Elementary Math Tools - Literacy In the Math Classroom

Don’t forget about Literacy, in your MATH classroom…

I think educators often overlook the importance of teaching literacy in your math classroom. You might be saying, “Wait, I am not the ELA teacher, I just teach math”. I hear you, I am the math focus teacher in my school too. Raise your hand if you noticed that in your math classroom, you have students that struggle with attempting word problems or explaining their thinking in written form. These are just some of the aspects of how literacy education impacts your math classroom. The mindset that it is the reading teacher’s job to teach the students how to read and the math teacher’s sole responsibility is to teach the math standards is an outdated mindset that needs to be addressed.

It is important to note that literacy instruction in a math classroom is going to look different in your math classroom. For starters, you won’t have separate, isolated literacy lessons, but you will implement strategies for your students within your math lessons. If you teach reading/literacy in the past or currently, some of the strategies are the same, however the representation or application of the strategies are modified for your math lessons.

Let’s dive in!

Math Vocabulary in the Classroom

An important literary element in math instruction is math vocabulary. Understanding a word’s meaning helps students understand the math concept just as knowing vocabulary in reading helps a student comprehend the story they’re reading. Let me walk you through how I enforce my students’ understanding of math vocabulary and how I have them apply it in their learning.

Introducing Math Vocabulary

Often I introduce a new math vocabulary word within a lesson that I am teaching. I imagine this is how many math teachers introduce math vocabulary as well. I explain what it means and in the past move on with the unit. I use the words orally in the lessons, but I was finding many students didn’t know the meaning if it was a question on a test. Something new I have started doing is having students fill out a word map.

Since I started using word maps, I have noticed that more students are starting to remember their math vocabulary. I have the students paste into a math journal so that they can access it as they need.

Word Walls

In the front of my classroom on the side of the whiteboard I have magnets holding up vocabulary cards that we are using in our current math unit. The cards themselves have the word and an image/example of the meaning. It doesn’t have a written definition on it, it’s meant to be a quick reference to the word with an example.

I think there is a lot of value in having math vocabulary that we are currently using front and center where we are learning. I can ask a question like “what model is this that I am doing?” and if not a lot of hands are going up I might point to the vocabulary cards and students have the reference to help them. I found these cards on TpT, but you could make your own or find already made similar resources.

I have a bulletin board in the back of my classroom where I move the words after we have completed the unit. I didn’t want to take the words away, but I only wanted the current words on the front board.

When possible, try to show your math vocabulary in the literal sense. I’ve done this in a few areas around my classroom and my students find them to be extremely helpful. They reference them a lot.

One of my favorite literal vocabulary examples is my area, perimeter, and volume examples. Every single year I have many students that can’t remember the difference between area and perimeter. Since putting up this visual they haven’t had any issues and they see it all year so

it becomes something they internalize and I am sure they will continue to remember their meanings for years to come.

Application of math vocabulary:

We don’t want to push math vocabulary so our students will just know it. They also need to apply it to demonstrate their understanding in either an oral or written response. Assignments that require students to respond to questions in a short answer format should have a word bank with words students could use in their response. From experience, doing this has helped my students construct a well written response to a math open-response question. Using the appropriate vocabulary in a math response helps students structure their thinking in written format in a clear way.

First, make sure you model the use of math vocabulary orally, all the time. It’s important when referring to the numerator in a fraction, that you say “numerator” not anything else. When you refer to a rectangular prism when teaching volume, you don’t say box, you say “rectangular prism”. Students listen to what you say and if they have the opportunity to see you refer to an image as a rectangular prism over and over, they will start to identify those images as a rectangular prism. This took me a while to get in the habit of doing, but I can’t stress the importance of doing it enough.

