teaching mathematics through problem solving prekindergarten grade 6

We will keep fighting for all libraries - stand with us!

Internet Archive Audio

• This Just In
• Grateful Dead
• Old Time Radio
• 78 RPMs and Cylinder Recordings
• Audio Books & Poetry
• Computers, Technology and Science
• Music, Arts & Culture
• News & Public Affairs
• Spirituality & Religion
• Radio News Archive

• Flickr Commons
• Occupy Wall Street Flickr
• NASA Images
• Solar System Collection
• Ames Research Center

• All Software
• Old School Emulation
• MS-DOS Games
• Historical Software
• Classic PC Games
• Software Library
• Kodi Archive and Support File
• Vintage Software
• CD-ROM Software
• CD-ROM Software Library
• Software Sites
• Tucows Software Library
• Shareware CD-ROMs
• Software Capsules Compilation
• CD-ROM Images
• ZX Spectrum
• DOOM Level CD

• Smithsonian Libraries
• FEDLINK (US)
• Lincoln Collection
• American Libraries
• Canadian Libraries
• Universal Library
• Project Gutenberg
• Children's Library
• Biodiversity Heritage Library
• Books by Language
• Additional Collections

• Prelinger Archives
• Democracy Now!
• Occupy Wall Street
• TV NSA Clip Library
• Animation & Cartoons
• Arts & Music
• Computers & Technology
• Cultural & Academic Films
• Ephemeral Films
• Sports Videos
• Videogame Videos
• Youth Media

Search the history of over 866 billion web pages on the Internet.

Mobile Apps

• Wayback Machine (iOS)
• Wayback Machine (Android)

Browser Extensions

Archive-it subscription.

• Explore the Collections
• Build Collections

Save Page Now

Capture a web page as it appears now for use as a trusted citation in the future.

Please enter a valid web address

• Donate Donate icon An illustration of a heart shape

Teaching mathematics through problem solving : prekindergarten-grade 6

Bookreader item preview, share or embed this item, flag this item for.

• Graphic Violence
• Explicit Sexual Content
• Hate Speech
• Misinformation/Disinformation
• Marketing/Phishing/Advertising
• Misleading/Inaccurate/Missing Metadata

plus-circle Add Review comment Reviews

58 Previews

Better World Books

DOWNLOAD OPTIONS

No suitable files to display here.

PDF access not available for this item.

IN COLLECTIONS

Uploaded by station47.cebu on March 16, 2023

SIMILAR ITEMS (based on metadata)

Teaching Mathematics Through Problem Solving: Prekindergarten-Grade 6

Frank k. lester jr..

269 pages, Paperback

First published October 1, 2003

About the author

Ratings & Reviews

What do you think? Rate this book Write a Review

Friends & Following

Community reviews.

Join the discussion

Can't find what you're looking for.

• Business & Investing
• Management & Leadership

Enjoy Prime FREE for 30 days

Here's what Amazon Prime has to offer:

Buy new: .savingPriceOverride { color:#CC0C39!important; font-weight: 300!important; } .reinventMobileHeaderPrice { font-weight: 400; } #apex_offerDisplay_mobile_feature_div .reinventPriceSavingsPercentageMargin, #apex_offerDisplay_mobile_feature_div .reinventPricePriceToPayMargin { margin-right: 4px; } $51.65 $ 51 . 65 FREE delivery May 27 - 29 Ships from: Amazon.ca Sold by: Amazon.ca

Save with used - very good .savingpriceoverride { color:#cc0c39important; font-weight: 300important; } .reinventmobileheaderprice { font-weight: 400; } #apex_offerdisplay_mobile_feature_div .reinventpricesavingspercentagemargin, #apex_offerdisplay_mobile_feature_div .reinventpricepricetopaymargin { margin-right: 4px; } $33.43 $ 33 . 43 $4.95 delivery may 31 - june 11 ships from: *house of treasures* sold by: *house of treasures*.

Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet or computer – no Kindle device required .

Read instantly on your browser with Kindle for Web.

Using your mobile phone camera, scan the code below and download the Kindle app.

Image Unavailable

• To view this video, download Flash Player

Teaching Mathematics Through Problem Solving: Prekindergarten-Grade 6 Paperback – Jan. 1 1800

Purchase options and add-ons.

