a variety of helpful resources, which you can access here. If you have any additional resources to share, please don’t hesitate to reach out. I will gladly add them to the collection and give you proper credit.
In the previous Mathematical Language Routine (MLR) discussions, we explored a variety of essential skills. MLR 1 focused on enhancing our understanding by revisiting and reinforcing key concepts, making our knowledge “Stronger and Clearer Each Time.” We then moved on to MLR 2, where we delved into the crucial skill of “Collecting and Displaying” data effectively. Building on this foundation, MLR 3 emphasized the importance of “Critiquing, Correcting, and Clarifying” our models and methodologies for optimal results. In MLR 4, we explored the “Information Gap” and how to use this strategy to be thoughtful of the information needed to solve problems. Continuing this journey, MLR 5 introduced the skill of “Co-Crafting Questions and Problems” collaboratively to foster innovative approaches and insights. Finally, in MLR 6, we explored the technique of “Three Reads,” emphasizing the significance of multiple reads in order to enhance student understanding. Let’s now embark on our next MLR discussion, MLR 7 Compare and Connect.
MLR 7: “Compare and Connect,” has the purpose of fostering students’ meta-awareness in their exploration of different mathematical approaches, representations, concepts, examples, and language. Through this MLR, students are encouraged to reflect on and verbally respond to these comparisons. This involves analyzing why certain mathematical actions or statements are done in a particular way, identifying and explaining connections between various mathematical representations or methods, and pondering how one idea relates to others in terms of both concepts and language. To support this learning process, teachers should model their thinking aloud when addressing these questions. This routine allows students to engage in rich mathematical conversations. We will explore two ways in which to accomplish this.
Getting students to engage in discussions about math, make connections, and consider different perspectives can be quite challenging. I often encounter students who simply say, “It was in my brain” or “My brain told me the answer.” However, by modeling and encouraging metacognitive awareness, students can begin to make connections on their own. One effective routine that focuses on linguistic skills is called ‘Which One Doesn’t Belong‘. This activity can be done in groups, in pairs, or as a whole class. Students are presented with four images, equations, numbers, graphs, or geometric shapes, and they are asked to identify a commonality among three of them and explain their reasoning. The interesting twist is that any combination of three out of the four options can be correct. For example, in the orange example, one could argue that the three triangles go together and the hexagon is the odd one out. Alternatively, one could justify grouping all the white-filled shapes while excluding the shaded shape. This activity is both enjoyable for students and provides the opportunity to hear and consider different viewpoints.
Another interesting activity that aligns well with this MLR is the Math EduProtocol Sous Chef from The EduProtocols Field Guide Math Edition (Chapter 9, page 56). In this activity, students are grouped together to solve a problem using different approaches and then present their work to the class. For instance, if students were given the task of solving 4 x 6 in third grade, one student might use equal groups, another could opt for repeated addition, a third student may create an array, while the last student represents the equation with the area model. Through this activity, students can establish connections with previously learned concepts and broaden their understanding. There are numerous ways to implement Sous Chef, but the central focus remains on fostering connections among ideas and encouraging students to share their thought processes orally.
In conclusion, incorporating this MLR into your math class will greatly benefit your students. It will help them enhance their meta-awareness, make connections between different concepts, and foster a deeper understanding of the subject. While we have explored two approaches to this MLR, there are numerous other equally powerful techniques available. In our next discussion, we will delve into MLR 8: Discussion Supports, which focuses on stimulating rich and meaningful conversations in the classroom.
Nowa Techie 