Apps for Numeracy and Planning for Differentiated Instruction is a resource that will provide an overview of various iOS apps and additional resources that may help to plan learning activities for students. The goal of this resource is to support educators in selecting and using apps to support their students’ numeracy development. This resource will explore a variety of iOS numeracy options, as well as how to both build implementation strategies to support inclusion, and plan for Differentiated Instruction (DI). Examples in the videos throughout the modules will demonstrate the capability and power of apps to support students in demonstrating mathematical concepts, with the opportunity to show their thinking in a creative and engaging way.

#### Module 1 - Numeracy and the Redesigned Curriculum

The redesigned BC Curriculum focuses on numeracy and literacy as foundations of learning and suggests that both are applied in all areas of learning. The redesigned curriculum offers opportunities for educators to engage students in new ways in order to develop skills and competencies to better interact with our changing, technology-driven world. The impact the redesigned curriculum may have on the design of learning experiences for students may result in student centered and flexible environments. Additionally, there is emphasis on deeper learning by way of concept-based and competency-driven methodologies.

Math Class Needs a Makeover (www.ted.com)

**Math and First Peoples Principles:**

- New App Uses Indigenous Basketry to Teach Math Concepts (Youtube.com)
- British Columbia Association of Math Teachers – First Peoples Resources (BC Association of Math Teachers)
- Five Principles for Supporting Literacy and Cultural Awareness in Mathematics; BC Association of Math Teachers (BC Association of Math Teachers)
- Raven by Gerald McDermott – Stories for Kids – Children’s Books Read Along Aloud (Youtube.com)

BC’s New Curriculum recognizes Differentiated Instruction as a “flexible approach to teaching in which a teacher plans and carries out varied approaches to address content, learning processes, learning style, practical procedures, presentation strategies, and assessment tools. It results in a more personal, proactive learning environment, inclusive of a wide variety of learners” (Curriculum Overview, Building Student Success).

So what is Differentiated Instruction (DI) anyway?

Differentiated Instruction (BCTF)

An important aspect of mathematics learning is engaging students. This is often a subject where students feel disconnected and thus give up easily. If we are able to peak their interest, this may be the key. Brain research tells us that there is a link between emotional state and the ability to learn. If students are interested and excited to do mathematics assignments they will likely learn more. We can facilitate this by giving students choice leading them to feel a sense of control (Dacey & Gartland, 2009). iPads are a tool that may give students choice and increase interest.

Dacey & Gartland (2009) list questions to think about when planning for mathematics instruction with differentiated instruction:

- What is the mathematics I want my students to learn?
- What do my students already know? What is my evidence of this? How can I build on their thinking?
- How can I expand access to this task or idea? Have I thought about interests, learning styles, use of language, culture, and readiness?
- How can I ensure that each student experiences challenge?
- How can I scaffold learning to increase the likelihood of success?
- In what different ways can my students demonstrate their new understanding?
- Are there choices students can make?
- How prepared am I to take on these challenges?

- Differentiated Instruction (BCTF)
- Building Student Success – BC’s New Curriculum (BC Ministry of Education)
- Mathematics-Building Student Success – BC’s New Curriculum (BC Ministry of Education)
- Instructional Samples – Building Student Success – BC’s New Curriculum (BC Ministry of Education)
- Core Competencies – Building Student Success – BC’s New Curriculum (BC Ministry of Education)
- New App Uses Indigenous Basketry Patterns to Teach Math Concepts SFU News – Simon Fraser University

#### Module 2 - Virtual Manipulatives

The iPad is a tool for utilizing virtual manipulatives in mathematics, especially if the classroom lacks physical manipulatives such as coins, base ten blocks, attribute blocks, colour tiles, fraction circles, fraction tiles, geared clocks, and geoboards. Students can explore a multitude of math manipulatives through free and paid apps on the iPad to enhance their numeracy experience. Having virtual manipulatives available helps educators to differentiate instruction to support levels of students in their classrooms. In this module we will explore the power of apps and look at examples to support numeracy in our classrooms with virtual manipulatives.

This video explores the use of math manipulatives.