• ISBN-10 0873535405
• ISBN-13 978-0873535403
• Publisher National Council of Teachers of Mathematics; edition (2003-11-04)
• Publication date Jan. 1 1800
• Language English
• Print length 269 pages
• See all details

Frequently bought together

Customers who bought this item also bought

Product details

• Publisher ‏ : ‎ National Council of Teachers of Mathematics; edition (2003-11-04) (Jan. 1 1800)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 269 pages
• ISBN-10 ‏ : ‎ 0873535405
• ISBN-13 ‏ : ‎ 978-0873535403
• Item weight ‏ : ‎ 454 g
• #3,421 in Business Decision-Making & Problem Solving (Books)
• #174,339 in Textbooks

Customer reviews

• Sort reviews by Top reviews Most recent Top reviews

Top review from Canada

There was a problem filtering reviews right now. please try again later..

Top reviews from other countries

• Amazon and Our Planet
• Investor Relations
• Press Releases
• Amazon Science
• Sell on Amazon
• Supply to Amazon
• Become an Affiliate
• Protect & Build Your Brand
• Sell on Amazon Handmade
• Advertise Your Products
• Independently Publish with Us
• Host an Amazon Hub
• Amazon.ca Rewards Mastercard
• Shop with Points
• Reload Your Balance
• Amazon Currency Converter
• Amazon Cash
• Shipping Rates & Policies
• Amazon Prime
• Returns Are Easy
• Manage your Content and Devices
• Recalls and Product Safety Alerts
• Customer Service
• Conditions of Use
• Privacy Notice
• Interest-Based Ads
• Amazon.com.ca ULC | 40 King Street W 47th Floor, Toronto, Ontario, Canada, M5H 3Y2 |1-877-586-3230

Teaching Mathematics through Problem Solving: Prekindergarten-Grade 6.

• MLA style: "Teaching Mathematics through Problem Solving: Prekindergarten-Grade 6.." The Free Library . 2004 National Council of Teachers of Mathematics, Inc. 21 May. 2024 https://www.thefreelibrary.com/Teaching+Mathematics+through+Problem+Solving%3a+Prekindergarten-Grade+6.-a0120354046
• Chicago style: The Free Library . S.v. Teaching Mathematics through Problem Solving: Prekindergarten-Grade 6.." Retrieved May 21 2024 from https://www.thefreelibrary.com/Teaching+Mathematics+through+Problem+Solving%3a+Prekindergarten-Grade+6.-a0120354046
• APA style: Teaching Mathematics through Problem Solving: Prekindergarten-Grade 6.. (n.d.) >The Free Library. (2014). Retrieved May 21 2024 from https://www.thefreelibrary.com/Teaching+Mathematics+through+Problem+Solving%3a+Prekindergarten-Grade+6.-a0120354046

Terms of use | Privacy policy | Copyright © 2024 Farlex, Inc. | Feedback | For webmasters |

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Graham Fletcher3-Act Tasks ]
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

Share This Book

• Скидки дня
• Справка и помощь
• Адрес доставки Идет загрузка... Ошибка: повторите попытку ОК
• Продажи
• Список отслеживания Развернуть список отслеживаемых товаров Идет загрузка... Войдите в систему , чтобы просмотреть свои сведения о пользователе
• Краткий обзор
• Недавно просмотренные
• Ставки/предложения
• Список отслеживания
• История покупок
• Купить опять
• Объявления о товарах
• Сохраненные запросы поиска
• Сохраненные продавцы
• Сообщения
• Уведомление
• Развернуть корзину Идет загрузка... Произошла ошибка. Чтобы узнать подробнее, посмотрите корзину.

Oops! Looks like we're having trouble connecting to our server.

Refresh your browser window to try again.

Product Key Features

• Author Frank K. Lester
• Publication Name Teaching Mathematics Through Problem Solving : Prekindergarten-Grade 6
• Format Trade Paperback
• Language English
• Publication Year 2003
• Type Textbook
• Number of Pages 269 Pages
• Item Length 9in
• Item Width 6in

Additional Product Features

• Lc Classification Number Qa13.T4315 2003
• Grade from Preschool
• Grade to Sixth Grade
• Copyright Date 2003
• Target Audience Scholarly & Professional
• Topic Teaching Methods & Materials / Mathematics
• Lccn 2003-021744
• Dewey Decimal 372.7
• Dewey Edition 22
• Illustrated Yes
• Genre Education

Teaching Paperback Textbooks

Mathematics paperback textbook, mathematics paperback textbooks, nonfiction teaching paperbacks books, mathematics paperback textbooks in english, vintage paperback mathematics antiquarian & collectible books.

