# data[].attributes.body
This article discusses the importance of a math program being taught with a focus on quality and true understanding. (McKay Wilson, 2013)

# data[].attributes.body
David McKay Wilson considers the updated Common Core standards and the importance of understanding in math rather than rote memorization. (McKay Wilson, 2013)

# data[].attributes.attachment_link.title
Search Results | Harvard Graduate School of Education

# data[].attributes.headline
Ami Agnetti - Why So Many Elementary Students Aren't Mastering Basic Math Facts

# data[].attributes.subject
Ami Agnetti - Why So Many Elementary Students Aren't Mastering Basic Math Facts

# data[].attributes.body
This article discusses long-range the effects of students not mastering their multiplication facts. (Diligent, n.d.)

# data[].attributes.headline
Ami Agnetti - Math Instruction for English Language Learners

# data[].attributes.subject
Ami Agnetti - Math Instruction for English Language Learners

# data[].attributes.body
This articles examines the importance of building background knowledge and a math vocabulary base for English Language Learners. (Robertson, 2009)

# data[].attributes.attachment_link.readable_text
My dad gave me one dollar bill
'Cause I'm his smartest son,
And I swapped it for two shiny quarters
'Cause two is more than one!

- Excerpt from "Smart," by Shel Silverstein

As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me. Looking back, however, I realize that I had an advantage that I wasn't even aware of - I understood the language in which the problems were written, even if I didn't understand how to solve them! Although it is easy to assume that many English language learners (ELLs) will excel in math because math is a "universal language" and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray.

As I spoke with teachers and did research for this article, it became very clear that making sure that students understand math vocabulary and have ample opportunities to use it are very important. Solving word problems, following instructions, understanding and using mathematical vocabulary correctly - 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).

For 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.

High school math teacher Hillary Hansen learned just how big a role language plays in math instruction when she taught her first Basic Math course for ELLs last year. She wanted so much to provide the students with the good foundation they needed, but she felt unable to reach the students or engage them in her lessons, and 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 to meet 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 this 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. I am happy to report that while Hillary still feels challenged and is working very hard, this year has been much more successful for her and her students.

As a result of more effective instruction, her students:

understand the content better and are working together to find creative ways to learn

discuss math more and know how to use the instructional supports their teacher has in place

are comfortable with math and asking questions to get the help they need.

Hillary feels that she is providing them with the foundation they need not only to understand the mathematical concepts, but also to successfully interact within a math classroom in order to continue learning more advanced concepts.

Following are some strategies that Hillary and some of the other teachers I spoke with found helpful this year, and that they recommend as best practices when teaching math to ELLs.

The importance of teaching academic vocabulary

Vocabulary instruction is essential to 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.

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 he/she "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:

Demonstrate that vocabulary can have multiple meanings. Help students understand the different meanings of words such as "table" and "quarter," as well as how to use them correctly in a mathematical context.

Encourage students to offer bilingual support to each other. 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. (Check the hotlinks for a list of bilingual translations of math vocabulary in multiple languages).

Provide visual cues, graphic representations, gestures, realia, and pictures. 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 ( ).

Identify key phrases or new vocabulary to pre-teach. This strategy will help students decide which math function they should apply. Example: "more than" means "add." (See hotlinks for more references about math vocabulary.)

The importance of reading and understanding written math problems

Written 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, a Colorín Colorado "From the Heart" article, 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.

The importance of building background knowledge

As the opening quote from the Shel Silverstein poem reminds us, background knowledge plays a critical role in math class! 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 building background knowledge of students.

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.

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.

The importance of increasing student language production in the content area

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.

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.

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 like math journals. This is an excellent way for students to process what they've learned and what questions they still have. The journal 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..." 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.

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.

Using technology

Technology can also be a powerful tool in math instruction for ELLs. Here are some ideas 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.

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.

Find out what's available online. Vital NY (Video Teaching and Learning for NYS Educators) on Teachers' Domain offers an online library of free media resources from the best in public television. 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 graphing calculator, 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 nor some of the more sophisticated models, such as the graphing calculator. 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.

