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For math teachers who will have students learning at home next year, fostering a feeling of community will be as important as building skills.

Over this past year, teachers have been continually stressed about how to effectively teach students in different locations simultaneously so that the class feels and operates like a cohesive unit. My district has adopted a hybrid format where some students learn remotely all year, some learn in class (roomers), and the remainder do a little of both (alternate weeks in class). In my role as instructional math coach collaborating with elementary teachers, I’ve been asked to both differentiate instruction and bring students together to provide a sense of normalcy.

Moving forward, how can teachers continue to engage students and encourage learning in school and at home while also sustaining best practices in mathematics? Here are three instructional teaching strategies that promote building community and math skills with a focus on promoting cohesiveness, positive self-efficacy about math, and student achievement.

Streamline Materials for Easy Access at Home

While online educational materials are definitely an essential option, it’s important to also offer parents print alternatives to online resources to ensure equitable opportunities for learning at home. Hundreds charts (to 100 or 120) are versatile and provide countless opportunities for learning and can be easily accessed online or drawn on graph paper. Either way, the charts can then be converted into game boards for different grade levels. Whether students are practicing adding up from any number or subtracting 10, they can use a hundreds chart as a game board for flexibility and ease of access to play games that strengthen foundational skills in mathematics at home and in class.

Many math games are available online, and here are examples of a few I’ve shared with teachers and parents to practice math skills using a hundreds chart with their elementary school children:

Race to 100 by beginning at 1. Rolling two dice (or using an online number generator), find the sum (or product) and practice with running totals. A player gets an extra turn if they land on a number that ends with a 0, switches with another player’s place if they land on a number that ends with a 7, or skips a turn if they land on a number that ends with a 5.

For young elementary school children, use an online spinner to choose a starting point. Practice counting up and skip counting to 120 using the chart.

Find the target number. Use the random number generator to choose a starting number. Players earn points by marking the number that is 10 more or 10 less than the starting number. Continue for 10 rounds. This activity cultivates math fluency in young children.

Using standard game boards at home is another way to play math games that will expand interest in math that goes beyond the daily routine of the elementary classroom. In addition to a hundreds chart, there are other standard math resources, such as easy-to-make number lines (also available online) and various sizes of graph paper that parents can use as game boards at home. Blank standard game boards are also available online at no cost.

Build a More Sustainable Math Community

Supporting regular communication that strengthens academics and also reaches beyond worksheet assignments or test prep contributes to building a more sustainable math community. Highlighting fun classroom activities and continuing that conversation at home sustains student interest and motivation in learning math while also solidifying skills and promoting achievement.

Pairing celebrating home involvement with engaging math can be as simple as students playing a game and capturing the “math moment” at home. I’ve collaborated with teachers to have parents send in photos of playing math games with their children. The photos are then organized into an online math slide show (set to music) for the class to watch. This fosters a cohesive, math environment both in and out of the classroom. During the follow-up class discussion, include dialogue about strategies for playing the math games. Additionally, teachers can inspire students to create their own games based on the standard game board and then share with the class.

Restructure the Math Conversation

Transform the math dialogue by looking for a literacy-math connection. For example, ask students to reflect on the probability of winning after playing a math game at home. Then have them write a “critic’s review.” Beyond academics, create a “math scavenger hunt” where students identify ways that math is an essential part of their everyday routine at home.

It’s important to strive to create opportunities that establish a sense of community regardless of the learning platform—remote, hybrid, in class. Easy-to-implement strategies can help bridge gaps by establishing a dynamic that’s engaging for students, no matter where they’re physically located. 

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Why This Algebra Teacher Has Her Students Knit In Class

One snowy January day, I asked a classroom of college students to tell me the first word that came to mind when they thought about mathematics. The top two words were “calculation” and “equation.”

When I asked a room of professional mathematicians the same question, neither of those words were mentioned; instead, they offered phrases like “critical thinking” and “problem-solving.”

This is unfortunately common. What professional mathematicians think of as mathematics is entirely different from what the general population thinks of as mathematics. When so many describe mathematics as synonymous with calculation, it’s no wonder we hear “I hate math” so often.

So I set out to solve this problem in a somewhat unconventional way. I decided to offer a class called “The Mathematics of Knitting” at my institution, Carthage College. In it, I chose to eliminate pencil, paper, calculator (gasp) and textbook from the classroom completely. Instead, we talked, used our hands, drew pictures and played with everything from beach balls to measuring tapes. For homework, we reflected by blogging. And of course, we knit.

