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Mr G Online

Archive for iPads in Mathematics

Jun 09

In my role as Maths Leader in Grades 5/6, I have many opportunities to work with groups of children in both Grade levels. Sometimes I find it hard to report back to the classroom teachers what learning took place during my lessons with their students. This year I have increasingly turned to Padlet, a collaborative, interactive Online Board, to record the teaching and learning experiences I facilitate. Here I share my lesson documentation through an embedded link to my Volume Padlet (note: Padlet needs the latest version of IE, Chrome, Firefox or Safari to view and use)

My basic use of Padlet follows this structure:

  • I post the outline of Tasks to be attempted during the lesson.
  • I add the initial image resources and models/examples that are needed for the task.
  • Students scan the Padlet-created QR Code to quickly open up the Padlet on their iPads.
  • The students start working on the task as outlined by the instructions on the Padlet and begin recording their responses. With the online Padlet wall visible to everyone on the iWB, students can start responding to what others are recording and as a teacher I can monitor from anywhere in the room or on my iPad and identify students to support or extend.
  • I pause during and after each task and invite students to share their responses. As they are already recorded on the shared Padlet, no time is wasted waiting for them to rewrite their work. As a class we can utilise all the time on collaborating, sharing, discussing and questioning.
  • If tasks involve using physical or digital resources, the students can quickly post screenshots, photos or images straight onto the Padlet wall on their iPads or laptops. Using a range of familiar iPad apps, children can record and/or annotate their working out and post it straight to the wall.
  • At ant time during the lesson, with constant access to all of the work being done by the students through the visible workspace on the iWB, I can reconnect with the students and offer feedback, teaching support or ask questions to call on children to explain their learning.
  • When the students leave me, I can immediately post the Padlet wall with all of the students’ learning documented onto their class blogs for their teachers and parents to view.

This particular lesson, embedded below, began with students viewing four rectangular prisms of varying dimensions. The students were asked to order the objects from largest to smallest and justify their decisions. In a traditional classroom setting, a teacher may call on 3-4 students to share their opinions and move on without having a true indication of the other students’ understanding. In using Padlet, I have an easily accessible, permanent record of all of the students’ understanding of volume concepts.

TASK 1

TASK 1

The next task was to verify their conjectures by calculating the volumes of each prism. This particular group of students were high achievers and needed little assistance in calculating the volumes ( the LxWxH formula was not the focus of the lesson, anyway but with a second group of students, I needed to do some revision and monitor progress). They were asked to record their working out directly to Padlet, with the option of recording the detailed calculations on Explain Everything and posting screenshots of the work. This group were able to simply write their calculations directly into Padlet. This provided a record of their work for their teachers to see later and was also a way for me to view their capabilities on screen in case I needed to assist. This was not needed with this group, but with the second group I was able to identify students with gaps in their learning simply by viewing their work on the Padlet wall.( At no stage did any student notice what others were doing – they were engaged in their own work.) What was also good to see was the variety of ways students calculated the volumes in terms of selecting which numbers to multiply first. This initiated a discussion about factors and the commutative/associative laws for multiplication. With all possible combinations visible rather than the 3-4 examples that would have been shared in a traditional setting, we were able to enhance the understanding of the range of dimensions that can result in the same volume. This also allowed them to refer back to their initial misconceptions of volume ( taller is bigger, etc) and led to a quicker transition into the final task.

TASK 2

TASK 2

Now that they had come to the realisation that there are many ways to construct a box of the same volume, we moved onto the final task which was constructing prisms of varying dimensions that would make a volume of 72 cubic units. At this point, they were introduced to an already completed example of the final product I was expecting of them ( which was already embedded on the Padlet wall, but out of view until needed) and the iPad apps available for the task – Think 3D and Skitch. They were also given the option of using physical blocks if they preferred a more tactile method. The simplicity of the apps required little instruction and the students were quick to start experimenting, further developing their understanding of the Volume formula by constructing rather than just calculating. The idea of factors were utlilised as they constructed layers based on the factors of 72. Again, with the use of the Padlet wall, students were able to post their annotated ( using Skitch)  constructions directly on to the wall, providing a record of their work that can be accessed in the future. Seeing other students’ constructions on the wall enabled students to consider other possibilities and further built on their understanding of different dimensions, same volume, which they were then able to reflect on later when the wall was embedded on their class blog. Having the lesson documented on line means that students also have the opportunity to add to the wall later on at home and explain their work to their parents.

TASK 3

TASK 3

I see many benefits in this process of documenting the learning and not just in Mathematics.

  • In this new era of collaborative teaching, it’s a great way of recording a lesson for other members of the team to view.
  • As a Maths leader/mentor, it’s a useful way of modelling a lesson for teams to discuss.
  • For students, it gives them access to previous learning that they can revisit at different times of the year to review/revise and support their learning
  • For assessment purposes, it can provide a record of the different stages of learning that took place during a lesson or series of lessons.
  • the use of Padlet itself opens up personalised access for students to work at their own pace ( not evident in this lesson as it was more of a guided lesson rather than an independent task)

This week, I was involved in a school based ICT Conference at my own school, during which several teachers led workshops on various ICT tools and practices. I presented this lesson structure and use of Padlet to the staff and they saw great possibilities. I am going to continue to develop a range of learning experiences using this documenting method. I see it having great benefits in enhancing the learning at our school.

Below is the whole Padlet wall as developed during this lesson. (If it is not displaying, it is likely you are running an old version of IE, as mentioned above)

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Apr 28


The two above images are good examples of purposeful thinking about iPad usage in schools.

One, a screenshot of an oft-used tool known as iPad As.. by edtechteacher.org, focuses on what the iPad can be used for and provides links to various apps that can be utilised for those functions. It goes without saying that it is a very useful website for schools thinking about iPads. It provides nutshell explanations of a number of apps that relate to each iPad as… category as well as pricing. It’s a good introduction into the functionality of the iPad that counteracts the misconception of iPad as consumption NOt creation tool.

The other, The Padagogy Wheel, is one of many variations on applying Bloom’s Taxonomy of skills to iPad apps. It develops from the general learning action verbs/skills we want our students to acquire to technology based activities that relate to these skills and finally to a selection of apps that can support this development.

Both tools have supported my reflection on iPad use in school and are worth checking out in detail. Having said that, though, I feel they both fall short in what is needed as a resource for implementing iPads in education. iPad as… does a good job at presenting uses for iPads in school – what they can be used for – but doesn’t really provide depth about the skill development that can arise from their use. It’s still action/activity emphasis rather than pedagogical/learning emphasis. It’s great to know that you can create videos, and it describes what the app can do,  but how will this improve learning and what learning will it improve is also a priority iPad schools need to address. I think it also pigeon-holes apps as one trick ponies – I’d like to emphasise the apps that can be used to develop many skills.

The Padagogy Wheel provides many links between skills and tech activities but doesn’t really address what iPad apps address which skills and activities specifically other than lumping them into a particular category. It too, tends to classify the apps as one trick pony options rather than seeing them as multiple category options.

Don’t get me wrong, I think both are great tools but there is room for improvement in creating a tool for supporting time poor iPads in Schools implementers in planning, selecting, justifying and integrating iPad apps in education.

Which leads me to attempt a herculean task… I’m going to try to blend the best of both of these resources and address the short falls I have mentioned by creating my own resource. But it’s going to be a work in progress for a while and I hope to get support from Mr G Online followers, subscribers, users and casual visitors.

I’ve started creating a table of important skills, some derived from the Padagogy Wheel, and actions, some derived from iPad As… What I am planning to highlight is that there are many apps that can be use for many purposes and for developing many skills. For example, I have already added “Explain Everything” to 9 categories as I see it as a multifunctional app and one worth its price because of the educational benefits it provides. Over the coming months I plan to add text descriptions to each category to explain how the apps listed address the skill or action they have been linked to and may also link them to other online sources that show them in action. I’ll also provide direct links to the App Store, as I always do on this blog when I mention apps so you can check them out yourself if you want.

Now this sounds like a big task and it is. So I do need some help. What do I want from you? Anything you can give. Just add them to the comments of this post.