Even 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 - and who knows? 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.
Common Core Math and ELLs: Blog Posts
These posts from our Common Core blog highlight resources that can be used in Common Core math instruction for ELLs.
EngageNY: Common Core and Math
The EngageNY website provides materials focused on curricular examples, standards for mathematical practice, and other materials for professional development.
Discovery Education: Puzzlemaker
Puzzlemaker is a free puzzle generation tool for teachers, students and parents where users can create and print customized word search, criss-cross, math puzzles, and more using their own word lists.
Catherine Snow: Word Generation
Catherine Snow's new website provides information and resources for educators who would like to learn more about Word Generation and how it is implemented. Includes links to comprehensive academic word list that students need to master to comprehend academic content.
Dave's Math Tables: English/Spanish Dictionary
List of English math vocabulary words with the Spanish equivalent.
PBS Teachers: Math Lessons
PBS Teachers offers a database of multimedia math lessons and activities that can be searched by grade or topic.
SMART Notebook Lesson Activities
Browse lesson activities and materials for classrooms using SMART notebook software. Choose the country or region, and search by curriculum standards, subject and grade level.
Colorín Colorado Webcast: English Language Learners in Middle and High School
This webcast features Dr. Deborah Short and discusses effective instructional strategies for teaching English language learner students in middle and high school, such as the SIOP model.
Translating Word Problems
This is a great site for teachers in the elementary levels, as it provides a list of keywords you can teach your ELLs to look for as they read word problems. Also included are useful ideas and tricks to better prepare students to understand written math problems.

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

# data[].attributes.attachment_link.description
My dad gave me one dollar bill'Cause I'm his smartest son,And I swapped it for two shiny quarters'Cause two is more than one! - Excerpt from "Smart," by Shel Silverstein As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me.

# data[].attributes.headline
Ami Agnetti - How to Help Students with Multiplication Facts

# data[].attributes.subject
Ami Agnetti - How to Help Students with Multiplication Facts

# data[].attributes.body
This article discusses the importance of students memorizing multiplication facts. It also includes methods to help them, along with video tutorials. (Mammoth Math, n.d.)

# data[].attributes.attachment_link.readable_text
In this article I discuss some general principles in helping students with their multiplication tables:

1. Why study the tables?
2. Learning tables in a structured manner
3. Example: memorizing the table of 3 in five steps
4. Are timed drills necessary?
5. Multiplication games
6. Music
7. What about mnemonic helps?
8. Math Mammoth Multiplication 1 book

Why study the times tables?

I feel that learning the multiplication tables is even more important than mastering addition and subtraction facts. Why? Because knowing the tables well facilitates the learning of basic division facts, multidigit multiplication, long division, most fraction math, and factoring. Even in algebra you will still need to be able to simplify rational expressions and factor polynomials, perhaps even multiply matrices.

Or, we can say it this way: if you child does not know the tables, he/she will have a terrible time mastering all those topics. I'm not saying children won't learn those topics conceptually-I mean they will have hard time completing the problems and exercises quickly, and can instead get all "bogged down" just by the multiplications.

That is why I feel every teacher/parent should put forth a good effort for their students to learn the times tables. Spend 1-2 months on it. It can pay off!

Yet, many students in today's world struggle with the tables. This article explains in detail how to go about teaching multiplication tables in a structured manner, followed by other helpful suggestions for practice.

Learning tables in a structured manner

This method is aimed at memorizing a certain times table by using so-called STRUCTURED DRILL. It differs from random drill substantially in the fact that we actually use the structure of each table in the drill. You should only start the drills after the child understands the concept of multiplication itself.

Please see this list of free VIDEOS (one for each table) that teach the tables from 2 to 12 using this structured drill.

When using this method (or any drilling), be sure to explain to the student that the goal is to memorize the facts-to recall them from memory-and not to get the answers by counting or any other method. Just like your child has probably already memorized your address and phone number, now she or he is going to memorize some math facts. You should expect the child to answer immediately when you are drilling. If he or she doesn't know the answer by heart (from memory), then tell him or her the right answer.

