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.
In our recent posts, we have explored various Mathematical Language Routines (MLRs) that aim to foster language development in the math classroom. We have covered MLR 1: “Stronger and Clearer Each Time,” MLR 2: “Collect and Display,” MLR 3: “Critique, Correct, Clarify,” MLR 4: “Information Gap,” and MLR 5: “Co-Craft Questions and Problems.” Each of these MLRs has offered valuable insights into different aspects of language acquisition in mathematics, be it written and oral expression, oral language proficiency, or the comprehension of tasks and word problems.
Building upon these discussions, let’s now delve into our next MLR, MLR 6: “Three Reads.” This routine plays a crucial role in enhancing reading comprehension and developing meta-awareness of mathematical language. By engaging in this exercise, students get the opportunity to practice navigating the intricacies of math-related questions, which often pose challenges for them. Through multiple readings, they can better understand the unique ways in which math concepts are presented and effectively plan their strategies for problem-solving. MLR 6: “Three Reads” serves as an important tool for supporting students’ grasp of mathematical language and equips them with the skills needed to tackle word problems with confidence.
The Three Reads protocol is a powerful tool designed to enhance students’ understanding of mathematical word problems. Its main purpose is to break down complex problems into manageable steps that students can comprehend, analyze, and solve successfully. This protocol is particularly beneficial for multilanguage learners and students with academic disabilities who often struggle with comprehending mathematical texts.
The Three Reads protocol begins by encouraging students to focus on the meaning of the problem. Instead of rushing to perform calculations, students are prompted to truly understand the context and situation described in the word problem. This initial step allows students to connect with the story or situation presented and reflect on its implications.
After gaining a solid understanding of the problem’s context, students move on to the second read. Here, they concentrate on identifying the units and quantities involved in the problem. By focusing on these key components, students can make sense of the mathematical concepts and relationships embedded within the word problem.
Finally, during the third read, students shift their attention to the specific tasks or questions asked in the problem. By this stage, students have already engaged deeply with the problem’s meaning and mathematical content. They are now able to formulate a plan of action and approach the problem in a strategic manner.
The Three Reads protocol supports not only reading comprehension but also sense-making and meaningful conversations around mathematical texts. By emphasizing understanding and meaning before diving into calculations, students are given the opportunity to reflect on different presentation styles, negotiate interpretations, and explore multiple solution strategies.
Let’s explore an example that demonstrates how to implement the protocol of co-crafting questions with students. In this example, Mateo’s M&Ms, the quantities are initially hidden to allow students to focus on comprehending the story. Once they grasp the concept that Mateo has a bag of M&Ms with different colors, the quantities are revealed. At this point, students can create a T chart to organize the quantities and their corresponding units. Although the units are the same in this particular example, it’s important to note that it may not always be the case in every word problem. Utilizing the T chart helps students effectively organize the information. Moving on, the final phase of this example involves students creating their own questions, reminiscent of MLR 5. By allowing students to differentiate and choose their own questions, they are encouraged to tackle more challenging problems rather than opting for the easy way out. This approach also fosters open-ended questions that lead to a variety of possible solutions. It’s a wonderful way to introduce this routine. Additionally, another teacher in my district, Nicole Garcia, and I have developed a recording sheet to assist students in organizing their thoughts and work. It’s worth noting that it’s also practical to address problems that already come with pre-established questions. In these cases, steps 1 and 2 of the protocol remain the same. However, in step 3, students formulate a plan to answer the provided question. Once again, reading the story multiple times allows students the necessary time to grasp and organize the given information.
In summary, the Three Reads protocol encourages students to engage actively with mathematical questions, reflect on presentation styles, and negotiate meaning. By following this routine, students can develop a deeper comprehension of word problems, improve their mathematical reasoning skills, and ultimately enhance their overall problem-solving abilities.
In this ongoing blog series on Mathematical Language Routines (MLRs), we have covered four essential routines so far. Let’s recap their key ideas: MLR 1: “Stronger and Clearer Each Time,” MLR 2: “Collect and Display,” MLR 3: “Critique, Correct, Clarify,” and MLR 4: “Information Gap.” These routines are designed to enhance oral and written communication skills in mathematics. Now, let’s turn our attention to MLR 5: “Co-Craft Questions and Problems,” which offers a unique approach. It encourages students to actively participate in the question and problem-solving process, enabling them to explore mathematical contexts before seeking answers.
