Review of MathTappers: Numberline

MathTappers: Numberline is a game that challenges players to find the locations of numbers on a number line.   This game is designed to help players to develop their understanding of our number system and their proportional reasoning skills (with whole, integer and decimal numbers).  MathTappers: Numberline is one of a group of educational apps designed to help children master mathematical concepts and skills.  Search for MathTappers on iTunes and download them all (they are all free and have no ads or upsells).

The development of this app has been supported through grants received from Constructivist Education Resources Network (CER-Net) and the University of Victorias Faculty of Education Centre for Outreach Education (CORE), along with the generosity of HeavyLifters Network Ltd. who have sponsoring the development and publishing of the MathTappers app collection. Visit www.MathTappers.com for additional information.  I’m one of the designers on these apps, so this review might be a bit biased, but I’m still doing my best to help you see the learning potential of various math apps. 

Playing the game:

The goal of this game is to accurately and efficiently locate the positions of numbers on a number line.

At the start of each round the player estimates the position of the requested number on the number line and taps on it, if the position they chose is correct (within +/- 2.5%), then the number is tagged, full points are awarded and the next challenge is presented.  If the estimated position is not close enough, the screen zooms in around the position selected, marks the location selected with the value, adds the nearest benchmark values, and gives the player a second chance to select the correct location.  If the number line currently presented on the screen is not centered on the proper range of values the player may simply swipe a finger along the line to shift it.

Ten rounds (numbers) are presented during a game and progress is marked on the bottom of the screen.

Useful Options

The app has six types of number lines. In each case the accuracy is set to the number of decimal places shown in the range. 

There are three levels of tick marks on the number line that you can choose between to assist you in estimating the position of numbers. 

• None - means that the only tick marks are those that appear under the two referent numbers on either end of the line.  Additional major ticks and referent numbers appear when you zoom in so that there are always two referent values available on the screen.
• Coarse provides a second level of tick marks that divide the space between the two referent numbers into obvious subdivisions (as the learner figures out the value of the tick marks they are assisted in efficiently estimating the position of the challenge numbers). 
• Fine tick marks subdivide the spaces between the coarse tick marks.  These are designed as training wheels - once the player understands the positions associated with these tick marks, then process of finding the right position can be reduced to counting.

Assessing progress

The Progress tab allows you to review the progress of players using the app.  Sliding the player name allows you to switch between players while the Clear button provides you with the ability to clear the database for that person. The Report button allows you (the parent/teacher) to forward all of the results to your own email address for further examination.  The Research button inside of the Report option allows you to forward a report with more comprehensive player action information to a second email address. The designers of this app encourage parents, teachers and other researchers to use this feature to support their own investigations.  No data is sent without the explicit action of the parent/teacher/student, and then it is only sent to the email addresses entered.

Advice for Parents and Educators:

The development of number sense is linked to the development of an accurate mental model of the number line and this model can be effective in keeping number sense alive as children master skip counting, fact retrieval, estimation tasks, arithmetic algorithms, proportional reasoning, measurement and fractions. A good grasp of a number line model also helps children to develop and use a wider variety of problem-solving strategies and better judge the reasonableness of solutions.

Even after mastering the counting to 100, young children placing numbers on open number lines from 0 to 100 typically plot values in a logarithmic pattern (i.e. distances between smaller numbers are exaggerated relative to distances between numbers closer to 100). As children gain experience with the number system, their number line placements become more linear and they start to manipulate numbers more accurately. In late primary grades children start to think proportionally about the number line and learn to use round numbers (benchmarks) to aid their placements.  This incremental process of refining their internal number line model repeats up through the grades as students develop their mastery of the numbers through 1000 (i.e., on a 0 - 1000 number line) and beyond and then again with decimal numbers, integers and real numbers.

The MathTappers: Numberline app has been designed to provide players with useful experiences as they explore a dynamic number line and receive automated feedback and encouragement through game play.  Don’t try to push learners faster than they are ready to go -  start with 0-100, then monitor progress.  At each level ask children to explain what they are doing as they select positions to confirm the current difficulty setting is accessible and appropriate.