  • Examples of apps that help to develop specific skills
  • Additional skills I haven’t listed here
  • Examples of apps that are multifunctional.
  • Explanations of good pedagogical practice with apps. Don’t worry, all credit will go to you when I include your suggestions.
  • Links to blog posts, websites, Youtube tutorials, open wikis, nings etc that promote good practice that I can link to from here.
  • Examples on add ons like bookmarklets for curation sites, websites that work well with iPads ( Flash-free) that can still be categorised under these headings for iPad use.
  • Spread the word regularly through Twitter, Facebook, Curation sites like Pinterest and Scoop-It to keep educators coming back.
This post will look messy for a while as new ideas get added. A blog may not be the best storage place for it in the long run. If I actually get the support – and it’s likely I won’t – and it grows I will probably move it to a separate website for better functionality. It may well be better as a wiki but  I didn’t want to move away from Mr G Online unless I needed. For easy access in the meantime, I will add this post to my main menu at the top of the blog so you can come back to check revisions. I will be planning weekly updates at least, more if I get regular contributions I can just copy and paste in from the comments.
I really hope I can get this off the ground. From reading so many blog articles, I can see there is a huge need for clarity in using tech like iPads. If you have been a regular reader of Mr G Online, you would know I am a big proponent of Pedagogy before Technology. That’s why I want iPurpose before iPad. Hope to hear from some of you soon.

 

IPAD AS VIDEO CREATION IPAD AS AUDIO RECORDING

iMovie Pinnacle Studio VideoScribe HD iStopMotion GarageBand  TagPad  Evernote  Notability

Explain Everything Art Maker Animation Desk iMotion HD AudioBoo
 Whether creating live action videos with iMovie and Pinnacle Studio, animated stories with iStopMotion, Animation Desk and iMotion HD or how to tutorials with Explain Everything, the iPad is a great tool for video creation. Creating videos with these apps develops organisation and planning skills, supports story telling skills in non writers and enhances creativity and problem solving in many ways.
 IPAD AS BOOK PUBLISHER  IPAD AS DIGITAL STORYTELLING
StoryWheel
Book Creator Creative Book Builder StoryWheel    Sonic Pics Explain Everything Toontastic Storify
Video Scribe HD
Providing opportunities for authentic writing with a real audience outside the classroom, publishing real books using the iPad can improve motivation and actual writing skills. With sufficient access, tech based writing can employ the editing capabilities to encourage children to write without worrying about rewriting from scratch. With the real possibility of publishing books online or in the iBookstore for others to read, children will be encouraged to put more effort into editing and improving their written work. The possibilities for multimedia additions allows for more creativit There are more ways of telling stories these days than text and pictures. Some students have stories inside them that don’t get shared because of a lack of writing ability. Let’s give them opportunities to tell stories orally until they are ready to write so that they can develop their imaginations and story telling for when they are ready to write. These apps all allow for alternatives to traditional writing texts, either through combining audio and images seamlessly in a variety of formats
 IPAD AS GRAPHIC NOVEL CREATOR   IPAD AS READING SUPPORT
 Strip Designer Comic Life     Book Creator iPrompter Creative Book Builder iBooks
Explain Everything
Creating stories with audio, highlighted annotations, vocabulary support through linked dictionaries, scrolling screens provides support for students who lack reading skills. Getting children to record themselves reading gives them feedback on their progress as well as support for independent practice.
 IPAD AS COLLABORATION TOOL  IPAD AS PRESENTATION TOOL
Edmodo VoiceThread Skype Evernote Keynote  VideoScribe  Haiku Deck   VoiceThread
Instapaper Whiteboard Popplet Comic Life  Explain Everything  Skitch   iPrompter
  IPAD AS A WRITING TOOL  IPAD AS BRAINSTORMING TOOL
Comic Life  Writing Prompts SpellBoard Tap Dictionary iMind Map 3D  Popplet  Skitch Inspiration Maps Lite
Notability
Notability Whiteboard
  IPAD AS INFORMATION COLLECTOR  IPAD AS INFORMATION MANAGER/ORGANISER
 
 Evernote Edmodo   PollDaddy Socrative   EverNote  Edmodo Pinterest  Instapaper
Notability
Notability  Notability
 IPAD AS NOTE TAKER  IPAD AS PROBLEM SOLVER
 Notability Hopscotch
 Skitch  Evernote  Notability    Wolfram Alpha Numbers  Hopscotch
 IPAD AS GRAPHING TOOL  IPAD AS RESEARCH TOOL
Wikinodes Notability
 Numbers  Wolfram Alpha  Doodle Buddy    Wolfram Alpha PollDaddy  WikiNodes Notability
 IPAD AS DATA COLLECTION TOOL  IPAD AS A ROLE PLAYING TOOL
 
Edmodo  PollDaddy   Socrative Numbers  Edmodo   Puppet Pals    
TagPad Evernote EasyTag
IPAD AS A CLASS MANAGEMENT TOOL IPAD AS AN ASSESSMENT TOOL
ClassDojo  Notability
Edmodo  Socrative   ClassDojo   Explain Everything   Edmodo Socrative  Notability 
IPAD AS A MAPPING TOOL IPAD AS A CALCULATING TOOL
Screen Shot 2013-04-23 at 8.32.29 PM   
Routes Explain Everything Skitch Geocaching Numbers Wolfram Alpha MyScript Calculator
MyMapsEditor
My Maps Editor
IPAD AS DEMONSTRATION TOOL IPAD AS COMMUNICATION TOOL
Skitch Explain Everything  Skype    Edmodo  Skype
IPAD AS AN ARTISTIC TOOL IPAD AS A DESIGNING TOOL
ArtRage Garageband Snapseed RoomPlanner
ArtRage GarageBand  Snapseed iStopMotion Skitch  Explain Everything   RoomPlanner iDraw
Phoster ScrapPad
IPAD AS AN EXPERIMENTING TOOL IPAD AS A DEBATING TOOL
Hopscotch  
Explain Everything   Numbers Hopscotch     Edmodo  VoiceThread Skype  iPrompter 

 

 

 

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Apr 01

20130401-120744.jpg

Around this time last year, I wrote a post about the lack of engaging Maths apps on the iPad that went beyond “skill and drill” number activities. Since then, developers have introduced a greater range of apps across all areas of the Maths curriculum that can be used to enhance the Maths teaching and learning in your classroom. Here’s a selection of 20 apps that cover Number and Algebra, Measurement and Geometry, and Statistics and Probability ( these are the Content strands (CS) Australia’s Mathematics curriculum has been categorized under ). They also cover the proficiency strands (PS) of Understanding, Fluency, Problem Solving and Reasoning. I’m sure other countries’ curricula are similar in many regards and you will be able to make the connections.

Undecided (free at time of writing)

A handy tool for probability experiments, Undecided comes with customisable dice ( up to six) with number of rolls, last roll and sum of rolls data, Heads/tails coin toss with cumulative tallies, a 1-10 spinner ( wish it was customisable) , Rock/Scissors/Paper and Short Straw simulations and a random number generator with customisable maximum number beyond thousands (although it’s time consuming to go beyond 1000).

CS  -Statistics and Probability  PS – Reasoning

Decide Now! ($0.99)

Does what Undecided doesn’t with spinner. You can create/edit unlimited numbers of spinners with any types of categories and combinations of categories. Minimum number of sections is 10. If you add less than that, it intelligently uses ratio to created the segments. Great for probability experiments, especially for increasing and decreasing chance of random events to occur.

CS  -Statistics and Probability  PS – Reasoning

DragonBox+ ($6.49 – expensive for multiple copies)

Despite the cost, which would be prohibitive for some schools with limited budgets, this is a clever app for building conceptual understanding of the principles for balancing algebraic equations. Presented in a game format, it builds up from simple to complex as you play through 5 levels and 300 individual puzzles. The object is to be left with a single object on one side by applying inverse operations to object on both sides. The final level introduces the alphanumerical symbols associated with algebra.

CS – Number and Algebra       PS – Understanding, Problem Solving, Reasoning

Dartfish EasyTag (free)

This app allows you to create data collection tools using panels as recording buttons for categories and and subcategories you create. Each time you touch a panel, it begins tallying results. It collects category totals and tracks the elapsed time by whole seconds, minutes and hours. Not only useful for data collection and statistics, but can be used as a simple timer as well. Results can be exported by email as a csv file which can be opened in Excel (not  iPad spreadsheet programs), although it records labels rather than numbers so editing the spreadsheet is necessary for tallying results.