Next, we ask our students all the time to explain their reasoning, both orally or in written format. This can be challenging for a lot of young children, I know it is for my 5th graders, especially my English Language Learners. Having the words front and center is helpful when students are explaining their reasoning and you’re encouraging them to use math vocabulary to help make their responses more clear. When we expect students to write their reasoning in some open-response/short answer format, we want their answer to actually have a clear explanation. I tell my students all the time, “you’re the teacher here, explain the math as if you’re teaching me”. You cannot expect students to do this on their own without ever seeing it modeled. I think the most effective way to do this is to list out a bank of words that they could use in their response to a particular question and then craft a response with the help of your students' participation. Tell students they don’t have to use every word, but give them a reasonable number of vocabulary to use. Here is an example of how I took a problem from our curriculum and scaffolded it to fit the needs of my students by providing a word bank and lines to write on. It occurred for me to do this when the responses I was getting from my students were not clear and not a full explanation of their reasoning. “Because I know the answer” was not going to cut it when the question asked them to explain their thinking.

One day, I decided to make a list of vocabulary words I wanted my students to use in their responses on board, then I modeled a response to a question for my students. As you can see part “b” of the problem is the same question with a new expression. I then asked my students to answer that question on their own using the format I just used in part “a”. I told them they would be using the same vocabulary words and to underline them.

It worked, my students were able to use my model to construct a response. From here, I knew I would have to provide some scaffolding in order for my students to get in the habit of using math vocabulary to help them structure a clear response.

I then took this problem that is asking students to show or explain your reasoning in addition to solving the problem. I added the word bank, sentence starter and lines. I told them they didn’t have to use all the words, but should use at least four. I didn’t model this one, as I had the previous day modeled the problem in the image above. I saw a drastic improvement in all my students' work. I decided since this was our first attempt using this scaffolded model to take it a step further and show my student examples of exemplary work and examples of work that have room for improvement.

Here is an example of an answer I considered exemplary. They used at least 4 vocabulary words correctly. On their unit test on volume, they had to answer an open response question like this and I included a rubric for them to reference. I did not provide the word bank on the test, but I did leave the word wall on the board. Most students scored a 3 or a 4. I was very pleased with the outcome of implementing this scaffold.

Using Visuals

How we represent the math we teach during our lessons has more impact on our students' understanding of numbers than we often think about. You may or may not have heard about the C-R-A model of teaching. It stands for three stages of learning, the concrete, representational, and abstract.

The image below is from a video by Christina Tondevold. She explains the C-R-A model and shows examples of each model.

Why is C-R-A so important when teaching mathematics?

Tondevold uses this great example, she asks you to visualize a cat. Go ahead, do it, close your eyes and imagine a cat. You probably saw something like this when you closed your eyes. You didn’t visualize the letters

C-A-T.

She then asks you to imagine the number 37.

Is this what you saw when you imagined the number 37? This comparison was eye opening to me, without representing numbers in model form, the only image of numbers our students have in their heads is the abstract number. As adults who have been working with numbers for all our life, we understand the abstract meaning of this number, but a young student does not, they just know it's a number. They haven’t connected it to the concrete meaning.

We want our young math students to know that 37 is equivalent to the image on the right.

Tondevold explains that students will move through the three stages at their own pace, they won’t all be on the same stage at the same time. It is important not to rush students off the concrete stage just because half of your students have moved on to representational or abstract. When students are working independently, let them use the manipulatives if they need them, let them draw out pictures if that is what is getting them through the problems. If they’re ready for the abstract, great, maybe they don’t draw out the models, but they should be able to identify them. Once a student reaches the abstract application of the content the models should not be forgotten, but already internalized. They should have a conceptual understanding of the content. Tondevold describes this place the “sweet spot”, where a student has brought it all together and now the abstract application has meaning behind it.

Let’s take a look at a teacher facilitating the C-R-A model in action. In the first 7 minutes of the video, a teacher does a warm up or a number talk with her students. The objective is for the students to subitize the dots in the image on the right. She shows the image just for a few seconds and then covers it again. She asks the students to draw what they saw. The teacher’s objective is to train her students' brain to see a number in the concrete form. She then gives the students one more quick look at the image. They can make sure their drawing is correct. Once her students have the drawing of the dots they saw on the board in the way they were represented, she asks them to tell her how many there are,

but to write an equation to match the picture. Students share the same solution, but with different equations. They take turns circling how they visualized the dots and came up with the number of 18 dots. From there she talks about pushing all the dots together and recognizing it as an array.