In MLR 5, students are given the opportunity to dive deeper into problem-solving by creating, analyzing, and enhancing mathematical questions, problems, and situations. The purpose of this routine is to provide students with a space where they can generate, analyze, and improve their understanding of mathematical concepts, specifically word problems. Through engaging in conversations, students refine their ability to formulate, select, and refine questions, allowing them to develop stronger critical thinking skills. This routine fosters students’ ability to use conversation skills effectively in the process of formulating, selecting, and refining their mathematical questions and problems. By actively participating in these conversations, students take ownership of their learning and become more proficient problem solvers. Curiosity Creator, found in The EduProtocols Field Guide Math Edition, Chapter 18, is one way to practice this routine.
Another excellent routine that can be seamlessly integrated into MLR 5 is the 3-Act Math Task, originally developed by Dan Meyer. This ingenious task involves providing students with minimal information, serving as a catalyst to foster curiosity and encourage them to generate questions that they can later answer with additional information provided in subsequent acts.
The first act, known as Act 1, is designed to stimulate students’ observation skills and curiosity. During this phase, students are encouraged to ponder upon what they notice and wonder about the given scenario. They also formulate questions that they will eventually solve. Act 1 is an opportune moment for students to engage in conversations regarding their observations, estimations ( too high, too low, or actual), a specific focus question, and the type of information they need to answer their own question.
In Act 2, students are provided with the necessary information to solve the problem or question they crafted in Act 1. Armed with this newfound knowledge, they embark upon the solving process. Finally, Act 3 allows students to review the actual answer and assess whether their initial predictions were correct.
This routine is highly engaging for students across all levels, offering a perfect balance between accessibility and challenge. It also offers a wide range of ready-made, standards-aligned tasks suitable for students from Kindergarten through high school. By incorporating the 3-Act Math Task routine into the classroom, instructors can empower students to actively participate in problem-solving, develop critical thinking skills, and cultivate a deep understanding of the importance of co-constructing problems and questions.
Recap of the previous MLR discussions: We have reached the halfway point in the series of Mathematical Language Routines (MLRs). So far, we have explored MLR 1: “Stronger and Clearer Each Time,” where the focus was on enhancing understanding and communication skills through the use of the “Convince Me That” technique. This was followed by MLR 2: “Collect and Display,” which aimed at expanding students’ academic vocabulary. MLR 3: “Critique, Correct, Clarify” was centered around improving both oral and written skills, utilizing the EduProtocol Nacho Problem. Now, let’s introduce MLR 4: “Information Gap,” a personal favorite, which promotes collaborative work and helps students identify critical information necessary for solving word problems. This routine plays a vital role in fostering meaningful interactions and communication in the realm of mathematics.
One of the biggest issues in math classrooms is the challenge of word problems, also known as story problems. These problems require students to go through multiple steps, including reading comprehension, deciphering the question, creating a plan, and solving the problem. However, students often struggle with knowing how to use the given information and which details are relevant to the solution.
To address this problem, Information Gap tasks have been developed to help students navigate this challenge. In these tasks, students are divided into two groups: one group has the data card, while the other group has the problem card.
The group with the problem card reads the problem silently and asks the group with the data card for the information necessary to solve the problem. It’s important that neither group shows their cards to the other. Before sharing the information, the group with the data card asks the problem group why they need that specific information. This process encourages the problem group to justify their reasoning and ensures that they have thoroughly thought out the solution process.
This collaborative process continues until all the required information is obtained. Once both groups have shared their cards, they can work together to solve the problem. The goal of this approach is to create a need for students to communicate and collaborate, as this type of task cannot be accomplished alone.
When starting this process, it is beneficial to demonstrate it to the class. Initially, I present the problem card to the entire class while holding the data card myself. I then instruct students to work in pairs and determine what information they need. They are encouraged to formulate questions to obtain the necessary information and provide a rationale for why they need it. I repeat this process several times until the entire class understands their roles. Gradually, I reduce the group size over time until they are working in pairs to complete this task. This routine helps students to slow down and approach their thinking more deliberately.
In summary, Information Gap tasks are designed to promote collaboration and problem-solving skills among students. By requiring them to share different pieces of information both orally and visually, these tasks facilitate effective communication and enhance their ability to work together towards a solution.
In the upcoming post, we will delve into MLR 5: Co-craft questions, where we explore how 8 P*Arts meets Word Problems, 3-Act Math Task, and Emoji Word Probz perfectly align with this approach. Join us in the next installment of the series to discover exciting examples and techniques that will surely ignite your interest and leave you eager to come back for more. Stay tuned!
Nowa Techie 