|January: Face the Facts||May: Fun for Two||September: Shape Up!|
|February: Building Operation Sense||June: Thinking Algebraically||October: Exercising Your Mind's Eye|
|March: Measurement||July: STEM||November: Recreational Math – not an Oxymoron!|
|April: Family Math||August: Fractions||December: Books for Mathy Kids|
Each month we share a few links to exemplary math websites that we think kids and adults can benefit from. Most are cataloged in the Mathlanding project that Claire and Bob worked on over the last several years. There you can read reviews of the resources we picked or check out some of the nearly 2000 others. Mathlanding has tools and other features for educators, so you might mention it to school personnel that you know. The site is open to all and searchable.
January: Face the Facts
We know it is important for children to know number facts. The most effective way to achieve that is through activities, games and problems that develop number sense, rather than through rote memorization. As an example, UB had a high school student who had all the multiplications memorized correctly except for three facts.The student had mixed up 7x4, 8x4 and 9x4. It was very difficult to convince this student that the three answers should be in increasing order – number sense!
The following two activities provide plenty of practice with numbers, but they also develop conceptual understanding, reasoning, and strategic thinking.
Number Puzzles: Requires only addition and subtraction. What strategies can you use to solve these? National Library of Virtual Manipulatives contains a wealth of fine applets.
Cows and Sheep: Can you develop a systematic way to approach these? NRICH is my hands-down favorite mathematics website!
For more about learning number facts, see the recent paper Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts by Jo Boaler, Professor of Mathematics Education, Stanford University.
By the way, what is going on with
February: Building Operation Sense
These selections give us more practice with number facts and also build understanding of the four basic operations (addition, subtractions, multiplication, division). We have ample evidence that rote memorization of these processes is not enough for long term learning. When memory fails, we are up the creek without a paddle. Children who have only learned procedures often do not know how and when to apply them to problem situations. Seeing three numbers in a problem, some children automatically add them, no matter what the context, especially if they have been taught to rely on "key words" rather than understand what is happening in the problem.
Factorize: The area model of multiplication is a powerful visual tool for understanding multiplication and division. With this applet you can explore the concept of factors by creating all possible rectangular arrays for a number. Use a randomly generated number or select your own number between 2 and 50. Which numbers have only one possible rectangle?
Rectangle Multiplication: This applet offers three different ways to visualize the product of two factors – a simple area model (Grouping), the lattice algorithm, and the traditional algorithm (Common). They make use of the commutative property of multiplication (4*5 = 5*4) and the distributive property, e.g., 23*8 = (20*8) + (3*8). The Lattice method is taught in some current math programs. Children who struggle with the traditional algorithm are often able to calculate accurately with this method. While it is an ancient procedure, used by various cultures as early as the 13th century, it flummoxes parents who attribute it to "that new math."
The Quotient Cafe demonstrates how the concept of equal sharing is represented in the partial quotients division algorithm. With this method children don't need to arrive at an exact quotient in one step, but can divvy up items in stages until all are equally dispersed. It reinforces the meaning of division more effectively than the traditional long division algorithm.
Dicey Operations is a series of paper-and-pencil games that reinforce place value, mental calculation, estimation, and strategic thinking. Being successful depends on understanding the relationships among operations and the effect of the size of numbers on the results of the operations.
This month we offer a selection of resources that help "uncover" some of the many concepts involved in measurement. They focus not on learning how to measure, i.e., how long is this line? (this is better accomplished hands-on), but rather on understanding relationships among measurements.
In Order asks you to think about the relative sizes of various measurements – temperature, speed, time, and sound – as well as what level of precision you need to consider in order to compare and order the examples. How close does your estimate need to be? The Teacher's Resources includes links to two similar activities with content that is less and more complex than this one.
Circle Tool is an interactivity that allows you to investigate the relationships among the measures of a circle (radius, diameter, circumference, area) and to use your recorded measurements to discover the circumference to diameter ratio we call pi. Test yourself out on the included problems!
Through the Window challenges you to determine the pricing structure of windows from the given examples. Solvers apply what they know about area and perimeter in a novel situation – and get lots of practice using number facts and calculation skills that we talked about in January and February.
Stop the Clock is an interactive game involving reading time on an analog clock and understanding fractional hour intervals. To be successful you also need to apply the kind of strategic thinking that you bring to Nim games.
April: Family Math
This month we highlight some resources aimed at helping parents support their children's math education.
Bedtime Math does a good job of advancing its mission: "helping kids love numbers so they can handle the math in real life." Each day the authors post interesting scenarios about everyday life and pose related math problems at several readiness levels. They believe that daily exposure to math experiences should be as common as the bedtime story. Amen, I say! There is a companion app for your mobile device.