CS – Measurement and Geometry, Number and Algebra  PS – Fluency, Reasoning

Pattern Blocks ($0.99)

A simple app that can be used for many purposes. The drag and drop geometric shapes can overlay translucently to create fraction models, supported by the grids. Tessellations can be created effortlessly and rotations can also be done. At junior levels, shape patterns can easily be created and continued. Relationships between different shapes can also be explored.

CS – Measurement and Geometry, Number and Algebra   PS – Problem Solving, Reasoning and Understanding

Room Planner (free)

Created with House planning in mind but can be applied for many measurement tasks. This app allows you to create and edit individual rooms or entire house plans. Each element ( room, architectural element or furniture) can have its dimensions adjusts though simple touch and drag, elements can be freely rotated and final plans can be viewed from all angles and views in 2D and 3D. Area and Perimeter investigations can be implemented and concepts of space can be explored through placing objects within the rooms. Text can be added and in 3D mode, creativity is encouraged though applying colours and textures for realism. Scale can be explored by creating models of actual rooms.

CS – Measurement and Geometry  PS – Problem Solving, Understanding and Reasoning

5 Dice Order of Operations (free)

A simple but engaging equation building game that builds understanding of order of operations rules. A target number is randomly selected and 5 dice are provided to use as the values to generate equations to reach the target. IT provides a whiteboard for experimenting with possibilities before dragging the numerals and operation symbols into place. There are options for using some or all operations and brackets to allow for different ability levels.

CS – Number and Algebra PS – Fluency, Problem Solving, Understanding and Reasoning

Foldify ($2.99)

Its whimsical nature and cost makes it appear superficial use of technology but it allows for an engaging exploration of 3D objects and nets. Can also be used to create patterns on dice faces that can encourage logical reasoning in building patterns.

CS – Measurement and Geometry PS – Reasoning and Problem Solving

Geoboard  (free)

Simply put, it is a Tech based Geoboard that allows for wide ranging angle and shape explorations as well as quick creations of arrays to build understanding of multiplication and division. Can also support fraction and decimal exploration with careful manipulation. Shapes can be rendered transparent or translucent for easier viewing and comparison.

CS -Measurement and Geometry, Number and Algebra   PS – Problem Solving, Understanding and Reasoning

Geometry Pad (free; $6.49 full features)

This app allows for exploration of shape, angles, co-ordinates, area, perimeter, circle properties,algebraic expressions on graphs and linear graphs ( functions in the paid version ). Free version is still quite functional but paid version has some compelling upgrade features for higher level mathematics.

CS – Measurement and Geometry, Number and Algebra    PS – Understanding, Reasoning

MyScript Calculator (free)

A screenshot doesn’t do this app justice. In a nutshell, this app converts your handwritten scrawlings into equations and calculates the answers. Recognises indices/roots, trigonometric functions, percentages and fractions as well as basic operations. YOu can edit equations on the spot by crossing out and replacing numbers and symbols and equations automatically update as you increase and decrease values on either side. Blank spaces are replaced with calculated values. A great app for exploring equations as well as a very functional calculator. Does have limits, which you will find as you explore but its free so explore at will.

CS – Number and Algebra    PS - Fluency, Problem Solving, Understanding and Reasoning

Friends of Ten  ($0.99)

A handy app for exploring subitising and the visual conceptualisation of 1-10, important number skills to develop in younger students. This app has six activities using Tens Frames to develop build to ten, how many and more than/less than.

CS – Number and Algebra   PS – Fluency and Understanding

Tens Frame Snap Lite (free)

This game based app consolidates the skills developed in Friends of Ten above using a 2 player Snap game.

CS – Number and Algebra   PS – Fluency and Understanding

Routes ($1.99) (My Maps – linked to Google Maps account – free but harder to use)

Using Google Maps as its base, this app allows students to build routes along maps by dropping waypoints along the way. It generates distances and estimated times along the route and between points and you can compare bicycle, car and walking routes to the same locations. It also creates instructions which can be tested by actually going out and following the routes created. The distances and times can also be tested by actually going along the route as well. Routes can be shared via email, Twitter/Facebook and printed.

CS – Measurement and Geometry, Number and Algebra     PS – Problem Solving, Reasoning and Understanding

Virtual Manipulatives! (free)

An app that provides manipulatives to explore the relationships between fractions, decimals and percentages. Limited to values from 1/2 to 1/12s ( no 1/7s or 1/9s)

CS – Number and Algebra    PS – Fluency and Understanding

Counting Board (free)

A simple but effective counting aid. Either show or hide numbers. Create visual number patterns. Use to develop count on/count to/ count backward strategies for counting, addition and subtraction. Has an option to say numbers as they are tapped.

CS – Number and Algebra    PS – Fluency and Understanding

Fraction Division ($0.99)

A very specific skill set for an app but great to see a conceptually difficult operation ( division of fractions) explained in a concrete way. I know teachers who don’t understand how to divide fractions or explore the rote learnt reciprocal concept. This app definitely helps

CS -Number and Algebra   PS - Fluency, Problem Solving, Understanding and Reasoning

Numbler Free (Free! – paid app $0.99)

A fun way to explore equations and practise calculations. Basically, this is a number based version of Scrabble. YOu are given a selection of tiles with numerals, operation symbols and an equality sign. The object is to make equations with the tiles you have and/or the tiles already on the board.  Easy to play, challenging to finish. Encourages experimentation by trying to score the highest possible score. Free version only allows for one player versus computer. Paid version allows two player game.

CS – Number and Algebra PS - Fluency, Problem Solving, Understanding and Reasoning

Logic Puzzles HD ($2.99)

I love Logic Puzzles. This app provides are large selection of puzzles to complete. While not easy to categorise under COntent strands, the logical reasoning developed throough these puzzles is essential for higher order thinking. I have successfully taught 7 year olds how to solve ( and create ) these types of puzzles which has encouraged thinking, problem solving, creativity and logic.

PS – Problem Solving and Reasoning

PollDaddy (free)

Others prefer SurveyMonkey but PollDaddy has its own iPad app that gives you a simple way to COLLECT data based on surveys created online. All you have to do is link the app to an account, download the survey and it creates an easy to use, question by question survey on the iPad. You can review the results and upload the surveys once done.

CS – Statistics and Probability

As you can see, most of these apps are free so you can easily try them out to see what you can do in your classrooms with them. The paid apps won’t exactly break the budget if you download one copy to try. While many have physical, old school versions that can be used instead ( just like they were pre-iPad), I am of the opinion that the iPad version are more user friendly are allow for more possibilities and instant, repetitive use.

Let me know what you think about these apps or maybe suggest some other apps I have left out.

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Mar 17

Algebra gets a ‘bum rap’. Then again, it has a lousy public relations manager. Whoever came up with the whole ‘letters and symbols’ campaign should be sacked. Yes, opening up to Exercise 7D and solving 50 variations of 2x + y = -7 is n0t anyone’s idea of fun. But as I said, Algebra needs a new PR campaign.

DISCLAIMER: I’m just a Primary/Elementary teacher without any official qualifications in High Level Mathematics – No Masters, no Ph.D, just an A+ Average in High School/College Maths and 25+ years teaching kids to enjoy,not stress about, Maths. I may be completely off base with the great mathematical minds out there in what I’m about to describe regarding Algebra but I make no apologies. my students get it this way – including the Year 7-11 students I’ve tutored at home to relieve the confusion caused at their schools. (WARNING: Bear with me, I’ll take a while to get to the point of this post’s title – skip ahead if you want to ignore my Algebra rant!)

Now we have that out of the way, back to my message for today. I have a certain belief about Algebra. I define it as a systematic way of organising, recording and explaining your mathematical thinking using numbers and symbols/letters instead of words and pictures. Where we seem to get lost is that we go straight to the symbol without developing the thinking through the words and pictures/objects. We provide no context or purpose; just a meaningless string of equations with Xs and Ys that need to be solved. I see Algebra as problem solving support, not equation solving.

Last week, I was called in to take a Grade 6 class to release a teacher for planning ( the usual release teachers were unavailable). Maths was on the agenda for the day and I had worked with some of the other Grade 6 students on a similar lesson earlier in the week as a support for some of the high achievers. This time, though, I was on my own and in control so I applied my full tech+Maths kit to the group of students I had for that session.

The lesson/task that preceded this actually had fractions as its focus. One of the teachers had introduced a task involving a a building pattern for shading in grids to make fractions.