Same but different

One crux of mathematical content is the equation, and crucial to this is the equal sign. An equation like x = 5 tells us that the dreaded x, which represents some quantity, has the same value as 5. The number 5 and the value of x must be exactly the same.

A typical equal sign is very strict. Any small deviation from “exactly” means that two things are not equal. However, there are many times in life where two quantities are not exactly the same, but are essentially the same by some meaningful criteria.

Imagine, for example, that you have two square pillows. The first is red on top, yellow on the right, green on bottom and blue on the left. The second is yellow on the top, green on the right, blue on bottom, and red on the left.

The pillows aren’t exactly the same. One has a red top, while one has a yellow top. But they’re certainly similar. In fact, they would be exactly the same if you turned the pillow with the red top once counterclockwise.

Rotating two square pillows. Sara Jensen

How many different ways could I put the same pillow down on a bed, but make it look like a different one? A little homework shows there are 24 possible colored throw pillow configurations, though only eight of them can be obtained from moving a given pillow.

Students demonstrated this by knitting throw pillows, consisting of two colors, from knitting charts.

The students created square knitting charts where all eight motions of the chart resulted in a different-looking picture. These were then knit into a throw pillow where the equivalence of the pictures could be demonstrated by actually moving the pillow.

A knitting chart for a throw pillow. Sara Jensen

Rubber sheet geometry

Another topic we covered is a subject sometimes referred to as “rubber sheet geometry.” The idea is to imagine the whole world is made of rubber, then reimagine what shapes would look like.

Let’s try to understand the concept with knitting. One way of knitting objects that are round – like hats or gloves – is with special knitting needles called double pointed needles. While being made, the hat is shaped by three needles, making it look triangular. Then, once it comes off the needles, the stretchy yarn relaxes into a circle, making a much more typical hat.

This is the concept that “rubber sheet geometry” is trying to capture. Somehow, a triangle and a circle can be the same if they’re made out of a flexible material. In fact, all polygons become circles in this field of study.

If all polygons are circles, then what shapes are left? There are a few traits that are distinguishable even when objects are flexible – for example, if a shape has edges or no edges, holes or no holes, twists or no twists.

One example from knitting of something that is not equivalent to a circle is an infinity scarf. If you want to make a paper infinity scarf at home, take a long strip of paper and glue the short edges together by attaching the top left corner to the bottom right corner, and the bottom left corner to the top right corner. Then draw arrows pointing up the whole way around the object. Something cool should happen.

Students in the course spent some time knitting objects, like infinity scarves and headbands, that were different even when made out of flexible material. Adding markings like arrows helped visualize exactly how the objects were different.

Knitting to learn. Carthage College

Different flavors

If the things described in this article don’t sound like math to you, I want to reinforce that they very much are. The subjects discussed here—abstract algebra and topology—are typically reserved for math majors in their junior and senior years of college. Yet the philosophies of these subjects are very accessible, given the right mediums.

In my view, there’s no reason these different flavors of math should be hidden from the public or emphasized less than conventional mathematics. Further, studies have shown that using materials that can be physically manipulated can improve mathematical learning at all levels of study.

If more mathematicians were able to set aside classical techniques, it seems possible the world could overcome the prevailing misconception that computation is the same as mathematics. And just maybe, a few more people out there could embrace mathematical thought; if not figuratively, then literally, with a throw pillow.

Sara Jensen is an assistant professor of mathematics at Carthage College. This article was originally featured on The Conversation.

Anonymous Inner Class In Java

Introduction to Anonymous Inner Class in Java

Anonymous Inner Class in Java is an inner class or nested class. An inner class is a class that is present inside an outer class. So an anonymous inner class is an inner class that has no name. It is either a subclass of a class or an implementation of an interface. So if we have to override a class or interface method, we can use an anonymous inner class. In the case of the anonymous inner class, a curly brace is followed by the semicolon.

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Below is the syntax, where the class can be an abstract class or a concrete class or interface.

class t = new class() { public void class_method() { /* some code of the class_method */ } }; Examples of Anonymous Inner Class in Java

Given below are the examples mentioned:

Example #1

An example where we create an interface and a simple class that implements that interface.


package p1; interface Emp { int eid = 2101; void geteid(); } class ImplClass implements Emp { @Override public void geteid() { System.out.print("Employee id is "+eid); } } class Demo { public static void main( String[] arg ) { ImplClass ob=new ImplClass(); ob.geteid(); } }


As in the above code, an interface Emp is created with the geteid() method and eid=2101. ImplClass implements an Emp Interface and provides the definition for the geteid() method. There is no need to write a separate class ImplClass; instead, it can be use as an anonymous inner class.