• NAEYC Login
• Member Profile
• Hello Community
• Accreditation Portal
• Online Learning
• Online Store

Popular Searches:   DAP ;  Coping with COVID-19 ;  E-books ;  Anti-Bias Education ;  Online Store

Mathematical Structure and Error in Kindergarten

You are here

There’s a growing literature on using children’s “misconceptions” to diagnose their thinking and to target interventions. While contributing to that literature, this article focuses on choices educators make and draws attention to a little-recognized benefit of children’s errors—errors can reveal strengths worth preserving, not just weaknesses to fix. With stories of two children, this article shows how wrong answers can reveal children’s attention to mathematical structure (NGA & CCSSO 2010). It also looks at how young children use patterns in language to make mathematical meaning.

Children’s errors—To correct or not to correct?

Young children are remarkable at generalizing the information they gather. For example, they quickly find the pattern of adding  s  to make a noun plural; but this correct conclusion also produces errors like saying “foots” instead of “feet.” Similar situations arise frequently in mathematics too, and these errors can reveal not just what students don’t know, but also what they  do  know. Therefore, teachers can use children’s ability and inclination to generalize and find patterns as an ally in teaching.

In general, the most useful place to intervene—adding challenge, correcting errors, or calling for reflection— is where the child’s intellectual growth is currently most rapid. Where the child is  working , that’s where the child’s attention is. Errors help us notice that place because they’re common when a child enters new intellectual territory and is actively building new ideas. Patterns of errors sometimes reveal correct structures that the child built but misapplied—structures to be preserved. Other times, they reveal incorrect structures built using correct principles.

Some errors, of course, are just slips, revealing nothing but a momentary lapse of attention. Should we correct them? That depends. If they hide a mathematical pattern from the child, correcting them removes a barrier. Or if such small slips are considerably more frequent for a particular child than for most others the same age, then the child might need help with directing attention, controlling impulses (perhaps just slowing down), or taking a second look. But in many cases, small slips are worth ignoring. Focusing on errors can be a distraction, drawing attention away from what the child is really working on and interfering with building and using more advanced ideas and structures. At those times, it is more useful to affirm and build on the correct part of a child’s work.

We, as educators, won’t always know exactly where a child’s intellectual focus is and when to intervene. Like all communication, teaching is complex. Fortunately, learning is robust and children are mostly resilient, so teachers can make slips and students will still learn well.

Ruth: Attending to structure but inventing an incorrect one

At the very end of kindergarten, a year in which the children learned about addition and subtraction, the children all create small thank-you cards for a regular visitor. Seth (top) draws a blue square with what looks like a red lollipop stuck to it, his representation of a wonderful logic puzzle. Many of the children entirely fill their cards with arithmetic of the sort they had been doing during the year, with some artistic elements and often with surprising extensions that they felt particularly good about. Sara (middle) adds a heart.

Six-year-old Ruth (bottom) creates this striking card with the math sentence 5 x 5 = 25 prominently displayed. The class didn’t study multiplication, but children pick up stray knowledge and love to show what they know. It’s not likely that Ruth knows yet what 5 x 5 = 25 means, but it’s fun to show off such a grown-up thing. 

Ruth’s card is a rainbow of colors, leaving deliberate space for her elaborately written 5 x 5 = 25. A complex pattern runs across the top. The card is already so full that there isn’t room for more, but just above the 5 x 5 = 25, she crams in 6 x 6 = 26, seemingly as an afterthought and perhaps a spur-of-the-moment idea.

How perfect! If  five  times  five  is twenty- five , then  six  times  six  must be twenty- six .

That’s wrong, of course, but what we learn is that Ruth’s attention was on  structure , not on random facts. Even though the structure she used is “wrong” (linguistic rather than mathematical), this is evidence of a fundamentally right idea about mathematics: she treated 6 x 6 not as another thing to remember, like the way she learns people’s names and the days of the week, but as something she could  figure out . The way she figured it out wasn’t grounded in the meaning of multiplication—we could hardly expect that of a kindergartner—but it shows that she sees mathematics as nonarbitrary, something that can be figured out and that should make sense.

Having noticed Ruth’s error, what might her teacher do? Three reasonable options would be (1) to tell Ruth that 6 x 6 = 36, (2) to show her what 5 x 5 means, and (3) to ignore it—after all, it is a drawing, not a math quiz. Ignoring it seems harmless. Ruth certainly doesn’t  need  correction right now; she’s not studying multiplication in kindergarten. And we need not worry that Ruth might, if uncorrected, memorize 6 x 6 = 26; if facts were that easily memorized, we’d all be ecstatic! Maybe telling her that the correct answer is 36 would also be harmless. Ruth is clearly a thinker, so she might seek a pattern in the new data and conclude that if  five  times  five  is twenty- five , and if  six  times  six  is thirty- six , then perhaps  seven  times  seven  is forty- seven . If she did, that would be another error, but no more harmful than the first. It would also be another example of Ruth seeking structure and (incorrectly) generalizing it.