Helping Children to Love and Do Well at Math: In this 10-minute audio interview Freeman A. Hrabowski, III (University of Maryland, Baltimore County), discusses his research on increasing student success and details ways in which parents can nurture a love of mathematics in their children.
A Maths Dictionary for Kids (and Parents): Animated and interactive, this dictionary from Australia (hence "Maths") explains over 600 common mathematical terms in simple language and provides examples and practice to help users gain understanding of mathematical terminology and symbols. A good homework helper!
Let's Play Math! Denise Gaskins' blog is "about the ongoing adventure of learning, teaching, and playing around with mathematics from preschool to pre-calculus." Intended to support parents who homeschool their children in mathematics, it contains many engaging activities and games. It's worth scrolling all the way down the home page to explore the full range of what's there.
A Family's Guide: Fostering Your Child's Success in School Mathematics helps parents understand current trends in mathematics education. Published by the National Council of Teachers of Mathematics, the document explains the thinking behind its Principles and Standards for School Mathematics and suggests ways that families can support their child’s learning of mathematics.
Helping Your Child Learn Mathematics is a publication by the US Department of Education that offers fun activities that parents can use with children from preschool age through grade 5 to strengthen their math skills and build strong positive attitudes toward math. It can be viewed online or downloaded as a pdf file.
Math at Home, published by Sonoma County, CA, presents information, suggestions, and resources for adults who wish to better understand the importance of mathematics and the role they can play in their children’s learning. It also includes an explanation of the Standards of Mathematical Practice from the Common Core.
May: Fun for Two
Now that testing season is over (we hope!), here are a few fun math activities for parents and children to enjoy together.
Nim-7: Nim is an ancient game for two players that develops strategic thinking along with number sense. This page offers directions to a basic version of this very adaptable game. The goal is to find a winning strategy. NRICH also provides an interactive customizable version which you play against the computer – Got It. Try ganging up on it with your child!
In Product Game opponents take turns selecting factors to claim products on a board. The first player to claim four in a row wins the game. It develops fluency with multiplication facts and strategic thinking. The factors and number needed to win are customizable. You can play against the computer, if you dare!
Square It: The object of this interactive game is to select the four corners of a square on a dot grid in some size and orientation. It develops the visualization skills and encourages use of systematic strategies (of course!). The dimensions of the grid can be adjusted.
Games from Around the World is a collection of 10 strategy games that can be played with simple materials. Easy to learn, difficult to master!
June: Thinking Algebraically
Part of the reason that algebra is so difficult for many students is that they lack earlier experience in thinking about the kinds of situations and problems that algebra can model. Long before children are introduced to the symbolic notation and rules of formal algebra, we want them to look for patterns and generalize scenarios, so that they understand the meaning and usefulness of the procedures when they encounter them. If these activities find you grabbing for your algebra "hammer," step back and try to think about how to use reasoning to approach them, as an elementary child might do.
Can You Balance? This activity helps young children understand equivalence – a major concept throughout math. Too many think the equal sign (=) means "Here comes the answer."
Pan Balance – Shapes applies the idea of equivalence and balance. It challenges solvers to determine the relative values of given shapes by comparing groups of them on a pan balance and finding the relationships among them.
Spotting Numbers Problems lets you explore a variety of growing dot patterns and try to predict successive iterations.
Double Function Machine is an open-ended tool for exploring input, output, and the rules that define their relationships.
Your Number Is ... displays a visual representation of what happens at each stage of a number trick to help you understand why the answer is always the same.
You've probably heard the acronym STEM a lot in the media, especially in reference to the need for future workers in those fields and how our educational system should be preparing them. You might also see STEAM, which adds arts/design to the mix of math, science, engineering and technology. The websites below integrate principals of the STEM fields, not necessarily all at once, in ways that challenge and engage learners of all ages. Warning: Some of these are addicting!
Cogs Environment: Use this interactive tool to explore how gears mesh and turn relative to each other as well as the relationship between the number of teeth on a set of cogs and the number of turns they make. When you're ready for a challenge, check out the companion problem, Counting Cogs.
Math Programming provides an introduction to programming. Use simple LOGO commands and geometric principles to navigate the turtle around the screen to create designs.
Golf: A SuperMath Game puts some of those same concepts of length and angle to practice in a nifty simulation.
Factory Balls involves choosing and sequencing the steps required to manufacture a given product.
Sugar, Sugar: Apply logic and rudimentary engineering skills to design paths that will deliver the required amount of sugar to given mugs.
Many children develop fraction phobia in upper elementary grades when they are introduced to operations and algorithms in isolation and without connection to their meaning. The resources below use a variety of visual models (not just pizza!) to reinforce the connection between concepts and symbols. Don't let anyone tell you fractions are no longer important; they are critical in algebra and beyond!
