Larry Ferlazzo’s Websites of the Day…

…For Teaching ELL, ESL, & EFL

July 14, 2018
by Larry Ferlazzo

Guest Post: In Math, 2 is company and 3 is never a crowd


Editor’s Note: This is the sixth in a series of guest posts that will be appearing on this blog about teaching math to English Language Learners.  I’ll be posting them over the next few weeks, and adding each one to The Best Resources For Teaching Common Core Math To English Language Learners.

The first in the series was “Speaking of Math: It’s time to talk in class” by Alycia Owen.

The second was “Support Reading, Support Mathematics Understanding” by Cindy Garcia.

The third was  Teaching Math To English Language Learners by Hannah Davis.

The fourth was Supporting ELLs in Math Instruction by Nicholas Pesola

The fifth was Quick tips for making ELL students comfortable in the Math Classroom by Sarah Peterson


Today’s post is by Alicia Fisher

Alicia Fisher began her teaching career in Indianapolis 8 years ago.  She has taught Kinder, third and fourth grades.  She has been teaching in Maryland for the last 3 years. Her passion is Math/Science and integrated lesson planning. She has one amazing daughter. Alicia loves cooking and gardening when not teaching at school. 


Eight years ago, I moved from the financial industry to the education sector.  For the last 3 years, I’ve had the opportunity to teach Math and Science to fourth grade students, many of whom are English Language Learners and Special Education students.  I’ve found that no matter the content, social-emotional learning is crucial in the classroom and essential for the ELL student. Social learning theory proposes that people learn through observation, imitation and modeling.  As educators, we know this to be true. We model the behaviors we want from our students. We model different ways to solve problems. When students don’t know how to respond or what to do, they look to their peers and/or imitate what they have observed.

In our classrooms, we are fortunate enough to have four adults; a general educator, a special educator, a paraprofessional and an ISEA (Itinerant Special Education Assistant).  This gives us the ability, after whole-group instruction, to break into several small groups with targeted direct- teaching. In schools with less adults per room, I would suggest three small groups/stations with at least one being independently run by the students themselves.  Whether you are a proponent for homogenous or heterogeneous grouping, we have found that AT LEAST two ELL students per group helps facilitate risk-taking and more active participation. To have a group of solely English language learners creates an environment where some introverted students sit back and let one or two students do all the talking and hence, all the learning.

Once your groups are established, it’s time to check the M.A.I.L. !


  • Meaningful


ELL students need to have meaningful content.  Link lessons to real life situations and interests.  Simple math tasks such as counting and graphing can be used to explore student preferences and point out cultural differences.  This provides an opportunity to have an inclusive discussion that enhances the classroom environment. Students may also have an opportunity to teach counting in a different language.  Posting visuals like Spanish counting cards is helpful to this end. For older children, incorporating games and sports increases engagement and affords opportunities to explore diversity in the classroom.

  • Applicable

Students need to see the application of lessons to their lives.  Alignment to Common Core Standards is important, as is the need for students to understand why they are learning specific content.  For primary students, saving an allowance to purchase a favorite toy requires addition and subtraction knowledge and money sense. The same is true for intermediate students with an entrepreneurial spirit.  Starting a business, even as simple as a lemonade stand, requires multiplication and division skills. The ability to calculate the price of a prom dress that is discounted 40% builds a connection between the classroom and everyday life.

  • Integrated

This is my favorite.  Our math class is NOT just a math class.  In order for ELL students to successfully gain English literacy they must have many and varied opportunities to engage with language, both English and their own.   Incorporating picture books and writing has proven to be a successful strategy in our math classroom. For any given topic, Marilyn Burns has a picture book to compliment the lesson.  The Greedy Triangle teaches about geometry and self-acceptance.  Spaghetti and Meatballs for All teaches about problem solving, multiplication and division and inclusion while also offering an opportunity to discuss customs and international foods.  Of course there are a plethora of books related to math concepts that can be used. Even fourth graders still love to be read to.

E.M. Forster is credited as saying, “How can I know what I think until I see what I say?”  Just as important as reading, writing for the ELL student reinforces patterns and language acquisition.  Many classrooms are full of reluctant writers. In the ELA classroom you may even hear groans when the teacher asks students to take out their journals.  In the Math/Science classroom there is a different response which makes integration so important (and so easy – in my experience).

The trick is BEFORE.  BEFORE we complete the Math problems, BEFORE we do the Science investigation, BEFORE we play the fraction game…you have to write in your journal.  Unlock the prompt in the word problem. What is the problem asking you to do? What strategies are you going to use? Write those down, and while you’re at it, write our objective as the title of the page.  When students are being scientists, they naturally write what they expect to happen and what really did happen. I’m currently teaching summer school, and my content is writing. During the first week of class, after all students had written 5 pages of notes and reflections, the teacher of robotics asked them what they had learned in their other class and the students answered with an emphatic ‘Science!’  She continued questioning them and said, ‘Haven’t you been writing?’ and they replied, ‘Oh yeah, that too’ as an afterthought. I was so pleased! Since I am ‘distracting’ them with STEM activities, they don’t even realize the extensive writing they are accomplishing.