Here’s the downside of correction. With no foundation to make sense of multiplication, telling Ruth that the correct answer is 36 would just give her another random fact. It wouldn’t aid her understanding of multiplication, and might harm her overall understanding of mathematics. Ruth tried to make mathematics make sense. If she is told “Nope, wrong answer”  without the knowledge and logic to understand why , she might change her perception of mathematics and believe it to be arbitrary. Ruth perceived mathematics as being about finding pattern and structure, even if her limited knowledge leads her to a spurious pattern or the wrong structure. That is the  right  perception of mathematics, and we don’t want to change it. Someday Ruth will make sense of multiplication and use that understanding to come up with a better result for 6 x 6. 

Focusing on errors can be a distraction, drawing attention away from what the child is really working on.

What about showing Ruth what 5 x 5 means? Without commenting on her error, the teacher could add meaning to Ruth’s correct statement by setting out dice while saying, “How cool! You’ve told us about five fives! Here’s a five, now two times, now three times, now four times, now five times. Twenty-five!

The teacher could stop there, without teaching multiplication. The dice would let Ruth see what her idea means, using a familiar image of 5 and a quantity she can count. Ruth might follow up, or she might not. She might try something like on her own. She’s still not “studying multiplication” (she’ll count or add), but she can attach a new meaning to a fact she’s already proud of knowing. And if she does not follow up, no time has been wasted, and she has not been criticized on the wrong outcome of her wonderful pattern-finding or pushed beyond her interest.

So, when should a teacher make sure children know the correct answer? And when is it more important for children to focus on pattern and logic, even if the outcome isn’t accurate? It depends on where the child’s attention is—where the child is actively  working . If Ruth’s error were related to concepts in which she was currently growing rapidly—addition and subtraction—it would be important to determine whether the mistake was systematic or just a slip. If systematic, her teacher might want to intervene. If just a slip, then asking Ruth to check two results (one correct, one wrong) to see if she can catch and fix the error herself might be sufficient. Still, correction is an option; we might decide that Ruth’s time would be better spent on something else.

Linguistic patterns, mathematical learning

Ruth’s invention of 6 x 6 = 26 illustrates a useful truth about young children’s mathematical learning. It’s connected to their learning of  spoken  language.

Children—especially young ones—are superheroes at language learning, many learning 10,000 or more vocabulary items in the first five years of life and developing a nearly complete  structure  (grammar) for their language before they reach kindergarten. This is a fact that even their errors reveal: they’ll say “I goed outside,” inventing an incorrect word by applying a correct rule that they themselves generalized. They also actively build many other kinds of structures, including categories (elephants and cats are different from chairs and tables) and subcategories (they find the concept of a pet elephant funny but don’t know what to make of a pet chair). They love to count things and compare sizes. Young children’s experiences are a riot of near-constant newness, and children are rapidly building new structures to organize them.

Teaching can capitalize on the fact that the spoken language of number can be such a strong support for mathematical learning, especially in the early grades, and can expose young children’s mathematical abilities that remain hidden when the approach is largely through writing. A recent discussion with a different kindergarten class provided an excellent example. To probe the children’s thinking, one of the authors asked them to name the biggest number they could think of. Most said things like “a hundred” or “a million.” Nobody said things like “ two  hundred” or “ seven  million.” But when we asked what they supposed two hundred plus three hundred might be, almost all readily answered five hundred. To better understand their responses, we asked another question: “What do you think a hundred plus a hundred is"? This question, which seems simpler, proved more of a puzzle for the children. Though some said two hundred, several gave other answers, like “a hundred,” “a million,” and “I have no idea,” even though just moments before they had correctly added to find five hundred!