Guiding beginning learners

If a child is unable to make sense of the number lines you are presenting, they may need to do further preparatory work with counting, comparing numbers and physical representations of number lines. Some activities to support the development of understanding of number lines might include having a child count out 30 or 50 pennies and line them up without gaps - then mark off where the line starts, where first group of ten pennies ends, and similarly for the second group of ten, etc. and followed by labeling the points on the resulting number line. Children will also benefit from examining, playing and working with rulers, measuring tapes or meter sticks and measuring objects of varying lengths. 

Accuracy first, fluency second

It is important to emphasize to children that we all learn differently, that accuracy is more important than speed and that the goal of this learning activity is not to have a race between students, but rather it is a challenge each of us to improve as individuals. The emphasis should always be on mastery of the concepts along with capturing the strategies or processes required to find the answers. 

Encourage learners to take their time to find the correct answers as they play the game.  Supervise children initially and challenge them regularly to give reasons and explain their strategies.  If at first they choose to count the ticks on the numberline to find their answer that is OK, but challenge them find and share additional, more-efficient strategies that also work for them.     If the challenge is too onerous (takes too much time), find an easier level - remember success breeds motivation.

Using this app in the classroom

This app may be used in several ways to support learning in the classroom. 

1. On an iPad connected to a data-projector it can be used as a tool to support introduction and class discussion.  Start with a physical number line, then move to the app.  Have the students share their strategies for making sense of the number line and finding numbers on it.
2. With an appropriately designed discovery learning worksheet a paired activity can be helpful.  Challenge the students to identify and articulate their understandings and strategies, then share their discoveries in a class discussion.
3. After completing lessons on a particular range of numbers, this app may be used at an activity center, or recommended to parents for home practice and discussion to support and reinforce the understanding that the learners are developing and to revisit previously learned relationships and strategies. 
4. In later grades this app, along with some moderate supervision and appropriate tutoring support, may also be effective review for students seeking to build/rebuild/remediate foundational skills.

A small set of iPod Touches (or even just one) with a variety of learning apps installed for individual practice can be an excellent and productive resource for learners to use during breaks between their other learning activities.   You might provide learners with a road map linking apps and levels within apps to curricular goals (from both current and prior years).  This may encourage children to take some responsibility for their own learning progress. Ask learners to use unique player names, and use the progress reporting function to assess individual growth and needs.

Summary

Obviously, as one of the designers of the app I am inclined to think that it is useful.  I believe that it can be used in the classroom or at home to support learners as they master different aspects of number, and that it offers something that you can’t do as well with paper and pencil.  Ultimately all I can say is – there is no cost to giving it a try.  I think you will like it too.   BTW, with this and indeed any other app, if you like the app, help the raise the profile of the app by giving it a good rating (and even a comment if you can spare the time).  

Review of Okta's Rescue - NCTM Illuminations

Okta's Rescue is produced by NCTM – essentially porting over an activity from their Illuminations website  – and it's free!

This app will support children in developing skills and fluency in counting, recognizing numerals, subitizing (counting groups automatically), recognizing odd and even numbers, and even addition and subtraction (as a children try to figure out how many more Oktas they need to collect on a second pass).  

Before using Oktas Rescue a child should first have mastered numerals and number words and be able to count physical objects accurately and efficiently.

Playing the game:

The game is simple – the player (child) is asked to capture a given number of Oktas (Octopus) and then click on  a pipe to suck them up (rescue them?). To capture the Oktas you draw a region around them – but you don’t have to capture them all at once you can continue to add Oktas until you believe that you have capture the required number. You repeat the process as many times as you can before a timer runs out.

The good:

The presentation of the captured Oktas in an organized form helps children to count by twos, and builds up their ability to discern larger sets of numbers with a quick glance.

The process to capture Oktas involves outlining a region using your finger (and although you get some funky pinched regions sometimes, the process works surprisingly well most of the time) . 

Option to turn on target – shows an outline of the number of Oktas that need to be rescued.    This supports children in building fluency with counting organized sets of numbers and recognizing odd and even numbers.

Custom settings allows parent or teacher to adjust the difficulty of the challenge to meet the needs of the child.

Presentation is interesting and engaging without too many extra details to distract (except timer ticking down)

You can turn the sound effects on or off.   And the sound effects provide some useful feedback (success or failure to capture the right number of Oktas)

Some Frustrations:

If you select too many Oktas you are stuck – I cant seem to deselect any. 

