Number 1: Educreations - With Educreations students can explain their thinking as they work through and solve math problems. When teachers/coaches watch a student's creation, they can figure out where a particular student needs assistance.
Student Example - In this example the teacher will be able to have a conference with her student about finishing the last step of the problem or she could ask her student to watch his Educreations creation in order to figure out what he might have missed.
How to Use Educreations Tutorial - A nice video tutorial covering how to use Educreations.
Alternative and similar Apps to Educreations(free): Explain Everything(paid) and ShowMe(free).
Number 2: ScribblePress - Using ScribblePress students can create word problem books, a book of algorithms, a book of shapes, etc. ScribblePress books can be shared to iBooks for anyone to read on the iPad or emailed to parents to open up on any device from home. A perfect way to share what students are learning while at school. In order to share, you will need to create a free ScribblePress account.
Student Example - In this example a third grade student explains the steps for multiplying two 2 digit numbers.
How to Use ScribblePress Tutorial - Tutorial by third grade students.
Alternative and similar Apps to ScribblePress(paid) - StoryKit(free) and BookCreatorFree(free).
Number 3: TenMarks - TenMarks(free) is not an app, but can be used through a browser on an iPad like Safari. The data that is provided to a teacher when students are using TenMarks is phenomenal. Almost every student I have worked with on TenMarks are highly motivated to do well because of the great way the site is set up. Students can earn games and certificates. Teachers can individualize assignments for each student and students don't really know that one student is doing something more difficult or easier than another student. If a student is struggling with a question, they can choose to read a hint or watch a tutorial video. It really helps students become independent learners. Plus TenMarks aligns their assignments with the common core standards.
Number 4: Sail Through Math(free) - Every once in awhile you need a good practice app and this one delivers. Sail Through Math is a fun way to become fluent with math facts. Students have to click on the correct answer to a fact problem as the answers shoot out of canons on a pirate ship. If teachers want evidence of learning, have students take a screen shot of their score and email it to their teacher or upload the screen shot into student digital portfolios, like Evernote(free) or Google Drive(free).
To take a screen shot hold down and release the home & sleep buttons at the same time.
Number 5: StoryMe(free) - With StoryMe students can create fun comic strips demonstrating their understanding of a math standard. I have yet to use this app for math, but I have used ToonDoo on laptops with students to create math comic strips and the students did a great job with demonstrating their understanding of fractions and other math concepts. StoryMe is one of my new favorite apps, but since it is still so new to me I have only used it with classes as a language arts tool.