Example #2

Here we rewrite the above java code to see an inner class working. The ImplClass class is accessible to other classes in the application. However, the ImplClass class’s functionality is not required by the other class in the application. Therefore we need not define an outer class. In addition, an instance of this class is used only once. Therefore, with the help of an anonymous inner class, the ImplClass class’s functionality can be implemented. In the below example, we created the object of the class directly by implementing the Empinterfece.


package p1; interface Emp { int eid = 2101; void geteid(); } class Demo { public static void main( String[] arg ) { Emp ob = new Emp() { @Override public void geteid() { System.out.print("Employee id is "+eid); } }; ob.geteid(); } }

As in the above code, an object of Emp is not created; implicitly, an object of ImplClass(may not be the same name) class is created. Note that the inner class has no name, so the compiler decides a name and creates it, implementing the Emp interface.

Example #3

Java code to create an anonymous Inner class that extends a class.


package p1; class Emp { void dispMesg() { System.out.println("This message is from main class"); } } class Demo { public static void main( String[] arg ) { Emp ob = new Emp() { @Override public void dispMesg() { System.out.print("This message is from child class"); } }; ob.dispMesg(); } }


Example #4

Java code to create an anonymous Inner class that extends an abstract class.

package p1; abstract class Emp { void dispMesg() { System.out.println("This message is from main class"); } abstract void abstrct_method(); } class Demo { public static void main( String[] arg ) { Emp ob = new Emp() { @Override public void dispMesg() { System.out.println("This message is from child class"); } @Override void abstrct_method() { System.out.println("Abstract Method"); } }; ob.dispMesg(); ob.abstrct_method(); } }


Example #5

Java code to create an anonymous Inner class that defines inside constructor or method argument where we will use the predefined Thread class and runnable interface to create and run thread.


package p1; class Demo { public static void main( String[] arg ) { Thread obt = new Thread(new Runnable() { @Override public void run() { System.out.println("Runing child Thread."); } }); System.out.println("Runing main Thread."); obt.start(); System.out.println("Runing main Thread."); } }



An anonymous inner class is an inner class that has no class name. There are different ways to create an anonymous inner class, like extending the concrete class, extending the abstract class, and implementing an interface.

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Binary Search Tree Class In Javascript

Here is the complete implementation of the BinarySearchTree Class −

Example class BinarySearchTree {    constructor() {             chúng tôi = null;    }    insertIter(data) {       let node = new this.Node(data);    // Check if the tree is empty    if (this.root === null) {             chúng tôi = node;       return;    }    let currNode = this.root;    while (true) {    if (data < {             if (currNode.left === null) {          currNode.left = node;             break;          } else {             currNode = currNode.left;          }       } else {          // Set the value here as we've reached a leaf node          if (currNode.right === null) {             currNode.right = node;             break;          } else {             currNode = currNode.right;          }       }    } } insertRec(data) {    let node = new this.Node(data);    // Check if the tree is empty    if (this.root === null) {             chúng tôi = node;    } else {       insertRecHelper(this.root, node);    } } searchIter(data) {    let currNode = this.root;    while (currNode !== null) {       if ( === data) {             return true;    } else if (data < {             currNode = currNode.left;    } else {             currNode = currNode.right;    } } return false; } searchRec(data) {    return searchRecHelper(data, this.root); } getMinVal() {    if (this.root === null) {    throw "Empty tree!"; } let currNode = this.root; while (currNode.left !== null) {       currNode = currNode.left;    } return; } getMaxVal() {    if (this.root === null) {       throw "Empty tree!";    }    let currNode = this.root;    while (currNode.right !== null) {       currNode = currNode.right;    }    return; } deleteNode(key) {    return !(deleteNodeHelper(this.root, key) === false); } inOrder() {    inOrderHelper(this.root); } preOrder() {    preOrderHelper(this.root); } postOrder() {    postOrderHelper(this.root); } } BinarySearchTree.prototype.Node = class {    constructor(data, left = null, right = null) {       chúng tôi = data;       chúng tôi = left;       this.right = right;    } }; function preOrderHelper(root) {    if (root !== null) {       console.log(;       preOrderHelper(root.left);       preOrderHelper(root.right);    } } function inOrderHelper(root) {    if (root !== null) {       inOrderHelper(root.left);       console.log(;       inOrderHelper(root.right);    } } function postOrderHelper(root) {    if (root !== null) {       postOrderHelper(root.left);       postOrderHelper(root.right);    console.log(;    } } function insertRecHelper(root, node) {    if ( < {             if (root.left === null) {          root.left = node;       } else {          insertRecHelper(root.left, node);       }    } else {             if (root.right === null) {          root.right = node;       } else {          insertRecHelper(root.right, node);       }    } } function searchRecHelper(data, root) {    if (root === null) {             return false;    }    if (data < {       return searchRecHelper(data, root.left);       return searchRecHelper(data, root.right);    }    // This means element is found return true; } /** * Takes root and key and recursively searches for the key. * If it finds the key, there could be 3 cases: * * 1. This node is a leaf node. * * Example: Removing F * A * / * B C * / / * D E F * * To remove it, we can simply remove its parent's connection to it. * * A * / * B C * / / * D E * * 2. This node is in between the tree somewhere with one child. * * Example: Removing B * A * / * B C * / / * D E F * * To remove it, we can simply make the child node replace the parent node in the above connection * A * / * D C * / * E F * * 3. This node has both children. This is a tricky case. * * Example: Removing C * * A * / * B C * / / * D E F * / / * G H I * * In this case, we need to find either a successor or a predecessor of the node and replace this node with * that. For example, If we go with the successor, its successor will be the element just greater than it, * ie, the min element in the right subtree. So after deletion, the tree would look like: * * A * / * B H * / / * D E F * / * G I * * To remove this element, we need to find the parent of the successor, break their link, make successor's left * and right point to current node's left and right. The easier way is to just replace the data from node to be * deleted with successor and delete the sucessor. */ function deleteNodeHelper(root, key) { if (root === null) { } if (key < { root.left = deleteNodeHelper(root.left, key); return root; root.right = deleteNodeHelper(root.right, key); return root; } else { if (root.left === null && root.right === null) { root = null; return root; } if (root.left === null) return root.right; if (root.right === null) return root.left; let currNode = root.right; while (currNode.left !== null) { currNode = currNode.left; } chúng tôi =; root.right = deleteNodeHelper(root.right,; return root; } }

