# data[].attributes.attachment_caption
Petti, W. (2019, October 21). Writing About Math | Education World. Education World. Retrieved November 15, 2021, from https://www.educationworld.com/a_curr/mat

# data[].attributes.body
"Writing is a complex process that requires a wide range of skills — a strong vocabulary; an understanding of genre, text structure, and voice; basic mechanical skills (grammar and punctuation); organizational skills; and higher order thinking" (Reading Rockets, n.d.). Cultivating mathematical writing skills requires critical thought and data analysis but should produce students that are able to thoughtfully explain trends and ideas. According to Petti, mathematical writing includes problem solving, reasoning and proof, communication, connections, and representations (2019). Engaging students in written mathematical discourse may seem daunting, but many ideas abound as to how to scaffold writing tasks to be engaging for all students.

Reading Rockets. (n.d.). Writing. Retrieved November 15, 2021, from https://www.readingrockets.org/teaching/reading-basics/writing

# data[].attributes.attachment_caption
Teachers pay teachers. (n.d.). Math Talk Sentence Starters: Math Talk Posters [Illustration]. https://www.pinterest.com/pin/650418371167203367/

# data[].attributes.body
Sentence starters are a great way to have students engage in spoken and written sentences. Scaffolding writing by giving students pertinent sentence starters to reference gives students a boost to use discourse specific language in a targeted way. As students become accustomed to writing in mathematical terms, this may need to be adjusted or omitted, but including sentence starters can be a powerful tool for engaging ELs and those struggling with mathematical discourse specifically.

# data[].attributes.attachment_caption
George Lucas Educational Foundation. (2019, March 28). Using Reading and Writing to Do Math [Video]. YouTube. https://www.youtube.com/watch?v=m-MOOuGDEL0

# data[].attributes.body
"We consistently ask children to make their thinking visible so that their ideas are clearly communicated with the world" (George Lucas Educational Foundation, 2019). Writing is arguably one of the most succinct tools we have to communicate with others. Building up students to be able to communicate how and what they are thinking is imperative to giving children a voice. This video details a word problem solving method that students at Concourse Village Elementary School use to be able to write about their mathematical thinking. By employing tools such as this to scaffold student learning and allowing students to engage in content discourse via both oral and written means, we are creating students competent in mathematics so they can make sense of and critique the world around them.

# data[].attributes.body
Vocabulary can be defined simply as "the body of words used in a particular language" (Oxford, n.d.) Vocabulary is the words that are available for a student to be able to express themselves, both orally and written.

Oxford languages and google - english. Oxford Languages. (n.d.). Retrieved November 4, 2021, from https://languages.oup.com/google-dictionary-en/.

# data[].attributes.body
There are three main tiers of vocabulary and teaching students each requires educators to be aware of each child's current level of English mastery. In middle grade mathematics, many of the tier one words will likely already be known to students. However, moving into tier two-students may need scaffolding to ensure that they understand the meaning of the words. This is vital if students are to be able to successfully read and comprehend word problems. Tier three words are discourse specific ones and are absolutely necessary to teach through direct instruction. Without access to and knowledge of tier three words, becoming users of mathematics is virtually impossible. This instruction must be explicit and engaging.

DeCarbo, C. (2021, June 25). The Three Tiers of Vocabulary for Classroom Instruction. Miss DeCarbo. Retrieved November 4, 2021, from https://www.missdecarbo.com/the-three-tiers-of-vocabulary/

# data[].attributes.body
Arguably the most succinct definition of comprehension is understanding. According to Young, "Reading comprehension is the essence of reading. In other words, it is the entire point of reading" (n.d., slide 4). Comprehension is the ability for students to be able to decode what the writing means and be able to personally digest that information.

Young, R. (n.d.). Understanding the process of reading comprehension - ppt download. SlidePlayer. Retrieved October 20, 2021, from https://slideplayer.com/slide/13694112/.

# data[].attributes.body
This graphic representation details the five levels of reading comprehension. In order for students to gain a deep level of understanding in any given text, they must be able to engage with the text at the lower levels before moving into the higher levels. At its most basic, comprehension is the ability for readers to "understand the key vocabulary in the text" (5, 2020). This can be done through making sure that the students are reading materials that are appropriate to their current level whether that be via Lexile leveling or some other means. Then, students move into the literal level of comprehension in which they can find and answer simple questions such as who, what, when, and where. At the third level of comprehension, interpretive, students can begin answering the why and how questions related to the text. Next, students are able to apply the knowledge gained from this text to be able to formulate an opinion or make a connection to their life. At the peak of the comprehension spectrum is affective comprehension. In order to get to this level of comprehension, students must be able to understand both the social and emotional contexts of the information and be able to derive more subtle information from the text including the motivations of the characters within (5, 2020).

5 types of reading comprehension. Hand In Hand Homeschool. (2020, May 27). Retrieved October 20, 2021, from https://handinhandhomeschool.com/teaching/reading/5-types-of-reading-comprehension/.

# data[].attributes.body
Building the skills necessary for students to comprehend texts is paramount to their success. This is a multi-faced issue however, as comprehension is "the ability to read text, process it, and understand its meaning and involves complex cognitive processes" (Oxford Primary, 2018, para. 1). This can be even more burdensome in mathematics classrooms as "maths texts contain more concepts per sentence and paragraph than any other type of text" (Oxford Primary, 2018, para. 5). In math, students are not wading through a litany of words, rather math is laid out pretty plainly, even when its a word problem. If a student does not understand one term in the sentence, the entirety of the passage could be misunderstood. This creates an obvious challenge when working with ELs but is not limited to just this population. This webinar attempts to explain and give some ideas for increasing comprehension across the board and advocates for an interdisciplinary methodology to surmounting this all encompassing issue.

Oxford Primary (2018, October 9). Comprehension for English and maths? taking a cross-curricular approach. Oxford Education Blog. Retrieved October 20, 2021, from https://educationblog.oup.com/primary/comprehension-for-english-and-maths.

# data[].attributes.attachment_link.title
Webinar: Considering Comprehension Across English AND Maths

# data[].attributes.body
Leading all students to comprehension of a topic is difficult as students bring a diverse amount of knowledge into the classroom. Having group discussions which attempt to further the entire classes understanding of a topic is important, however "how the questioning unfolds impacts whether or not students view themselves as respected members of the learning community" (Larson, 2018, para. 5). It is imperative that students view themselves as consumers and users of math and not just bystanders. All students have to be brought into the fold with meaningful conversation about complex topics and their voices be heard and validated-even when the "wrong" response is given. Rather, teachers should use that incorrect input to uncover the misunderstanding in the student's mind and help process the student's prior knowledge to help them get to understanding.

Larson, D. M. R. (2021, June 10). Equity is more than access. Education and Learning Resources. Retrieved October 20, 2021, from https://www.hmhco.com/blog/equity-is-more-than-access.

# data[].attributes.attachment_link.readable_text
Many of us, when we think of addressing equitable student outcomes in mathematics education, think of advocating for each and every student to have access to a rigorous and coherent curriculum. And this does remain a significant concern in the U.S.

Research indicates that students from marginalized groups continue to have less access to college preparatory pathways in math than non-marginalized students. Our work to ensure that all students have access to a meaningful math curriculum that supports both their professional and personal aspirations remains a critical component of our equity work. But, if we are truly committed to equitable learning outcomes in mathematics education, we must do more than just provide access.

Below are three other important aspects we must keep in mind in our efforts to better math education.

1. Fostering Positive Math Experiences and Identities

It is not enough to simply get students into the mathematics "pipeline." We must also attend to how students experience math once they are in the "pipeline" because it is their dissatisfying experiences in the classroom that cause some students to shy away from math as a subject. Simply put, equitable math instruction must provide access and empower students to see themselves as capable learners, users, and doers of mathematics.

How students see themselves and how they believe others see them as learners of mathematics greatly impacts their success. By implementing research-informed and equitable instructional practices, teachers can significantly enhance the experiences and learning outcomes of students in the math classroom.

2. Cultivating Equitable Math Discussions

Consider one research-informed instructional strategy from NCTM's Principles to Actions: "Effective teaching of mathematics uses purposeful questions to assess and advance student reasoning and sense making about important mathematical ideas and relationships." This is an effective instructional strategy, but simply asking students more questions in the classroom is not enough. How the questioning unfolds impacts whether or not students view themselves as respected members of the learning community. As a teacher of mathematics you should reflect on the following:

Are all student ideas heard, valued, and pursued?

Who does the teacher call on to answer questions?

Whose ideas are selected for further inquiry?

Whose thinking is disregarded in the classroom?

Asking ourselves these questions reminds us that the implementation of our strategies shapes how students see themselves and their competence. As math teachers, we are all identity builders, and we have to take that role seriously as we design instructional experiences and interact with students.

3. Inspiring the Citizens of a Math-Centric Future

We are also engaged in something much bigger than the daily tasks related to curriculum, instruction, and assessment. Mathematics is an essential analytical tool that students can use to help them better understand their context, experiences, and the world-and, potentially, to make the world a better place. Never has this been more important. Mathematics is deeply embedded in many aspects of society today- social media advertising, medicine, finance, security, technology, and the increasing use of "big data" across many disciplines.

As a result of this math ubiquity, societal challenges are increasingly formulated in mathematical terms, and the proposed solutions frequently depend on mathematical understanding. In other words, math is essential-not just to college and career readiness but also to informed, active engagement in all aspects of life. The future of our democratic society depends on every student having both access to a rich math curriculum and positive classroom experiences that ensure each student is truly empowered by the opportunities that mathematics affords.
The views expressed in this article are those of the author and do not necessarily represent those of HMH.
***

# data[].attributes.attachment_link.description
Many of us, when we think of addressing equitable student outcomes in mathematics education, think of advocating for each and every student to have access to a rigorous and coherent curriculum. And this does remain a significant concern in the U.S.

# data[].attributes.body
According to Reading Rockets, fluency is defined as "the ability to read a text accurately, quickly, and with expression. Reading fluency is important because it provides a bridge between word recognition and comprehension" (Fluency, 2020, para. 1). Fluency in math is no different. As a teacher, it is my job to ensure that all students are able to speak in mathematical terms as it relates to the content at hand.

Fluency. Reading Rockets. (2020, October 16). Retrieved September 28, 2021, from https://www.readingrockets.org/teaching/reading-basics/fluency.

Sedita, J., & *, N. (2015, June 2). Resources for fluency instruction. Keys to Literacy. Retrieved September 28, 2021, from https://keystoliteracy.com/blog/fluency-instruction/.

# data[].attributes.body
These three levels are especially important to note. I was surprised at how low the level of understanding is at which students would feel frustrated. This is especially important to understand in the context of a math classroom. Children can get bogged down in the wording and although they may be able to understand the concept, they may not have the fluency to be able to discern what the question is asking. This is vital when it comes to assessments as many state sanctioned exams are not written in alternative languages and this could lead to skewed results and the presumption that a child is "not good" at math.

Fluency: An introduction. Reading Rockets. (2013, November 7). Retrieved September 28, 2021, from https://www.readingrockets.org/article/fluency-introduction.