  • Lasting

Lastly, see what I did there?  Having a student-centered approach to language acquisition and learning will have a lasting effect.  Teaching content with materials and strategies to meet varied learning styles helps students navigate developmental and learning challenges.  As I previously mentioned, our math stations always include a center with manipulatives, a center focused on reteaching (RTI) or extensions for SpEd students, and an independent center focused on using technology.  These centers are specifically chosen to aid the students that need tactile lessons, need additional time or content, and the students that would rather work alone using the computer. It is of note that even the students that prefer the computers enjoy sharing their work with their group.  Learning is social. Encourage group work and observe how students rise and help each other. You’ll discover strengths in your students and joy in your classroom.

July 13, 2018
by Larry Ferlazzo

Guest Post: Quick tips for making ELL students comfortable in the Math Classroom


Editor’s Note: This is the fifth in a series of guest posts that will be appearing on this blog about teaching math to English Language Learners.  I’ll be posting them over the next few weeks, and adding each one to The Best Resources For Teaching Common Core Math To English Language Learners.

The first in the series was “Speaking of Math: It’s time to talk in class” by Alycia Owen.

The second was “Support Reading, Support Mathematics Understanding” by Cindy Garcia.

The third was  Teaching Math To English Language Learners by Hannah Davis.

The fourth was Supporting ELLs in Math Instruction by Nicholas Pesola

Today’s post is by Sarah Peterson

Sarah Peterson is a mathematics teacher for 10 years in the New York City Public Schools.  She has taught Algebra 1 and Geometry to the ELL and SPed population throughout her career.  Sarah can be reached via email,


When I tell someone outside of Teacher World that I teach math to English Language Learners I continuously hear the same two responses. The first is “You must speak Spanish ” Well the truth is, I don’t. Not even after six years of taking it in school, and teaching math to ELL’s for ten years. Even if I did, it wouldn’t help me too much because I have students from Ghana, Yemen, Albania and Bangladesh in the room. The second is  “Well isn’t math just numbers, the universal language?” I wish this was true. But numbers in Arabic look different, many countries use a comma as a decimal separator, and most math problems these days are contextual. Word problems full of non content specific vocabulary words. What’s an ELL math teacher to do?

Have compassion

Put yourself in their shoes. To keep me humble I will have students explain a math problem to the class and in their native language. Woah, to hear the slope formula explained in French is a real eye opener. Their brains are overloaded all day with all the new content and learning the language. Give them more time to think and formulate an answer.

Get them talking

Data shows that ELL talk much less in class than their non ELL peers. I break the ice in the beginning of the year by reading a math problem that I put into Google Translate to one of the languages a student in the class speaks. By the time I am done they are laughing so hard, and teasing me about my pronunciation and accent. It shows them we are all learning a new language together and it’s okay if it doesn’t sound perfect every time we speak.

I create a list of sentence starters that the students can use when answering or posing a question. The sentence starters are posted in the classroom and are written in the students notebooks for reference. Some examples are: I agree with you because…., another strategy that can be used is ……, I can connect this to when we learned about…. Until they have the confidence to speak on their own this is very helpful to them.

Modify your speech

When I was in my Masters program in Secondary Math Education, there were no ELL math classes offered. So I had to find my own way. I speak slowly and use more wait time than I would with a native English speaking class. I repeat and paraphrase throughout the lesson. Often having the students paraphrase what I just taught.  But I do not shy away from using the same math vocabulary as I would for native English speakers, I have the students use the vocabulary when answering questions, and they have a vocabulary section in their notebook to reference. I gesture throughout the lesson and reference pictures, graphs, examples whenever possible.


It’s easy to be overwhelmed by the seemingly endless hurdles your students face. But by taking the time to put them at ease in your classroom, you will see great gains in their understanding of the English language and mathematics.


June 15, 2018
by Larry Ferlazzo
1 Comment

Guest Post: Supporting ELLs In Math Instruction

Editor’s Note: This is the third in a series of guest posts that will be appearing on this blog about teaching math to English Language Learners.  I’ll be posting them over the next few weeks, and adding each one to The Best Resources For Teaching Common Core Math To English Language Learners.

The first in the series was “Speaking of Math: It’s time to talk in class” by Alycia Owen.

The second was “Support Reading, Support Mathematics Understanding” by Cindy Garcia.

The third was  Teaching Math To English Language Learners by Hannah Davis.