The lesson was differentiated to allow for a range of responses. Some needed to build the patterns with counters to discover anything. And then there was “Sheldon” ( not the boy’s real name) whom I walked in on to find him showing his mate the formula for the relationship between square and triangular numbers! When I confronted “Sheldon” to explain his formula and why it worked, he didn’t know how. So began my challenge and the rationale behind the lesson I’m about to recount. In the end, Sheldon actually discovered the key to this lesson I led in the class I took later in the week.

SO…this fraction lesson turned into a pattern and algebra exploration. All the children were able to discover the growing patterns in both number sequences and could describe the change. Square number differences increased by +2, the triangular number differences increased by +1. But that additive thinking was as far as they got. They needed more support to think multiplicatively, to think ‘Algebra’.

Enter (finally we get to the title of this blogpost!) the iPad and AirServer. Yes, I could have done all of this without the technology. I had done so earlier in the week with my small group of advanced students. But the engagement and ease of use was no comparison between the ‘sheets of paper and coloured marker’ group and the iPad and AirServer. If you are unaware of AirServer, I explained its significance in a recent post. Basically it projects multiple iPad screens onto a computer connected to a projector/iWB.

We started with creating the fraction grids using the iPad App Hands On Maths Color  Tiles ( I reviewed this and others in the Hands On Maths collection last year ). Again, we could have hand drawn grids or made them with counters but I had the students more engaged by getting them to make 1 grid each using Color Tiles and getting multiple students to project their grid onto the whiteboard using AirServer. This took 1 minute instead of 10 and allowed us to move straight into discussion with all the visuals needed on the screen – created by students, not me.

We then discussed the three properties visible in these tiles – side length, square size and the shaded (red here) area ( they hadn’t recognised them as triangles yet). I introduced the problem solving strategy of ‘Make a table’ – a strategy that should be embedded in their thinking by now, but it wasn’t. I created the table on my iPad and projected it on the screen. The students then created their own tables, using Numbers, on their iPads and filled in the side lengths, square sizes and shaded areas. Once they had the numbers in tables, they could start looking for relationships in numbers across the properties, rather than just look at the isolated number sequences. It was at this point that some students were able to recgognise that the shaded area numbers increased by adding on the next side length.

From that discovery, some children then saw that by adding the side length e.g. 4 to the square number 16 ( by this time we had recognised these as square numbers, not just square size), 20 the shaded area was half the size – 10. Here we talked about the importance of proving our theory by testing with other numbers. EVERY child in the class then tested this out with the other numbers, using Explain Everything as a whiteboard to quickly write out equations and project them on the screen to show their proof. Again, this could have been done on paper but by spotlighting everyone through the AirServer iPad mirroring it engaged those children who more often than not pretend to do the work and then let the teacher pleasers to put their hands up and call out the answers. This process really had everyone involved at all times. Some of the less than stellar mathematicians were excited about this discovery. But we were not finished.

I wanted them to see what type of numbers they were creating with the shaded areas – most still didn’t realise. This time I went back to old school methods -

counters. AirServer and my iPad still played a role. I asked the group to use the counters to create the sequence of numbers in the shaded area column in rows. As they began, some weren’t sure what to do. Instead of telling them what to do, I used my iPad’s camera to spotlight pairs who were building triangles onto the screen, thus giving support to others who needed a hint. Every group then wanted their triangles on the screen as well! This idea of spotlighting using iPad and AirServer can work in many ways to maintain engagement – kids like to be on show and recognised .

Once this was done, the students realised they were creating square and triangular numbers and that there was a relationship between them. Children started to recall the rule we had discovered – square the side plus the side then half it gave us the triangular number. But I posed one final challenge – why does this work and how can we show it with our tiles to explain the relationship? Back to Color Tiles we went. We recreated our two coloured square tile pattern. Then we added an extra column/side length. Bingo! The students recognised that this created two equal halfs, a red and yellow half- two triangular numbers!

4×4 Square with extra column of 4 results in two equal shaded areas- triangular numbers!

The final step in the process now was to put all of these theories into one explanation and come up with a formula – finally Algebra was coming into play. The important thing here is that they were thinking algebraically all along – I just didn’t tell them because Algebra is such a dirty word. Now they were quite excited that they were doing algebra.

I asked them to take screenshots of the tiles and the table and import them into Explain Everything. Then we looked at the table again. I explained that the only difference between what we had been doing and algebra was that we needed to replace our words and ideas with letters and symbols. What was the starting point? The side lengths. What will we call them – we decided on s ( could have been x,y, l etc). What is the square number? s x s or s^2. What did we do next? +s. Finally we halved the total ÷2 . With all these symbolic represenations students were able to create a formula for finding a triangular number: (s^2 +s)/2

Now thinking they were expert mathematicians, the students were able to record their understandings in Explain Everything AND find any square and triangular number without creating a long sequence. And they got it because we started with the thinking and investigating, not the formula that “Sheldon’ told us about. By the way, he worked this out independently and actually helped out my thinking with the idea of adding the extra side to the square grid – that’s the first time I had visualised the two triangular halves. This shows that our high achieving students can support the learning in the class – they just need a biy of guidance in their thinking, He was happy with knowing the formula. Now he UNDERSTANDS the formula and why it works. His discovery helped the less able students to also understand the thinking behind it all. And the iPad, the apps  and AirServer kept them engaged long enough to get there.

Oh, one more thing. I mentioned earlier context and purpose. I put this whole task in the context of a tile designing company. I talked about how the construction of Federation Square ( a modern structure in the City of Melbourne laden with geometric designs ) was not a random design. It was very mathematical. I put to them the scenario of customers wanting a design like the one we investigated created at a size of their own choosing. As employees of the company, we needed a method for quickly calculating how many of each tile we would need – the formula we discovered would get the job done.

Algebra need not be hard. It’s just logical thinking written down in an organised, symbolic way. Taking students through the right process can demystify it all. And it doesn’t hurt to use a bit of tech like my good friends the iPad and AirServer to help them along the way.

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Nov 28

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This week, Grade 5 began a unit on Volume, Capacity and Surface Area. On a weekly basis, I take combined groups from the 4 grades consisting of the higher achievers, while the classroom teachers concentrate on the mainstream group and students needing more individual instruction to achieve success. I made a conscious decision this week to focus on using iPads with my group to explore both volume/capacity as well as surface area.

I chose 3 apps to assist me in this learning experience – Think 3D ( free version) and Skitch, which are both free apps and Numbers ($9.99- $4.50 through the Volume Purchasing Program if 20 or more bought). Note: you could substitute the currently free CloudOn app, which is basically a server based Office app, or Google Spreadsheets, a free component of Google Docs/Google Apps for Education.

In the past I would have run this lesson using a limited number of connecting blocks and would have asked the students to record their observations in their exercise books. In using the iPads and the selected apps, I wanted to trial how this type of investigation could be enhanced and improved upon by using technology rather than traditional tools.

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The lesson began with the following premise. Each pair of students ( didn’t have enough iPads for 1:1; would probably work in pairs regardless to encourage collaboration and discussion) was to create a cuboid or rectangular prism with a volume of 72 cubes using Think 3D. In the past, students would have used a limited supply of blocks and would only have had enough to make one model. Using the iPad app, they were able to explore multiple ways of making a 72 cube prism with a limitless supply of cubes with a simple touch of the screen adding or deleting  a cube to the prism each time.

Another advantage is that, while there are many benefits in physically seeing and touching a real 3D object rather than a 2D representation of one on a screen, the ability to rotate the prisms on the iPad to view the different surfaces with a simple swipe made for easy investigation and no chance of the object falling apart and needing to rebuild, thus saving time for more analysis.

Using Reflection on my Macbook ( also available for PCs), the children were able to mirror their iPad screens on our interactive whiteboard and share all of the possible prisms and cuboids. This allowed for easy comparison and discussion without having to move our models around as we would have in the past.

The next step was to save the models as images in the Photo library on the iPad so that we could import them into Skitch, (an annotation app linked to Evernote.) As you can see from the image below, the students were able to clearly label the dimensions of their prisms and record surface area measurements as well. The use of this app enables easy collection of data for assessment rather than the rather difficult alternative of taking photos with a camera and writing notes about each photo. It also makes it easy for the children themselves to keep records of their work and thinking, an improvement on the lesson for both teacher and student. They were also able to swipe back to Think 3D to manipulate the prism to investigate the dimensions closely during the annotation stage.