# data[].attributes.body
This guide was a great resource into the importance of fluency in math for all learners. Although it does tend to emphasize procedural fluency, it also delves into verbal fluency in the discipline of mathematics as well. In this guide, the author states that number talks, discussed more later, can be used to "figure something out rather than be told the steps to follow, explain what they think rather than wait for the teacher to explain, explain why rather than just know how, test new ideas and embrace mistakes as part of the process, and see wrong answers and opportunities to learn something new" (Middle, 2021, p. 5-6). This important communication tactic is one of many that could be implemented but would allow me to hear student's voices and how they use mathematical language.

Middle school math Fluency Guide. UnboundEd. (n.d.). Retrieved September 28, 2021, from https://lessons.unbounded.org/other/8609.

# data[].attributes.attachment_link.readable_text
Description

The UnboundEd Middle School Fluency Guide offers strategies and rationale for busy teachers who want to develop procedural fluency in Middle School math. Connect and build on your students' conceptual understanding through structured practice that will help students make sense and expertise with procedures. Learn how to effectively support student development of their productive disposition toward math.

Downloads

There may be cases when our downloadable resources contain hyperlinks to other websites. These hyperlinks lead to websites published or operated by third parties. UnboundEd and EngageNY are not responsible for the content, availability, or privacy policies of these websites.

Tags

There are no tags for this resource

# data[].attributes.attachment_link.description
The UnboundEd Middle School Fluency Guide offers strategies and rationale for busy teachers who want to develop procedural fluency in Middle School math. Connect and build on your students' conceptual understanding through structured practice that will help students make sense and expertise with procedures.

# data[].attributes.body
Phonics can be difficult for some students to masters, especially ELs whose first language has a more intuitive phonics system. Unfortunately English has more phonemes than it has letters to represent them (44 vs 26), thus leading to the deep orthography of the language (Pauline, 2021). By being aware of language differences in each of my students, I can be more aware of the potential for phonics issues and help students decipher meaning of vocabulary words.

Pauline, S. (2021, July 7). What is Phonics? And Why is it IMPORTANT? ⋆ think Beyond Literacy. Think Beyond Literacy. Retrieved September 28, 2021, from https://thinkbeyondliteracy.com/what-is-phonics-and-why-is-it-important/.

# data[].attributes.body
This website sponsored by the University of Texas at Austin has a great program and videos showing how to implement number talks in a middle school math classroom.  Adding these types of talks to the classroom can be beneficial to all students and for the teacher in that they can use these number talks to surmise where the students are in their learning. The students use number talks as a way to voice their thought processes and engage with the mathematical vocabulary, thus becoming more fluent consumers and producers of math.

Borda, D., & Villarin, A. (n.d.). 7Th grade math - What's the Savings?: Inside Mathematics. The University of Texas Dana Center. Retrieved September 28, 2021, from https://www.insidemathematics.org/classroom-videos/number-talks/7th-grade-math-whats-the-savings.

# data[].attributes.attachment_link.readable_text
In this Number Talk, Debbie Borda and Antoinette Villarin engage their 7th grade students in mentally calculating the savings received by a student who paid $54 for an item that was discounted 40%. The students in the class first think their way through the problem, then share their strategies with a partner, then share strategies and answers with the class as a whole.

# data[].attributes.attachment_link.title
7th Grade Math - What's the Savings? | Inside Mathematics

# data[].attributes.attachment_link.description
In this Number Talk, Debbie Borda and Antoinette Villarin engage their 7th grade students in mentally calculating the savings received by a student who paid $54 for an item that was discounted 40%. The students in the class first think their way through the problem, then share their strategies with a partner, then share strategies and answers with the class as a whole.

# data[].attributes.body
This blog breaks down many different strategies for aiding in phonics instruction. Although in middle school, not all of them are relevant, I found strategy to be particularly useful. On this website, the author breaks down vocabulary words on the board into their specific phonemes. This could be useful with vocabulary terms in math as some of these disciplinary specific words may be difficult for students to grapple. For instance, when a student was having a hard time spelling the word wren, the author "wrote the word 'write' on the board, suspecting she knew it. She did, immediately. So I underlined the wr and said, 'Use this part of write to help you with this word you don’t know'" (Elkins, 2021, para. 2). By using other, simple words that my students are familiar with to show how the different parts of the unfamiliar word sound, I can give all students the opportunity to engage in the correct pronunciation of mathematical words.

Elkins, C. (2021, February 15). Phonics Part 7: Word Analogy Strategy. Ok math and reading lady. Retrieved September 28, 2021, from https://cindyelkins.edublogs.org/tag/phonics/.

# data[].attributes.attachment_link.readable_text
by C. Elkins, OK Math and Reading Lady

Manipulating phonemes refers to the abilty to add, delete, or substitute them orally. This is a prerequisite skill to visually manipulating letters in order to read and write (the phonics part). First grade programs should include both of these routines on a daily basis - and at other grade levels with struggling students. Skill with manipulating letters relates to a child's ability to use a known word (or word part) to read and write new words. Stay tuned for 10 ways to work on manipulating letters (whole class or small group) as well as some free resources to accomplish your goals. Phonemic awareness falls under the Phonological Awareness umbrella.
Add phonemes - phonemic awareness (examples - letters inside / / represent the sound):

"Say /at/. Now add /k/ before /at/. What do you hear?"

"Say /un/. Now add /f/ before /un/. What do you hear?"

Add letters - phonics (examples):

"Write at. Now add the letter c before at. What word did you make?"

"What other letters can you add before at to make new words?" (suggest b, f, h, m, p, r, s)

"Write un. Now add the letter f before un. What word did you make?"

"What other letters can you add before un to make new words?" (suggest b, g, n, r, s)

Delete phonemes - phonemic awareness (examples):

"Say fat. Take off the /f/ sound. What part is left?" (at)

"Say run. Take off the /r/ sound. What part is left?" (un)

Delete letters - phonics (examples):

"Write track. Take off the letters tr. What part is left?"

"What other words can you write with the letters -ack?" (suggest back, smack, lack, tack)

"Write sting. Take off the letters st. What part is left?"

"What other words can you write with the letters -ing?" (suggest sing, ring, fling, thing)

Substitute phonemes - phonemic awareness (examples):

"Say like. Take off the /l/ sound and trade it for the /b/ sound. What word do you hear now?" (bike)

"Say hop. Take off the /p/ sound and trade it for the /g/ sound. What word do you hear now?" (hog)

Substitute letters - phonics (examples):

"Write like. Change the letter l to a b. What word did you write?" (bike)

"Write like. Change the letter i to the letter a. What word did you write?" (lake)

"Write like. Change the letter k to the letter f. What word did you write?" (life)

You will notice that manipulating letters also links directly to identification of onsets and rimes.

With onsets and rimes, we want students to be able to identify the rimes. This means they have to separate it from the beginning of the word. Example: sh + op

We also want students to be able to recognize and produce the rimes or common "chunks" in other words. This involves deleting and substituting letters. Examples: hop, mop, drop, helicopter. If they can't hear it, they most likely can't read it or write it.

Here are 10 ways to work on letter manipulation in the classroom or with small groups:

1. Utilize some of the above examples as part of your daily word work routine.

2. Use sound boxes and magnetic letters or letter tiles to model manipulation of the letters.

3. Use post-its to practice manipulating words.

4. Provide word family practice like these rimes -op, -and, -it, - un, -et, eet, -oat, -ine, and so on.

5. Give students letter cards to hold and arrange to make / change words.

6. Break words apart (and put back together).

Break apart letter by letter.

Break apart onsets and rimes

Break apart beginning / middle / ending

7. Explore words by making a word chain. Write it or use letter tiles to help. How long can you make the chain go on?

Start with one word such as cat.

Change just one letter at a time to make a new word (this can be the beginning letter, middle letter, or ending letter): hat

Change it again (just one letter): hit

Change it again: bit

Change it again: big

Change it again: dig

Change it again: dog

Change it again: dot

Change it again: hot

Change it again: pot

Change it again: pet

Can you keep going???

8. Use spelling words to make changes. After all, the purpose of spelling is not to just spell the words on the list, but to apply the generalization to other words. Spelling word is wide? Try these: slide, hide, bride . . . Spelling word is eat? Try these: meat, seat, treat . . .

9. Give pairs of students word family lists to partner read (-ake: bake, cake, lake, take, make, rake, snake, shake, wake . . .)

10. Check out some of these FREE resources from TPT:

As always, please share your favorite manipulating phonemes or letters activities!!

# data[].attributes.attachment_link.description
by C. Elkins, OK Math and Reading Lady Manipulating phonemes refers to the abilty to add, delete, or substitute them orally. This is a prerequisite skill to visually manipulating letters in order to read and write (the phonics part). First grade programs should include both of these routines on a daily basis - and at other grade levels with struggling students.

# data[].attributes.body
Cognates are especially helpful to keep in mind for ELs whose first language is Spanish. Cognates can be used as "a tool for understanding a second language" and are merely words that are similar in meaning and sound between two languages (Colorin Colorado, 2021, para 3). It is important, however to not use these two languages to create an interlanguage, but to use cognates as a tool to facilitate the knowledge transmission of math from Spanish to English. Other languages have cognates to English as well, but Spanish is the focus here as there are so many overlapping words and Spanish speaking ELs make up the majority of what I may encounter in Florida schools.

Colorín Colorado. (2021, June 14). Using cognates to Develop comprehension in English. Colorín Colorado. Retrieved September 28, 2021, from https://www.colorincolorado.org/article/using-cognates-develop-comprehension-english.

# data[].attributes.body
This website explicitly defines oral language as "is the system through which we use spoken words to express knowledge, ideas, and feelings (Audet, 2021, p. 2). It individually separates the five components of oral language and explains each with examples.

Audet, L. (n.d.). What is oral language? What Is Oral Language? Retrieved September 14, 2021, from https://blog.heinemann.com/what-is-oral-language.

# data[].attributes.attachment_link.description
What Is Oral Language? Developing oral language means developing the skills and knowledge that go into listening and speaking-all of which have a strong relationship to reading comprehension and to writing.

# data[].attributes.body
"Phonological awareness is the ability to recognize and manipulate the spoken parts of sentences and words" (Phonological, 2020, p.1). This includes the ability for students to distinguish between different parts of a word or sentence and the abilities to rhyme and identify alliteration (Phonological, 2020). It is important to recognize that not all students in my classroom will be fluent in English or may not understand these topics that are typically introduced in elementary school. Learning the 40-44 different phonemes and the associated letters that represent them will allow me to help break down complex mathematical terms to my students.

Phonological and phonemic awareness: Introduction. Reading Rockets. (2021, April 27). Retrieved September 14, 2021, from https://www.readingrockets.org/teaching/reading101-course/modules/phonological-and-phonemic-awareness-introduction.

# data[].attributes.attachment_link.readable_text
Learn the definitions of phonological awareness and phonemic awareness - and how these pre-reading listening skills relate to phonics.

Letter of completion

After completing this module and successfully answering the post-test questions, you'll be able to download a Letter of Completion.

Phonological awareness and phonemic awareness: what's the difference?