Short drill sessions are usually best. For example, you might drill for five or ten minutes at a time, depending on the attention span of the child.

However, try to have at least two sessions during the day as your schedule permits. Research on how the brain learns has shown that new memories are forgotten soon and that new information is best retained when it is reviewed within 4-6 hours of the time it is initially learned. (By the way, this principle applies to anything new a person is learning.)

Pencil and paper activities that the student completes alone do not work really well for memorizing facts because the child can get the answers by counting and not from memory. Proper drill requires an investment in time from the instructor. If you can, utilize older siblings, too, in the task of drilling.

Here's a five-step method for memorization, taken from my book Math Mammoth Multiplication 1. Normally only a few of the steps would be included in any one session, depending on the child's concentration and ability.

Example: Memorizing the table of 3 in five steps

You can view a short video explaining the main points of the structured drill here:

Have the table to be learned already written on paper. Here we will use the table of three as an example.

11 × 3 = 33
12 × 3 = 36

1. The first task is to memorize the list of answers (the skip-counting list). Have your child study the first half of the skip-counting list (3, 6, 9, 12, 15, 18), saying the numbers aloud while pointing to the answers one by one with a finger or a pen. This technique uses the senses of seeing, hearing, and touch simultaneously to fix the information in the brain. After he has gone through the list a few times, ask him to repeat it from memory.

Expect your child to answer, and don't give her the answers too easily, because ONLY by putting forth an effort will she memorize the facts. Just like the muscles, the mind needs exercise to become stronger.

Require her to memorize the skip-counting list both forwards and backwards. Keep practicing until she can "rattle off" the first list of 3, 6, 9, 12, 15, 18. With some tables, like the tables of 2, 5, and 10, it helps to point out the pattern in them. The pattern in table of 9 is more subtle but still useful.

2. Then tackle the last half of the list: 21, 24, 27, 30, 33, 36. Do the same things you did with the first half of the list.

3. Next, work with the whole list of answers (the skip-counting list). Practice the list going up and down until it goes smoothly and easily. These steps may be enough for one day. But be sure to review again later in the day.

4. Next, practice individual problems randomly while keeping the whole list of problems visible to the student (without the answers of course). Cover the answers. You can ask orally ("What is 5 times 3?"), pointing to the problem on the chart. I recommend reading the question aloud while simultaneously pointing to the problem because, again, using multiple senses helps to fix the information in the mind better. Watch the video to see how this is done.

The goal at this stage is to associate each answer 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, with a certain multiplication fact (such as 7 × 3).

You can also mix facts from earlier tables that she already knows with these new problems and drill them together with flashcards.

5. The last step is to do the drill the other way around. Now you say the answer ("21"), and the student has to produce the problem ("3 × 7"). Keep the table visible, hide the problems (but not the asnwers), and point to the answers in a random order. This technique can also work the other way around, where the student says the answers, and you produce the problems. Give wrong answers sometimes, too, to check them out.

As an extension, you can say answers from several tables that you've studied and have the student give the corresponding problem. Sometimes there are several answers. For example, 36, 30, 24, and 20 are in several different times tables. This is an especially good exercise as it prepares for the concepts of division and factoring.

The memorization probably won't happen overnight. On subsequent days, you can mix steps 1-5 (normally you won't need to concentrate too much on steps 1 and 2 after the initial practice). This kind of drilling takes a little time and effort from the teacher, but it can be very effective. Homeschoolers can obviously do some of it while going about other tasks, such as traveling in the car and so on.

While you are doing this table by table, you can also try to teach the process to your child, so that she will learn how to do the memorization herself. She can hide the answers and try to reproduce the list in her mind.

Check out also my series of free videos for the multiplication tables, one video for each table, using this structured drill.

Other helpful ideas

Hang on your wall a 12 by 12 grid (or 10 by 10) with the skip-counting patterns written in it. Remind your child to glance at it several times a day. It can work wonders for visual students!

Hang another poster beside it, with an empty grid, in which the child fills in those facts he has mastered. See an example of how I use the 12x12 grid at the end of the lesson for the table of 3 (taken from my book Math Mammoth Multiplication 1).