This paradox vanishes when we figure out what’s really going on in their bright, active minds. The child who said “a hundred plus a hundred is a hundred,” interpreted hundred to mean “big number.” Given that, his statement was perfectly correct: a big number plus a big number is a big number! The child who responded “a million” was saying that a big number plus a big number is a bigger number. In contrast, adding two hundred plus three hundred is a linguistic act to which the children applied their knowledge of 2 + 3. (It reminds us of a child who, when asked what five-eighths plus five-eighths was, confidently answered “Ten ayfs! What’s an ayf"?) Though hundred just meant “big number,” the children were processing “two hundred plus three hundred” the way they’d process “two goats plus three goats.” The  numberness  of hundred was not really used; most of the children seemed to have only a vague idea about it as a quantity and maybe not even complete certainty that it was a specific number. (Similarly, in grade 1 or 2, many children who know that half of eight is four confidently assert that half of eight million is four million and half of eighty is forty, if these are  spoken  computations. The children may not yet have a solid mathematical concept of those numbers—for that matter, few adults have a clear picture of exactly how huge a million really is—but making  verbal  sense of manipulating large numbers in this way helps children when they later formalize their structure and notation.)

While kindergartners have little to no concept of a million as a number, Ruth and her classmates had gotten lots of experience counting, investing precision in the initially vague meanings they had for counting words. The counting words they used and their experiences  building  the number hundred from 10s gave at least partial meaning to these numbers and to addition. Importantly, the linguistic patterns they started with provided a structure for their counting. Language and mathematical meaning codeveloped, each supporting the other.

Ruth’s “6 x 6 26” comes from a (wrong) use of linguistic pattern; but the attention she and her classmates give to linguistic patterns in number names, along with the experience they gain  using  that language in context, allows them to do— with understanding and competence and confidence —many mathematical things not usually expected in kindergarten. (For more about how mathematical habits of mind relate to language, see Goldenberg, Shteingold, & Feurzeig 2003.)

Sonya: Basing a correct structure on an incorrect first step

Sonya already knows that if she repeats an addition that got her from 24 to 34, it will get her from 34 to 44, then to 54, then 64. Having shown her mathematical strength throughout the year, she clearly knows (and says) that to get from 34 to 44 you must add 10, not 6. She also shows facility with the sequence of 10s and with pairs like 4 + 6 that make 10. In this instance, she has correctly counted to 30—but noticing her slip of writing 34 is another story. And using her knowledge to catch the slip would require a self-reflective slowness that is rare among kindergartners. So, when she rolls the die a second time and again changes it to get 6, she just continues her pattern without a moment’s thought.

The last line on her activity sheet shows the same kind of (super common!) slip she started with. Sonya meant to write 64, got distracted by her starting number, 54, and wrote 54 instead. This time, she did spontaneously catch and correct her slip.

So, should her teacher point out the other errors? There are both nonmathematical and mathematical issues to consider.

››  Nonmathematical considerations : Sonya filled in the page wrong, every line of it. Children should learn that just writing stuff on a page is not the goal. And if the paper will be seen by parents or others, it should not look like the child was slapdash and the teacher didn’t even notice. Correction can come without fuss. Sonya might even enjoy seeing the “oops” and fixing everything in order to have a perfect paper to show. And seeing that what she wrote is not what she knows might help her become more careful. Such self-monitoring is rare at this age. Is it learnable for  this  child at  this  time, or is intervention before more maturity just wasteful and distracting?

››  Mathematical considerations : Was this error at Sonya’s growing edge? Yes. Her current intellectual focus is on structure in addition. What did Sonya’s error reveal about her thinking? It showed that she’s mathematically reflective; rather than just moving on, she noticed her first result. She did not notice that it was wrong, but she did notice the increase by 10 and applied that structure. Many of her earlier lessons and activities indicate that the error was just a slip, not a mathematical misunderstanding. Sonya has nothing mathematical to learn from correcting this mistake.

We have choices about where to focus and about whether and when to correct.

So, should Sonya be corrected? This is where teacher judgment is important. There’s no general answer. Happily, a general answer is not needed. We teachers teach specific children, not general ones— specific children with specific strengths and specific relationships with their teachers. As in all human interaction, we make a guess—but an educated one. And we watch. If Sonya’s teacher misses the mark, she might try again using a different strategy. Or we wait. Not everything needs to be corrected; and not everything that does need to be corrected needs to be corrected immediately. (For two stories about how what appear to be errors can indicate deep understanding, see Goldenberg 2017.)

Focus on structure versus focus on accuracy

No, not  versus ! That’s the point. We have choices about where to focus and about whether and when to correct. In fact, this is very familiar to many early childhood teachers for subjects other than mathematics—for example, teaching young children to write. At first, letter reversals, seemingly random capitalization, and lack of punctuation are not a great concern; grasping the concept of representing words with letters is the primary goal. Invented spellings are evidence that children have moved beyond the random letter-spray of preschool and have grasped that the letters one uses to write a word should be related to the sounds of the words. Choosing what to emphasize and when is a big part of teaching.