Phonological awareness is the ability to recognize and manipulate the spoken parts of sentences and words. Examples include being able to identify words that rhyme, recognizing alliteration, segmenting a sentence into words, identifying the syllables in a word, and blending and segmenting onset-rimes. The most sophisticated - and last to develop - is called phonemic awareness.

Phonemic awareness is the ability to notice, think about, and work with the individual sounds (phonemes) in spoken words. This includes blending sounds into words, segmenting words into sounds, and deleting and playing with the sounds in spoken words.

Phonological awareness (PA) involves a continuum of skills that develop over time and that are crucial for reading and spelling success, because they are central to learning to decode and spell printed words. Phonological awareness is especially important at the earliest stages of reading development - in pre-school, kindergarten, and first grade for typical readers.

Explicit teaching of phonological awareness in these early years can eliminate future reading problems for many students. However, struggling decoders of any age can work on phonological awareness, especially if they evidence problems in blending or segmenting phonemes.

How about phonological awareness and phonics?

Phonological awareness refers to a global awareness of sounds in spoken words, as well as the ability to manipulate those sounds.

Phonics refers to knowledge of letter sounds and the ability to apply that knowledge in decoding unfamiliar printed words.

So, phonological awareness refers to oral language and phonics refers to print. Both of these skills are very important and tend to interact in reading development, but they are distinct skills; children can have weaknesses in one of them but not the other.

For example, a child who knows letter sounds but cannot blend the sounds to form the whole word has a phonological awareness (specifically, a phonemic awareness) problem. Conversely, a child who can orally blend sounds with ease but mixes up vowel letter sounds, reading pit for pet and set for sit, has a phonics problem.

44 phonemes

There are 26 letters in the English alphabet that make up 44 speech sounds, or phonemes.

# data[].attributes.attachment_link.description
Learn the definitions of phonological awareness and phonemic awareness - and how these pre-reading listening skills relate to phonics. Letter of completion After completing this module and successfully answering the post-test questions, you'll be able to download a Letter of Completion. Phonological awareness and phonemic awareness: what's the difference?

# data[].attributes.body
This blog is ripe with current, relevant content as to how to break down and teach phonological awareness to children.

What is phonological awareness? And how do i teach it? How Wee Learn. (2021, March 1). Retrieved September 14, 2021, from https://www.howweelearn.com/what-is-phonological-awareness-and-how-do-i-teach-it/.

# data[].attributes.body
This research seeks to determine the link between math and reading literacy by means of phonological awareness. "Our study found that phoneme awareness was related to verbal arithmetic when children were asked to solve a single-digit addition or subtraction task without time limits, but not to single-digit arithmetic fluency tasks that were timed and answered on paper with no verbal response needed" (Amland, et al., 2021, p. 47). Thus, it is important to continue having students express themselves orally and that phonological awareness does contribute to their ability to do so. It also found that this awareness did not seem to correlate into written skills.

Amland, T., Lervåg, A., & Melby-Lervåg, M. (2021, January 7). Comorbidity between math and reading problems: Is phonological processing a mutual factor? Frontiers. Retrieved September 14, 2021, from https://www.frontiersin.org/articles/10.3389/fnhum.2020.577304/full?&utm_source=Email_to_authors_&utm_medium=Email&utm_content=T1_11.5e1_author&utm_campaign=Email_publication&field=&journalName=Frontiers_in_Human_Neuroscience&id=577304.

# data[].attributes.attachment_link.readable_text
Department of Special Needs Education, University of Oslo, Oslo, Norway

Department of Education, University of Oslo, Oslo, Norway

There is a relationship between reading and math skills, as well as comorbidity between reading and math disorders. A mutual foundation for this comorbidity could be that the quality of phonological representations is important for both early reading and arithmetic. In this study, we examine this hypothesis in a sample traced longitudinally from preschool to first grade ( N = 259). The results show that phonological awareness does not explain development in arithmetic, but that there is an indirect effect between phoneme awareness in preschool and arithmetic in first grade via phoneme awareness in first grade. This effect is, however, weak and restricted to verbal arithmetic and not arithmetic fluency. This finding is only partly in line with other studies, and a reason could be that this study more strongly controls for confounders and previous skills than other studies.

Introduction

Mastering reading and math is vital for not only academic performance but also important life skills critical for employment and participation in society. Researchers have long known that there is a rather close relationship between reading and mathematics. The two skills correlate moderately to highly ( Peng et al., 2020), and a large number of children with dyslexia also struggle with math difficulties. Reversely, many children with dyscalculia also have difficulties in reading ( Joyner and Wagner, 2020). A question yet to be answered is what kind of foundational skills might be a common factor underlying reading and arithmetic. One hypothesis that has gained support is that phonological processing underlies not only early reading skills but also foundational mathematic skills. In this study, we investigate this hypothetical cause of the relationship between early reading and mathematics skills and thus focus on the relationship between math and early reading.

To What Extent Are Reading and Arithmetic Correlated?

Although correlation does not imply causation, a correlation is often a starting point for disentangling causality. Studies have shown that reading and mathematics are correlated both at the genetic level and in brain activation patterns ( Shaywitz et al., 1998; Temple et al., 2001; Ashkenazi et al., 2013; Pollack and Ashby, 2018). These correlations are also reflected at a behavioral level: on a broader level, a recent meta-analysis illustrated that the mean correlation between language and math across 392 independent samples was moderate ( r = 0.42; Peng et al., 2020). This correlation, unsurprisingly, is also reflected in children at the lower end of the distribution with math and reading problems. A recent review showed that children with a math disorder were over two times more likely to also have a reading disorder than those without a math disorder (odds ratio = 2.12; Wagner et al., 2019). This result coincides with findings from other meta-analyses (see Swanson et al., 2009; Landerl and Moll, 2010).

Phonological Processing as A Foundation for Arithmetic as Well as Word Decoding

Theoretically, the "triple code model," the most influential framework for understanding early number processing, assumes a pathway from language to numeracy ( Dehaene, 1992). An important question then is which aspects of language might be causally related to mathematics. One candidate is skills related to phonology. The consensus concerning reading is that phonological processing of sounds in words (often measured with phoneme awareness tasks) is an important causal precursor of growth in word decoding and the primary cause of reading problems such as dyslexia ( Melby-Lervåg et al., 2012; Caravolas et al., 2013; Snowling and Melby-Lervåg, 2016). In a meta-analytic comparison of the effectiveness of reading interventions, phoneme awareness programs were found to be among the most successful approaches to boosting children's reading skills over time ( Suggate, 2016). To master such phonological awareness tasks, a child must be able to encode, maintain, and reproduce accurate representations of words from memory. Performance on phonological awareness tests is therefore considered to reflect the quality of phonological representations stored in memory. Thus, the more fine-grained and detailed the phonological processing and representations are, the better the performance on phonological awareness tests.

However, performance on such phonological processing tasks is highly correlated with not only decoding but also early mathematics skills. In particular, performance on such phonological processing tasks is strongly correlated with arithmetic, the part of mathematics that concerns numbers and basic operations on them-addition, subtraction, multiplication, and division. A review by Peng et al. (2020) showed an average moderate correlation between phonological awareness and arithmetic ( r = 0.35).

The quality of phonological representations can be important for arithmetic problem-solving in several ways. First, to solve a computation problem, a child must first transform the numbers and operators in the problem into a speech-based code ( Dehaene, 1992; Hecht et al., 2001). Studies have shown that this Arabic-to-verbal translation appears to be routinely used by children not only to solve simple arithmetic problems but also for more general math computation problems, such as long division and fractions. A second stage during which phonological representations might be important is after the Arabic-to-verbal translation. The child must then process the phonological information using a specific task-solving strategy. For a simple arithmetic problem (4 + 3 = ?), one must retrieve the answer directly from long-term memory, and so the ability to solve such a problem is dependent on the storage of phonological information. Another alternative can be to use a counting-based strategy to retrieve the answer. The phonological system is then employed when the child uses the phonological codes for the number names in counting. Thus, there are several ways in which phonological processing may also yield a causal influence on numeracy. Phonological processing tasks are likely important for both decoding and arithmetic since both tasks depend on mental processes that use sound-based representations.

As for previous studies, some observational investigations have offered support for this theory. One study followed children from second to fifth grade ( N = 201; Hecht et al., 2001). The researchers found that phonological processing was uniquely associated with the development of early arithmetic skills. Importantly, phonological processing (as measured by digit span, rapid naming, and phonological awareness) almost entirely accounted for the relationship between reading and math. However, the authors found no unique influence of phonological processing on arithmetic fluency speed from fourth to fifth grade. There are several potential explanations for this result. For example, the children could have been too old for phonological processing to have an influence. Another explanation might be that phonological processing (or, more specifically, the quality of phonological representations) influences accuracy but not speed. A study by De Smedt et al. (2010) of fourth- and fifth-grade children ( N = 37) yielded results in line with those of Hecht et al. (2001): they found that the quality of phonological representations was particularly important for the retrieval of existing arithmetic facts (e.g., 2 + 3 = 5). Such single digits problems are easier and faster to retrieve compared to larger problems requiring procedural strategies.

Recently, a larger 1-year longitudinal study of Finnish 7-year-olds ( N = 200) followed up on Hecht et al. (2001) and examined predictors of the covariance between reading and arithmetic fluency ( Koponen et al., 2019). The results showed that phonological awareness is a unique longitudinal predictor of this variation. Along the same lines, a 1-year study of students in kindergarten and first grade by Cirino et al. (2018) examined predictors for the mutual variance between reading and math ( N = 193). This study found that phonological awareness, together with other linguistic naming tasks, accounted for nearly the same amount of overlap as all predictors together. This finding matches that of an earlier concurrent study ( N = 233) from the same research environment, which showed a relationship between both timed and untimed arithmetic tasks ( Child et al., 2019).

Furthermore, a recent concurrent study of 5-year-old kindergarteners ( N = 188) also suggests that phonological awareness is related to early arithmetic ( Vanbinst et al., 2020). The main finding was that phonological awareness was related not only to early reading but also to early arithmetic. The relationship notably remained after controlling for early reading- and arithmetic-specific cognitive correlates. Finally, Singer et al. (2019) also found, using concurrent data on children aged 9-11 years ( N = 262), that phonological awareness was uniquely related to simple arithmetic fluency computations.

Still, other observational studies have found no unique relationship between phonological awareness and arithmetic, despite high correlations between the two. Purpura et al. (2011) found that letter knowledge-but not phonological processing-uniquely predicted development in arithmetic from 1 year to the next. Moll et al. (2015) reported that children with arithmetic difficulties only, and not comorbid reading problems, had difficulties restricted only to the arithmetic domain, while children with only reading difficulties also had problems with tasks related to arithmetic exercises that involved a verbal code. The comorbid math/language group exhibited difficulties with both. Thus, in this study, only the result of the comorbid group was in line with the hypothesis that children with arithmetic difficulties have phonological problems. In a previous study by Landerl et al. (2009), findings suggested that dyslexia and dyscalculia have separable cognitive profiles; when comparing groups of dyslexic, dyscalculic, and dyslexic/dyscalculic children, the researchers found a phonological deficit in both dyslexic groups, but not in the dyscalculia-only group. Contrastingly, they found difficulties in symbolic and non-symbolic number processing in both groups with dyscalculic children, but not in the dyslexia-only group.