Difference Between Class And Object In Oops

Key Differences between Class and Object

A class is a template for creating objects in a program, whereas the object is an instance of a class.

A class is a logical entity, while an object is a physical entity.

A class does not allocate memory space; on the other hand, an object allocates memory space.

You can declare a class only once, but you can create more than one object using a class.

Classes can’t be manipulated, while objects can be manipulated.

Classes don’t have any values, whereas objects have their own values.

You can create a class using “class” keyword, while hand you can create an object using “new” keyword in Java.

Class vs Object

What is Class?

A class is an entity that determines how an object will behave and what the object will contain. In other words, it is a blueprint or a set of instruction to build a specific type of object. It provides initial values for member variables and member functions or methods.

What is Object?

An object is nothing but a self-contained component that consists of methods and properties to make a data useful. It helps you to determines the behavior of the class.

For example, when you send a message to an object, you are asking the object to invoke or execute one of its methods.

From a programming point of view, an object can be a data structure, a variable, or a function that has a memory location allocated. The object is designed as class hierarchies.

Class vs Object – Difference Between Them

Here is the important difference between class and object:

Class Object

A class is a template for creating objects in program. The object is an instance of a class.

A class is a logical entity Object is a physical entity

A class does not allocate memory space when it is created. Object allocates memory space whenever they are created.

You can declare class only once. You can create more than one object using a class.

Example: Car. Example: Jaguar, BMW, Tesla, etc.

Class generates objects Objects provide life to the class.

Classes can’t be manipulated as they are not available in memory. They can be manipulated.

It doesn’t have any values which are associated with the fields. Each and every object has its own values, which are associated with the fields.

You can create class using “class” keyword. You can create object using “new” keyword in Java

Understand the concept of Java Classes and Objects with an example.

Let’s take an example of developing a pet management system, specially meant for dogs. You will need various information about the dogs like different breeds of the dogs, the age, size, etc.

You need to model real-life beings, i.e., dogs into software entities.

Moreover, the million dollar question is, how you design such software? Here is the solution-

First, let’s do an exercise.

You can see the picture of three different breeds of dogs below.

Stop here right now! List down the differences between them.

Some of the differences you might have listed out maybe breed, age, size, color, etc. If you think for a minute, these differences are also some common characteristics shared by these dogs. These characteristics (breed, age, size, color) can form a data members for your object.

Next, list out the common behaviors of these dogs like sleep, sit, eat, etc. So these will be the actions of our software objects.