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We then opened up Numbers to systematically record and calculate the measurements using spreadsheet formulas. Being capable students, they already knew how to use the L X W for area and L X W X H for volume formulas. I wanted to skill them up in using spreadsheet formulas to make quick calculations so that more time could be used for analysing the measurement data and the 3D models.

The spreadsheet was laid out so all possible dimension combinations discovered by the students were recorded. We then inputted a volume formula to verify each prism had a volume of 72 cubes. We then used formulas of our own creation to calculate the surface area of each prism. Once one formula was created, we were able to copy and paste that formula for each prism to calculate each prism’s surface area. Once we had all of the volumes and surface areas, combined with the 3D models, students were then able to make informed conjectures, observations and proofs about why certain prisms of the  same volume had varying surface areas.

While I am not saying I haven’t taught this lesson successfully in the past, using these apps and the iPad allowed for more direct and focused engagement from all students. Previously, the recording of data would have been a whole class event, which I always feel has the potential for disengagement as children watch others do the work. Having limited resources in terms of blocks, early problem solvers are left waiting for others. With the use of Think 3D, they were able to continue on with their own investigations rather than waiting for another pair to make an alternative model.

With today’s lesson, the children were actively involved in all aspects. They had opportunities to explore as many options as they had time for, they inputted all mesurement data, they annotated all of their images, which enabled them to consolidate and record their thinking more efficiently. The technology used also enabled them to save a permanent record of all the work they did today, whereas in the past, it was lost once the cubes were packed up. I  think this is a good example of how technology, and the iPad in particular, can be used for greater engagement and deeper thinking in Mathematics. Yes, all of the steps in the lessons could have been done without tech or iPad specifically, but I don’t think it is as effective.

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Sep 28

Earlier in the year I wrote a post titled “Maths Extension/Enrichment with Edmodo“, outlining my plans for an enrichment/extension program for high achievers in Maths at school. It took longer than anticipated to get started but from the start of Term 3 (July), I met with 5 Grade 6 students, 8 Grade 5 students and a couple of very bright Grade 4 boys on a weekly basis for an hour. ( Another teacher does the same with Grade 3 and 4 students ).While we can argue that research suggests mixed ability groupings are more beneficial ( for the rest of the week, these children work in that environment), I am in no doubt that the program has been a resounding success and a great sense of engagement and enjoyment has been felt by all involved, including the Maths teacher!

Whether it is enrichment, extension or a mix of both, which was a point of contention with some readers back in the original post, I am not sure. Regardless, some great mathematical thinking is taking place every week between an enthusiastic, engaged group of students.

The weekly lesson itself takes no time to plan. I simply upload a problem to the MEP (Math Extension Program) Edmodo group at the start of the week so the students can check in for some preparation time before we meet. Don’t get me wrong, I know exactly what I want out of the lesson when I select the problem and I send a post lesson report to the classroom teachers outlining what we did. The beauty of what we do, though, is that we don’t know what will result from the lesson until it is over. There is no chalk and talk, no pre-task explanation of what to do, no expectations that we have to solve it at the end of the hour. What you will see is a group of mathematicians sitting around together, sharing strategies, discoveries, questions verbally, through demonstrations on the whiteboard or via iPads or by posting on Edmodo.

What has improved throughout the term has been their problem solving skills, collaborative discussions, use of technology aids to organise and simplify the process ( Numbers on the iPad  has been a real winner, using formulas to test and monitor conjectures, as has Explain Everything to record ideas and share via the whiteboard) and most importantly, their ability to articulate their thinking and learning, both their successes and failures ( something they haven’t experienced much beforehand).

A great example of the whole process is our last learning experience, which lasted over two weeks. Most of our problems have come from the well established Maths Enrichment website, nrich. ( another worthwhile site is New Zealand Maths ). The beauty of nrich is the incentive to have your solutions published on their website, giving bragging rights to those who succeed, either partially or fully ( more on that later) Our last problem before the holiday was Summing Consecutive Numbers. The problem is presented via an introductory video that explained the nature of the  task. Each student had their own iPad ( its only a small group – we could have used the laptops) so watched it independently. After a two minute debrief to make sure everyone understood the task, we went straight into solving the problem. Beforehand, though, we made a pact that we would publish our solution on nrich, which always had to be posted by the 21st of each month, which just happened to be the last day of Term 3 ( we had previously missed deadlines or solved old problems, so this was our first chance.)

What was great about this particular problem was that the task itself was simple to start with – just adding numbers – but discovering and proving patterns and formulas was a real challenge that need real arguing and collaboration. During the first hour, the students were so focused on discovering patterns. Every idea they had, no matter how small, was posted on Edmodo. This proved to be an important step as the following week we were able to refer back to all of our discoveries. LEt me interject here and state that I was an active part of this as well. Before the lesson started, I was none the wiser about the solutions so I became an authentic learner with my group, making conjectures and testing theories side by side with them. (I talked about the importance of being a learning role model in a previous post). Some children used Numbers spreadsheets to arrrange the numbers into common sets as we investigated, others jsut used pen and paper while others used Explain Everything to brainstorm every idea they had. At the end of the sessions, we had over 60 posts on Edmodo and had made some amazing progress and they continued on over the weekend and into the following week determined to meet our deadline.

The following week, we met with all of our discoveries articulated on Edmodo and we were ready to write our Proof of the Summing of Consecutive Numbers. The final result was exceptional and is published below for your viewing pleasure.
Consecutive Numbers Proof
I showed their classroom teachers and my fellow MEP teacher and they were blown away by the depth of articulation and understanding in the submission. I merely guided them through the process of writing the proof but it is all their work (some sentence structures needed some modelling). To a person, they all requested a copy to put in their blogs and digital portfolios and now wait excitedly for the news it is posted on nrich’s website next month. Regardless, I am going to showcase their effort at the School Assembly, much to their satisfaction of being recognised for being mathematicians.

Being such a successful and rewarding experience, I then started thinking – should this just be the domain of the MEP group? Why can’t the other students in their grade follow the same process? It’s not as if they don’t do problem solving based tasks. This task in particular could have been entered into by ALL the students at different levels and the MEP students could have worked with the others to extend their thinking. The more I work with my group, the more I realise this model of collaborative problem solving should be done more at school. Sure, some of the less able students would not have arrived at the sophistication of thinking these high achievers attained but they could have contriubted to the adding and would have discovered some of the lower level patterns.

I think we have to stop thinking that not all students can enter into these tasks. Nrich is full of problems for all ability levels. Its my new goal to attack at school. I still think these MEP students deserve their time together to work with like minds. But I also think everyone deserves the experience they are getting. It’s what a differentiated curriculum is all about.

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Jun 09

Earlier in the year, I wrote a couple of posts on the iPad and Maths Apps. I questioned whether there were apps out there that went beyond number facts drills and calculation games. One of my readers of those posts, Melissa,  let me know about a group of apps called Hands on Maths. This set of apps provide a range of digital versions of hands on manipulative tools that are needed to develop important Mathematics concepts and skills. I am in no way suggesting that they replace the physical tools entirely but they do provide always available, easy to manipulate tools that are linked to independent investigations generated by the app itself.

These apps include digital versions of geoboards, counting charts, Base 10 blocks, attribute blocks, fraction strips, grids, coloured tiles, abacuses and other maniuatives that support the development of basic number and spatial concepts.They would be particularly useful in supporting individual and small group learning plans for students who need visual aids and teacher aide intervention. Each app is customisable and allows for different skill levels and different types of tasks within the same app through a simple user interface. The settings are changed through the “cog” icon, the activities are accessed via the arrow icon and there is a home button to return to the beginning. There is also a tutorial included to explain the use of each app.

What follows is a brief overview of some of the Hands On MAths apps available on the iPad used on how I have used them. For a more expensive look at the apps before purchasing them ( each app is $1.99 AU or the equivalent in your country) the company Ventura Educational Systems has an excellent website providing detailed information about all their apps, including downloadable PDF instruction guides. I wish other app creators would provide this much information about their apps so that you could make informed decisions about purchasing.