I have done some of these subitizing exercises with my class. I purchased a resource on Teachers Pay Teachers called Quick Writes, I have also done a lot of Graham Fletchy’s subitizing and progression videos. There are a lot of resources on his website.

Using Visuals to Increase Communication

Another way I have recently started using math visuals to teach math is to use them to foster math discourse to understand math concepts. I took two images, one of a decimal square model with a hundredth shaded in. The other image was the same size model broken into hundredths. However only 1/10 of the hundredth model was shaded in. I asked the class to tell me what they noticed was the same and what was different. They were given sentence starters, but had to give me the rest of the sentence. The math word wall had decimal related vocabulary on the board for their reference.

This short activity was a great gateway to discussion on how the model accurately represents the concepts of a hundredth to a thousandth. Students had the opportunity to discuss with their peers what they noticed and shared their thoughts about the compare and contrast activity. Through giving my students the opportunity to have these discussions, we are expanding their math understanding and their ability to communicate their thinking. This is part of math literacy.

Creating Discourse Through Number Talks

If you’re not doing number talks in your classroom as part of your math routine, make this the first change in your teaching practice immediately. If you’ve never heard of a number talk or maybe you’ve heard of it, just don’t know how to implement one, let me go over the basics.

A number talk is a 5-10 minute routine that is carried out either before or after your math lesson. I personally like to start my lessons with a number talk, it helps students “warm up” their math thinking and it helps naturally get your students settled into learning. The objective of a number talk is essentially to build number sense, but in turn it encourages math discourse in the classroom, which is so unbelievably important in a math classroom.

Number talks don’t all look the same, there are several different approaches you can take when implementing a number talk in your classroom. Ideally, one or a small set of math problems are provided to solve, usually students are not given any writing tools, encouraging them to solve the problem mentally. This method forces students to use previously learned strategies to solve the problems given.

This video shows an example of a teacher doing a number talk with her students as a warm up to her lesson. Watch as she accepts everyone's answers whether or not they’re correct or not and each student that shares is asked to share how they solved the problem. By writing down a solution that is incorrect, students are now encouraged to share their answers even if they’re unsure if it is correct or not. It is important to note that so many students in our class do not participate in our lessons when they may have a solution purely based on the premises of being embarrassed if their answer is incorrect. When a teacher routinely implements number talks, they create a respectful math community that students feel safe to participate in class discussions.

Not every student may have used some of the created strategies the students used in the video like making a friendly number. When students share their strategy and the teacher writes it down, other students are listening and learning from their peers.

I carry out a number of talks every day at the start of my lessons. What I do to help those students that don’t have the level of number sense that others have is to put up similar problems every day that will encourage strategies used and shared to be used again. Usually, by the end of the week I have students participating and sharing that were not in the beginning of the week. This is why it is important to make number talks part of a routine and not sporadically implemented.

As seen in the video, number talks can be done at the front of the board, typically students are gathered on a rug together. I prepare mine on a slide and use my iPad to annotate their answers, therefore, I can share some number talks I have done recently with my students.

Here I put the number problem 25 x 16 and I asked students to solve it without paper. Students put up their thumbs when they have an answer. Every strategy that was shared I put on the board. You can see the different thinking that is happening. One student used an area model to solve. Another one knew that 25 x 4 is 100, so they divided 16 into 4 groups of 4 and added the products of 100 together. Another student rounded up to a friendly number of 20, multiplied and then subtracted that group of 4 that was added on away from the estimated product.

Not all number talks look like these though, here are a few more annotated number talks I have done in my classroom and are from John SanGiovanni’s book Daily Routines to Jump-Start Math Class, Elementary School: Engage Students, Improve Number Sense, and Practice Reasoning.