Today’s post is by Nicholas Pesola


Nicholas Pesola has been a mathematics teacher for 14 years at Marble Hill School for International Studies, a public school in the Bronx, NY.   He has taught all levels of high school mathematics to native speakers of English and ELL alike, and currently teaches Algebra 1 to ELL and SpEd populations, and AP Calculus.  If you have questions or would like to open a dialogue, he can be reached at


My work has run the gamut of teaching experiences, but a fair amount focuses on a 9th grade math class with our lowest level ELL population. I have often found the pursuit of professional development in teaching math to ELL’s frustrating because I find colleagues across the city reluctant to join me in this pursuit, and often the workshops I attend are geared towards the idea that language barrier is our only issue.  This strikes me as disingenuous when my classroom contains many Students with Interrupted Formal Education (SIFE) and those who have not been diagnosed for Special Ed needs for various reasons. It is difficult to find the right support and guidance for teaching mathematics to this ELL population.

For teachers who feel the pressure of high-stakes testing and expectant administrations, pushing a Common Core curriculum on a beginner ELL population only to have less than 30% of your class pass the state test, I would like to argue that starting the year teaching such a curriculum can have deleterious effects on their math education.  Pushing students prematurely down this path creates greater holes in their conceptual understanding, with residual academic and psychological effects as they move from course to course, teacher to teacher. Even if you are extremely thorough, you are likely to find the class’ progress stalls for long periods of time. There are too many variables your new ELL class enter with: Different social and academic cultures, different years of experience with any academic subjects, different exposure to the English language and literacy in any language, on top of the general diversity of any classroom.  The teacher needs to control some variables first before any substantial learning can begin, and this can take the majority of the first semester.

I would argue that we should see ourselves foremost as facilitators of the transition to learning at our grade level in this first semester, and this is the general blue-print I propose to get there:  Consider a 3-Unit structure where Unit 1 sets the culture, Unit 2 practices the culture on an essential prerequisite topic, and Unit 3 applies that topic at the next level to support their transition into the core curriculum.

Unit 1 is most essential and has the most to achieve in my opinion, so I would suggest taking as much time as needed.  Here I have been heavily influenced by the work of Jo Boaler at Stanford University, and much of her literature and resources can be found at  If you have a grade team, I see this kind of unit as a worthy push for all classes. These are the main goals:

  • Create an environment of inclusion for all ethnicities, languages, and levels in your classroom.
  • Develop ‘Our Math Culture,’ which emphasizes growth mindset and rewards exploration and collaboration, and helps students to transition to academics in your school.
  • Establish structures and protocols for soft skills that students can use to enhance their own learning and literacy for their entire academic experience.

This includes numeracy games and puzzles that develop the language of numbers and operations for basic classroom discussion. One such game I call ‘Dib Dub,’ where you select a digit for ‘Dib’ and a second digit for ‘Dub.’  Students take turns counting, but when a number is a multiple of those digits or contains the digits, the student substitutes the words Dib and Dub accordingly. If your classroom has multiple languages, sharing words in different languages becomes a great way to ease all students in the majority and minority into an inclusive classroom, and everyone feels their background is valued when the class repeats their language.  Days are devoted to math puzzles that are open-ended, have a ‘low floor / high ceiling’ to engage and give all levels a chance to contribute. One such activity is ‘1-2-3-4’ or ‘Four 4’s’, where groups seek ways to combine these numbers in full arithmetic expressions that equal all numbers from in an arbitrary range. There are many ways to extend the learning to multiple sessions and bring about more discoveries with each puzzle, illustrating and discussing fundamentals of pre-algebra.  Every classroom structure and protocol should be repeated and applied so students can see the value added to their learning experience, whether it be group interactions and presentations, or helping them to develop notebooks that improve their literacy and study habits. For the latter, I have the students use color-coding, and develop activities specifically pushing them to reference and develop literacy with their own notes, otherwise notebooks become a mindless classroom procedure that students do without reflecting or studying.  These are just a few ideas to address Unit 1 goals.

Unit 2 should be a topic selected from previous years that is fundamental to the current grade level and course, but also represents a stretch for the students.  Lately, I have been developing a unit on ratios, fractions, and percentages – a widely misunderstood topic that adds a lot of fundamental depth to numeracy and concepts like Rate of Change in Algebra 1.  My goals for this unit are to:

  • Explicitly demonstrate Unit structure and the purpose of each phase in their Unit learning process.
  • Reinforce ‘Our Math Culture’ with group and individual structures directed towards more specific math goals.
  • Develop depth of understanding and application of prerequisite math concepts.

Class activities and discussion still center around basic math vocabulary, but generally I keep the English literacy load as light as possible.  I lean towards math without words, with situations that are as visual and observational as possible.

Unit 3 begins to expose students to the core curriculum concepts.  My progression from my Unit 2 is currently Proportional Relationships, and we begin to use variables in meaningful ways.  The content directly relates to Unit 2 so that the same examples can be recalled for a deeper dive. Ideally, your Unit 3 allows you to access grade-level material without placing high literacy demands on the students yet.  They gain experience with minimal language-based confusion.

By the time you are three units in, you will find the class making significant leaps in their overall abilities across subject areas, so that they are ready to engage in grade-level material and make better sense of it.  The patience will have paid off because they can learn quicker with fewer misconceptions, and they will have the confidence to take on bigger challenges. For teachers who are afraid of investing so much of the year to off-curriculum learning, I would consider that a smaller amount of the course learned effectively will ultimately lead to better results and math futures.