BASE 10 BLOCKS

 Hands On Maths:Base 10 Blocks is a virtual mamipuative app that allows you to explore both whole number and decimal place value using the familiar base ten blocks, known in some countries as MAB. It also allows for addition and subtraction of numbers with and without regrouping. It is limited to 3 digit numbers from 100s through to hundredths. It works through simple dragging and dropping of block into a work space and the values are automatically generated as you build the numbers. A useful feature is built in that allows for groups of smaller values to automatically transfer into the higher value accompanied by an arrow that shows where the values transfer to. ( e.g when you make 12 tens in the tens place, it will change 10 tens into a hundred and leave the remaining 2 tens intact). this works in the decimal format as well. As I said in the introduction, I’m not suggesting we do away with the physical block usage as many younger mathematicians in training need to manipulate physical models. Where digital virtual manipulative excel is in instant feedback, quick turnaround of use, instant access and reuse and unlimited resources ( we often run short of blocks in whole class settings). Together with discussion with a teacher on a one to one or small group basis while manipulating the virtual blocks, I see this as a good tool for working with at risk students. I like that the app allows for the use of decimal place value as well, even though here is a school of thought that we should use different models for decimal place value. Me personally, I like to maintain the link between the base 10 system across whole and decimal numbers to show the consistent relationship.

INTERACTIVE HUNDREDS  CHART

The Hands On Maths Interactive Hundreds Chart is a counting board which you can set up starting from 0 or 1 and use to investigate, explore and discover number patterns and sequences. Users can mark out multiple counting sequences using different tools including crosses, ticks, circles and squares( transparent, opaque and solid) of different colours. Using these tools, students can discover patterns, common factors and multiples, predict the next few numbers in the sequence by studying the pattern show so far. They can create their own or follow sequences given by the teacher or other students. Used effectively, much discussion can be generated about number sequences as a precursor  to Algebraic patterns through visual representation. Again the advantage of the digital tool is the quick turnaround in exploring patterns and the instant reuse of the board.

INTERACTIVE COLOUR TILES

Hands On Maths Color Tiles has a huge range of options for developing important Mathematical concepts. The tiles can be used to create arrays for exploring multiplication and division. Addition and subtraction can be explored by adding or subtracting tiles by dragging on or off the workspace. These operations mentioned are supported by a built in pad that supports the calculations being done with the tiles. This pad can be customised to show fractional. decimal and percentage proportions of tiles on the workspace as well. There are also built in grids that can be used to support calculations or be used as graphs or co-ordinates. Symmetry can also be explored through symmetrical grids that create duplicate reflections vertically, horizontally or both as you place tiles on the grid. By exploring this app you will find more and more applications for the range of tools it provides. Read the PDF guide that is available on the website listed above. It gives further ideas. The moe I explore it the better opinion I form on this app. Check it out.

OTHER NUMBER BASED APPS AVAILABLE IN THE RANGE

There are a number of other apps in the Hands On Maths Range that address number concepts. I’ll provide the links here and direct you again to the company’s website so you can check out for yourself what these apps offer.

Number Sense provides ways for exploring whole numbers, fractions and decimals

Number Balance can support the introduction and development of equality, equations and algebraic thinking by providing a balance tool that enables you to crate equations that equal different value combinations on either side.

Tangle Tables and Multiplication Toolkit both give many opportunities to explore basic multiplication concepts in a hands on, concrete way.

Hands On Maths also has a number of apps that support the teaching of geometry and other spatial concepts. I’ll discuss them in a later post.

When I first discovered these apps, I thought they were nice little activities for the juniors to explore. As I explore them deeper, though I can see their applications in higher grades as well, used creatively and in context. In tutoring middle/high school children on the side, I get frustrated by the lack of hands on explorations of concepts by teachers in these schools. I can see a place for some of these apps in the right  context.  I recommend that certainly elementary/primary school teachers give these apps a go. Even if you don’t buy them, check out the company’s website ( I have absolutely no affiliation with them – I just discovered the site today file researching for this post). You might find some great applications for using the real versions of these virtual manipulatives that you can use to improve your maths teaching.

 

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Mar 24


Addressing the needs of all students in your Maths Classroom can be a real challenge. Do we stream based on ability? Do we use collaborative mixed ability groups? What’s the role of rich,open ended questions and differentiated curriculum? How do we pitch to the middle 50% but still cater for the upper AND lower 25%? It’s a challenge I’ve been grappling with for 25 years. Recently, I’ve been considering the use of Edmodo to provide access to extension and enrichment Mathematics opportunities for the more able students in the classroom. ( For those unfamiliar with Edmodo, click here for a description) This is my plan. I would be interested in feedback on its potential effectiveness before going further with it.

Identifying the target group
This is not a simple task. The standard method these days seems to be the standardized test. In Australia, we have NAPLAN, the yearly national assessment task targeting Years 3,5,7 and 9. Debating its merits here is not my intention today. I see its usefulness in quick identification of the higher achievers in a current group of students. I would then administer the next level test to these able students to gauge how far their abilities extend beyond the current class level. For example, after selecting a group in Grade 6 based on Grade 5 results from the previous year, I would give them the Year 7 test. Using data analysis, I’d identify their strengths and learning needs for future programming and targeted areas for extension and enrichment.

This would only be a starting point. Standardized tests are a narrow form of assessment that don’t necessarily identify fully the student’s need for extension in Mathematics. I’d continue to evaluate the children within and outside the extension group. I’m sure during the year I would identify children who could join the group for extension in specific areas they excel in. The beauty of using an open, collaborative, independent learning platform like Edmodo is that students can opt in and out of specific tasks or units of work.

The Edmodo Extension Maths Program
This is how I envisage setting up and running an Extension Program in Mathematics within the standard classroom environment.

First I would set up a Maths group for every student in the class. I wouldn’t want the Extension group to stand out from the crowd by having sole access to Edmodo for Maths. I would use this area to post problem solving tasks that the whole class could engage with, links to quality Maths sites that students could use to consolidate understanding in current units and revise past lessons as well. I would provide opportunities for discussion of strategies used, allow children to share their understanding, ask questions that both teachers and students could answer and share with the class. I’d allow for the possibility of using iPad apps like ShowMe or Explain Everything to post audiovisual explanations or lessons created by teachers or lessons. I would also post resources children could access to support them while working independently. The extension group could have a lead part in sharing their expertise with other classmates in this main Edmodo group. they could even create their own mini “Khan Academy”.

I would then create a subgroup within the main Edmodo group for my Extension/Enrichment group. I envisage this group being formed from able students across all classes in a particular grade level, possibly across several if there are able students in lower Grade levels who could qualify. I would plan for this group to access materials and concepts beyond what is available to the main group but accessible through the same platform as everyone else.

Obviously there would need to be some significant planning and negotiation with all class teachers to ensure this worked within their programs. consideration would have to be made about how these students would participate in both the extension sessions and regular class lessons. I see this happening in a number of ways.

Option 1. The students begin the lesson with the rest of the class. When they have received enough instruction on what is expected of them, they move on to completing required work for their class teacher independently, leaving their teacher to work with those who need support. When they complete the set task, they submit it on Edmodo through the assignment section and then enter their Edmodo Extension Sub group to collaborate on the higher level tasks assigned by me. They communicate with each other either personally if in the same grade or via posting their strategies, solutions, suggestions, questions, comments on Edmodo for the rest of the Extension group to respond to. Their work will be completed digitally and submitted through the Assignments section of Edmodo so that I can feedback and collaborate with them on the tasks.

Option 2. Alternatively, for one session a week, the group would meet with me and work on high level problem solving tasks and extension work related to the unit of work their class in currently involved in. Using online enrichment programs like the website nrich, the group would be collaborating on problems, sharing their possible solutions and strategies not only with each other but by submitting group or individual solutions on the nrich site for other like minded students to collaborate on through global forums. I envisage opportunities for the students to use technology such as screen casting computer programs or iPad apps I previously mentioned like ShowMe to record their solutions and strategies audio visually. Using a site like nrich, which would allow them to self select problems to solve would give them the freedom to challenge themselves both individually and in teams. It would also give them the option to opt in and out to return back to their class if they choose to.

Option 3. A third model could be a choice of making daily decisions to complete regular class work as homework and deciding to work in their extension groups or individually on Edmodo on a daily basis. As their test results would have already indicated in being selected for the program, they have most likely mastered the skills being taught in the regular class program and a simple completion of the tasks for homework would satisfy their class teacher’s need for evidence they have understood that area so they can report on it later in the year. This option fits a Personalized Learning model commonly encouraged in today’s schools and would allow the student to remain engaged in Maths at or beyond their level rather than going through the motions of completing simple tasks.