In mathematics, teachers often seem more reluctant to make such choices. Maybe that is because so many people (including teachers, administrators, and parents) believe that the final written answer  is  the measure of mathematics learning—after all, even if your thinking is perfect, the bridge collapses if you write the wrong number at the end. Or maybe it is because, unlike in a piece of writing, we often do not see a story in the math, just a final number; so, if it is wrong, well, it is wrong. But the work of Sonya and Ruth shows how even a single number can sometimes tell a story.

Accuracy matters, but not always. How do we teachers know  when  to correct or where to focus? We won’t always choose perfectly—there may not even be a choice with no risks. As long as teachers are  thinking  about whether or not to correct and what is communicated with that correction, as long as they are  deciding  what to do and not simply reacting to what was written, as long as they are considering the  particular  child, they increase their chances of making a good instructional decision.

Sonya’s teacher chose not to correct her. In  that  instance, supporting Sonya’s inclination to use a pattern seemed more helpful for Sonya’s mathematical development than drawing her attention to incorrect results. It is not possible to know for sure whether pointing out the slip might have had a better outcome. Fortunately, children and circumstances provide corrective feedback over time. Like Sonya, who will get other chances to add more accurately, we teachers too will get other chances to decide what responses are best. As long as teachers are guided by thoughtful responses, they build models about each child’s learning that help guide interactions far better than any generic list of do's and don’ts could.  

Authors’ note:  The work reported here was funded in part by The National Science Foundation (award numbers 1119163 and 1348564) and by the Heising-Simons Foundation (award number 2015-023). Views expressed here are the authors’ and do not necessarily reflect the views of the foundations.

More for the Interested Teacher

Hansen, A., ed. 2014. Children’s Errors in Mathematics . 3rd ed. Exeter, UK: Learning Matters.

Ryan, J., & J. Williams. 2007. Children’s Mathematics, 4–15: Learning from Errors and Misconceptions .

Goldenberg, E.P., N. Shteingold, & N. Feurzeig. 2003. “Mathematical Habits of Mind.” In Teaching Mathematics through Problem Solving: Prekindergarten–Grade 6, ed. F.K. Lester. Reston, VA: National Council of Teachers of Mathematics.

Goldenberg, E.P. 2017. “Correct Answer/Wrong Question: Theory of Mind in a Mathematics Class.” EDC Learning and Teaching Blog, Jan. 6.  http://ltd.edc.org/wrong-answers-or-wrong-questions .

Hansen, A., ed. 2014.  Children’s Errors in Mathematics . 3rd ed. Exeter, UK: Learning Matters.

NGA (National Governors Association) Center for Best Practices & CCSSO (Council of Chief State School Officers. 2010. “Look for and Make Use of Structure.” (CCSS.Math.Practice.MP7.) Common Core State Standards in Mathematics. Washington,DC: NGA & CCSSO.  www.corestandards.org/Math/Practice/#CCSS.Math.Practice.MP7 .

Ryan, J., & J. Williams. 2007.  Children’s Mathematics , 4–15:  Learning from Errors and Misconceptions . Berkshire, UK: Open University Press.

Photographs: 2, 4, courtesy of author; 1, 3, 5, © iStock

E. Paul Goldenberg has taught mathematics from early elementary through graduate school and now develops mathematics curriculum at Education Development Center (EDC), in Waltham, Massachusetts. He especially likes building mathematical connections and fostering students’ questions and problem posing leading to depth, surprise, and new ideas.  [email protected]

Sharon J. Miller has taught kindergarten at The Rashi School, in Dedham, Massachusetts, for 18 years and has worked with young children for three decades. She enjoys helping students recognize and pursue mathematical sense through exploration and use of familiar and new materials.  [email protected]

Cynthia J. Carter teaches mathematics in sixth, seventh, and eighth grades at The Rashi School. She especially loves helping students think like mathematicians, building systems and posing—as well as solving—problems.  [email protected]

Kristen E. Reed,  project director at EDC, has worked as a teacher, curriculum developer, professional development facilitator, and researcher. For more ways to make math engaging, challenging, and fun, visit ym.edc.org .  [email protected]

Vol. 72, No. 3

Print this article

• 1st Grade Math
• 2nd Grade Math
• 3rd Grade Math
• 4th Grade Math
• 5th Grade Math
• 6th Grade Math
• 7th Grade Math
• 8th Grade Math
• Knowledge Base
• Math for kids

How to Teach Math to Kids: 6 Awe-Methods

reviewed by Jo-ann Caballes

Created on May 14, 2024

Updated on May 17, 2024

When teaching a kid math, you should focus on the three pillars: practicality, patience, and support.