Fraction Models: Explore different representations for fractions including improper fractions, mixed numbers, decimals, and percentages. Choose from several models (length, area, region, and set) to illustrate the connections between visual representations and their symbolic/numeric forms.
Visual Fractions: This website is one-stop shopping for fractions. It includes lessons, explorations, practice and games, all focused on using visual models to enhance the meaning of fractions and operations with them.
Thinking Blocks with Fractions provides a visual tool for modeling six different flavors of word problems involving fractions. Videos instruct you in how to use the tool.
Kids and Cookies: While not glitzy by today's standards, this Flash applet develops fraction concepts (partitioning, equivalence, and unit) in the familiar context of sharing cookies fairly among friends. It incorporates multiple representations of fraction models with numerical symbols.
Ordering Fractions: Compare and order the given fractions from smallest to greatest. Go to the Testing Room to see a visual comparison of the their values.
Fraction Number Line develops the concept of fractions on a number line and allows a user to find (and understand) equivalent fractions and to compare fractions.
Fraction Game: This game reinforces the number line representation of fractions. The object is to get all of the markers to the right side of the game board, using as few cards as possible.
GapZappers: Apply your knowledge of addition of fractions and equivalent forms to navigate 15 levels of play.
Fraction Feud: Use the given numbers to create a fraction that is larger – or smaller – than the one created by your opponent (the computer). [You may need to select "Guest Pass" and then "Challenge Yourself" under Fraction Feud.]
September: Shape Up!
Long before children are expected to master geometric definitions and formulas, they need to play with and manipulate shapes in two and three dimensions. This month's recommendations provide that experience. While virtual manipulatives cannot completely replace real ones, they do offer special advantages as well as being a bridge between concrete materials and static pictorial representations.
The first three resources are Java-based and come from the National Library of Virtual Manipulatives (NLVM) at Utah State University. Each one includes tabs for Instructions, Parent/Teacher, and, once you've played to your heart's content, challenging Activities. By sliding, flipping and turning the pieces, Pattern Blocks and Tangrams can be made to tessellate – to fit together without gaps or overlapping to cover a surface like pieces of a puzzle. Mathematicians refer to these motions as transformations and label them translation, reflection, and rotation. This type of geometry has applications in how we navigate through everyday life.
Pattern Blocks: The wooden kind are a staple in elementary classrooms and one of the most versatile learning tools I know of. These virtual ones are even more flexible and don't have to be picked up off the floor! Children can use them to explore patterns, fractions, ratios, tessellations, symmetries, perimeter and area.
Tangrams: This electronic version of the classic ancient Chinese puzzle develops spatial reasoning and understanding of geometric relationships. Fourteen puzzles are included, with hints if you need them.
Geoboard: Use the bands to explore concepts of shape, area, perimeter, and more.
Geometric Solids from NCTM lets learners explore the Platonic solids and their properties. They can manipulate and color each shape and investigate the relationship among the number of faces, vertices, and edges.
GeoGebra is an open-ended tool for constructing and manipulating 2-D geometric objects. The software can be downloaded or run through a browser. Once you've created a shape, drag on its sides and vertices to find out what changes and what stays the same – interactions not possible with paper and pencil.
October: Exercising Your Mind's Eye
The following interactivities help develop spatial sense and visualization. Both are critical for success in geometry and in life. They involve imagining how something might look when manipulated or viewed from a different perspective. This is particularly tricky in 3 dimensions, and here's where technology helps. Do you know someone who claims to have no sense of direction? They can improve with practice!
Sliding Puzzle: This version of the classic puzzle allows you to change the size of the grid and tells you how many moves you've made. Your goal is to solve the challenge in as few moves as possible – and generalize a strategy.
About Space: This set of three interactive challenges involve visual perspective and changing points of view in three dimensions. The background information describes the importance of visualization skills and provides examples of how we use it in everyday life.
CubeNets: A net is a 2-D figure that can be folded into a 3-D object. Which of these nets could be folded to make a cube?
A Puzzling Cube displays the adjacent faces of a decorated cube and asks you to create a net of the cube. The Getting Started page links to a printable sheet you can experiment with.
Big Seed: In this free (and addicting) iOS app, Jiji the penguin needs you to reflect seed squares (horizontally, vertically and diagonally), growing larger pieces to perfectly fill all the empty cells.
Plumber Game: Use 90-degree rotations to connect a network of pipes to deliver water. You earn points for speed and economy of steps.
November: Recreational Math – not an Oxymoron!
I've been saving up some just-for-fun math gems. A recent article in the NYT, The Importance of Recreational Math, inspired me to offer them this month. All involve serious math content, but in a playful format. The last two resources build the case for puzzles in the math curriculum – not as dessert, but as the main course!
Duck: Think Outside the Flock: This set of 25 interactive challenges provides practice in problem solving (problem = a situation for which a solution is not readily apparent), logical thinking, and spatial orientation. Your challenge is to figure out what the problem is!