I gave this number talk to my students when we were starting our decimal unit. I wanted it to connect that big numbers can be small things: connecting to decimals. The concept of decimals can be a struggle for some students and through a class discussion, we were able to find examples of when the number “580 and 2,000 are small numbers when…”. They were able to formulate a great list, but one of my favorite connections was the 580 pennies and I asked, "How much money is 580 pennies? They were able to tell me that was $5.80 and that is not a lot of money. This was a great segway into the value of decimal numbers.

I used this image of a ferris wheel and asked the student to tell me how many people they thought could fit on the ride. The numbers on the right are all the different estimates I got. I asked students to share their reasoning and you can see by the different annotations I made on the image of their thinking. Some thought 4 people per gondola and others said 6. They told me it was multiplied by the number of gondolas. Some students didn’t see the gondolas at the bottom. I have circled and changed their answer.

This number talk is a fun and quick way to get students thinking about easier ways to solve problems without actually having to solve them. The “this or that” number talks have one side that has problems that may be challenging to compute in their heads without using a particular strategy. The students don’t need an exact answer to solve the problem, they just need to know if their answer will be bigger or smaller than the “that” column. You can see by my annotations students told me the strategy some of them used to know that the “this” column was bigger or smaller than the “that” column.

I do make up some of the number talks I do, but most of the ones I do either come from Peggy Parish’s book, Number Talk or John SanGiovanni’s book, Daily Routines to Jump-Start Math Class, Elementary School: Engage Students, Improve Number Sense, and Practice Reasoning. You can also find some great subitizing warm ups from Graham Fletcher's website.

Teacher Modeling and Think Alouds

When using any literature in math, students should be able to transfer the same skills they learn to understand text in reading class to their math class. Some of these strategies math teachers can model for students are the before, during, and after reading strategies that are taught during reading class. Strategies such as previewing vocabulary, making diagrams, and/or accessing prior knowledge are going to translate well when reading math text.

Almost daily, students come up to me for help when they are given a task to complete that is in word form. Whether it is a story problem, or is more language based than a skill sheet. It doesn’t matter that they already understand the math the problem entails, they don’t apply any strategies to break down the paragraph and understand what it is saying. I try to model these strategies, but I’ve never really made the connection that they’ve existed already. When the students approach me for one-on-one help I do my best to help them and I try to share some strategy to help them.

Review key terminology: words that imply addition, subtraction, multiplication, or division.
Read problems slowly and more than once. Pause at each period and check for understanding.
Break up the problem into short segments.
Act the problem out.
Use manipulatives or drawings to help with understanding.

I started using the “3-read protocol” . The first time I read the problem to them without numbers. Students are automatically searching for numbers to compute when they read a problem and it confuses them and they don’t even know what the problem is about. We talk about the main idea of the story problem, discuss what it is about.

The second time they read it and identify important key vocabulary words or numbers that are needed to solve the problem. I think it’s important to note that I try to incorporate problems that sometimes have numbers in the problem that are not needed to solve the problem. Exposing students to problems written this way ensures that they will get in the habit of understanding what the numbers in the problem are representing. This prevents students from not reading problems and just taking numbers and multiplying or adding them. This is the very reason I do not like the CUBES strategy. The last read through is a fluency read, we read the problem together. The students at this point have analyzed the meaning of the problem and one more read puts it all together.

Literature to Teach Math

Read-alouds in the math classroom is a great way to aid student comprehension of math concepts. Ultimately they are a great way to improve students' ability to do problem solving. Math is introduced in an engaging story that is relatable. It also allows students the opportunity to share their thoughts or understanding with less pressure.

I will admit I do not do enough of them, but a few I have used in my classroom before has been a highly engaging way to help students connect math to literature. One of my favorite read-alouds is Spaghetti and Meatballs for All!, by Marilyn Burns. The students are connecting a real life problem of perimeter and with every page the students have a slightly new problem to solve. There is a great extension activity the students have fun doing after the story is finished. I found an example lesson transcript from Math Solutions that uses this book.