At school, some students feel like the next Carl Friedrich Gauss. The others freeze in horror when thinking about math. Still, all kids can master main math concepts — no magic required!

Keep reading if you want your child to stop saying “I’m bad at math”.

What is a quick and easy way to teach your child math?

You can quickly teach your child math by:

Using math daily

Playing math games, singing math songs.

• Math illustrative content

Reading math books for children

• Math and Cooking

Now let’s analyze these ways step by step — tick them off on your fingers!

Teaching children mathematics can be effortlessly woven into your daily routine. You will be surprised how seamlessly math is integrated into our lives.

Board games like Monopoly or Uno involve counting, adding, and recognizing patterns.

Remember how you learned the ABC — math can be as easy as ABC, too! Check out catchy tunes for children that teach counting, addition, subtraction, and even fractions.

Math illustrative content 

Although there are different ways to teach math, the list can’t be complete without visual materials. With visual aids like pictures, kids will grasp math concepts more easily.

Modern publishers suggest a huge number of math books for kids. Read a fairytale with a Number as the main character. It can be even more engaging and thrilling than a Cinderella story!

Math and cooking – we’ll take two!

If you want to diversify your everyday practice, try baking or cooking together. Measuring ingredients can assist your kid in understanding fractions. The whole recipe is like a math task: getting it right leads to the correct result.

Remember that learning is tough, and children are very sensitive. One put-down (e.g., “You know nothing” or “I’m disappointed”) can leave an adverse mark that will be difficult to erase. Support your child to keep a positive attitude for a long time. You can praise them for even the least success — for example, when a child remembers the count from 1 to 10. That will inspire kids not to give up and make them wide open for new discoveries.

And of course, practice, practice, practice. The more you play, the better you slay!

How to make my child understand math

The best ways to teach your child math concepts should combine such points as:

• Building the right attitude to math
• Making learning math fun and engaging
• Calming the math crisis with support
• Offering additional assistance

Let’s take a closer look at each point.

Build the right attitude to math

An article’s headline. Well, if 30% love math and 30% are neutral, we have 40% of people who loathe the subject

Hans Freudenthal, a mathematician and teacher, believed that school math turns the exciting world of mathematical discovery into the “fossilized remains” of reasoning processes.

So, teaching kids mathematics should start with the proper attitude. Math is not a boring, useless school subject l, invented to make kids suffer. It’s a trainer to prepare for adult life. Think about an inattentive market salesman or fraudster who tries to scam. You need it when counting items, comparing prices, cooking, or playing sports. Math is crucial for financial literacy, from budgeting and understanding interest rates to calculating taxes and making sound investments. Understanding the real role of math makes it relevant and appreciable.

Make learning math fun and engaging

The most efficient way of teaching children math concepts is by connecting them to their interests. Try personalized learning and different games.

For example, a kid who loves sports cars will have more interest in solving a math task on the topic:

• A car engine uses cylinders to create power. Powerful sports cars often have more cylinders. Let’s say a regular car has 4 cylinders. How many cylinders will your super speedy sports car have?
• Two cars, a Jaguar and a Ferrari, cover 400 kilometers. Jaguar’s speed is 62.1 mph; Ferrari’s speed is 96.5 kph. How much time does it take for both?

Traditional math lessons can leave students bored and frustrated. But “traditional” is not the only option. At Brighterly, tutors are experts at turning frustration into fascination. Our interactive lessons use games, activities, and real-world examples. That will help develop an in-depth understanding of math and love for learning.

Your child will fall in love with math 
after just one lesson!

Choose 1:1 online math tutoring.

Calm the math crisis with support

Stay cool if your child doesn’t understand a simple rule. In your school days, math was not any easier. You just forgot all the bitter tears and long hours with a math textbook.

No matter which ways to teach your child math concepts you choose, create a “safe zone” for your kid. It’s OK not to understand something and sometimes to get upset. Learning should be a comfortable process. Otherwise, in the word “studying” your child will see only the part without “stu”.

Explain new concepts in a clear, intuitive way and connect them to what they already know. If they learn to count, ask them to bring you three tomatoes and two bananas in a supermarket. If a kid is on the way to mastering fractions, ask to pour a third of a glass of milk and a quarter cup of flour when cooking.

Offer additional assistance

Remember that teachers are not foes. They are a great resource for specific strategies and recommendations. Like Brighterly’s tutors who make lessons interactive and personalized. They follow the intended curriculum and know how to bring math concepts to life. With joint efforts, you can boost your child’s progress.