Thus, taken together, findings on the role of phonological processes as a foundation of arithmetic as well as word decoding are inconclusive. Two possible reasons for discrepancies in the findings are differences in age of the children studied as well as differences in tasks used to measure such phonological processes. It is possible that detection of this relationship depends on the degree of automated fact retrieval and whether or not the task used to measure phonological skill is a phonological awareness task or a different measure aimed at capturing other related processes. Phonological skills have been measured by different tasks reflecting different components, such as rapid automatized naming (RAN) and verbal working memory ( Wagner and Torgesen, 1987; Hecht et al., 2001). However, results concerning rapid naming and phonology tasks tapping verbal working/short term memory can be difficult to interpret since it is not entirely clear what these components measure. Because the construct phonological processing has evolved over time, and recent theory relating to arithmetic has focused on phonological awareness, this is also the aspect of phonological processing assessed in the present study. Beyond phonological processes, it is also possible that other domain-general abilities can account for the observed relationship, as previous studies vary in the extent to which they include broader abilities.

The Correlation Between Phonological Awareness and Arithmetic: Is It Spurious and Caused by Underlying General Cognitive Skills?

The first potential explanation for the correlation between phonological awareness and arithmetic is that general cognitive ability is the third variable that underlies both. Possible cognitive candidates that have been suggested to underlie this correlation are nonverbal abilities ( Bull and Scerif, 2001; Geary, 2004) and general language skills ( Moll et al., 2015). If this is correct, controlling for cognitive abilities in the relationship between phonological awareness and arithmetic would weaken the correlation or make it disappear entirely. A recent review by Peng et al. (2020) demonstrates that general intelligence explains a large proportion of the variance in the relationship between mathematics and language.

However, previous studies of phonological awareness and arithmetic have varied with the extent to which they take such variables into account: Hecht et al. (2001) controlled for general language skills and found that the relationship between phonological processing and arithmetic held. They did not, however, control for nonverbal abilities. Koponen et al. (2019) controlled for various memory-related variables but not verbal and nonverbal abilities, while Cirino et al. (2018) controlled for processing speed, visuospatial working memory, and nonverbal abilities. As for the concurrent studies, Child et al. (2019) controlled for working memory and processing speed; Vanbinst et al. (2020) controlled for nonverbal abilities; and Singer et al. (2019) controlled for both verbal and nonverbal abilities. Regarding those studies that did not find a relationship between phonological awareness and arithmetic, Purpura et al. (2011) controlled for nonverbal intelligence, while Moll et al. (2015) controlled for both verbal and nonverbal abilities.

Phonological Awareness-More Important for Some Aspects of Arithmetic than Others?

As mentioned, the quality of the phonological representations can be important for arithmetic problem-solving in several ways. This also implies that the relationship can stronger for some types of computation than others. However, only one previous study has examined whether there are differential effects for different types of arithmetic types. This study found a relationship between phonological awareness and addition/subtraction fluency, but not with number knowledge or word problems ( Singer et al., 2019). This distinction is potentially significant because it implies that the general language ability required for word problem-solving is separate from the phonological processes involved in efficient arithmetic fact retrieval.

The remaining studies have used a variety of tasks: Hecht et al. (2001) used a latent variable with a range of tasks such as untimed simple digit addition and subtraction, multi-digit addition, multiplication, division, fractions, and algebra as well as addition and subtraction fluency. Koponen et al. (2019) used a latent variable with addition and subtraction fluency and basic arithmetic, while Vanbinst et al. (2020) measured basic addition and subtraction accuracy. While such tasks are likely to trigger arithmetic fact retrieval due to small problem sizes and frequent administration with a time limit, they are rarely compared to other types of arithmetic tasks that may or may not also rely on phonological processes. Comparing additional types of arithmetic concerning their ties to phonological skill could provide clarity on which aspects of early number ability account for the observed relationships. In addition to procedural vs. fact-retrieval demands, task characteristics such as the degree of vocalization, working speed, and use of digits vs. number words might matter in terms of whether a relationship is detected.

The Current Study

Taken together, several studies have pointed to phonological awareness as a candidate for explaining why there is a relationship between early reading and arithmetic. However, previous results have been inconclusive. Moreover, the studies have varied in whether they have controlled for confounders in their outcome measures. Furthermore, some of these studies have included tasks in their phonology construct tasks that would now be considered rather unconventional, such as rapid naming and verbal short-term memory measures. In the present study, we deal with these issues and examine the relationship between phonological awareness (measured with widely used task formats) and arithmetic in a large sample, controlling for measurement error and verbal and nonverbal abilities. We examine the following research questions:

* Can phoneme awareness in preschool predict both reading and arithmetic skills in first grade when controlling for early reading and number skills, general language skills, and nonverbal abilities?

* Does phoneme awareness predict reading and arithmetic concurrently, and are there indirect effects of phoneme awareness in preschool on arithmetic and reading outcomes through phoneme awareness in first grade?

* Does phoneme awareness relate differently to verbal arithmetic and arithmetic fluency?

Materials and Methods

Participants

We recruited a cohort of 259 Norwegian-speaking children (135 boys and 126 girls) with a mean age of 5.5 years (range 4.9-6.1 years, SD = 0.3) when the study began. The children were recruited from a district in southeastern Norway that compared to the national average on aspects relating to educational level and socioeconomic status ( Statistics Norway, 2020a, b). Children diagnosed with severe learning or developmental disorders, such as autism, sensory impairments, or an intellectual disability, were excluded from the study during data collection.

We obtained informed consent for the children's participation in the study from their parents. Additionally, the children gave verbal consent at each time point of the data collection.

At the first time point, the children were in their final year of preschool. Norwegian children enter primary school and begin formal literacy and numeracy instruction the year they turn six. No formal instruction in reading or math is given in preschool before children enter first grade in primary school. At the time of the testing in grade one, the children in our sample had had about 6 months of formal instruction, with some children yet to master reading and calculations. However, due to the early assessment point, there might be lower-performing children in our sample who will receive diagnoses as they age.

Design and Procedure

The children were assessed individually on a range of measures of numeracy, literacy, and general cognition with a 12-month time interval in their respective kindergartens and later schools. The tests used in the data collection are mainly internationally established and widely used measures that have been adapted to Norwegian. See Table 1 for the descriptive statistics and reliabilities for all measures.

The tests were administrated individually in a fixed order by trained research assistants. The research assistants visited the children's kindergartens and later schools three times in the weeks between early January and late March 2 years in a row. The tests were part of a larger test battery involving a range of cognitive abilities, and each of the three sessions lasted about 35-55 min, depending on the child's working speed and needs.

Table 1. Descriptive statistics and reliability for all observed variables.

Measures

The following indicators were used.

Preschool, Age Five

General language skills were measured with two indicators: first, receptive grammar skills were assessed using the Norwegian version of the Test for Reception of Grammar (TROG; version 2; Bishop, 2009). The Norwegian version of the TROG has been standardized and normed with a Norwegian sample ( Lyster and Horn, 2009). In this task, the child hears sentences of increasing complexity and selects a picture for each sentence. Example sentences include the following: "The girl is sitting on the table" and "The blue cup is on top of the small yellow book." The test contains 80 four-choice items and is discontinued when the child makes one or more errors in five consecutive blocks.

Second, receptive vocabulary skills were assessed with a translated version of the British Picture Vocabulary Scale (BPVS; Dunn and Dunn, 2009). The Norwegian version of the BPVS has also been standardized and normed with a Norwegian sample ( Lyster et al., 2010). The test requires the child to match a spoken word with one of four presented pictures. This instrument has a total of 144 items of increasing difficulty, and test administration ends when the child makes eight or more errors in a block of 12 items.

Nonverbal general abilities were also measured with two indicators: first, Raven's Coloured Progressive Matrices (CPM) is a test designed for children aged 5-11 ( Raven, 2000). The instrument measures nonverbal intelligence and abstract reasoning abilities, and very limited verbal instruction is given by the test administrator. The child sees a set of illustrations with a piece missing and is then required to identify the construction pattern of increasingly complex geometrical figures by selecting the piece required to complete the set. There are 36 items organized in sets, and the test is administered without time constraints.

Second, the matrix reasoning task from the Wechsler Preschool and Primary Scale of Intelligence, 4th edition (WPPSI-IV) is also an index of nonverbal abilities ( Wechsler, 2012). In this task, the child sees an incomplete matrix and is required to select the missing alternative that completes the matrix. Picture or figure analogies must be nonverbally identified for the child to select the correct response. An example of such a task is a matrix with a horse, a barn, and a fish, as well as an empty square. Among the alternatives for the missing piece is an empty fishbowl, which would be the correct choice. As the child progresses, items become gradually more abstract and complex.

Number word knowledge in preschool was measured with a number-naming task. The child was shown a series of printed numbers and asked to name them: "Which number is this?" The numbers increased in size and range from single- to four-digit numbers.

Verbal arithmetic skills in preschool were measured with a verbal addition task. The task consisted of linguistically simple addition problems presented orally and calculated mentally (e.g., "What is seven plus nine?"). The child responded verbally. For 5-year-old children, we only used addition as an indicator of arithmetic skills due to the large number of children who are unfamiliar with subtraction at this stage.

Letter knowledge in preschool was assessed by having the child give the name or sound of letters in the Norwegian alphabet shown on a sheet of paper, ordered alphabetically. In task one, the printed letters were vowels (e.g., A, E, I ...) and in task two, the letters were consonants (B, D, F ...).

Phoneme awareness at age five was measured with a phoneme isolation task. The child was asked to identify a phoneme in a word read aloud by the test administrator (e.g., "What is the first sound in boat?"). We used two blocks where the child was asked to identify the first sound of a given word in block one and the last sound of the word in block two.

The four first items in each block were accompanied by pictures for support. The items consisted of three or four-letter words [either consonant-vowel-consonant (CVC), CCVC, or CVCC].

First Grade, Age Six

Number word knowledge in first grade was measured by two tasks assessing the child's knowledge of the correspondence between digits and number words.

Number naming. The first number-knowledge task was the same as that used in the previous year but extended to include more difficult items (i.e., larger numbers with up to four digits). The child was shown a series of printed numbers and asked to name them ("Which number is this?").

Number identification. The children were also given a number identification task, similar to the task used by Göbel et al. (2014). Number identification entailed drawing a circle around one of five printed numbers for each number read aloud by the test administrator. For example, the test administrator might read "163", and the child would then mark one of five options presented on a sheet (136, 10,063, 13, 163, 16). The target numbers ranged from one to three digits.

Phoneme awareness was assessed as follows: two phoneme deletion tasks were administered, one with words and one with non-words. The tasks required the child to delete sounds in the beginning, middle, or end of the words. A translated example is "Say 'cat' without saying '/k/'."

Arithmetic fluency was measured with two tasks, one with addition problems and one with subtraction problems. The subtasks were taken from the Test of Basic Arithmetic and Numeracy Skills (TOBANS; Brigstocke et al., 2016). The child completed three practice items for each set to ensure that they understood the mathematical operation required and was then asked to solve as many problems as possible with pencil and paper within 60 s. The number of correct answers was recorded.