Recite the skip-counting lists or multiplication facts aloud just before going to bed. This can turn them into mastered facts by the next morning.

Use this free online practice program by David Chandler that helps you practice and master the multiplication tables - using the same order of learning the tables as in my book and my curriculum!

This tool also practices the tables both forwards and backwards at the same time - which is exactly the approach I've taken in my books and in my videos for the multiplication tables. Practicing them both ways is so great, because it allows the students to master division facts "in disguise" at the same time.

Are timed drills necessary?

I feel that timed drills are just one tool among many when it comes to learning math facts. Some children will thrive on them; in other words, they will learn quickly when they are used. Perhaps they like the challenge of racing against the clock. There are a number of timed computer games that can work very well for drilling facts. Here are some online ones:

You can also use regular, printable multiplication worksheets for the multiplication tables

Yet for other students, timed drills may be counterproductive and they may end up in tears and frustration. The proof is in the pudding: just try it and see how it goes.

Multiplication games

Games are very helpful once you get to the "random drilling" stage (after having completed the structured drills). I have compiled a LONG list of online multiplication games and software here.

Computer-based drilling can be very rewarding to children when they begin to notice that they are truly learning the facts and are becoming able to complete the drills successfully. They can actually come to enjoy the process.

This simple card game is also noteworthy (the "Product War" version). Essentially, each player is dealt two cards face up, players multiply those, and the person with highest product captures all cards in that round.

Or, try a dice game called Damult Dice.

Music

Music (songs) can be very effective in helping children to learn their multiplication tables. It works with the brain in manners that we are not yet fully aware of, but it is certain that music enhances learning and brain function.

You can simply use a familiar tune (Row, Row, Row Your Boat; Happy Birthday, B-i-n-g-o, or any other tune the child already knows), and just add the numbers in place of the words. Some children may like rap better, and those tunes work, too. Or, if you'd like to have songs that are especially written for multiplication tables, check out my review of Multiplication Mountain CD by Hap Palmer.

Another useful idea is to use little 'dance' routines (jumping, twirling, touch-the-floor, marching, etc.) along with the music for the kinesthetic learner. The girls can see it as 'dance' and the boys can see it as 'movement'. Most kids will love being out of their seats. A 'hand-jive' game can be a means to teach the skip-counting list, such as 4, 8, 12, 16, 20...

Using music and movement can be especially helpful for students who have math anxiety because it doesn't seem like "math" to them. By circumventing their 'mental block', they can learn the tables without difficulties-they see it as fun, and the 'math triggers' that the mind uses to 'block' the brain and produce the 'fear response' are never stimulated. Also, younger siblings often learn the songs (and should be encouraged to learn) at the same time.

What about mnemonic helps for multiplication tables?

Mnemonic devices in themselves are not bad. We use them all the time, in everyday life situations. Maybe you have a phone number that you divide into 2-digit numbers, and remember that it has successive numbers, or doubles, etc. I once memorized a certain 4-digit pin number by thinking of it as two 2-digit numbers, and remembering that the latter was 9 less than the first... but after a while I remembered it without that.

For example, Times Tales is a mnemonic program for multiplication tables. It associates a silly story and picture with each "difficult" upper times tables fact. For example, if a child learns 8 × 7 by using Times Tales, the child will remember a silly picture of "lady eight" and the character for seven driving in a car, and remember "It's 56." That is not too much different from using a rhyme such as "5, 6, 7, 8, fifty-six is seven times 8." Such a program can be a confidence booster as well.

Now, for some children these silly stories do not work because they don't easily remember the stories, and then it can cause frustration. Also, older children may not enjoy the silly stories anymore. All such "helps" are fine in their place, but you have make sure that the mnemonic "help" does not end up being an additional burden in itself.

Keep in mind that mnemonic helps are of course only additional (supplemental) resources, and don't replace studying the multiplication concept itself or studying the structures in the tables.

