Maths Extension – engagement with nrich and Edmodo

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.

The iPad and Maths – Are we there yet? Pt 1


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.

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. 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.


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.


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.