Verbal arithmetic skills were assessed in first grade with linguistically simple arithmetic problems presented orally and calculated mentally. The test had two parts; part one consisted of increasingly difficult addition problems, and part two contained subtraction problems. The child responded verbally and the number of correct responses was recorded.

Reading efficiency was assessed by requiring children to read aloud two lists of words from a Norwegian translation of the Test of Word Reading Efficacy (TOWRE; Torgesen et al., 1999). Children are given 45 s to read aloud as many words as they can from each list. The score is the number of items read and pronounced correctly.

Results

Descriptive Statistics and Correlations

Table 1 shows the means, standard deviations, minimum and maximum scores, and reliabilities for all measures, and the correlations between all observed variables are available in online supplement ( Supplementary Table 1). Table 1 shows that all variables had decent reliability. All further analyses were done in Mplus version 8 ( Muthén and Muthén, 1998-2017) using the full information maximum likelihood approach to handle missing values.

Confirmatory Factor Analyses

To test the relations between the latent constructs and their respective observed indicators (i.e., the measurement model) and between the latent constructs themselves, we estimated a confirmatory factor analysis (CFA) in which we included all variables. Six constructs were measured in preschool (nonverbal abilities, language, letter knowledge, phoneme awareness, number knowledge, and verbal arithmetic) and five constructs were measured in first grade (phoneme awareness, number knowledge, verbal arithmetic, arithmetic fluency, and word reading). See Table 2 for the indicators of the constructs. This model had an excellent fit to the data, = 109.902, p = 213; RMSEA = 0.021 (90% CI = 0.000, 0.040); TLI = 0.993; SRMR = 0.026. Table 2 shows the factor loadings and factor correlations. As illustrated, the factor loadings were relatively strong for most observed indicators, except matrixes ( λ = 0.407). Regarding the hypotheses, strong correlations were found between phoneme awareness and both reading and verbal arithmetic. The correlations between phoneme awareness and arithmetic fluency were somewhat lower.

Table 2. Standardized factor loadings for all latent constructs and their respective indicators plus factor correlations between the latent constructs.

Does Phoneme Awareness Predict Reading as Well as Verbal Arithmetic and Arithmetic Fluency?

To test this, we first estimated a structural equation model (SEM) in which we regressed reading, verbal arithmetic, and arithmetic fluency in first grade on phoneme awareness in preschool together with the control constructs letter knowledge, number-word knowledge, verbal arithmetic, language, and nonverbal abilities in preschool. Also, phoneme awareness, letter knowledge, number-word knowledge, and verbal arithmetic skills in preschool were regressed on language and nonverbal abilities in preschool.

As shown in Figure 1, phoneme awareness in preschool predicted reading but not verbal arithmetic and arithmetic fluency in first grade. Number-word knowledge in preschool predicted both reading and the two arithmetic constructs in first grade; verbal arithmetic in preschool predicted both verbal arithmetic and arithmetic fluency in first grade. Also, there was a negative suppression effect of letter knowledge in preschool on arithmetic fluency in first grade. Language in preschool directly predicted phoneme awareness, letter knowledge, number-word knowledge, and verbal arithmetic in preschool and indirectly predicted reading and the two arithmetic constructs in first grade. The same was true for nonverbal abilities in preschool, with the exception that this construct did not predict phoneme awareness in preschool beyond language. This model had an excellent fit to the data, = 59.392, p = 353; RMSEA = 0.015 (90% CI = 0.000, 0.042); TLI = 0.997; SRMR = 0.023.

Figure 1. Structural equation path model showing the longitudinal relations between Time 1 preschool preliteracy and numeracy skills, and Time 2 first grade reading and arithmetic. Single-headed arrows between the latent variables are standardized regression weights. Double-headed arrows indicate correlations (covariances). * p < 0.05, ** p <0.01.

Next, we added phoneme awareness and number naming in first grade as additional predictors of reading, verbal arithmetic, and arithmetic fluency in first grade. This was done to see if phoneme awareness predicted reading, verbal arithmetic, and arithmetic fluency concurrently, and if there were indirect effects of phoneme awareness on these three outcomes through phoneme awareness in first grade. As shown in Figure 2, and partially in line with the phonological hypothesis, phoneme awareness in first grade did predict both reading and verbal arithmetic but not arithmetic fluency in first grade. In addition to phoneme awareness in first grade, both phoneme awareness and number-word knowledge in preschool predicted reading in first grade, and verbal arithmetic and number-word knowledge in preschool predicted verbal arithmetic in first grade. Only number-word knowledge in first grade predicted arithmetic fluency in first grade beyond all other potential predictors. The indirect effect of phoneme awareness in preschool on reading and verbal arithmetic was 0.086 (bootstrapped 95% CI = 0.001, 0.191) and 0.080 (bootstrapped 95% CI = 0.001, 0.173), respectively. However, it should also be noted that phoneme awareness in preschool had a direct effect on reading so that the total effect of PA in preschool on reading in first grade was β = 0.279 (95% CI = 0.098, 0.429). This model had the same excellent fit to the data as the full CFA explained above.

Figure 2. Structural equation path model showing the longitudinal relations between Time 1 preschool preliteracy and numeracy skills, and Time 2 first grade phoneme awareness, number word knowledge, reading, and arithmetic. Single-headed arrows between the latent variables are standardized regression weights. Double-headed arrows indicate correlations (covariances). * p < 0.05, ** p <0.01.

In both these models shown in Figures 1, 2, there was an unexpected negative path from preschool letter knowledge to either first-grade arithmetic fluency ( 1) or first-grade number-word knowledge ( Figure 2). These paths seemed to be caused by the high correlation between preschool letter knowledge and preschool number-word knowledge ( r = 0.731). Deleting number-word knowledge from the model resulted in these two paths becoming positive but nonsignificant ( β = 0.080, p = 0.462 for first grade arithmetic fluency in Figure 1 and β = 0.178, p = 0.078 for first grade number-word knowledge in Figure 2). It should however be mentioned that all of the eight latent predictor variables in Figure 2 were within the tolerances for commonly suggested thresholds for potential collinearity problems (e.g., all tolerances were above 0.2) and that the main results concerning the hypothesized relationships between phoneme awareness and arithmetic and reading did not change as a function of deleting the preschool number-word knowledge variable.

Discussion

This study has revealed several interesting findings that add to the literature about the relationship between phonological awareness and arithmetic. First, we did not find that phoneme awareness can predict the development of arithmetic from preschool to first grade when early reading, arithmetic, general language skills, and nonverbal abilities are controlled for. However, we did find that phoneme awareness is concurrently related to arithmetic in first grade and that there is an indirect effect of phoneme awareness in preschool on arithmetic in first grade via phoneme awareness in first grade. Moreover, the relationship between phoneme awareness and arithmetic is restricted to verbal arithmetic-simple arithmetic problems presented orally and calculated mentally-and not arithmetic fluency.

Phoneme Awareness in Preschool Does Not Predict the Development of Arithmetic Skills in First Grade

The finding that phoneme awareness in preschool does not predict the development of arithmetic skills in first grade when controlling for early reading and number skills, general language skills, and nonverbal abilities contrasts with the two other longitudinal studies in this area. Hecht et al. (2001) found a uniquely direct relationship between phonological awareness and the development of early arithmetic. However, that study did not control for nonverbal abilities, and the researchers employed a rather broad phonological processing construct that also integrated verbal short-term memory and rapid naming. Thus, these two factors can perhaps explain why they found a different result. Koponen et al. (2019) also reported that phonological awareness is a unique longitudinal predictor of the covariance between reading and arithmetic fluency. Similarly, they did not control for verbal or nonverbal intelligence, and this might explain the differences in our findings. Thus, our study overall employed stricter controls, in particular for previous skills, but also confounders such as verbal and nonverbal abilities. We also controlled for measurement error using latent variables. These factors together could explain why we found that phoneme awareness did not predict development, while other studies have come to the opposite conclusion.

An Indirect Effect of Phoneme Awareness in Preschool on Arithmetic in First Grade

While we did not find a direct effect of phoneme awareness in preschool on arithmetic development, we did observe an indirect effect from phoneme awareness preschool via phoneme awareness in first grade. Notably, our study is the first to examine such indirect effects. As for the size of this effect, the indirect effect implies that being one standard deviation above the mean in phoneme awareness in preschool is associated with being 0.08 standard deviations above the mean in verbal arithmetic at age six. Thus, the effect is rather limited in size. Language-related interventions have been suggested as a tool for ameliorating math problems. For instance, for language comprehension, Fuchs et al. (2020) recently demonstrated the effects of a vocabulary and language intervention on word problem-solving. If we had found a large indirect effect, one could perhaps suggest something similar for phonology-that number-related phoneme awareness interventions in preschool could ameliorate arithmetic difficulties. However, since the effect size was rather limited, the prospects for training effects are also rather limited.

Phoneme Awareness Is Related to Verbal Arithmetic but Not Arithmetic Fluency

Our study found that phoneme awareness was related to verbal arithmetic when children were asked to solve a single-digit addition or subtraction task without time limits, but not to single-digit arithmetic fluency tasks that were timed and answered on paper with no verbal response needed. No other studies have examined differential effects on arithmetic accuracy vs. fluency; the two previous longitudinal studies had both types of tasks mixed in a latent variable. Among the concurrent studies, one had fluency tasks and found a relationship ( Singer et al., 2019), and the other had accuracy tasks and found a relationship ( Vanbinst et al., 2020).

While efficient fact retrieval should be expected to contribute to success in both these task variations, there are some interesting differences in terms of the additional skill needed in each format. First, fluency tasks are timed and require faster fact retrieval and symbolic number-knowledge. In this process, Arabic-to-verbal translation-thought to evoke phonological processes-is likely an integral aspect. Second, the verbally presented math problems are connected to language in a broader sense than is the case with arithmetic fluency. In addition to calling for an accurate representation of number-word knowledge rather than Arabic digit knowledge, they might also evoke a greater degree of procedural strategies due to the working memory load since there is no permanent visual information. Third, the verbal arithmetic tasks were somewhat more complex, implying less fact retrieval and more procedural strategies for the more challenging items.

Thus, one reason that we found a relationship between verbal arithmetic and phoneme awareness could be that this task was presented orally to the children and that the children then used the phonological codes for a counting-based strategy to retrieve the answer. In contrast, the arithmetic fluency tasks were presented visually to the children. These children had just started to receive numeracy instruction and were unlikely to be able to use stored phonological information to retrieve the answers directly from long-term memory. Thus, they more likely used, for instance, strategies such as finger-counting or writing down dots to find the answers, and since the tasks were presented visually, the visual presentation format did perhaps not induce a lot of language processing. While confirmatory factor analyses indicated that these two constructs are best treated separately, the obvious similarities between the tasks call for further investigations of how distinguishable the two processes are over time and whether they evoke different arithmetic strategies.

As mentioned, the degree to which fact retrieval is established could determine the degree of reliance on phonological codes necessary for efficiency in the fluency tasks. Therefore, it cannot be ruled out that the outcome would have been different had the second assessment been carried out a year later.