The balance

There is a BALANCE, as in everything. If you've already expended a considerable amount of effort and the child is not retaining the tables, please don't make the multiplication tables to be the reason why your student hates math. You can back off and try again later. Some give their students a "crutch"-tables written out-and eventually the child notices how slow it makes him/her having to check the answers in the table instead of knowing them, and decides to memorize them.
By Maria Miller
See also

Multiplication Tables Online Practice
Simple and free online practice here at MathMammoth.com - you can choose timed or not timed practice, the number of questions, and the tables you want to practice.

Structured drill videos for the times tables
A set of videos that employ the structured drill method for the multiplication tables.

Multiplication Tables Matching Hidden Picture Game - choose which exact tables you want to practice. This game is right here at MathMammoth.com!

Comments

I just wanted to thank you for this curriculum! We started homeschooling this year, 4th grade. My daughter started out slow, and I wasn't sure this was the right product for her. Then I figured out that she didn't know her math facts well enough to move on. So I just went back to the math facts chapters in earlier grades and am having her work on those. Your multiplication worksheets are amazing! Your way of teaching them is so great that she's really picking them up quickly and we'll soon be able to move on to the 4th grade curriculum. You really DO teach the concepts, not just problem solving. Unlike many homeschoolers, I will never have to spend more money on math program after math program, trying to find one that works. Thank you!!
Michelle

By the way, I just want you to know what a Godsend your materials have been. As a Math lover myself I was thrilled to come across a curriculum that taught Math exactly how I "think" it. Your explanations are wonderful, and the amount of practice perfect. After struggling with Math for 5 years, my 6th grader has grown in leaps and bounds. She actually enjoys Math now. Prior to getting your Math, she spent three years working on her times tables using flashcards, drill, computer games, mnemonic devices and programs with rhyming stories in conjunction with her textbooks. All to no avail. After 5 months using your products and methods, she has memorized all her multiplication facts, and zoomed through to long division. My 5 year old has been listening and actually knows skip counting twos, threes, fours and fives. Apparently your methods work. THANK YOU so very much for all your work.

Many Blessings,

Christine Swanson

Just want to thank you for Math Mammoth! Very affordable. I have thrown everything else out and this is all we are using. My math-hating daughter who just finished third grade is doing the multiplication from the blue series to prepare for 4th grade next year. She is doing this on her own over the summer because she likes it! Never thought I'd see that. She has really struggled with multiplication and she says this is making it so much easier for her to understand. I love the ability to use the blue series to fill in the gaps of what they don't "get" the first time around. This is so customizable for different kids. I love the pdf format also. I can put the book together in any order I want. I love the no teacher's manual thing. With three kids it is hard to shuffle around 3 books. This is so much better because when each kid has questions they already have it all with them when they bring it to me. I was so excited to find this. Wish I had found it a couple of years ago! Thank you for this curriculum and thank you for making it affordable. I bought all 6 levels at once. I could never afford to do that with any other curriculum. This has ended the struggle (for the most part) with my daughter. Thank you for making teaching my kids math affordable, easy, and even enjoyable!

Sara

Maria, I just wanted to send you a quick message to say a heartfelt thank you for giving me the tools to teach math so well to my children. It is early days for me with Math Mammoth materials, having just ordered the Light Blue 1A and 3A/B texts from Rainbow Resource (found them because of a recommendation by one of their q&a people).

Before staying home with my children, I taught 1st and 2nd grades in the public schools, mostly as reading/writing teacher. Since I want to go back to teaching when my youngest is older, my children are in public school but I afterschool them to fill in the gaps or enrich. Even with all my "training," I have been really puzzled about how best to teach math to my older boys. The curriculum in their school just doesn't seem to be systematic or comprehensive enough. I think the teachers do their best with the materials they are given.

However, my 3rd grader is challenged by mastering his multiplication facts and your email today, along with the video, is so illuminating! I never would have thought to teach the tables this way. Thank you so much! I wish all public school teachers had access to such wonderful materials as you provide (well they do if they have an internet connection but you know the school systems are throwing so much else at them). I hope you are getting lots of positive feedback from many people for your efforts because I am sure your efforts are benefiting many. Thank you for sharing your knowledge!!!

