Larry Ferlazzo’s Websites of the Day…

…For Teaching ELL, ESL, & EFL

April 29, 2018
by Larry Ferlazzo
1 Comment

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.

January 1, 2018
by Larry Ferlazzo

Great Clip From “Lady Bird” On A Growth Mindset

My daughter, granddaughter and I went to see the movie, “Lady Bird,” this afternoon.

It’s a great movie, and I say that not because it’s very Sacramento-centric.

There’s also a great clip in it about a growth mindset.

Here’s the quote itself in text form, followed by the video clip itself (I’ve embedded the trailer, and it should be set to start at the 2:17 mark when the growth mindset piece takes place):


I’m adding this to:

The Best TV/Movie Scenes Demonstrating A “Growth Mindset” – Help Me Find More

The Best Apps, Online Tools & Other Resources For Math

December 24, 2017
by Larry Ferlazzo
1 Comment

The Best Apps, Online Tools & Other Resources For Math

I haven’t created many “Best” lists for math, since I don’t teach it, but thought it would be worth bringing together what I have shared about that subject into this post.

Please feel free to let me know if you think I’m off-base on some, or if I’m missing others:

All my Education Week Teacher posts on Math Instruction.

The Best MATH Sites That Students Can Use Independently And Let Teachers Check On Progress

The Best Multilingual & Bilingual Sites For Math, Social Studies, & Science

The Best Resources For Teaching Common Core Math To English Language Learners

The Best Places To Find Theatrical Movies On Science, Math & History

The Best Resources For Writing In Math Class

The Best Posts About The Khan Academy

The Best Resources For Helping Beginner ELLs Learn About Numbers

Three Apps That Solve Math Problems Through a Picture is from Richard Byrne.

With both Jo Boaler and Dan Meyer endorsing Super Math World, I can only assume it’s a great math learning game.

You Can Now Create Your Own Activities With Amazing Math App Desmos

New “Volley” App Looks Like A “PhotoMath” For…Everything

PhotoMath Is Now Available For Android

PhotoMath & Reactions To It From Around The Web

“Mathpix” Solves Handwritten Math Problems

Visual Math Learning Pre-Algebra Lessons offers audio with text support and illustrations on a variety of math topics.  The audio is clear and at an accessible pace.  It has links to many good interactive math activities but, unfortunately, they don’t have audio.

Harcourt’s That’s A Fact game  reinforces elementary lessons, provides audio support to its text, and students like playing it.

Villainy Mission One and Villainy Mission Two teach geometry and algebra through a story “game” about bad people taking over the world.  Players have to stop them.  Besides it being a fun way to learn math, a lot, if not all, of what the characters speak is shown in text as well as heard.  It’s been developed by Thinkport in Maryland.

The Learn Alberta organization has three math sites called Math Under The SeaMath 5 Live, and Spy Guys Math.  Instead of explaining each one, I’m going to suggest that they’re definitely worth the time to just go and check them out.

SAS Curriculum Pathways, one of my favorite online sites, offers a free Math 1 course. You can read more about it here.

“Equations That Changed The World”

10 Tweaks That Can Deepen Math Tasks is from Middleweb.

Students Must ‘Engage in Math Problem-Solving’ & not Just ‘Follow Procedures’ is the headline of one of my Education Week Teacher columns.

Great Clip From “Lady Bird” On A Growth Mindset

Using Jilk’s (2016) “It was smart when…” statement to name and notice students’ mathematical strengths is from Embracing Life With Major Revisions.

Finding the Beauty of Math Outside of Class is from Edutopia.

Author Interview: ‘Motivated – Designing Math Classrooms Where Students Want to Join In’ is the headline of one of my Education Week Teacher columns. In it, Ilana Horn, author of “Motivated: Designing Math Classrooms Where Students Want To Join In,” agreed to answer a few questions about her book.

Five Ways To Shift Teaching Practice So Students Feel Less Math Anxious is from MindShift.

The MTBoS Search site is a search engine for posts from Math teachers.  It’s pretty impressive.

September 27, 2017
by Larry Ferlazzo

New TED-Ed Video: “A brief history of banned numbers”

A brief history of banned numbers is a new lesson and video from TED-Ed.

Its content reminded me of when the then-military dictatorship governing Argentina banned the use of Venn Diagrams in school. You can read more about that story at The Best Multimedia Resources For Introducing Students To The Advantages Of Charts, Graphs & Infographics.

September 26, 2017
by Larry Ferlazzo

The Best Resources For Helping Beginner ELLs Learn About Numbers


As I shared a few months ago (see Beginning A New “Best” Series Of Interest To ELL Teachers), I’m beginning to create an updated series of “Best” lists for Beginning English Language Learners. They will ultimately replace the resources I have on my outdated website.

I’ve done quite a few so far, and you can find them all at All My Thematic “Best” Lists For Beginning ELLs – In One Place!

I’ve got several more to go, however, and here’s the latest one – on numbers:


Number Exercises

Numbers 1 to 10 ESL Vocabulary Interactive Board Game

Numbers 10 to 100 ESL Vocabulary Interactive Board Game

Months and Ordinal Numbers ESL Vocabulary Game Activity Online

Numbers From The British Council

Numbers 1 to 10 Vocabulary – Listening Memory Game

Numbers 10 to 20 Vocabulary – Listening Memory Game

Numbers Game

August 9, 2017
by Larry Ferlazzo

Good Short Growth Mindset Video

Here’s a nice video from Jo Boaler on a growth mindset and its impact on the brain. It specifically talks about math, but would be useful in any subject.

I’m adding it to:

The Best Resources On Helping Our Students Develop A “Growth Mindset”

The Best Resources For Showing Students That They Make Their Brain Stronger By Learning

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