June 14, 2018
by Larry Ferlazzo

Guest Post: Teaching Math To English Language Learners

Editor’s Note: This is the third in a series of guest posts that will be appearing on this blog about teaching math to English Language Learners.  I’ll be posting them over the next few weeks, and adding each one to The Best Resources For Teaching Common Core Math To English Language Learners.

The first in the series was “Speaking of Math: It’s time to talk in class” by Alycia Owen.

The second was “Support Reading, Support Mathematics Understanding” by Cindy Garcia.

Today’s post is written by Hannah Davis

Hannah Davis has spent the last four years teaching high school math exclusively to ESL students in the Newcomer Academy at Connally High School in Pflugerville ISD, outside of Austin, Texas. Her students were in their first two years of schooling in the US and came in with a variety of languages and math ability. Here are some of the things that really helped her as a teacher and helped her students.


Vocabulary is such a large part of mathematics and understanding the vocabulary is key to passing the Texas State STAAR Algebra test that students are required to take at the end of their first year in High School. The two strategies I used that substantially helped my students learn the vocabulary were choral reading and including visuals on anchor charts or word walls. We started each day by choral reading the pertinent vocabulary for that day’s lesson. Students would track the text and listen as I read the 3-8 words for the day. We could then use these words to predict their meanings, recognise cognates, or review from the previous lesson. After listening to me, students would then read the word aloud at least twice. Doing the whole group choral reading first helped the students feel more confident when they used the words later on in partner talk or during a quick write because they had already read, listened to, and spoken the words.  I added choral reading for vocabulary in this last year and saw a dramatic improvement in student’s use and understanding of vocabulary. It only took a few minutes each lesson and the students also really enjoyed it, often repeating the phrases like “same letter, same exponent” or “writing is thinking” to me in the hall in following years. Choralling is one of my and my students favorite strategies. Using the choraling strategy and including visuals with all anchor charts really helped my students over the years.



I used visuals as much as possible in my presentations and on the walls in my classroom. For each unit or topic, I created an anchor chart and corresponding vocabulary words with matching visual images for each. I also used visuals for non content related instructions so students could concentrate on the content and not get lost in the directions. For example, if the students required scissors or colored pencils for an activity or lesson, then those images would be included in presentations. For students struggling with solving equations, I included  pictures of scales to represent equations and for SIFE (students with interrupted formal education), I would often draw out an equations with shapes to represent variables and tally marks to represent the numbers.

3x + 5 = 3   may become ⍰⍰⍰ + IIIII = III

I love Noun Project for visuals –  there is a free version and an educator account is $19.99 a year, I especially love the add-on for Google Slides.



As much as possible try and model the mathematics.  I use ‘Hand on Equations’ with scales and pawns and I created a lifesize coordinate grid with tape in the classroom which we could use when students are learning to plot points and graph functions. We used ‘clothesline math’ examples using index card, strings and magnets  to solve equations. Human number lines and many more. Hands on activities with movement work so well with our English Language Learners.

Total Physical Response

Using gestures really helped my students and having them repeat the words with actions helped them remember what they learned. It also creates a fun and engaging atmosphere in the classroom. For example, my students and I would never say the words “above” or “numerator” without also making the actions.  Or we would trace shapes, like a parabola as we said the word. These total physical responses really helped the students understand the lessons because they physically connected with the concepts.



Students enjoyed using DESMOS especially for graphing activities. There are so many great activities included and many of them are editable so it’s easy to take an activity and customise by simplifying the English to make it more accessible to ELs. I love being able to easily create digital card sorts in DESMOS as another great way for students to practice their vocabulary. An additional great resource for math is the Khan Academy website because it has excellent videos and practice problems. I would often assign videos to students who want extra practice at home or in tutorials. The entire Khan Academy website is available in Turkish , German, Norwegian, Polish , Hindi, Bangla , Georgian , Spanish, Portuguese, French and the majority or videos have subtitles in many more languages so students can read along in their native language as they listen to the video in English.



A large percentage of my students came with low numeracy skills. Everyday for our warm up students complete a five minute warm up from Numeracy Ninja’s free program with printable workbooks, powerpoints with timers etc. Students compete against themselves to get to the next level “ninja belt.” This has been a huge hit in my class and we have a large display with certificates celebrating their growth at least every 9 week period.  This resource is also great during our intervention period where students may do peer tutoring with other students that had not had schooling. I print workbooks aimed at K-4th grade and students work with those manipulatives and multiplication skills for students to work on outside of the regular class time.

Color Coding

I color code as much as possible and it is especially helpful with my lower learners. For example,  I assign red to ‘X’ and blue to ‘Y’ for the year so the horizontal x-axis will be red and the Vertical Y-axis will be blue this goes for Domain and Range values and can be referred to all year long. When problems involved multiple steps I would color code each step so that students could clearly see which order to perform the steps.