A book that I learned about that I will read to my students in my division unit is A Remainder of One by Elinor J. Pinczes. The lessons I teach my students about interpreting remainders can be challenging for many of my ELL students. In order to understand the object of the lesson and be able to interpret the remainder accurately, the student must understand what a remainder is in the first place. To them, it's just a number they can’t divide anymore. The book The Remainder of One describes a remainder clearly in story form what a remainder is and how one bug’s journey to no longer be the remainder when he and the 24 other bugs make an equal formation in a parade.

Creating Story Problems

Through this course and in the many years of my teaching, I have thought and tried as much as I could to connect math to a story. Present math in a real world “story problem”. It can be challenging connecting math to realistic story problems, especially how we write them. A great way to help your students learn how to read story problems is to have them create them. One activity I have implemented in my classroom is a “Mad Lib” style problem creating activity. Students would pick a card for the subject and pick a verb. They would then create a math problem using those two words. For example, a student might pick the subject card “chickens” and the verb card “run”. When they are required to use random subjects/verbs in their problem, they start to create some funny problems. My students have found this activity highly engaging. They submit them to me and I use the ones I want on a worksheet or test.

Another project that I gave my students was to create a story problem in a comic. Recently, I taught my unit on evaluating and writing numerical expressions. This can be a challenge for many of my students, first they need to understand the key math words that help them identify the operations needed to write an expression that would match the problem. The other day I displayed the following math problem and had them work in pairs to discuss how to write a numerical expression for it.

Mary’s weekly allowance is $8 and David’s weekly allowance is $5. Every week they each spend $2 for lunch. Write a numerical expression to show how many weeks it will take them together to save enough money to buy a video game for $45.

I listened as they discussed their thinking, some pairs were able to identify that the 8 and the 5 in the problem were money gained and others were not able to. One of the bigger struggles I witnessed was students not understanding how to incorporate the 45 in the problem. Many of the students that did recognize that they were subtracting 2 from 8 and 5 and then adding the difference together, were then using multiplication in their numeric expression to get to the 45. It took some time for them to understand they needed to divide 45 by the rest of their expression. As a summative project to this unit, I had the students create their own story problems and draw a comic to accompany it. The numerical expression had to be written at the top and they created a story in the order it would be solved according to order of operation. This application gave them a chance to use their writing skills to represent their understanding in this math standard.

Quick Writes

We want to give our students the opportunity to write in math class. This is a skill we want them to develop and be able to do with ease. Giving students time to write down their thoughts, questions, general understanding or misunderstandings about a specific math concept. It can be done any time during a lesson, as a warm up or closure to a lesson. As a preview to some of my math lessons I have had students write down things they notice and wonder about the new math concept. In Julianna Matherson’s blog, Math With Mrs. Matherson, describes how she uses these templates to get her students writing in math. I actually have students write their thoughts down independently before we share. I believe it gives them a chance to actually see things they may not see right away and delay their understanding.

I’ve also implemented a writing activity called “2 truths and a Lie”, also from Julianna Matherson’s blog. I give the students an image or a math expression and ask them to write down two truths and a lie. They then have the opportunity to share with their peers and compare their truths and lies. Students have fun with this and most importantly, they’re writing using math language.

Writing in the Guided Math Classroom

If you’re still reading this blog, many of you may be asking yourself, when do we find the time to weave all this class discussion, reading and writing into my math block. It is a lot of information, but the bigger question is, what does your math block/period look like? Is it teacher led instruction with independent practice? Many teachers feel this is the “tried and true” method, but as we enter the 21st century of teaching we are learning this method is tired and ineffective in many ways.

That being said, many teachers are implementing a guided math model in their classroom. The basic premise is that in some form students at some point during the math block rotate to different stations and work on a variety of tasks or activities while the teacher works with a smaller group of students. With that all being said, guided math instruction allows the teacher to offer a variety of activities for their students including tasks that involve writing.