Jennifer Nowicki
WAIT!

Receive my monthly collection of math tips & resources directly in your inbox - and get a FREE Math Mammoth book!

# data[].attributes.attachment_link.title
How to help students with multiplication tables?

# data[].attributes.attachment_link.description
A comprehensive article explaining in detail how to go about teaching multiplication tables in a structured manner. Includes a video (about structured drilling), suggestions for games, and other helpful hints. This will be the LAST article you need to read about learning multiplication tables!

# data[].attributes.headline
Ami Agnetti - Making it Stick:  Memorable Strategies to Enhance Learning

# data[].attributes.subject
Ami Agnetti - Making it Stick:  Memorable Strategies to Enhance Learning

# data[].attributes.body
This article examines different methods and strategies for improving memory and recall of information. (Richards, 2008)

# data[].attributes.attachment_link.title
Making It Stick: Memorable Strategies to Enhance Learning

# data[].attributes.attachment_link.description
We all use strategies throughout our day to remember the variety of facts and ideas we need to retain. It is valuable for teachers, therapists, and parents to understand the memory process in order to become better equipped to help our students understand and use strategies.

# data[].attributes.headline
Ami Agnetti - Underachievement of Verbally Gifted Children

# data[].attributes.subject
Ami Agnetti - Underachievement of Verbally Gifted Children

# data[].attributes.body
Discusses how the different styles of learning affect multiplication

(Bainbridge, n.d.)

# data[].attributes.attachment_link.readable_text
The term verbally gifted is used to refer to children who have strong language skills. Verbally gifted kids become competent in language before other kids their age do. They also perform better on verbal and general information tests and tests of English expression than mathematically gifted children do.﻿﻿

Why Do Gifted Kids Underachieve?

The lack of challenging work, combined with the learning style and temperament of verbally gifted children, can lead to a loss of motivation and the loss of motivation leads to underachievement.

It's often difficult for teachers to understand this about young, gifted children because of what they have learned about child development, primarily Jean Piaget's stages of development. Basically, Piaget did not consider children capable of true abstract thinking until they are around 11 or 12 years old.

Some gifted children, particularly those who are extrinsically motivated, are able to complete whatever work is required for excelling in school. However, the anxiety verbally gifted children feel when they are given one tedious task after another is often more than they can bear.

All too often, the only way they know to cope with anxiety is to not do the work at all. They will spend more time trying to get out of doing the work than they would spend if they just sat down and did it.

But sitting down to do the work can produce anxiety as well. Avoiding it and finding novel ways to avoid it not only helps them escape the anxiety but gives them a challenge.

Understanding the characteristics of verbally gifted kids may help adults support them so that they can be successful.

Verbal skills include the ability to understand language easily. This includes grammar, as well as creative uses of language such as poetry. Learning languages tends to come easily to the verbally gifted and they generally have a good ear for the sounds of a language.﻿﻿ The verbally gifted also have the ability to understand and manipulate language symbols like alphabets.

Most people would no doubt say that verbal skills-reading, writing, and speaking-are among the most important skills needed for success in school. So it seems logical to believe that verbally gifted children are at an advantage since they tend to be good readers and are good with language.

Learning Style

There isn't a ton of recent research on verbally gifted children, but one reason verbally gifted children may be more at risk of underachievement is because of their learning style. These children tend to be holistic or global learners. This means that they want to understand the big picture first and get the details later.

They look for meaning and want to understand concepts and what those concepts imply. They aren't motivated to memorize detail, which is usually what is found on tests, and they're likely to see rote memorization as meaningless.

For example, holistic learners aren't motivated to memorize the multiplication tables. They would prefer to learn multiplication facts in a meaningful context.

However, most schools expect children to memorize the details first. These are the facts that are seen as the essential building blocks of learning. After all, one needs to know multiplication facts before using them to work out problems.

Holistic learners, though, need to understand why the facts are necessary before they will bother learning them. It may seem odd to use math as an example of the problems verbally gifted children have, but remember that this is about learning style rather than subject matter.