I have many students who may be shy to participate and speak English in front of their peers. My aim is to have as many questions as possible answered by all students either using ABCD response cards, whiteboard, or gestures that we created as a class. For other questions I rely on a spinning online name picker or popsicle sticks to randomise students in a transparent way. I also use a ticket system where students earn tickets by reading objectives, helping each other, etc.

Make It Visual – Make It Collaborative – Make It Meaningful – Make It Hands On – Talk Slowly and Repeat

My favourite resources: You Cubed, Khan Academy, Desmos, Math Equals Love, Twitter, Kahoot, Quizizz, TalkingPts (a multilanguage remind), Stanford’s University Mathematical Mindset Course (with English and Spanish subtitles), Noun Project, Numeracy Ninjas

June 5, 2018
by Larry Ferlazzo

Guest Post: “Support Reading, Support Mathematics Understanding”


Editor’s Note: This is the second in a series of guest posts that will be appearing on this blog about teaching math to English Language Learners.  I’ll be posting them over the next six weeks, and adding each one to The Best Resources For Teaching Common Core Math To English Language Learners.

The first in the series was “Speaking of Math: It’s time to talk in class” by Alycia Owen


by Cindy Garcia

Cindy Garcia serves as the district wide instructional specialist for Bilingual/ESL Mathematics PK-6 in Pasadena ISD. Cindy previously served as campus mathematics coach and bilingual third grade teachers in PISD.

Mathematics much more than good computational skills.  Students’ reading skills can support or hinder their understanding of mathematics embedded in context. In order for mathematics to be more accessible to English Learners (ELs) it is critical for students to be explicitly taught how to read mathematics and how that differs from reading in other situations.

A recommendation is for teachers to make the time for quick spotlight lessons focusing on the following high leverage strategies:

  • Understand Language Organization

In mathematics, text is not just words read left to right and top to bottom. For example, when analyzing graphs it is common to read a combination of words, numbers, and symbols from bottom to top and right to left in order to understand all of the data presented.

Assumed directionality when reading could lead to students committing an error when reading that breaks down their understanding of a foundation concept.  For example 21<57> 13 could be read by a student as 21 is less than 57 greater than 13 rather than 21 is less than 57, 57 is greater than 13. The second statement is precise and supports students understanding of the meaning of the comparison symbols.

  • Interpret the Meaning and Purpose of Prepositions

Prepositions such as on, after, for, below, at, to, over, and in can lead to student misconceptions when reading because they might not understand that these small words can change the implied meaning of another word or add precision to a phrase. Take at the following problem situation.

Sam save a total of $6 over 3 weeks.

In this example the meaning of over refers to extend periods of time.  Without spotlighting prepositions, students might not understand that the $6 must last all 3 weeks.

  • Recognize Proper Nouns

In mathematics, proper nouns rarely make a big impact on a mathematical concept. Students should spend their time grappling with the concept rather than trying to read Shioban’s Sundae Shop. Students can practice locating proper nouns and renaming them with easier words to read and pronounce or just use the initials of the proper nouns.

  • Use Cognates (Spanish Speakers)

Cognates can be a powerful tool to use with vocabulary previously learned in Spanish. Students need to be aware that they already know the meaning of words such as angle, perimeter, area, volume, minute and decimals because it looks and means the same as the words ángulo, perímetro, área, volumen, minuto, and decimales.

  • Recognize Polysemous Terms

In mathematics we encounter words that have various meanings in mathematics, other subject areas, and everyday life. Students need the opportunity to find these words in order to make sense of their meaning in mathematics. Sometimes if students do not fully understand a concept they define important terms with definitions or examples unrelated to the concept being taught.  For example, the word degree can refer to unit of measurement of angle (mathematics), unit of measurement for temperature (science), or an academic rank (everyday life).


  • Chunk the Text

For some students mathematics can seem overwhelming before getting to the math just by looking at piece of text that mixture of words, numbers, symbols, and graphics.  Students can practice reorganizing text by creating bullet points. Bulleting mathematics text makes it easier to understand the math because each action, step, question, and piece of data is its own line of text.

The complexity of the English language when students are reading independently can increase the difficulty level of mathematics. Students can feel defeated by the words before they are able to dig in deep into the math. Short (3-5 minute) spotlight lessons that focus on a feature of the English Language will impact students greatly and make mathematics more accessible.

June 3, 2018
by Larry Ferlazzo

Guest Post: “Speaking of Math: It’s time to talk in class”


Editor’s Note: This is the first in a series of guest posts that will be appearing on this blog about teaching math to English Language Learners.  I’ll be posting them over the next six weeks, and adding each one to The Best Resources For Teaching Common Core Math To English Language Learners.


By Alycia Owen

Alycia Owen is an international educator, instructional coach, and EAL specialist who has implemented the co-teaching model in math, science, and language arts. She has provided professional development for schools in the US and abroad and has been a workshop presenter at NESA, AASSA, and EARCOS international teachers’ conferences. She currently lives in China where she serves on the faculty of the American International School of Guangzhou.