When you go to flush the Oktas away (suck them up?) if you have the wrong number waiting above the tube – they slide sideways instead of down when you open it – this form of feedback will be a bit too subtle for some kids (the sound effect is more effective). 

The timer – having a timer tick off  in front of you can be distracting.  Granted the app is designed to help children develop efficient counting skills, but  I am not a fan of counters adding stress to a game.  Some children may overfocus on the timer and end up performing poorly and gain nothing for their time.  An alternative would be to present 10 screens/challenges and report both the accuracy in terms of the number captured and the speed.

The instructions/help screens are extremely limited – it would be nice to have a page somewhere that explained why we were moving the Oktas over to an area where they are sucked away.  How is this related to saving them?

The region selection tool appears to degrade over time – to the point that sometimes the region just doesn’t show at all.  Some might count this as a feature – but really it appears to be a glitch – shutting it down and restarting (in OS 5.0) seems to restore the feature to its original form.

There is no record keeping or user names so the parent or teacher must observe the child as they play (or immediately after they finish a round) the app in order to assess them.   

My list of frustrations is a bit long, and I hope they might be helpful to the designers of the app if they choose to generate a second version - but don't let it dissuade you from downloading this app.

Summary:

Although the app has some quirks and limitations, I believe that Okta's Rescue is a worthy app providing a sound learning experience for children.

Watching a child play with this game will give a parent or child an idea of how well the child has mastered number words, symbols, counting, subitizing, and even addition and subtraction to a degree.  

Learning can be enhanced by asking the child to describe what they see on the screen and to share their strategies.  Mastery should always be the primary goal - if the child is struggling - it might be best to move back to counting sets of physical objects for a while.

Addition Coach - a simple app to encourage the exploration of addition

With this app there is no stress and no failure, just success. This app is ideal for children from Pre-K through grade 1, although it can also be useful as a support for older children who are still struggling with mastering number and basic addition facts.   In presenting an ‘equation’ of block sets with the total presented in numerical terms, the app helps to introduce the idea of the symbols for addition and equals and their use in arithmetic expressions and equations.  

Playing the game

The child is presented with a number to create and a menu of colored block sets (1, 2, 3, 4 & 5 blocks), and is able to manipulate the block sets on the screen by tapping on them to add them to and take them away from their collection.   Big fingers may be a little frustrated by the narrowness of the target space for the block sets, but this will not be an issue for most children (for special needs - try it on an iPad!).

 As a child taps on a colored set from the menu on the bottom of the screen, it is added to their new collection in the centre of the screen (an expression with a series of block sets added together) and a new total is calculated and presented numerically on the right hand side.   If they choose to tap on a block set already in their collection, it disappears, and the total is adjusted again.  This process continues until the total in their collection matches the total in the challenge (top right) at which point there is a celebratory cheer and everything disappears in a flash of sparkles.  The app nicely starts with full size clones of the block sets and scales the collection down if room is needed to allow for up to 10 sets of blocks to be combined – although the child can create a sum as high as 50, the challenge values don’t seem to go higher than 15.

Helping children to learn

Before working with this app, parents/teachers might invite a child to play with some beans (or some other convenient counter), challenging them to count small sets (up to 5, then 10 then 15), and confirming that they are able to count correctly.  If they have not yet mastered counting small sets, it may be better to spend some additional time with physical counters, or some other app to build up this understanding before introducing this app.

As a child begins to play with Addition Coach, parents/teachers might first focus on helping the child understand how the blocks can be added to the collection and removed from the collection and then focus on supporting the child in making sense of what is being presented.  Ask the child to tell them how many blocks are in each set in the menu and help them to recognize number as an attribute of each set (just like color is an attribute).  Then, as they add sets to their collection, ask them to count the total number of blocks in their collection.  Guide them to recognize that number beside the collection is the same, and discuss that equals means the same as. 

Addition Coach could also be effectively used with a small group in an activity centre. Invite one child at a time to manipulate the app on the screen and encourage the group to discuss what happens.

Suggestions for the authors.