How Edmodo would help me implement this program

  • All links to nrich and teacher/student created work would be posted on Edmodo, with individual entries tagged or saved in libraries so that students could always have easy access to the tasks.
  • The collaborative nature of leaving instant comments and feedback allows the group to stay in contact with each other outside of school to continue their problem solving together. This could become engaging homework, with the teacher able to remain in contact and feedback on the work they post on Edmodo.
  • Each member of the group can work on their own problem solving and submit it to me or their teacher independently of the group for personal feedback before sharing with group if they choose to.
  • The function of the Assignment process in Edmodo allows for children to receive private feedback and allow the teachers to collect, collate and mark each submission, enabling effective assessment to occur at all points in the program. Teachers can submit rubrics and criteria for marking the work on Edmodo so the students know what is expected of them. I have had success with such use last year working with a Literature Circle group.
  • The fact that all students from the classes are also using Edmodo for their Math work as well means that all students can easily be given the opportunity to opt in to or out of the Extension group at any time without any extra planning or organisation by the teachers. I think this would be an important option as it would encourage other students to take on the challenge of extension tasks if they choose to.
  • Other teachers can be given co teacher status and become involved in the program, either as observers or contributors. This would allow for professional feedback on the suitability and effectiveness of the program.

These are my initial thoughts and obviously this kind of radical change to the status quo of primary schools as I have experienced them would involve leadership, class teacher, parent and student discussion. I need to think through this more and would appreciate feedback from others on how they have managed the needs of the more able students in their classrooms. I would really appreciate readers leaving a comment and contributing to the conversation of extending and enriching the learning of the able mathematician.

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Mar 03
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Can ubergeeks like me find Mathematical applications in the non Maths apps on the iPad? Skitch app and Leaning Tower of Pisa meets angle measuring

My last post focused on the state of apps related to Numeracy education and concluded that the iPad has a fair way to go in how it is being utilized to develop number skills through the current crop of Maths applications. On the whole, there are too many apps focused on rote learning and simple number facts and equations. Having said that, I actually believe the iPad has much to offer teachers and students in Mathematics. However, don’t go looking for the apps in the Education section of the App Store.

I’ve always been a big proponent of making Maths real in the classroom. Yes, every teacher out there wants to make Maths interesting and relevant for their students but sometimes this just comes in the form of “Real life Maths” textbook problems. The danger of Apple’s foray into textbooks on iPads is that we will continue to get the traditional textbook experience with the bells and whistles of interactivity and ‘magic tablet dust.’ The good news from my perspective is that the iPad already is ready to replace textbooks and make Maths an engaging real world experience. What it takes of course, as I have been pushing in this blog since its inception, is quality teaching practice coupled with great, easily accessible technology. And it comes not from Maths apps but the already available, successful apps across all walks of life in the App Store.

Screencasting/Whiteboard apps
Whether you choose Explain Everything, ShowMe, Educreations or ScreenChomp (depending on price, features or usability), using these apps to record children’s learning is a killer feature of the iPad for me. Yes, we can sit them in front of Khan Academy or YouTube and get them to watch someone else teach them how to do Maths but any good educator knows that students learn more by participating in the teaching and learning process rather than being passive receptors of information. With these apps, children can record visually and with audio the strategies they use to solve equations, real problems, geometric tasks, statistics, virtually any Maths task and share with others in the classroom.

If you are lucky enough to have access to an AppleTV and a full set of iPads, the whole class can switch from one student’s iPad screen to another to participate in an engaging real time discussion without waiting for their turn on the interactive whiteboard. Their recordings can be saved and posted to blogs or other online options to share with outside audiences, with possibilities of feedback or collaboration.

This kind of use of the iPad would do far more for developing deep understanding and granting access to authentic assessment than any textbook. It would also encourage the less confident students to share their knowledge without getting up in front of the class. You could just watch their recordings and comment later.

Drawing Apps
Doodle Buddy is a popular app already being used by many students for a variety of reasons. It has many possibilities in the Maths Classroom. Younger students can create pictographs by using the stamp tools to record their survey results. It can then be used to develop more “mathematical” visual representations by drawing around those pictures to make column graphs. The app comes with special backgrounds, one of which is a dot grid. Students could use this to draw paths following directions, create shapes and angles and use it to create line graphs as well.

A more complex drawing app is iDraw. This application can be used for shape exploration using its tools to create transformations through rotations, reflections and resizing. Scale concepts can be introduced and area and perimeter explored through drawing plans of houses, gardens, playgrounds and the like.

iDraw

Further engagement could take place through a specific house design app called Home Design HD, which not only gives you a wide range of tools for precision drawing and measuring of house plans but also converts the plans into 3D models.

With the Skitch app, students can import pictures then use annotation tools to divide them into grids to make arrays, plot a path on a map screenshot, annotate graphs, draw angles on an image then measure with a real protractor. They can use this for real trigonometry problems instead of detached examples from textbooks.

Video Apps
Children can become engaged in sharing their Maths understanding by making learning videos with iMovie, recording entire lessons with concrete materials as support for presenting what they have learnt.

iMotion HD is a time lapse/stop motion animation app that could be used to record shape transformations made with real materials then played back and slowed down to discuss the transformations as they happen. Time lapse photography with this app could be used to record experiments related to measurement and time relationships then played back at different speeds to explore how long it took for ice to melt, for food to cook or a snail to travel over a certain distance. Possibilities are endless for authentic learning here.

Navigation Apps
Imagine exploring average speeds by looking over the results of a car trip, run or walk recorded with an app like MotionX GPS. The app can present its findings in graphs showing distances, elevations, speeds and maps (with wifi). By looking at the break up of the distances travelled and the time taken, engaging authentic discussions can take place about what caused the fluctuations in times taken to travel certain distances, differences in elevations and how that affected the journey, all real life applications of Maths. At a simpler level pedometer apps can be used just to track distances and times on shorter walks around the school to support earlier estimations of distances/lengths

Other Measuring Apps
The Theodolite app mimics the function of a real theodolite and can be used for real angle readings to determine the height or distance away from an object. Just like my earlier Skitch example, this could be use for practical applications for geometry and trigonometry that written problems in a textbook can’t compete with.

Clinometer, Gigantic Compass, Multi Protractor and the Best Ruler are tools that can replace traditional tools for measuring angles, directions and lengths while also providing quick reference to real numbers that can be used for relevant equations. Clock Pro HD provides a range of time recording tools that can be used for many tasks in a creative Maths environment. There are a range of conversion apps that can be used to support work in measurement tasks.

Wolfram Alpha is a workhorse app (Google on steroids)  that can do just about everything in Mathematics information that will support  all areas of Maths. Just read the description on the store to find out more. Here’s a detailed review from YouTube

Cat Physics

Games
Use physics based games like Angry Birds (does anyone need a link for this!?!), Cat Physics, PocketTanks and Super Stickman Golf to explore the use of angles then apply them to the creation of real life models. YouTube already has examples of Angry Birds Physics lessons here. The beauty of Cat Physics is that it shows the path after completing each level so you can actually measure the angle you used and then try out the real model you build. Pocket Tanks actually gives you the angle you are using. Playing these games can inspire students to test out other real applications for angles like golf club design, ramps, pinball machines, mini golf courses and so on.

Yahtzee, Monopoly and Solitaire games all have great possibilities for developing number concepts. There are also dedicated dice and decision making apps like DiceBag, Wheel of Chance and Undecided that can be used to enhanced Probability lessons.

Maths is all pervasive in our lives. So are iOS devices, be they iPods, iPhones or iPads. All it takes for the iPad to become an essential tool in the Maths Curriculum is creativity in finding Maths in our daily lives and using the related apps available. So while I bemoaned the lack of creativity in Numeracy related apps in my last post, I finish this entry confident and excited about how the iPad can make Mathematics an exciting experience for students across all grade levels. A lot of what I have mentioned here may not be staring at you in the face when you look at your country’s Curriculum standards or national standardised tests. But I don’t mind that. Get kids doing real maths, make the links to what the Education Departments want and it will all take care of itself. Be brave, but above all, be creative and real.

This is by no means an exhaustive list of apps or ideas. I would like to hear from others how they have used apps in Maths lessons. I haven’t mentioned spreadsheet/graph making apps like Numbers or Office like apps because we’ve been using them for years. I’m looking for something new.