- Math Manipulatives – Youtube.com (Learning Disabilities of Ontario)

The following 10 free apps come from The Math Learning Center :

Fractions

Geoboard

Math Vocabulary Cards

Money Pieces (American money)

Number Frames

Number Line

Number Pieces

Number Pieces Basic

Number Rack

Pattern Shapes

**Additional Free Apps:**

Virtual Manipulatives (Fractions, Decimals, Percents)

In *Good Questions, Great Ways to Differentiate Mathematics Instruction*, Small (2012) discusses two core strategies for differentiation in math instruction: open questions and parallel tasks (pg. 6). When we ask open questions like “Describe the following picture by creating an equation” rather than a very specific, direct question like “To which fact family does the fact 3 X 4 = 12 belong?” we allow for students to show what they know and be creative, producing many different answers. Students learn from each other and have the opportunity of seeing how others solved the problem. The opportunity for collaboration is greater when we ask open questions. Asking questions this way accounts for varying abilities, rather than eliminating students who don’t how to complete the problem through multiplication.

Look at the pictures demonstrating answers to an open question, “Describe the following picture by creating an equation”. Students can use various manipulatives apps to demonstrate how they would describe an image of twelve objects. See examples below. Farmyard (addition), Pieces Basic (multiplication), Number Frames (division), and Pattern Shapes (combined operations).

This image uses Farmyard (addition) to describe twelve objects. Students use the pen tool built into the iPad to markup their screenshot from Farmyard.

This image uses Pieces Basic (multiplication) from The Math Learning Centre to describe twelve objects.

This image uses Number Frames (division) from The Math Learning Centre to describe twelve objects.

This image uses Pattern Shapes (combined operations) from The Math Learning Centre to describe twelve objects. Students with a more complex understanding of fact families may choose to demonstrate combining operations like this.

When students have completed marking up their picture, the teacher can project the different examples to demonstrate twelve using an equation. Another option for students is to use the built in screen recording to add an audio description of how they demonstrated their answer.

Besides open questions, the other core strategy that Small (2012) discusses are parallel tasks, which allow students at all developmental levels to participate. It also creates a community learning environment in the classroom where all students are able to contribute. The example below elicits this strategy because students have choice in which numbers they would like to divide. Afterwards students can all answer questions like:

- Which numbers did you choose to divide?
- What made your problem a division problem?
- What was your problem?
- How did you solve it?

**Division using Base 10 Blocks** – Students in Grade 3/4 were having difficulty with division concepts, specifically the Learning Standard in the Grade 4 content: division of two- or three- digit numbers by one digit numbers. They used base 10 blocks in either Manipulatives by Braining Camp (paid) or Pieces Basic by The Math Learning Centre (free) to show their chosen division statements. After completing this activity those students that were having the most trouble were better able to grasp the concept of division with larger numbers as they could physically see the objects grouped. Students could choose their own equation so if they had a better understanding of division they chose larger numbers. The Grade 3 students who had a good understanding of their Learning Standard in Content: Multiplication and Division Concepts, could easily work on the Grade 4 content piece. It was important for students to show their steps and take screenshots to put in a math story, shown below.

## App Smashing

Math Stories using Virtual Manipulatives and Book Creator

The following videos were created using the screenshots the grade three and four students took using their division statements with base ten blocks and Book Creator. The cross curricular application between literacy and numeracy created an engaging learning experience for students.

## Your Turn!

**Try this: **

Using the Number Frames app from The Math Learning Centre, show different combinations of making the same number. Use the blue and red counters (inside of the number frames) to distinguish numbers being added together. Test out the pen or text tool and write your statements under your diagrams. For a full lesson on this take a look at *Innovate with iPad* (Lirenman & Wideen, 2016, p.20) and the lesson titled “Combinations of Twenty”.

**Try this:**

Using the Geoboard app from The Math Learning Centre to show equivalent fractions.

- Use the rubber bands to show different fractions equivalent to one-half.
- Shade the fraction inside the rubber bands using the pen tool.
- With the pen tool, write the fraction next to each picture.
- Take a screenshot of your work if you would like to save it to the camera roll for later use.

For a full lesson on this take a look at *Innovate with iPad* (Lirenman & Wideen, 2016, p.28) and the lesson titled “Equivalent Fractions”.

## Augmented Reality- Creating Virtual Manipulatives

Using Quiver, students can experiment with augmented reality after colouring a handout. (available for download http://www.quivervision.com/coloring-packs/). There are free and paid colouring sheets, and examples of how students’ art can come alive. After saving the video file that is recorded in Quiver, it can be put in another app like Book Creator or Explain Everything. Many of the colouring sheets can be adapted for a mathematics problem. The pack of Platonic Solids (approx. $3.99) is great for exploring geometric shapes (http://www.quivervision.com/education-coloring-packs/#quivervision-platonic-solids).