Conclusion and Prospects for Future Studies

This study revealed that phonological awareness does not explain development in arithmetic, but that there is an indirect effect between phoneme awareness in preschool and arithmetic in first grade via phoneme awareness in first grade. This effect is, however, weak and restricted to verbal arithmetic accuracy and not fluency.

To rule out third variables that might have caused the detection of a relationship between phoneme awareness and arithmetic in other studies, we controlled for general language and nonverbal abilities. Future studies should aim to also include a working memory component as working memory has been established as an important aspect of both phonological processing and early arithmetic ( Peng et al., 2016).

We examined a sample of typically developing children and focused on correlations between continuous variables, and not on co-occurrences of categorical diagnoses. As pointed out by Krueger and Markon (2006), the term "comorbidity" could legitimately refer to either phenomenon. In light of this, an interesting question is whether these findings would have been replicated in a sample with children with reading disorders, arithmetic disorders, or both. As mentioned, Moll et al. (2015) found that only their comorbid group and the reading-disabled group had phonological awareness difficulties; those with only a math disorder did not. In general, both reading disorders and math disorders are defined using rather arbitrary cut-offs on continuously distributed variables. This perhaps explains these results, and predictors of individual differences may not be any different at the lower end of the distribution.

Furthermore, there is little evidence of a qualitative difference between those who have a learning disorder and those who are above the threshold ( Snowling and Melby-Lervåg, 2016). A recent study adopted a transdiagnostic approach to examine whether brain differences relate to cognitive difficulties in childhood ( Siugzdaite et al., 2020). The results indicated patterns suggesting that cognitive strengths and weaknesses cut across disorders and difficulties. According to Siugzdaite et al. (2020), "This stands in contrast to theories that specify a particular cognitive impairment as being the route to a particular diagnosed learning problem but is consistent with earlier ideas that developmental difficulties reflect complex patterns of associations rather than highly selective deficits" (p. 7). Thus, based on the continuous nature of reading and arithmetic, their correlation, and the limited support for qualitative differences between them, a similar pattern for children at the lower end of the distribution is perhaps likely. Future studies seeking to examine comorbidity might benefit from approaches where testable models can be used to identify the functional nature of disabilities and the need for intervention, as opposed to forming categories of atypical and typical development based on cut-off criteria ( Branum-Martin et al., 2013).

Data Availability Statement

The original contributions presented in the study are included in the article/ Supplementary Materials, further inquiries can be directed to the corresponding author.

Ethics Statement

The studies involving human participants were reviewed and approved by Norwegian centre for data research https://nsd.no/nsd/english/index.html. Written informed consent to participate in this study was provided by the participants' legal guardian/next of kin.

Author Contribuitons

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnhum.2020.577304/full#supplementary-material.

References

Bishop, D. V. (2009). Test for Reception of Grammar. Version 2. TROG-2 manual [Norsk version]. Bromma, Sweden: Pearson Assessment.
Google Scholar
Brigstocke, S., Moll, K., and Hulme, C. (2016). Test of Basic Arithmetic and Numeracy Skills. Oxford: Oxford University Press.

Dunn, L. M., and Dunn, D. M. (2009). The British Picture Vocabulary Scale. London: GL Assessment Limited.
Google Scholar
Lyster, S. A. H., and Horn, E. (2009). Test for Reception of Grammar-2nd edition. TROG II manual [Norwegian ed.]. Windsor, UK: GL Assessment.
Google Scholar
Muthén, L. K., and Muthén, B. O. (1998-2017). Mplus User's Guide, 8th Edn. Los Angeles, CA: Muthén and Muthén.
Google Scholar
Wechsler, D. (2012). Wechsler Preschool and Primary Scale of Intelligence, 4th Edn. Bloomington: Pearson.
Google Scholar

# data[].attributes.attachment_link.title
Comorbidity Between Math and Reading Problems: Is Phonological Processing a Mutual Factor?

# data[].attributes.attachment_link.description
There is a relationship between reading and math skills, as well as comorbidity between reading and math disorders. A mutual foundation for this comorbidity could be that the quality of phonological representations is important for both early reading and arithmetic. In this study, we examine this hypothesis in a sample traced longitudinally from preschool to first grade (N = 259).

# data[].attributes.headline
Orthographic mapping and phonological awareness

# data[].attributes.subject
Orthographic mapping and phonological awareness

# data[].attributes.body
This video details research that determined that phonemic awareness is not just a visual concept in which seeing the word repeatedly increased understanding, but rather that this is a visual-perceptual-spatial processing task that requires multiple inputs in order to deeply understand. This is another instance of why it is so important that, as teachers, we present information to students in various formats to cement their comprehension.

Gramenz, J. (n.d.). Orthographic mapping part i. Ms. Jane's Tutoring & Dyslexia Services; An Advanced Certified Tutor, Advocate, Consultant, Screener, & Professional Presenter. Retrieved September 14, 2021, from https://www.msjanestutoring.com/dyslexia-vlog/orthographic-mapping-part-i.

# data[].attributes.attachment_link.title
Orthographic Mapping Part I: How We Store Words For Proficient Reading

# data[].attributes.headline
Phonics Instruction For Middle and High School ELLs

# data[].attributes.subject
Phonics Instruction For Middle and High School ELLs

# data[].attributes.body
This article gives some great strategies that I can implement into practice in a middle or high school mathematics classroom. Some of the strategies mentioned including integrating phonics into the content area, helping bridge the language between the learners first language and English regarding vocabulary terms, finding appropriate materials that are more accessible, using music to aid in understanding, and creating games (Robertson, 2021). All of these practices are essential in ensuring that math material is as accessible to all students. This ensures equity within the classroom when all students have what they need to be successful and become fluent in the language of math.

Robertson, K. (2021, June 14). Phonics instruction for middle and high school ELLs. Colorín Colorado. Retrieved September 14, 2021, from https://www.colorincolorado.org/article/phonics-instruction-middle-and-high-school-ells.

# data[].attributes.attachment_link.title
Phonics Instruction for Middle and High School ELLs

# data[].attributes.attachment_link.description
I recently had the opportunity to teach summer school, and two of my sixth-grade students barely read at a first-grade level. This was very challenging for me, and I struggled to find the right way to address their need for phonics and comprehension instruction. Little by little, however, I began to…

# data[].attributes.headline
Listening and Reading Comprehension by Age (Stricht & James,1984)

# data[].attributes.subject
Listening and Reading Comprehension by Age (Stricht & James,1984)

# data[].attributes.body
Notice the wide gap between comprehensions, which is still there (although decreasing) during middle school. This is the reasoning behind why it is vital to continuing to encourage and develop oral language skills in the classroom. In middle grade mathematics this will be crucial to students as they approach algebra as they will need a good base mathematical vocabulary to understand the more complicated topics that await in high school.
This is also an example of a visual aid which seeks to further learning by supplementing oral language to foster connections in the material.

Stricht, T.G., & James, J.H. (1984). Listening and reading. In P.D. Pearson, R. Barr, M.L. Kamil, & P. Mosenthal (Eds.), Handbook of Reading Research (Vol. 1, pp. 293–317). White Plains, NY: Longman.

# data[].attributes.body
The five components that make up oral language are phonological skills, syntax, morphological skills, pragmatics, and semantics (Moats, 2010). Understanding the multiple facets of oral language can help in teaching students by focusing on vocabulary terms and ensuring that each student has opportunity to use them aloud.

Moats, L. C. (2010). Speech to print: Language essentials for teachers (2nd ed.). Baltimore, MD: Brookes Publishing.

# data[].attributes.body
This is a great example of how to incorporate the triad method into lessons in order to foster oral language in all students. You have one student make a statement, a second questions the statement, and then a third that then would summarize the content of the first and second students (Gilbert, 2015). This would allow the material to be more fully realized by asking students to talk about their thought processes. This could be used in my classroom to not only bolster understanding but also promote collaboration and use of native languages.

Gilbert, C. (2015, October 26). Conversations with a purpose. ZONES MATH - A MATH WORKSHOP MODEL FOR TEACHING BALANCED MATH. Retrieved September 13, 2021, from http://zonesmath.weebly.com/blog/conversations-with-a-purpose.

# data[].attributes.headline
The Importance of Teaching Vocabulary to ELs

# data[].attributes.subject
The Importance of Teaching Vocabulary to ELs

# data[].attributes.body
This breaks down why teaching math vocabulary to students is so important and gives several techniques as to how to implement good practices in the classroom through demonstration, allowing students to support each other through use of their native language(s), pre-teaching new vocabulary terms, and incorporating visual aids and graphic representations (Robertson, 2020).

This will be very important in my classroom as many children are not as developed in math as others and breaking down vocabulary terms and using multi modal ways of presenting these terms will make daunting vocabulary more accessible to children. This is especially true with students that are ELs as although they may understand the concept, they may not be able to effectively communicate these words into English. Hence, this is why it is vital that we empower children with the tools (in this case, mathematical vocabulary) in order to develop critical consumers of math.

Robertson, K. (2020, February 18). Math instruction for english language learners. Colorín Colorado. Retrieved September 14, 2021, from https://www.colorincolorado.org/article/math-instruction-english-language-learners.