So far we have defined following things,

Class: Dogs

Data members or objects: size, age, color, breed, etc.

Methods: eat, sleep, sit and run.

Now, for different values of data members (breed size, age, and color) in Java class, you will get different dog objects.

You can design any program using this OOPs approach.

Classes and Objects in Java

In the below program, we have declared a class called Dog. We have defined an object of the class called “maltese” using a new keyword. In the last statement System.out.println(maltese.getInfo()); we are displaying dog information like Breed, Size, Age, Color, etc.

class Dog { String breed; String size; int age; String color; public String getInfo() { return ("Breed is: "+breed+" Size is:"+size+" Age is:"+age+" color is: "+color); } } public class Execute{ public static void main(String[] args) { Dog maltese = new Dog(); maltese.breed="Maltese"; maltese.size="Small"; maltese.age=2; maltese.color="white"; System.out.println(maltese.getInfo()); } }


Breed is: Maltese Size is: Small Age is:2 color is: white

Types of Class

Following are the important types of class:

Derived Classes and Inheritance

A derived class is a class which is created or derived from other remining class. It is used for increasing the functionality of base class. This type of class derives and inherits properties from existing class. It can also add or share/extends its own properties.


A superclass is a class from which you can derive many sub classes.


A subclass is a class that derives from superclass.

Mixed classes

A mixed class is one more functionality that helps you to inherit the properties of one class to another. It uses a subset of the functionality of class, whereas a derive class uses the complete set of superclass functionality.

Uses of Class

Here are the important uses of class:

Class is used to hold both data variables and member functions.

It enables you to create user define objects.

Class provides a way to organize information about data.

You can use class to inherit the property of other class.

It can be used for a large amount of data and complex applications.

Use of Object

Here are the important uses of an object

It helps you to know the type of message accepted and the type of returned responses.

You can use an object to access a piece of memory using an object reference variable.

It is used to manipulate data.

Objects represent a real-world problem for which you are finding a solution.

It enables data members and member functions to perform the desired task.

Apple Updates App Store Guidelines In Response To Class

Apple is announcing a trio of updates to the App Store Guidelines today that focuses on giving developers more flexibility for communicating with their customers. One of the changes specifically comes to address Apple’s settlement in a class-action lawsuit from small developers in the US, first announced back in August.

As background, Apple promised a handful of small App Store changes back in August as part of its settlement with small developers in the United States. One of the most notable changes announced at the time was that developers would be able to communicate outside of their applications about alternatives to Apple’s in-app purchase system.

In line with this settlement, Apple says that it is updating guideline 3.1.3 to remove its previous restriction on using customer communication information obtained with an application to communicate outside the application about payment methods other than in-app purchase. Apple says this update satisfies the terms of the settlement.

Here is the wording of guideline 3.1.3 prior to today’s update (emphasis ours):

The following apps may use purchase methods other than in-app purchase. Apps in this section cannot, within the app, encourage users to use a purchasing method other than in-app purchase. Developers cannot use information obtained within the app to target individual users outside of the app to use purchasing methods other than in-app purchase (such as sending an individual user an email about other purchasing methods after that individual signs up for an account within the app). Developers can send communications outside of the app to their user base about purchasing methods other than in-app purchase.

And here is the wording of guideline 3.1.3 following today’s revision:

The following apps may use purchase methods other than in-app purchase. Apps in this section cannot, within the app, encourage users to use a purchasing method other than in-app purchase. Developers can send communications outside of the app to their user base about purchasing methods other than in-app purchase.

To go along with this update, Apple is also adding a new guideline saying that apps may request basic contact information from users, with some restrictions:

Apps may request basic contact information (such as name and email address) so long as the request is optional for the user, features and services are not conditional on providing the information, and it complies with other provisions of these guidelines, including limitations on collecting information from kids.

In practice, this means that developers will be able to collect contact information from users (as long as it’s not required), then use that contact information to communicate outside of their application (such as with email), about more purchase and payment options that are perhaps cheaper than in-app options.

It’s important to keep in mind that changes are solely in response to the class-action lawsuit from US developers that Apple settled in August. These changes do not put Apple in compliance with the Epic Games ruling or the ruling from the Japan Fair Trade Commission. While the Epic Games case is being appealed, Apple says that it will update its App Store guidelines to accommodate the JFTC ruling in early 2023.

Apple is also making tweaks to the App Store guidelines to accommodate In-App Events, as we reported earlier this week. You can read the full App Store Review Guidelines on Apple’s Developer website here.

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