“We are all teachers of language.”   As educators, we know this to be true.  It’s fairly easy to identify why a science teacher would teach students how to format lab reports, for example. Or, the value of the history teacher addressing how to differentiate between primary and secondary sources.  There are countless examples of language objectives that support learning outcomes in the content areas. But how do we give our English learners rich experiences in math class that will help them dive deep into processing content while they’re acquiring language skills?   

The secret is in speaking.

Math class requires students to conjecture, reason, argue, prove, critique, explain, analyze, interpret, model, question, discern, compare, describe, and justify.  While reading, writing, and listening are integral to the learning of mathematics, it is the speaking tasks that allow students to engage more fully in mathematical practices.  The classroom should reflect the vital role that speaking plays in math. It should facilitate collaboration, make specialized math language accessible and visible, and provide a wide variety of opportunities for students to speak the language of math.

Set the Stage for Collaboration

The organization and layout of the physical space in the classroom is an important first step in establishing the types of discourse that are valued and utilized in that environment.  English language learners construct meaning by using language in context as they collaborate with peers. The classroom must support this by offering flexible seating arrangements that allow for partners and small groups to collaborate often.  Space for students to use manipulatives and model their thinking should also be considered. Next, procedural norms should be explicitly modeled and reinforced for students. For example:

  • How and why to make eye contact when someone is speaking. This will look different during whole class vs. partner activities.  Make these differences transparent to students.

  • Introduce a quiet signal that includes what to do with both the voice and hands when the teacher needs students’ attention during discussions and group work. Students will be talking while working with manipulatives and need a signal that helps them seamlessly move from high activity into focused attention on the teacher.

Lastly, and most importantly, construct lessons in which you are doing less of the talking and students are doing more.  Facilitate this by modeling “talk moves” and using sentence frames that enable English learners to collaborate with peers in full sentences that include academic vocabulary.


The Language of Math

To speak the language of math requires the use of specialized vocabulary and symbols to build meaning and to understand concepts.  Make sure students have the opportunity to contribute to interactive word walls they can refer to when talking. Take care to identify words that have different meanings in and out of math class and model their varying uses.  Here are a couple of my favorites: He was a great ruler of his kingdom. vs Use a ruler to measure the book. Or, I heard about that product on a commercial. vs The product of 5 and 7 is 35.  It’s also helpful to translate number sentences into words and post the two versions. 58 + 3 = 61/Fifty-eight plus three equals sixty-one.  Lastly, have students create a math dictionary as the school year unfolds (alternatively, a mini-dictionary for each unit). This can be a small spiral notebook or even some stapled sheets of paper with a construction paper cover.  As a term is introduced and added to the word wall, students update their dictionaries with the new word, its definition, and an example or picture. This way, when a word is removed from the wall to make space for others, students still have a reference tool they can use independently.  The dictionary has the added benefit of providing a tangible reminder of how much has been learned, which goes a long way toward helping English learners feel proud of their efforts and motivated to keep talking.


Time to Talk

If teachers should be talking less and students talking more, what should they be talking about?  How do I fit speaking into lessons that are already packed with content and activity? I’ve discovered it’s almost always a good time to talk in math class.  Here are some opportunities for speaking practice that require no extra preparation because you’re probably already doing them. With academic vocabulary visible in the room and a sentence stem or two to guide them, use a pair/share or small-group format so your students can talk to each other about:

  • Word problems (What key words can you find to help us solve it? What strategy should we use? Why did you choose that strategy?)
  • Warm-ups (Did you and your partner get the same result? Use the same strategy?)
  • Using Manipulatives (This model shows _____.) 
  • Graphic Organizers (Use your _____ to explain _____ to a partner.)
  • Math Dictionaries (Look at your partner’s dictionary and find something to compliment.) 
  • Math Journals/Written Reflections (Tell your partner what you plan to write. Read your journal entry aloud to your partner.)
  • Projects (Share your progress with a partner.  What have you accomplished so far? What do you still need to do?)
  • Check-ins (The most challenging thing about today was ______.  I now understand ______.)

Any time students are talking, circulate to listen in on conversations and participate when you think it’s appropriate.  You’ll get immediate formative data and even more opportunities to model math vocabulary, including how to use any sentence frames you’ve provided.  It’s also a nice time to validate students’ thinking, ask guiding questions, encourage elaboration and clarify misunderstandings.


Speaking of math… If in every class meeting, every student has a chance to talk using the language of math, their confidence will grow right along with their competence.  Even math anxiety can be made less severe or eliminated when students know they have peers to talk with, ideas worth sharing and when the language of math is made transparent and accessible to all.  Now THAT’s something to talk about!