# data[].attributes.attachment_link.title
Math Instruction for English Language Learners

# data[].attributes.attachment_link.description
Several years ago, I met a high school math teacher named Hillary Hansen who had just taught her first Basic Math course for English language learners (ELLs). She wanted so much to provide the students with the good foundation they needed, but she felt unable to engage them in her lessons. By the end of the year, she was exhausted and frustrated.That summer she had an opportunity to join a district Sheltered Instruction Observation Protocol (SIOP) cohort to receive professional development and support the needs of ELLs in content classes. She learned about the importance of language acquisition, building background knowledge, increasing student language production, and explicitly teaching academic language. She began the following school year with a new set of tools and a deeper understanding of the instructional scaffolding ELLs need in order to learn the content while also learning English.While Hillary still felt challenged and was working hard, the second year was much more successful for her and her students. She felt she was providing them with the foundation they needed not only to understand the mathematical concepts, but also to successfully interact within a math classroom in order to learn more advanced concepts.As a result of more effective instruction, her students:Understood the content better and were working together to find creative ways to learnIncreased their participation in discussions and learned how to use instructional supports in the classFelt more comfortable with math and asking questions to get the help they neededYou can learn more about some strategies that Hillary and other teachers recommend for teaching ELLs math in the information below, along with reflection tips and ideas for getting to know your students’ math backgrounds.ELL Math Resource CollectionThis resource collection features several ideas for teaching math to ELLs.Reflection QuestionsHow much math instruction have your ELLs received in their prior schooling?How much do their levels of math background vary?What are some of their strengths related to math?What are some of their challenges?What are some practical ways your ELLs are using math in their daily lives?What kinds of strategies have supported their success or engagement in other areas?What are some ways that collaboration or co-teaching might support your ELLs?What are some leadership opportunities you might give ELLs in pair work or small group work?Learn About Students’ Math BackgroundsTeaching ELLs: Tips for Math TeachersPrint PDF >It’s important to remember that ELLs’ math backgrounds may vary significantly. Some students may have had consistent schooling where they developed strong math skills. Others may have limited or no formal schooling, but may have practical skills related to time, money, or other daily topics. If possible, find out whether students have attended school regularly and what kind of experience they have had with math instruction and with math in general. Look for ways to tie instruction to student interests (like sports) and responsibilities at home that involve math, such as helping with family finances, working in family businesses, and holding other jobs. For example, students from migrant farmworker families may be using math related to agriculture, such as weighing or selling produce. Also keep in mind that students may use other numeric and measurement systems students, such as metric system, Celsius, military time, etc.In addition, Pennsylvania educators Sally Lopez and Erin Minick also encourage educators to:Look for leadership opportunities for all students in math classProvide students with opportunities to “teach” others, work together, and demonstrate what they know in multiple waysGroup students intentionally and with sensitivity, avoiding the designation of specific students as “needing more help” in their grouping, partner assignments, or placement in the classroom(Ms. Lopez and Ms. Minick conducted a book study focused on Teaching Math to Multilingual Students, which you can learn more about from their website.)Keep in mind that veteran educators of ELLs also stress the importance of getting to know students as a key part of supporting student success. For example, in this NCELA webinar, Dr. Haiwen Chu, a national expert on mathematics education for English Learners at WestEd, recalls that visiting his high school students’ neighborhoods provided valuable insights on their experiences and familiar topics.See additional ideas for getting to you know your ELLs in 10 Strategies for Building Relationships with ELLs.The Language of MathAlthough it is easy to assume that many ELLs will excel in math class because math is a “universal language” and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray. While computational skills may transfer, ELLs may still need support with language-focused tasks, such as solving word problems, learning (and using) math vocabulary, and writing explanations. All of these skills require a language proficiency that sometimes exceeds our expectations. We tend to think of mathematics as a subject that does not require a strong command of language. In reality, however, mathematical reasoning and problem solving are closely linked to language and rely upon a firm understanding of basic math vocabulary (Dale & Cuevas, 1992; Jarret, 1999), especially when considered along with the demands of college- and career-ready standards.Co-Teaching and CollaborationFor many educators, the challenge of bringing language and math instruction together is a relatively new one. ELL teachers who hadn’t taught content areas previously are now being asked to lead or support instruction in the math classroom, and many math teachers who don’t see themselves as language instructors are now responsible for providing effective math instruction to ELLs. When educators collaborate, however, they can better understand each other’s expertise and combine their expertise for the benefit of students.For example, when planning a lesson, a math teacher and an ELL teacher could collaborate to identify both the target content, such as key standards, concepts, or specific topics, as well as key vocabulary, academic language, and the language-focused tasks students will need to complete the lesson. These tasks might include writing out a proof, reading a math problem, explaining how to solve a problem, or writing out numbers as words. The ELL specialist could walk the math teacher through this process several times so that it becomes a more integrated part of the math teacher’s planning over time. After the lesson, the teachers could identify other areas of focus or clarification needed and share these insights with their colleagues.Another tool content teachers can use is language objectives, which help target the language demands of the lesson. You can learn more about them in the videos featuring Mr. Jesus Ortiz below.Teaching Academic VocabularyVocabulary instruction is essential for effective math instruction. Not only does it include teaching math-specific terms such as “percent” or “decimal,” but it also includes understanding the difference between the mathematical definition of a word and other definitions of that word, as well as understanding the concept a vocabulary word describes.The following example, used in a presentation by Dr. Judit Moschkovich of the University of California at Santa Cruz, underscores why vocabulary must be introduced within the context of the content (Moschkovich, 2008): In this problem, the student is instructed to “find x.” The student obviously knew the meaning of the word “find” because they “found” it on the page and circled it. The student even put a note on the page to help the teacher in locating the lost “x”. The student understood the meaning of “find” in one context, but not in the appropriate mathematical context.I recently helped a math teacher create a sheltered lesson, and I was surprised to find that there were some vocabulary words that I didn’t understand. My lack of familiarity with the words hindered my ability to do the math problem and gave me a deeper empathy for ELLs who struggle in the same way with vocabulary and comprehending math assignments.Following is a list of tips for explicitly teaching mathematical academic vocabulary.Teach key academic vocabularyAfter you identify your target content, identify the academic vocabulary words that students will need to understand that content. Some of these key phrases or new vocabulary words may also be helpful to pre-teach in small groups or mini lessons. Think about:Academic vocabulary (equilateral)Words and phrases needed to solve a problem (less than, given, arrange)Common words needed to solve word problems (train station, dolphin, favorite)Homophones (eight/ate, plane/plain, one/won, two/to/too, four/for, some/sum)Words that students are likely to encounter in other settings (formula, prove)Words that represent a numerical value necessary for solving the problem, such as measurement words (foot, feet, meter, pounds, kilometers)After you identify the vocabulary you wish to teach, use scaffolds for academic vocabulary, such as: Visuals: These can include visual representations of concepts, photos, and illustrations of key vocabulary words, posters, and number lines.Realia (real objects) and manipulatives: These might include hands-on objects related to a topic or small items, such as coins or small blocks for counting, sorting, weighing, etc.Graphic organizers: Graphic organizers are tools that allow students to organize information in a visual or hands-on way.Anchor charts: Anchor charts are posters that students and teachers co-create to highlight key concepts and vocabulary that students will need for reference.Offer students the chance to work with objects and images in order to master vocabulary. If there aren’t enough items for each student, use manipulatives on the overhead or posted throughout the classroom, and demonstrate the vocabulary in front of the students. For example, Hillary created a Math Word Wall that has three parts: key vocabulary, “in your own words” definitions, and a variety of ways to portray a function. For example, multiplication is portrayed by the following symbols: x, *, and ( ).In addition, it can be helpful to ask students which strategies support their learning. Students often can provide valuable information on what is working for them!For examples of how vocabulary can impact students’ understanding of math problems, see the related video with ESL teacher Anne Formato below.Teach words with multiple meaningsDemonstrate that vocabulary can have multiple meanings. Help students understand the different meanings of words, as well as how to use them correctly in a mathematical context. This is also a good opportunity for using visuals and realia. Examples include:YardPlotTableProofAreaEvenPlanePlaceTimesBillQuarterChangeDraw on students’ home languagesUse cognates, which are related words in two languages. English and Spanish have many cognates within academic terms, like fraction/fracción.Use translated words or phrases strategically. For example, New York University has published bilingual glossaries for several topics.Allow (and encourage!) students to use their languages in group work to solve problems, brainstorm, and review material. Students will understand material better if they explain it to another student, and the new student will benefit from hearing the explanation in their first language.Increasing Student Language ProductionThe language demands of math are complex. Working with an ELL specialist can be a good starting point for identifying some of the math demands of the lessons you are teaching, as well as for planning how to teach and scaffold those language demands.As I’ve worked with content area teachers in my district to develop Sheltered Instruction lessons and activities to enhance ELL learning, I’ve told them, “If a student doesn’t say it in your class, they’re never going to say it.” This is a bit dramatic, but it’s true to some extent. When students learn new vocabulary, the opportunity to use it must be presented in class, because students are unlikely to try it out on their own — especially academic words like “parallelogram” or “function”!Here are some tips to increase student-to-student interaction with academic language in the math classroom:Have students translate symbols into words, and write the sentence out. Hillary used this strategy to check students’ comprehension of problems before they solved them. For example, 3x + 4 = 16 would be written out, “Three times X plus four equals sixteen.” This helps students process the operations involved in the question and gives them an opportunity to think through how to solve it. It also gives students a chance to familiarize themselves with important vocabulary words. Writing out the answer to a problem is a very important skill to develop because many state math tests require a constructed response to questions.Create a “sentence frame” and post it on the board. Write the format of the sentence you would like students to use in discussion and then hold them accountable for using it. For example, “The answer is _______ degrees because it is a _________ triangle. (Learn more in our sentence frame and sentence stems strategy.)Have students share problem-solving strategies. This involves asking a simple question, such as, “Did anyone else get the answer in a different way?” Then allow enough wait time so students can think through how their problem-solving process was similar or different to the one offered.Allow students to discuss how they are thinking about math. This is a way of redirecting the lesson from teacher-to-student to student-to-student. For example a student might ask a question, “How do you know what kind of triangle it is?” Instead of the teacher answering and going to the board and pointing out the names and different triangles, the teacher can simply ask, “Does someone have an answer? Or “Would someone like to offer help to Mario?” Allow students to share how they think about the math concept and any tips they have for remembering the information.Incorporate writing activities, such as entrance/exit tickets or math journals. This is an excellent way for students to process what they’ve learned and what questions they still have. The journal or exit/entrance ticket prompt could start with simple prompts such as, “One thing I learned today...” “One thing I still don’t understand…” “One way I can get the help I need…” “The answer to this problem is…” “Something that helped me was...“Challenge students to create their own math problems. This can be a fun activity if students create a problem similar to the ones you have used in class and they exchange problems with a partner. By creating the problem and checking the answer they are reinforcing their own learning.Teach students how to read, interpret, and summarize data. Provide students with explicit, step-by-step instructions in how to read charts, graphs, tables, and infographics, as well as how to describe what they observe in visual representations of data.Model the kinds of writing students are expected to complete (e.g., written explanations). Students need to see and understand what they are expected to do and produce, such as explaining how they solved a math problem. This is a great opportunity for the “gradual release” strategy -- I do, we do, you do.Try AI tools. Consider using AI tools to rewrite math problems for students at early levels of proficiency or to generate math problems with familiar vocabulary that achieve your target objective. Bilingual Glossaries in Content AreasNew York University has published several bilingual glossaries in multiple languages that cover different content areas, including elementary math, middle school math, Algebra 1 and 2, Calculus, and related math terms.Reading and Understanding Written Math ProblemsWritten word problems present a unique challenge to ELL students and teachers alike. In Reading and Understanding Written Math Problems, Brenda Krick-Morales writes, “Word problems in mathematics often pose a challenge because they require that students read and comprehend the text of the problem, identify the question that needs to be answered, and finally create and solve a numerical equation — ELLs who have had formal education in their home countries generally do not have mathematical difficulties; hence, their struggles begin when they encounter word problems in a second language that they have not yet mastered” (Bernardo, 2005).Teacher Xiao-lin Yin-Croft has encountered this pattern in her classroom of bilingual Chinese students in San Francisco. She has developed a very creative way to use her students’ background knowledge of math as a stepping stone for other language learning. She does this by accelerating math instruction at the beginning of the school year and then building on what students have learned in math in reading and other content areas. In Building Bridges for the Future, Xiao-lin explains her strategy:First, we read math word problems; I demonstrate the logical thinking process while translating words into pictures and, finally, into number sentences. Soon, they start to explain their own thinking after reading complicated word problems that involve several steps. They correct each other and argue about which number sentences they should use to arrive at the correct final results. As they sharpen their math skills, I capitalize on their enthusiasm to teach them how to extract the most important information from texts, and move them toward the oral and reading fluency they need to understand and discuss more challenging texts.Even if you aren’t accelerating math instruction, however, there are a number of ways to help students master word problems. Krick-Morales offers suggestions in the previously mentioned article, such as explicit instruction of key vocabulary, daily practice of problem solving, repeated readings of the word problem together as a class, and hands-on activities such as movement, experiments, or drawing to help students comprehend the problem. As students become more familiar with math vocabulary, they will be able to solve problems more easily.Identify key background knowledge in math problemsIt’s important to identify what kinds of background knowledge students will need to understand a math problem. For example, Ms. Lopez and Ms. Minick used this activity with a conference session: Ms. Lopez chose four math problems from her son’s math textbook and asked session attendees to brainstorm the kinds of background knowledge students might have questions about, some of which were relevant to solving the problem and some of which weren’t. Here are some things that attendees came up with:A math problem about a coupon and sales tax at a dog groomer: What is a dog groomer? Why do people take their dogs to get groomed? What is sales tax? What is a coupon? What is “deluxe”? What is an estimate and a discount?A math problem about people receiving a voucher for a free water bottle at a football game: Is football different from soccer? What is a voucher? Does the water bottle come with water inside it?A math problem about a fishing pole on sale: What is “percent off”? What is a sale? What is “regularly priced”? What steps have to be followed to solve this problem?A math problem about calculating a tip in a restaurant: What does “leave a tip” mean? What is a tip? Does the fact that there are four sisters (Emilia and three sisters) impact the outcome of the problem? Why aren’t they using an app like Venmo?As seen from the example above, this can be a powerful area in which math teachers and ELL teachers can collaborate.The Role of Background KnowledgeWhen it comes to background knowledge that impacts students’ understanding of new concepts, students may have relevant experiences to what you are teaching that just need to be drawn out (as in the vignette below). In other cases, students may not have the needed background knowledge. My colleague Hillary found that sometimes her students would get “lost” in a problem simply because they didn’t understand the context. Following are some tips to help in tapping into and building background knowledge of students.Identify connections to your students’ experiences. When planning the lesson, consider whether they might have relevant experiences, and as you introduce the lesson, find out what they know, giving them an opportunity to make connections. They may come up with connections you never had imagined!Modify the linguistic complexity of language and rephrase math problems. Students will understand the problem better if it is stated in shorter sentences and in language they understand. This may be an area in which you wish to experiment with AI tools that can rewrite problems.Guide students to cross out the unnecessary vocabulary in word problems. Doing so allows students to focus on the math function required. For example, one problem Hillary’s students came across referred to a “school assembly.” Even though the meaning of that phrase wasn’t important in the solving of the math problem, students didn’t know it wasn’t important, and the lack of understanding contributed to their confusion.Build knowledge from real world examples. Try to reinforce concepts with examples that students can picture and talk students through the situation. For example, if you need to paint a room, you need to know how much area will be covered so that you know how much paint to buy. Look for familiar ideas or props that can be used to engage students such as recipes, news stories about the economy, or discussion of personal spending habits.Use manipulatives purposefully. This is important at all grade levels. Hillary has found math cubes to be very useful in having students represent the numbers in the problems and then manipulate the cubes to get the answer. She used the cubes and the terms “hot” and “cold” numbers when teaching with the concept of negative numbers. Students would use the red cubes as “hot” or positive numbers and the blue cubes as “cold” or negative numbers. As students laid out the number of hot cubes and cold cubes represented, they could easily see if the answer would be a positive or negative number by which color had the most cubes. A problem such as -2 + 1 = -1 would look like this: The student then removed pairs of cubes — one red, one blue — until they could no longer remove any blocks. The remaining blocks represent the answer. Classroom vignette: A moment of humilityA middle school math teacher was trying to teach points on a grid and vertical/horizontal axis to her ESL students, three boys from Vietnam. She tried many approaches, including visuals, games, simulations, and manipulatives, and was just about to give up after very little progress. Suddenly, a student said, “Oh, I know — same same you drive a boat!” (“The same as you drive a boat!”) It turns out the child had helped his uncle use the stars to navigate a raft when they were fleeing Vietnam and trying to reach refugee camps in Thailand. He explained the concept to the other boys in Vietnamese with gestures and drawings on the board, who knew exactly what he was talking about. The teacher reflected, “I felt so humble. These boys knew a lot more about math than I had given them credit for. I had failed to activate their prior knowledge, and in the process had almost given up.“Source: This vignette is presented in Chapter 4 of Special Education Considerations for English Learners: Delivering a Continuum of Services. Using TechnologyTechnology can also be a powerful tool in math instruction for ELLs. Here are some ways you can play with technology in a math lesson:Look for educational resources that accompany your school’s technology tools and programs. There may be online or software programs available for teachers. For teachers who have an electronic whiteboard in the classroom, there are many resources available on links that can be easily accessed and brought into the classroom.Find out what’s available online. Both PBS Learning Media and Vital NY (Video Teaching and Learning for NYS Educators) on Teachers’ Domain offers an online library of free public media resources. Teachers’ Domain resources include video and audio segments, Flash interactives, images, documents, lesson plans for teachers, and student-oriented activities. (Free registration required.)If students are using a calculator or other device, ensure that they learn how to use it. Based on background and prior educational experience, students might not be familiar with how to use a calculator, graphing calculator, or math apps on devices such as phones or tablets. Give students a chance to practice solving problems with their calculators once you have reviewed different functions. Texas Instruments offers numerous activities and product tutorials in their educational materials.Learn more about how to use AI tools in math. AI tools can be used to generate, differentiate, or rewrite a math problem; create a summary of a lesson plan; and many other tasks. Keep in mind that AI-generated content always need a careful review and edit. (See more in Using AI in Math from Edutopia.)Look for interactive games that offer students a chance to practice their mathematical skills. Nintendo DS has an educational game called Brain Age, which is not language specific. The game provides excellent mathematical training for numbers and tracks results, showing student improvement over time.Books About Math and MathematiciansIn addition, there are lots of books about math and mathematicians that you can incorporate into your classroom and classroom library. You can see a wide range of titles for different ages in our Math Resource Collection.Closing ThoughtsEven if it doesn’t come easily at first, there are ways to get ELLs excited about math. By keeping their language skills and needs in mind when planning mathematical instruction (and by helping your colleagues do the same), you will be taking important steps in helping students master mathematical concepts and skills. You never know — your students may be the next generation of economists, rocket scientists, and math teachers just waiting for the tools they need! Note: I’d like to thank my Minnesota colleagues, Hillary Hansen at Burnsville Senior High School and Kim Olson at Hidden Valley Elementary, as well as Xiao-lin Yin-Croft from San Francisco, for providing many of the helpful math teaching tips in this article. It is inspiring to know that there are talented, creative teachers who are always finding better ways to teach and are willing to share the knowledge.Related VideosHow vocabulary can get in the way of solving a word problemELL Specialist Anne Formato talks about the trouble a student had in understanding a written math problem based on a vocabulary word. Teacher Alejandra Rojas: Why cognates are your “best friend” in math classElementary teacher Alejandra Rojas, who teaches in a dual-immersion program, explains why cognates support vocabulary development in her Spanish-language math class. Teacher Alejandra Rojas: When bilingual students translate for their parentsAlejandra Rojas, a teacher at a dual-immersion school in Arlington, VA, shares some of her experiences as a child and teenager translating for her mother. How to Write Language Objectives: Tips for ELL EducatorsIn this video from Syracuse, NY, Jesus Ortiz, a bilingual teacher, learns how to write a language objective from Areli Schermerhorn, a peer evaluator with expertise in ELL and bilingual education. Teacher Observation Cycle: A Math Lesson on PictographsThis video highlights Mr. Ortiz’s math lesson on pictographs. While the main topic of the video is teacher evaluation, numerous best practices and strategies related to math instruction for ELLs are mentioned in the video. Recommended ResourcesNCELA Teaching Practice Brief: Math (A related podcast is available as well)Supporting English Learners in Math and Science: Effective Instructional Practices and Examples from NCELA’s Teaching Briefs PBS Learning Media: Math LessonsEngageNY: Common Core and MathCommon Core Math and ELLs: Blog PostsDiscovery Education: PuzzlemakerCatherine Snow: Word Generation SMART Notebook Lesson ActivitiesTranslating Word ProblemsColorín Colorado Webcast: English Language Learners in Middle and High School