KickBox: Remember JiJi™ the penguin from October? He's back! The goal of this free and addictive iOS app is to position lasers and mirrors to knock all balls out of the penguin's path. The puzzles start off easy, but get progressively more challenging as you move through seven levels.
Bear vs Bee: This interactive Java applet challenges you to solve problems by using logic and rudimentary engineering skills. The goal in each case is to create a sequence that gets the bear to the pot of honey, avoiding the bees. The game has 32 stages of increasing complexity.
Games from Around the World: These 10 strategy games develop spatial skills and strategic thinking. Two of them include interactive versions, but all can be played with simple materials, indoors or out.
Frogs: The goal of this interactivity is to swap the pink and blue fogs to the opposite sides in as few moves as possible –– and to discover a rule for the minimum number of moves necessary based on the starting number of frogs. This version allows the user to adjust the numbers of frogs independently from one to nine. It encourages players to work systematically and look for patterns.
Puzzles and Recreational Math: This collection of 43 challenging and diverse puzzles will keep you busy for a few hours!
Puzzles: This article provides justifications and implementation suggestions for using puzzles in the classroom (or the living room!). It provides links to an article about puzzles and examples of some common puzzle types.
Why Math Education Needs Puzzles: This slide presentation by artist and puzzle designer Scott Kim provides compelling rationale for the incorporation of puzzles into the math curriculum, referring to puzzles as the "literature" of math as contrasted with the "grammar" embodied in the rules and symbols of algebra.
December: Books for Mathy Kids
With the holidays approaching we're departing from our usual suggestions for online resources and recommending some of our favorite children's books with math connections. We've included some suggested ages but in most cases, they can appeal to a wide range, so don't feel constrained.
Why do we encourage using children's trade books to enrich math learning?
If You Hopped Like a Frog, by David M. Schwartz, illustrated by James Warhola; Scholastic.
This book introduces children to concepts of scale and proportion in a fun context. The authors present interesting facts about the physical abilities of various animals and extrapolate them to human proportions. The book includes straightforward mathematical and zoological explanations for each scenario and encourages readers to explore some challenges of their own. (ages 5-12)
Counting on Frank,
by Rod Clement; Houghton Mifflin.
A curious boy and his dog (Frank) create amusing challenges for themselves involving counting, scale, and estimation. We learn that math is useful – and fun! Adults will find plenty of entertainment here. (ages 6 and up)
The Number Devil: A Mathematical Adventure, by Hans Magnus Enzensberger, illustrated by Rotraut Susanne Berner; Henry Holt & Co.
Robert is a young boy suffering from math anxiety in school. Over the course of twelve dreams, the Number Devil teaches Robert interesting mathematical principles that have developed over the course of history. (ages 10 and up)
The Doorbell Rang, by Pat Hutchins; Mulberry Books.
We all know "no one makes cookies like Grandma." Two children are about to share 12 cookies, but then a friend arrives, and then another, .... This book introduces division as equal sharing and offers practice in counting, calculation, and problem solving. (ages 6-9)
Anno's Counting House, by Mitsumasa Anno; Putnam. (ages 5 and up)
A wordless picture book that invites questions about counting and combinations of 10.
Also by Anno and from Putnam:
Anno's Counting Book(ages 6 and up)
A wordless picture book that challenges the reader to observe carefully and try to understand the mathematical relationships illustrated visually.
Anno's Mysterious Multiplying Jar (ages 7 and up)
An imaginative exploration of multiplication and factorials.
Cook-a-Doodle-Doo! by Janet Stevens and Susan Stevens Crummel; Harcourt Brace & Co.
Big Red Rooster (great-grandson of The Little Red Hen) gathers some funny friends to help him make strawberry shortcake. A mess ensues in this hilarious tale. Children will learn about cooking measurements and terms.
G is for Googol: A Math Alphabet Book, by David M Schwartz, illustrated by Marissa Moss; Scholastic.
A witty smorgasbord of math concepts and trivia that will intrigue your budding young mathematician at home, and everyone else as well.
Children are fascinated by big numbers. These books give them meaning:
Math Curse, by Jon Scieszka, illustrated by Lane Smith; Viking. (ages 6 and up)
The teacher, Mrs. Fibonacci, says "You know, you can think of almost everything as a math problem." Thus begins the heroine's adventure.
Where the Sidewalk Ends, by Shel Silverstein; Harper & Row. (ages 6 and up)
This volume contains many mathy poems that lend themselves to problem solving and investigations. Check out "Smart":
My dad gave me a one dollar bill
'Cause I'm his smartest son,
And I swapped it for two shiny quarters
'Cause two is more than one! ...
Several books (all good) make use of a folktale to introduce the concept of doubling (powers of 2). (ages 7 and up)
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