April 29, 2018
by Larry Ferlazzo
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Guest Post: Integrating Writing Into Math Classes


Editor’s note: I did a series on writing in math classes over at my Education Week Teacher column but, due to my disorganizational skills, I overlooked this contribution.  The authors have graciously given me permission to publish it here.  I’ll add it to The Best Resources For Writing In Math Class

Laura Bolton, Ben Avila, and Kelly Mahoney are teachers in the Central Unified School District in Fresno, California. Laura is at Saroyan Elementary School, has been teaching for 15 years, and has taught math using Cognitively Guided Instruction for the last 11 years. Ben is at Teague Elementary School, has been teaching for 15 years, and has taught math using Cognitively Guided Instruction for the last 11 years. Kelly is at River Bluff Elementary School, has been teaching for 12 years, and has taught math using Cognitively Guided Instruction for the last 10 years. They are members of the Instructional Leadership Corps, a collaboration among the California Teachers Association, the Stanford Center for Opportunity Policy in Education, and the National Board Resource Center at Stanford.

Writing has become an essential part of a 21st century math class. It is no longer enough for students to merely solve twenty problems on a worksheet and regurgitate answers. Students need to explain their thinking and justify their answers. According to Common Core Mathematical Practice Standard Three students must construct viable arguments and critique the reasoning of others.

“They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.” (p. 6, CCCSS, 2014)

Writing can take on many purposes and forms. Writing for mathematics can promote a deeper understanding of concepts and procedures. By explaining their thinking students are more apt to make connections and solidify their thinking. Explaining their thinking to others is a good way to sort through their understandings and misconceptions.


“Students might demonstrate deep conceptual understanding of core mathematics concepts by solving short conceptual problems, applying mathematics in new situations, and speaking and writing about their understanding. Students who lack understanding of a topic may rely on procedures too heavily.” (Overview of the Frameworks, p.11, 2015)

The new standards ask students to apply mathematics to everyday life and situations. Students can write to explain how their strategy works in a real world context. They can explain why their solution strategy is the best and most efficient path to an answer. Or explain why their answer given a multitude of possible answers would be the best choice in a given scenario.

Writing in mathematics classrooms is also a good form of assessment. By having the students write you can get to know what students are thinking. Teachers may know if a student got the correct or incorrect answer, but that doesn’t always tell you the full picture. By having students write their explanation for their answers you can better formatively assess their understanding and tailor lessons to your students’ strengths and weaknesses.

At the kindergarten and first grade level, writing in mathematics will and should look different than in second grade and above. Young children can begin writing in math with simple drawings with labels.  Labels could be words and/or pictures.  Sentence frames and math word banks can help young students with basic descriptions of their solution strategies.  Annotation apps such as Seesaw and Explain Everything can help kinder and first grade students record explanations of their thinking in greater detail before their writing skills have fully developed.

Some ideas to get your students writing in math class could be:

  • Describe how to solve a problem
  • Explain why an answer or solution is reasonable
  • Write a journal entry about what they learned in class or what was difficult or easy
  • Explain whether or not they agree with a given solution or statement and explain why
  • Examine errors in other students’ work and explain what could be done to correct the mistake
  • Describe how two solution strategies are alike or different

Students at all grade levels can and should write during math class. Younger students can benefit from learning to label their work and writing sentences to clarify their answers and strategies. The written rehearsal of strategies helps older students consolidate their thinking and strategies, pushes them to think reflectively about their own and their classmates’ explanations or strategies, and can provide teachers with a means of assessing a student’s understanding and misconceptions.


California Department of Education. California Common Core State Standards: Mathematics. Edited by CDE Press. Sacramento, CA, 2014.

Instructional Quality Commission. Mathematics Framework for California Public Schools: Kindergarten through Grade Twelve. California Department of Education. Sacramento, CA, 2015.


April 10, 2018
by Larry Ferlazzo

Guest Post: Using A “Jigsaw” In Math Class

Editor’s Note: I’m a big fan of using the Jigsaw instructional strategy (see This Is Interesting: Hattie Says Jigsaw Strategy Hits a Homerun and Two Useful TOK Class Resources: Jigsaw Instructions & Allegory Of The Cave Videos/Evaluation Forms).I saw a tweet from Katie Elder about a math jigsaw she recently did, and invited her to write this guest post about it. I’m adding this post to The Best Posts On Helping Students Teach Their Classmates — Help Me Find More.

Katie Elder has been an instructional coach for Rowland Unified School District for six years. She also teaches one period of 11th grade English and reads everything she can get her hands on.

Jigsaws in Algebra II

When my partner, Lily Thio, and I were assigned as instructional coaches to one of our district’s comprehensive high schools, we had some trouble getting into classrooms. Teachers were filled with hesitance, for varied reasons, at the suggestion that they work with a coach. In order to be useful (as well as ornamental) on campus, Lily and I devised many avenues through which we might reach teachers. One of those structures is our Prep Period PD, held once a month on a different topic, and for which teachers come to our classroom during their prep period for some informal learning, discussion, and strategy sharing. Our first was on checking for understanding, and next we addressed differentiation. These topics were derived from a staff survey that we had shared with teachers earlier in the year.