# data[].attributes.body
Vocabulary can be a powerful tool for students to successfully engage in the discourse of mathematics. One way students can participate in vocabulary instruction is by the inclusion of a word wall. This can be done through a variety of means, in this pictorial representation, it is a educator curated word wall. However, student participation is fundamental to a social-constructivist pedagogy and elicits better recall for students when they are able to actively participate in the creation of a word wall (Bruyere, 2019). Another way to facilitate this tactic that is even more inclusive, would be to have each student have their own, small word wall within their mathematics notebook in which they add words to it with small post it notes as they encounter unfamiliar words (Uscianowski, 2019).

Bruyère, J. (2019, April 22). Building a better word wall. Edutopia. Retrieved November 4, 2021, from https://www.edutopia.org/article/building-better-word-wall.

Uscianowski, C. (2019, June 28). Use these 5 actionable strategies today to grow your students’ math vocabulary. Luminous Learning. Retrieved November 4, 2021, from https://luminouslearning.com/blogs/sped-math/teaching-math-vocabulary

# data[].attributes.attachment_caption
Urquhart, V. (2009). Using Writing in Mathematics to Deepen Student Learning. Retrieved November 15, 2021, from https://files.eric.ed.gov/fulltext/ED544239.pdf

# data[].attributes.body
This chart could be a fantastic resource to explore inside of a middle grades classroom, perhaps it should even be part of an anchor chart. Clearly defining and explaining what each of the terms is asking students to do is imperative to their writing  success, especially when it pertains to standardized testing. The referenced booklet also contains explicit ways in which to teach these new vocabulary terms and argues that "writing fosters community in a classroom and, because writing is a social act, it is a vehicle for students to learn more about themselves and others" (Urquhart, 2009, p. 6).

# data[].attributes.body
Students learn vocabulary both directly and indirectly through exposure in conversations both inside and outside of the classroom setting. Ensuring that all students within the classroom are privy to " instruction in specific words that are important to students’ content learning or understanding of a particular text" and "teaching students more general word-learning strategies that they can apply to a variety of words, such as analyzing parts of words (e.g., root words)" is imperative to granting access to those words to all (Reading Rockets, n.d.). Creating an equitable classroom requires educators to explicitly teach the pertinent discourse vocabulary in palatable ways to students.

Reading Rockets. (n.d.). Vocabulary. Retrieved November 4, 2021, from https://www.readingrockets.org/teaching/reading-basics/vocabulary

# data[].attributes.body
High-low books are a great way to engage students in math literacy while these students are still learning and working on phonics. "Effective high/low books provide supports for struggling readers, including illustrations to support the text, carefully chosen vocabulary, simple sentences, compelling stories, and characters that interest the reader" (Sublett, 2020, para. 3). These books are high interest and low difficulty which lends itself to being able to engage students while keeping the text simple and easier to read. Just because a student may not speak English well does not necessarily correlate to mathematical insufficiencies. This would allow me to engage with my students and be aware of where they are and what they know in my content area. This website on reading rockets has links to many high low books that could be used as a great resources to ensure equitability to all students.

Submitted by S.Sublett (not verified) on November 8. (2020, October 15). High/low books for children. Reading Rockets. Retrieved September 28, 2021, from https://www.readingrockets.org/article/highlow-books-children.