This is the structure of my guided math classroom:

Class meets on the rug for a short number talk (5-10 minutes)
A new or review lesson is taught (15-20 minutes)
Students break into their center work (30-40 minutes)

If you typically have anywhere from 50+ minutes for your math block, guided math is very realistic practice.

Guided math centers/stations can look different. Some are timed rotations with students in assigned groups. They switch after a timer goes off and attend different stations. Others are menu or task based. Students have a checklist of tasks to complete and move about the centers at their own pace. This year I have implemented the math menu, student paced model and it is working well for my students.

What tasks are my students doing that involve literacy and writing?

Many of my quick writing activities I mentioned above are part of their math centers. Students have assigned “must do” tasks every day. Two of the “must do” assignments are various tasks that they would find in one of these buckets. Most of these activities are written based tasks that ask students to create and communicate math problems or ideas. Most of the quick writes I use are from Julianna Matherson’s TpT store. What I love about these writing tasks is that they do not ask for long writing pieces from the students. They’re meant to foster students' critical thinking and share their math thinking. In “ask and answer” students are given an image or math expression and students need to create two questions with corresponding answers. Depending on my students' math abilities, I push some students to create more higher leveling questions.

Therefore, these activities rigor depends on a real expectation I have for that particular student, allowing me to differentiate upon each student's needs. “Same, but different” has students compare and contrast two images or math expressions, this is creating math collaboration and communication that students are asked to share in written form. Often similar discussions are done in whole class warm ups or during a lesson or number talk. Absolutely, those oral discussions are important, but we also need students to be able to write down their thoughts as well. It’s a skill that is expected out of them and this is an approachable way for students to be asked to do so. If you have a student that doesn’t like writing, asking them to write a paragraph or essay is a daunting task that increases anxiety. These small tasks are not meant to create huge tasks for your students, but as small as they are they are still powerful.

Math anxiety, yes it’s a real thing

We all have those students in our classroom that don’t want it to be “math time”. You initiate the transfer to the start of math and you hear some students give the sigh of dred. It’s not a warm and fuzzy feeling when you see a student not like anything you do in the classroom. The question is, what role as teachers are we playing in causing that dred, hesitation, that fear? Yes, many students come to you that way, maybe that have struggled with math for years and it has never been a likable subject for them. Well, I am here to tell you it’s possible to turn this mindset around and make math class an enjoyable experience for all learners in your classroom.

What math anxiety does to a person…

The higher level of math anxiety is linked to lower performance in math skills.
Decreases the number of people that end up finding careers that entail the usage of math skills.
Disable one's ability to memorize math facts.
Make multi-step math problems difficult to understand/solve.
Cause stress
Projects one’s anxieties onto others (their own children).

Creating a growth mindset:

Jo Boaler has been an advocate for the math open mindset for years. Upon years of research, it has been discovered that people with fixed mindset about their math abilities do in fact struggle to learn. Brains actually work differently for someone with an open mindset vs. a fixed mindset. People that have an open mindset are able to learn from their mistakes in a topic.

While watching a TED talk given by Boaler, she describes someone with a fixed mindset believing they can’t do something, therefore their belief becomes a reality for them and they don’t actually learn anything. I thought about my journey in trying to learn a foreign language. I have struggled to learn any new language fluently. I have heard statistics about learning new languages and the younger you are the easier it is for you to acquire it and become fluent. While I am not discrediting this idea, I do think it has created a fixed mindset and I made the assumption that because I didn’t take any foreign language classes until I was in 7th grade is the very reason I just can’t learn a new language. In actuality, I can learn a new language when I open up my mindset and put forth effort to learn from mistakes. My mindset started to change when I was in Italy. I took an Italian 1 course before going and struggled with conjugating verbs, just as I had in French or Spanish class years before. I took what little vocabulary I learned and while shopping in Florence, I was mistaken for someone that was fluent in Italian more than once. This gave me confidence and opened up my mindset to learning new languages. I have since learned more Italian and the proper pronunciation than other languages since then. Now, I am not claiming I have mastered the Italian language quite yet, because I haven’t, but my mindset has changed and I know if I put forth the effort I can get there.