This is a great app, but it would be helpful if the authors could provide some options to allow parents/teachers to choose the level of difficulty and information presented to the child. Although, given the limited screen space, some of these suggestions may be difficult to implement.

For instance it would be great to be able to choose the challenge range or maximum challenge value  - very young children could start with sums to 5, then 10, etc..

It would be nice to have the option to have a symbolic expression/equation presented either in concert with the blocks or at the end before the blocks disappear.  Similarly it would be helpful to allow the sum to be represented as a complete collection of blocks to allow the child to see how their collection of block sets might be represented as a whole – perhaps grouped in sets of fives or tens.

It might also be helpful to have the option of choosing where the equal sign and the total is presented (right or left – to reinforce the idea that the equals sign is used to signify a relationship rather than an operation).

An option to require a double-tap or ‘tap and dwell’ or ‘tap and drag’ to add or remove a set of blocks might help reduce the frustration of some children who like to touch the blocks as they count them.

Finally it might be nice to have the name of the app fade away after the first round to reduce the clutter presented on the screen – but it is already fairly subtle, so that is just a minor quibble

Final word

Addition Coach is a fun app to support children in exploring the nature of addition – and it is free!  I recommend it to parents and teachers – both as a tool to support guided exploration in the classroom and as a tool to support anytime anyplace microtutoring. It is a great app and deserves to be much farther up the list in the app store.

Building fluency and learning problem solving with KENKEN

KENKEN captures the fun of a Sudoku logic puzzle and augments it with arithmetic making it a great game to support learners as they build fluency with basic facts and mental math as well as develop logical thinking skills, problem solving strategies and perseverance.

There are 3 versions of KENKEN on the AppStore - and the are all based on the same puzzle – here I am focusing on the $0.99 version by Capcom.

  

Playing KENKEN:

In a nutshell you are presented with a grid ranging in size from 3x3 to 6X6 (up to 9X9 on paper versions) and like Sudoku you are challenged to place a numbers in the cells so that no row or column contains any duplicate numbers.  The unique twist with KENKEN is that the grid is broken up into a series of cages (connected cells marked out with bold lines) which are marked with a goal value and an operation (left out in the hardest cases).  The challenge is to make sure that the numbers used within a cage combined with the operation produce the goal value. 

As you complete your first 3x3 grid, KENKEN may appear to be quite easy – but very quickly you will find yourself drawn in to puzzles that are very challenging.  Make sure you make use of the note feature to keep track of plausible values in each cell to support you in your puzzle solving endeavors.

Keeping it simple:

The game is easy to get into, all of the controls are intuitive, you don’t get lost, and it is nicely focused on a single puzzle type

The options page allows you to manage the music, change the visual theme (all of which are visually appealing and not too distracting), reset the game and choose a language (English, French, German, Italian and Spanish).

The quick-start ‘how to play’, rules, control descriptions and other help features are an easy read and straightforward.

Learning math with KENKEN:

Although players can start right away with a few free-play puzzles at any level, they need to go through the academy to unlock each of the levels before they will have access to all of the puzzles.

Guess and test can be a good way to start, but players (you and/or your children/students) will get a lot more out of the puzzles if you have an opportunity to think through and talk about the choices you are making and elaborate on your strategies.    Look for a KENKEN buddy, use a notebook, or work in a group.  When working with others, ability levels don’t need to be the same.  Like-ability players can be very productive in co-generating strategies, but unlike-ability players will also benefit from collaboration.  The ‘masters’ will benefit from thinking through, articulating and explaining their strategies, and the ‘strugglers’ will benefit from being able to ask questions and having a buddy to check their moves, provide clues and guide them when they are stalled (preventing excessive frustration). 

The ‘Academy’ provides a nice short series of three lessons and an exam for each level of difficulty.   If you are already a master of the arithmetic and pick things up quickly, you may dash to the ‘Master’ level – in a few hours.  But if you are a leisurely player who prefers to refine your mastery of each new skill, you will appreciate that there are dozens of puzzles at each level to practice on in the ‘free play’ section

I suggest that if you find that you (or your children/students) are getting too frustrated by puzzles at a given level, then you might back up to the preceding level, review the strategies again and practice some easier puzzles.  Success with appropriately challenging puzzles will breed motivation and encourage perseverance – but too much frustration is not helpful. 