Apart from the school curriculum, there are a variety of educational websites and apps. Find interactive math games and lessons. Try a few of them to find yours.

How to teach low-performing kids math

The efficient learning strategy for low-performing kids should let you:

• Focus on understanding
• Divide challenging topics into small pieces of information
• Use math games, bright flashcards, books or applications
• Praise your kid for each success they make
• Hold brief math lessons
• Repeat the covered before you start a new topic

A study ( PLOS ONE, 2012 ) found that simply anticipating math triggered areas of the brain linked to threat and pain in people with high math anxiety. This may explain why 20-25% of kids experience moderate to high math anxiety.

Framing the problem’s boundaries is the initial step towards resolution. No one said that teaching kids about mathematics would be smooth. But we’re not afraid of challenges, are we?

Here are some tips for effective math teaching to low-performing kids.

Focus on understanding of math concepts

Jumping into complex issues can be a mistake. We suggest breaking down concepts into smaller, manageable steps. Ensure your kid grasps each stage before moving on.

You must pay attention to the logic behind the steps; the result of resolving should be a bonus.

If you clearly explain “Why”, your kid will gain a solid foundation for future learning.

Use supportive approach

Positive reinforcement is always “it”. Don’t be shy — say some pleasant words about your kid’s success in math. Maybe today is the day when your child remembers what the number  5 looks like. Or maybe today your child gets a clue about percentages.

Next, tailor math to your student’s hobbies or interests. Incorporating math games, puzzles, and online educational tools will make learning interactive. And children love interactives! Use them to teach your kids mathematics. 

Address math gaps with the right learning style

Sometimes, difficulties stem from a weak basis. Thus, you need to review and solidify basic math facts and operations.

When teaching kids math, give preference to short, focused practice. Such brief sessions will make a bigger contribution to your kid’s math progress.

A win-win method is pairing your kid with a stronger peer for additional support and explanation. Friendly rivalry can foster a child’s growth and achievement.

How to teach children math? Let’s sum it up!

Math develops critical thinking skills and logical reasoning applicable to various aspects of life. Teaching math to kids should be an exciting, life-changing journey.

We provided just a few tips to start teaching your kids about math quickly. Keep in mind that learning math demands patience and a positive attitude. Celebrate your child’s progress, no matter how small it is. 

Brighterly offers a fresh approach to learning. With an interactive platform and modern learning tips, your child will experience math in a whole new light.

Sign up today to reveal your kid’s math potential.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

Homeschooling has become increasingly popular, offering a unique and potentially enriching learning environment. However, it’s a hard decision for any parent since you become the only one responsible for your child’s grades and socialization. Check out this article to learn the pros and cons of online homeschooling and determine whether this studying approach fits your […]

May 13, 2024

SAT is a unified test colleges and universities use to assess students’ skills and decide if they are qualified enough to enter. The examination includes testing of core writing and reading skills along with math proficiency, and it’s where most students stumble.  But indeed, each applicant can prepare for what is expected. All they need […]

May 14, 2024

Book 1 to 1 Math Lesson

Kid’s grade

After-School Math Program Boost Your Child's Math Abilities! Ideal for 1st-12th Graders, Perfectly Synced with School Curriculum!

After-School Math Program

Related posts.

Best Math Word Problems for Kids

Our day-to-day activities require math, and we would struggle with the essentials without a basic understanding of maths. The main goal of teaching kids math is so that they can solve real-life problems that require math. The easiest way to ensure that your kid can understand math when they encounter it outside of the classroom […]

Aug 25, 2023

Mental Math Practice Is a Must for Your Kid

Mental math is a great way to keep your mind sharp! Mental math practice improves  overall mathematical skills. In the end, it helps to enhance kids’ counting and verbal arithmetic skills. This article will discuss how to learn mental math and how the practice can improve the ability to solve problems mentally. What Is Mental […]

May 06, 2022

30 Best Math Games for Kids: Online, Classroom, and Outdoor Fun

Math requires high mental activity. It’s often the primary reason many children are reluctant to study this subject. But you can make learning funnier with dedicated math games and applications. Keep reading to discover simple math games for kids for outdoor, in-class, and indoor implementation. What are some fun math games? Some fun math games […]

Jul 13, 2022

We use cookies to help give you the best service possible. If you continue to use the website we will understand that you consent to the Terms and Conditions. These cookies are safe and secure. We will not share your history logs with third parties. Learn More