Three steps to helping your students avoid math anxiety:

Help them develop a growth mindset: If you the teacher has a growth mindset, then you will project that onto your students. It is important to truly believe that all your students can learn in your classroom. If you don’t feel that way, you won’t push those students that struggle and will only further create or establish their fixed mindset.

Teach your students to embrace their mistakes: We can so so much more for our students then just tell them it’s ok to make mistakes. For starters, show it’s ok to make mistakes by making mistakes ourselves. I love it when I make an error and my students catch it and are able to point it out to me. Some teachers might become embarrassed by that, but I shrug it off, we’re human. I say, “that’s how we learn boys and girls”. When they see us model making mistakes as no big deal and moving on with new understandings, they start to embody that same attitude.

Don’t make math about the solution, but about the journey: If we are only providing math problems that only have a right or wrong answer then how can we expect our students to think positively when they get a lot of problems incorrect. Teachers can create a positive math experience by providing problem based learning in their classrooms. Students learn through discovery and collaboration. Learning isn’t focused on the solution anymore, but how you get there. By providing rich math tasks like this in your classroom, students are encouraged to participate because they're not afraid to get something wrong. By doing this it also removes the emphasis on speed, it takes stress to get an answer quickly and removes that anxiety which in turns opens the mind instead of blocking it.

References

Coleman, Greg. “Using Concept Maps in Math Class” Who’s Who and Who’s New 15 March. 2015 http://whoswhoandnew.blogspot.com/2015/03/using-concept-maps-in-math-class.html

Illustrative Mathematics (2023) Illustrative Mathematics K-5 (Grade 5) [Program of studies]. https://illustrativemathematics.org/

Build Math Minds. (2019, March 7) What Is The Concrete Representational Abstract Approach [Video] YouTube. https://youtu.be/8Kl3CvIKytc

Massachusetts DESE. (2015, October 27) 4th Grade Math [Video] YouTube. https://youtu.be/Nw4R3HwwiUY

Fletcher, Graham. “Subitizing to Foster Multiplicative Thinking” Gfletchy. https://gfletchy.com/2021/06/17/multiplication-subitizing-cards/

Moore, Kimberly. (2015, October 28) Number Talks Video [Video] YouTube. https://www.youtube.com/watch?v=WAhkbSFtvAI

Parish, Peggy. (2014). Number Talks Whole Number Computation, Grades K-5: Helping Children Build Mental Math Computation Strategies Math Solutions

SanGiovanni, John J. (2019). Daily Routines to Jump-Start Math Class, Elementary School: Engage Students, Improve Number Sense, and Practice Reasoning Corwin; First Edition

Krick-Morales, Brenda. “Reading and Understanding Written Math Problems” (2006). Reading Rockets

https://www.readingrockets.org/article/reading-and-understanding-written-math-problems

Eutopia. (2019, April 5) 60-Second Strategy: 3-Read Protocol [Video] YouTube.

https://www.edutopia.org/video/60-second-strategy-3-read-protocol

Burns, Marilyn. (1997). Spaghetti and Meatballs for All! Scholastic Bookshelf

Pinczes, Elinor J. (2002) Remainder of One Clarion Books

Matherson, Julianna. “Using Math Menus In Your Classroom” Julianna Matherson: Math with Mrs. Matherson 20 March. 2020 https://mathwithmsmatherson.com/using-math-menus-in-your-classroom/

TEDxTalks (2016, May 22) How you can be good at math, and other surprising facts about learning | Jo Boaler | TEDxStanford [Video] YouTube. https://www.youtube.com/watch?v=3icoSeGqQtY&list=PLtk7pLRHt3gsa6QYlLNujNrXECbMBkI7R

Lake, Bethany “5 Powerful Ways to Help Kids Develop a Growth Mindset” August. 2009

https://biglifejournal.com/blogs/blog/5-ways-kids-develop-growth-mindset-in-mathematics

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