For our third Prep Period PD, we decided to tackle flexible grouping. Our teachers have been operating under the assumption that changing the seating chart every grading period, or putting students into trios or groups once in a while, fit the bill. Some even practiced intentional grouping based on their general impression of their students as middle, high, or low, and they created mixed groups, sprinkling in their English Learners and RSP students. However, we didn’t feel that our teachers had a solid understanding of how to use data to inform grouping decisions, nor how to use this data and these groups to provide truly differentiated instruction. As Lily and I talked about what this workshop might look like, we decided that a unique approach was called for. After looking over our master schedule, we determined that, in working with two of our Algebra II teachers, we could offer our Prep Period PD on one date, every period, in a math class with actual students. Talking about intentional grouping, and many other issues in education, is much more effective in practice than in theory.

After discussing our idea with the teachers, Mr. Dawdy and Mr. Wong, Lily worked with them to develop a formative assessment that the teachers would give to their students as an exit ticket at the end of the period, the day before our lesson. Because two of the class periods were honors-level Algebra II, Lily created two different assessments, each aligned to that teacher’s pacing and taking into consideration the teachers’ goals for the period. Our Algebra II classes were reviewing for a chapter test on rational functions, and the Algebra II Honors students were beginning to learn how to graph sine and cosine. We were interested to see if our flexible grouping strategy worked better for one lesson or the other–at the beginning of a unit or at the end. Spoiler alert: it worked well for both!

The next step was to score the assessments in a way that grouped students based on their readiness with the topic. In some classes, there were four clear levels, and in others, only three. As the math expert of our coaching team, Lily sat with each teacher and talked carefully through what each level of student should be able to do, and once they had begun, the sorting went quickly. Then prep began for the next day. Lily grouped the students into both “home” and “expert” groups. Each home group had at least one student of each level, and each expert group contained students of homogeneous readiness. Lily then determined which math problems each expert group would tackle together.

On the day of the lesson, with a few teachers observing in each period, Lily discussed with students the learning objective and reviewed the academic vocabulary they would need to have their group discussions. She then passed out to each student their own organizer with one problem on it, the one that their homogeneous group would become “experts” at solving. The students worked silently and independently on their problems and then met together in their pre-assigned expert groups. Together, the students discussed the problem, shared their answers, and looked for errors in their work, making sure that by the end of the assigned time (about five minutes), each student felt sure that they could explain to their home groups the procedure for solving the problem. Lily walked from group to group, paying special attention to the groups who had received the lowest-level problem, but answering questions and guiding all students toward correct solutions.

When students moved again, this time into home groups, each student took a turn using a small white board to write out their problem, wait while the other students copied it onto their organizer, and then “work the problem” on the board, discussing each step. To lower anxiety and increase confidence, the students considered low in their groups went first (we grouped students into X, Y, and Z expert groups, so these were our X students). Then our Y students taught their problem, and then our Z students. Again Lily visited each table, listening to explanations, offering suggestions, and answering questions.

At this point in the lesson, I went into an adjoining empty classroom with the visiting teachers to discuss the lesson objectives and the lesson’s effectiveness, focusing primarily on the flexible grouping strategy to differentiate content based on readiness level. Teachers had many questions about how the formative assessment was developed, how the students were grouped, and how to structure the activity to ensure students equitable talk time and smooth transitioning between groups.

The lesson went well. Students responded positively, and there was 100% engagement in each of the classes. Although we have not had time to check in with the teachers to look at test score data, the math teachers whose classes we worked with both commented on how glad they were to see some of their low students speak with added confidence and some of their quiet students speak at all! One teacher even commented on how a student who perpetually “checks out” was working well beyond his normal contribution.

As an added bonus, Lily and I actually got to do real coaching with two unsuspecting math teachers who thought they were helping us out!

Was it perfect? No, of course not. What lesson is the first time through? Here are a few thoughts on what we will do differently next time:

  • The Algebra II Honors students received parts of one problem: identifying the characteristics (amplitude, period, and transformations), creating a coordinate table, and graphing the function. Next time, we will have all students do the problem up to and including their own assigned part. As it turned out, graphing a function from a completed table was not challenging enough for our “Z” groups.

  • We found that many students teach like their teachers! They write the problem on the board, complete it, and then look for confirmation, all without saying a word! The students definitely needed guidance in how to explain. Next time we will give home group students sentence frames that demonstrate how to use required academic vocabulary as a tool to aid in their scripting of the explanation that expert group members will use once back in their home groups.

  • Finally, we found out that the members of some of our “X” groups were able to come up with the same wrong answer, or that they could persuade less strong students to believe their incorrect answer. For this reason, next time we might try an “ambassador” structure, in which expert groups swap members in order to review their work and practice their scripts for explaining.


Although there are some challenges associated with organizing a jigsaw activity, teachers who do a thorough job of pre-planning will find that they have class time to visit with and assist some of their neediest students, and that the reciprocal teaching gives students practice with multiple problems, builds confidence, and sharpens their skills.

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