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Feb 25

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My last two posts on iPads and good teaching have focused on teaching and learning writing. Now I’m moving on to my favorite subject as a teacher – Mathematics.

I love Maths – both learning and teaching it. For those who don’t know me ( which is obviously most of you reading ), I am a Primary ( Elementary ) School teacher but I have spent most of my 25 year teaching career also tutoring High School Maths on the side, supporting many children who have missed out on understanding important Mathematics concepts.

Maths involves an incredibly diverse range of processes, ideas, skills and concepts. Both children and teachers alike enter into Maths teaching and learning at different levels, depending on their personal experiences, successes and challenges with the subject. Over the years I’ve been involved in the conversation about teacher-led process/algorithm based teaching which I ( along with the parents of our children we teach) experienced as a child vs a more student centred, understanding and multiple strategy based learning approach more prevalent today.

The challenge facing us is that, while education training is rightly focused on the latter approach, the traditional process system still holds sway in many homes and can sometimes be a fallback for teachers who are unsure in Maths. It is also, in my experience, very much the default teaching method in many secondary/high schools in my part of the world. I won’t get into the pros and cons of the two models – that’s for a later post. This post I again focus on where the iPad (and other iOS products) sits in the world of Maths Education and whether, along with good teaching practice, it can have an impact in developing skilled mathematicians of the future.

The state of Maths apps on iPads at the moment
There’s a lot of potential in the apps available on iOS devices for Mathematics but overall I think they fall a little short of what I would like. Many of the apps are more directed towards the traditional memory/algorithm/procedural methods of teaching or drill practising of number facts and operations. I think where they may fall down is in the fact that the app developers are not necessarily involved in education and are basing their app concepts around traditional Maths they were exposed to.

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Maths Bingo

The number fact/ 4 operations apps serve their purpose of consolidating learning and improving automatic recall but they don’t necessarily support the initial teaching and learning required to develop understanding of concepts.
Here are some examples of apps in this category.
Drill/Number fact games
Freefall Maths - drill practice drag and drop
Factor Samurai- focuses on recognizing prime and composite numbers in a Fruit Ninja style game
Painless Algebra – practises +/- rules in operations
Maths Bingo – a popular app focusing on calculation involving the four processes at different levels of complexity
Math Hero - equations needing order of operations to solve in a game setting
Math Mago – a large grid of numbers 1-9 to eliminate by solving 2 number equations. At least there is more than one answer and it makes you think about which numbers to choose so you have options left. Sort of open ended but still just basic equations.
Math Kid – more number facts to solve but at least you get a visual aid to support your thinking after a few seconds instead of just timing you out. An improvement on others above in terms of mathematics teaching an learning.
Operation Math – lots of engaging bells and whistles based around secret agent missions but behind it all jut another number fact time limited practice app.

Procedure based apps
Fraction Basics – provides step by step instructions for working with fractions and the four operations as well as steps for working out equivalent fractions and improper/mixed numbers
Mathboard – a very popular app that has strengths. It generates equations involving all operations for solving at different levels and provides support in how to calculate the answer if the student can’t solve it. I would love this app if the solutions provided were a range of mental and written strategies rather than just the vertical algorithm. Again for those comfortable with this as the one method for solving equations, it is a winner. For those wanting more than that, it falls short.
YourTeacher.com and Khan Academy apps – two apps for the latest trend in Maths Ed – Flipped Teaching. Again I applaud these apps for the step by step support for children to follow to achieve success. The down side for me is the one solution fits all approach without involving the children in the discussion. At the end of the day, they tell the student what to do, but tend not to ask why.
Algebra Touch - a nice app that allows you to manipulate numbers by clicking and dragging, splitting them into simplified forms to aid in solving algebraic equations. Requires discussion to develop understanding of what is actually happening but the interactivity has potential.

They are all Fun games or sound step by step procedural apps for practising newly developed skills, monitoring progress or challenging students to improve. However, the games don’t teach or develop new understanding and the procedure apps don’t give scope to different strategies that may be better options. Some may argue that these apps enhance the users’ number knowledge through practice and repetition. This may be the case for some like myself who responded well to this method when I was in school. However, just as many of my friends and colleagues past and present have not developed the same understanding and find themselves relying on written methods they can’t necessarily apply mentally or to complex problems they have not exposed themselves to since college. Drill and practice 20 years ago has not developed their permanent understanding. This is why I support a more multiple strategy/understanding based model.

Having said that, I can still see a place for the apps mentioned above. The immediacy of results gives feedback to the students and shows gaps in their ability they can work on. These apps could be used as a tuning in session to introduce strategies to make playing them easier and improve mental calculations. However, you can do that with an interactive whiteboard and computer or web based software. I’m looking for innovation that makes the use of the iPad better than previous technology. For many years there have been websites offering fun ways to practise number facts and operations without showing great improvement in student understanding. Either these sites haven’t been used enough because of lack of computer access or the method of drill games is not effective. We need to find new ways to use technology with Maths Education

The procedural apps can be effective if followed up with discussion to ensure the students have developed understanding, not just followed steps. These apps could easily be improved if they included a range of strategies. One app that does that is School A to Z, although I’m not sure it is available outside of Australia, as it is developed by the NSW Education Department. It includes instructions for a range of computational strategies based on Australian methods and curriculum standards. I would love to see more apps being developed by actual education experts rather than just app developers who like Maths their way.

Mathemagics

Another app I love because it presents a strong mathematical strategy focus in Mathemagics. Its main purpose is to present a wide range of mental calculation strategies. It provides the methods and tricks and then allows you to practise them within in the app.

Open ended apps
There are a range of apps that allow for more open ended problem solving. These apps allow for critical thinking to take place, for choices to be made. They offer problems with multiple solutions and allow opportunities for students to choose the operations they need to solve the problems. As a result, conversations can take place between users to discuss how best to solve the equation.

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Aydox

Examples of these apps include;
Aydox- a challenging mental arithmetic game that involves strategy, thinking ahead, multiple calculations to think of possibilities. This is the type of Maths App that can encourage a lot of mathematical thinking. At its simplest level, it can be used to create equations that equal or nearly equal given numbers on the matrix. At the highest level complex thinking takes place to try to score the lowest possible score.

3D Math – basic equation creation involving problem solving and critical thinking. It allows you to alter your initial choices and think about the possibilities in front of you to solve the problems. It is time based which still makes it difficult for weaker students to engage in.

Number Pyramid – involves finding missing numbers using operations and partial answers in a pyramid format
24 Challenge Lite -based on the 24 game, using numbered playing cards to find equations that equal 24 using four numbers and any of the 4 operations.

Manipulative Apps
This is where the iPad’s touch interface should excel. Apps where users can drag and drop objects to sort, count, group, divide, increase, decrease etc, should be n abundance. Again though, I haven’t found a massive range of apps to mimic the interactive features of electronic whiteboards or websites like NLMV. There seems to be toomuch of a focus on number facts on the iPad. Nevertheless, here is a sample of apps that fit the description.

Virtual Manipulatives – an app that lets the user drag and drop fractional parts representing decimals, fractions or percentages.
Number line – allows the user to place whole and decimal numbers on number lines
Motion Math HD – physically manipulate by tilting iPad a fraction in a ball to land on a blank number line in its estimated position. Fractions are presented as decimals, percentages, fractions and visual models. Quite engaging and challenging, developing visual estimation of fractional size.

SketchPad Explorer

SketchPad Explorer – offers a range of manipulatives in Number and Geometry. Potentially a very good app if more variety added.

Think 3D – in the app, the user gets to explore and build 3D objects, rotating the shape for different views.

Montessori Place Value – Students can move place value cards around to create 4 digit numbers or less

These are the kinds of apps I want to see more of on the iPad. It sells itself as a magical touch device. It should be full of possibilities for building, altering, creating. Its features shouldn’t be wasted on just pushing numbers to answer simple number facts.

As there are over 1/2 million apps in the App Store, I’m sure there are hidden gems I haven’t discovered. I would love to hear from others out there on the Net about Maths apps they have found useful. If you want to agree or disagree with me about the current crop of apps I’ve discussed here, I’d appreciate any comments.

In my next post I’ll discuss Maths reference apps, the role of screen casting apps in Maths and how to use apps not specifically labelled as Maths apps to enhance the Maths program. I’ll also look beyond Number and talk about apps for other areas of Maths.

;I

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