It's easy to think that verbally gifted children resist memorizing the multiplication tables because they aren't interested in math or because they have less ability in math than in language, and while that can be true, their resistance can also be caused by their dislike for meaningless learning.

Intrinsic Motivation

Many gifted children are intrinsically motivated.﻿﻿ Smiley faces, star stickers, and even good grades aren't likely to motivate them. These children may sacrifice such external rewards in order to work on more appealing tasks, those they find interesting, challenging, and relevant to their lives. It's the challenge they find rewarding, not external rewards. Rote memorization of facts and concrete details is neither challenging nor rewarding.

If these children are not given sufficiently challenging work, they will make it more challenging. For example, they might give themselves time limits when no time limits exist. They may do this even though it means they may risk performing well on an assignment or on a test. Although they could get an ​A with little effort, they find this type of challenge more inherently rewarding.

Verbally gifted children also tend to be impulsive.﻿﻿ When they're impulsive, they don't pay attention to detail; they don't have the patience for it. As early as infancy, gifted children prefer novelty, which basically means that they need mental stimulation. This preference for novelty makes it very difficult for gifted children to continue working on tedious tasks, and the tasks that are too easy are often tedious for them.

Emotional Temperament

Verbally gifted underachievers can also tend to be high-strung and anxious. Most people might find it hard to understand why a child would resist completing easy work, but when the work is not challenging and a child is not motivated to do it, they can become anxious about it.

In fact, a gifted child can become so anxious when trying to complete what they feel is tedious work that they will just avoid doing it altogether. Unfortunately, many teachers will determine avoidance as a sign that the child doesn't understand the material or is too lazy or disorganized to do it.

It doesn't help that young children might not be able to correctly express the reason for their anxiety. For example, a young child might tell their parents or teacher that the work is just too hard. But the child and the adults aren't using the word "hard" in the same way.

For the adults, "hard" means that the work is beyond the child's abilities or that the child hasn't yet mastered the concepts needed to do the work. What the child actually means, though, is that having to continue working on a too-easy task is causing them a great deal of anxiety.

# data[].attributes.attachment_link.title
Are Verbally Gifted Children at Risk for Underachievement in School?

# data[].attributes.attachment_link.description
The term verbally gifted is used to refer to children who have strong language skills. Verbally gifted kids become competent in language before other kids their age do.

# data[].attributes.headline
Ami Agnetti - Procedural Fluency:  More Than Just Memorizing Facts

# data[].attributes.subject
Ami Agnetti - Procedural Fluency:  More Than Just Memorizing Facts

# data[].attributes.body
Discusses and describes procedural fluency (Ginsburg, 2012)

# data[].attributes.attachment_link.readable_text
Procedural fluency or conceptual understanding--math educators have debated for years which is more important. I sided with conceptual understanding until my colleague Angela McIver helped me see

the value of procedural fluency in terms of stamina. Like all of us, students have finite energy. The more energy they use for procedures, the less energy they have for problem solving. And the less energy they have for problem solving, the less likely they are to gain conceptual understanding. A lack of procedural fluency can therefore contribute to a lack of conceptual understanding.

The issue, then, isn't if we should target procedural fluency but rather how we should target it. And, how to do so while also targeting conceptual understanding, as called for by the Common Core State Standards (the "dual intensity" instructional shift I mentioned in my last post).

Of course, it's hard to help students achieve something if you don't know what it is. A lot of teachers think procedural fluency is synonymous with "knowing your math facts," but it's more than that, as defined in the National Research Council book, Adding It Up: Helping Children Learn Mathematics:

Does procedural fluency involve memorizing multiplication tables and other facts? Sure, but it also involves thinking. Students must know when, as opposed to just how, to use a procedure. And they must not only be able to perform procedures accurately, but also flexibly and efficiently--again, think stamina.

To illustrate the thinking aspect of procedural fluency, I ask teachers at workshops how their students would solve the following problem: 7/11 = ?/121. Middle and high school teachers say most of their students would cross-multiply, which provides the correct solution of 77 (7 x 121 / 11). Elementary teachers say most of their students would use the equivalent fractions approach, which also provides the correct solution (121/11 = 11; 11 x 7 = 77). Which approach is more efficient? Well, which is easier: multiplying 7 x 121 or 7 x 11?