Student examples:

## Youtube Videos

- The Number Pieces App (The Math Learning Center)
- Modeling Subtraction with the Number Pieces App (The Math Learning Center)
- The Number Frames App (The Math Learning Center)
- The Math Vocabulary Cards App (The Math Learning Center)
- The Number Line App (The Math Learning Center)
- The Number Rack App (The Math Learning Center)

## Task Card:

#### Module 3 - Interactive Whiteboard Apps

Interactive whiteboard apps are powerful tools for all grades. In this module we will explore free and paid apps that can support students’ numeracy experiences. Apps of this type provide opportunity for students to be creators using multimedia. The use of pictures and videos in these apps engages students on many levels. However, the question exists – “how to move from having kids pinch, swipe, and tap on consumption apps to creating evidence of learning” (Lirenman & Wideen, 2016, Forward). A blank whiteboard may intimidate a student; however, through this module you will see that we can support students along the way with these types of apps, especially in the area of numeracy.

**The Power of Templates – Explain Everything**

In this Kindergarten example using Explain Everything and the iPad, we see the power of creating a multi-lesson project using templates with an interactive whiteboard. Students are able to build their understanding of the numbers one to ten as they discover these numbers in the world around them.

As shown in the following video and pictures, the first step for students was to create their assigned number with objects in their classroom. They took pictures of their number with the iPad camera and inserted the picture on top of the number in the template.

When students are ready, they explore their school and outdoors to gather pictures of their numbers. Once these are gathered and added to the project, results can be reviewed as a class. This is a great opportunity to discuss if the numbers gathered look exactly like the text numbers in the template. In the above example, you will see that students at this age will often reverse numbers like 2, 3, or 5. Having them side by side in the project, students can clearly see if it is correct or incorrect. At this time, teachers can model the proper writing of the number with the whole class and then students can edit their projects and add the correct numbers. Later in the year, students can take pictures of groups of objects aligned with their given number. You can create a class book and export the final project as a PDF and print a good copy for all students. If you prefer a digital file, students can add audio descriptions for each number and save the file as a video.

Educators can create similar multi-lesson projects that can include pictures and videos taken on the iPad throughout the year. Students can add items to a project in Explain Everything throughout the year.

## The Power of Interactive White Boards

The following samples show how using open questions allows for students to be creative in their answers and all students in the class can answer the question.

Question: In groups of 2-3, use an iPad app to create a word problem where the response is 35.

Note: Both respondents decided to use the app Show Me.

## Explaining Mathematical Thinking

Interactive whiteboard apps are a powerful tool for students to explain their mathematical thinking. Traditionally this has been done on paper with students writing out each step in an equation. The following example demonstrates Grade 8 students explaining their process of solving linear equations of their choosing.

#### Module 4 - Other Math Apps

This module will explore more math apps designed to help differentiate instruction and engage students.

The following apps may support mathematics instruction and planning for differentiation.

- Desmos – Beautiful, Free Math – an Interactive Graphing App
- GeoGebra Math Apps – an Interactive Graphing App
- Photomath – Scan.Solve.Learn – a Photomath App
- Popplet – Organization and Mind Map Creator
- Pic Collage – a Photo Collage Editor App
- Draw and Tell – Draw, Tell, and Share a Story
- Prodigy Math Games – Math Games Grades 1-8
- Khan Academy – Free Online Courses and Practice

**Examples:**

Representing a number using the app Draw and Tell.

Using the app Pic Collage to represent a number- note the duplicate button.

Using the app Pic Collage to represent a number.

Using the mind mapping app Popplet for students to demonstrate a number. Lesson idea taken from Innovate with iPad; lesson title Getting to Know a Number (page 11).

- Desmos – Beautiful, Free Math – an Interactive Graphing App
- GeoGebra Math Apps – an Interactive Graphing App
- Photomath – Scan.Solve.Learn – a Photomath App
- Popplet – Organization and Mind Map Creator
- Pic Collage – a Photo Collage Editor App
- Draw and Tell – Draw, Tell, and Share a Story
- Prodigy Math Games – Math Games Grades 1-8
- Khan Academy – Free Online Courses and Practice