One reason students don't always choose the most efficient method is that they don't really choose any method. They just apply an algorithm without thinking. They know how to use procedures but not always when to use them. They perform procedures accurately but not always flexibly or efficiently. In other words, they lack procedural fluency.

This brings us back to the Common Core challenge of helping students develop procedural fluency while also helping them develop conceptual understanding. And here are a few suggestions for meeting this challenge:

Will these ideas ensure procedural fluency for students? No. They still need to know their facts in order to use procedures flexibly, accurately, and efficiently. In the above problem, for example, the equivalent fractions approach is only more efficient than cross-multiplying if students have memorized multiples of 11.

I'll address the facts-fluency connection further in a future post, since I don't have the stamina for it now. Check out Jo Boaler's recent Ed Week article, Timed Tests and the Development of Math Anxiety, in the meantime.
Image by Vlue, provided by Dreamstime license Join my mailing list for announcements about webinars and the work I do to improve teaching and learning.

# data[].attributes.attachment_link.title
Procedural Fluency: More Than Memorizing Math Facts (Opinion)

# data[].attributes.attachment_link.description
Procedural fluency or conceptual understanding--math educators have debated for years which is more important. I sided with conceptual understanding until my colleague Angela McIver helped me see the value of procedural fluency in terms of stamina. Like all of us, students have finite energy. The...

# data[].attributes.headline
Ami Agnetti - Total Recall:  Helping Our Students Memorize  Multiplication Facts

# data[].attributes.subject
Ami Agnetti - Total Recall:  Helping Our Students Memorize  Multiplication Facts

# data[].attributes.body
Methods to help students master facts (Zimmerman, 2012)

# data[].attributes.headline
Ami Agnetti - Why Children Have Difficulties Mastering the Basic Number Combinations and how to Help Them

# data[].attributes.subject
Ami Agnetti - Why Children Have Difficulties Mastering the Basic Number Combinations and how to Help Them

# data[].attributes.body
Discusses the reasons so many students have difficulties mastering the facts (Baroody, 2006)

# data[].attributes.body
Is it appropriate to use

information from past classes?

# data[].attributes.headline
Ami Agnetti - Assessing Elementary Understanding of Multiplication Concepts

# data[].attributes.subject
Ami Agnetti - Assessing Elementary Understanding of Multiplication Concepts

# data[].attributes.body
Emphasizes the importance of understanding the underlying concept

of multiplication (Smith, Smith, & Smith, n.d.)

# data[].attributes.body
This causes students to have difficulty

with long division, word problems,

problem-solving, basic algebra, etc.

# data[].attributes.body
Possible Resources

# data[].attributes.headline
Ami Agnetti - Teaching for Mastery of Multiplication

# data[].attributes.subject
Ami Agnetti - Teaching for Mastery of Multiplication

# data[].attributes.body
Includes studies and reasons that prevent students from mastering facts (NCTM, 2005)

# data[].attributes.body
What is the best way to obtain

consent? Will it be sufficient

to only obtain my principal's

consent, or will I also need it

from the parents and students?

# data[].attributes.body
When trying new multiplication

strategies, how will I monitor

and show progress to show whether

a new method caused students

to internalize multiplication facts?

# data[].attributes.body
It could be difficult to gather

supporting data from other

classrooms.

# data[].attributes.body
Possible Issues:

# data[].attributes.body
Math builds on certain skills. Multiplication

facts are the stepping stone to higher-level

math skills and without them, a student's

learning will be hindered.

# data[].attributes.body
Many students memorize one set of

facts for a short-term, and even do

well on the timed tests. They do not

transfer this to long-term memory.

# data[].attributes.body
Potential Research Topic:

A high majority of students are

not truly learning their multiplication

facts, which has a direct result on

their understanding of future

math concepts. What is the barrier

to their learning, and which methods

would be more effective?