Suggestions for the authors:

The app takes 5 seconds to load on a 2^nd generation iPod Touch – if removing the (non) talking head would reduce this by 1 second it would be worth it.

The numeral 4 is a bit difficult to read in the markers and totals – the tall stroke shouldn’t diminish in the middle above the horizontal stoke.

The arithmetic that you are practicing does not get very difficult until you are fairly far into the game – to make it more useful for motivating the mastery of basic facts it would be great if the numbers used in the grids could focus on higher number values at lower levels (i.e., I don’t believe there is any reason not to make a KENKEN 3X3 with 7, 8, 9 as the selectable values)

Accomplishments are tracked for only one individual and only by game board – it would be nice if there was an easy way for a parent or teacher to easily review the progress of multiple players.

Final word:

I recommend this game – both as an entertaining pastime and as an educational tool to support the mastery of some basic facts (see note above about improving this) and mental math skills along with encouraging logical thinking and problem solving.  

Learning to tell time with ‘ClockMaster’

I was involved in the design of this application, so I won’t call this an unbiased review.   However, as a parent, teacher, and teacher educator, I have an opinion on how to help kids learn to tell time, and couldn’t find the perfect app to support this process - so we had to build one.  Inside the app there are information tabs to support parents and teachers in helping children to learn to tell time using the app – I’ll present a condensed version of these ideas here.

There are several challenges associated with mastering time: children have to come to understand that there are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day, they need to learn to read time on a traditional analogue (or analog in US spelling) clock, and they need to recognize that the hour names are recycled twice each day.  All of these can be a bit of a struggle for young children who have just mastered the number system where each place value was 10 times as big as the last and 100 follows 99 in the natural order of things.  

We have designed clock-master as a game to help children to make the connection between hours and minutes and to help them to become fluent in both reading and setting time on digital and analogue clocks.  Parental or teacher/tutor involvement in the process is essential.

Preparation

Before focusing on this application it is recommended that children have some basic skills.  Your child should be familiar with the number system and be able to read numbers fluently up through 100 and recognize the relative size of numbers (e.g., that 43 is greater than 37).  Your child should also have had some exposure to both digital and analogue  clocks in the real world – to help them to make connections between the clocks in this application and the real world.    If your child struggles too much to play this game then it may be too soon for them to be using it - set it aside until they are ready.

Practice

This mode is designed to give children an opportunity to explore the relationship between time shown on an analogue clock and time shown on a digital clock.   Any adjustment made to either of the clocks is echoed on the other (move either of the hands on the analogue clock or spin the digits on the digital clock).

Work one-on-one with your child in this practice mode until they become comfortable with the relationship between hours and minutes on both clocks and are able to read each hour is a good place to start.  Some discussion may be necessary to help your child understand that the hours are ‘recycled’ so that after 12, the next hour is 1 and after 2:59 is 3:00.

Settings

Sound – A positive tone is heard whenever your child does well.   When things are going less well, a more neutral tone is played. The sound allows you to monitor your child’s progress from a short distance and be ready to intervene if success is elusive.

Difficulty – ‘Easy’, ‘Medium’ and ‘Hard’ –  The ‘Easy’ level challenges children to work with the easier times with minutes set to 0, 15, 30 or 45, and constrains the movement of the clocks to these values. The ‘Medium’ level challenges children to show times to the nearest 5 minutes and ‘Hard’ requires children to show times to the nearest minute.  It is suggested that children should master the ‘Easy’ level in each of the play modes – find digital, and find analogue –  before moving onto the next level of difficulty.

Play Mode – Find Digital and Find Analogue – The process of translation needs to work both ways for children to be fully prepared to work with clocks in the world.  In ‘Find Digital’ mode the child is challenged to read the analogue clock and adjust the digital clock (hours and minutes move separately) to match.   Similarly in ‘Find Analogue’ mode the goal is to read the digital clock and then accurately position the hour and minute hands of the analogue clock.

Clock Face – This setting allows the you to adjust the face of the analogue clock to show a) a regular clock with 60 tick marks, b) the same clock with every 5 minutes shown on the outer bevel, and c) the same clock with every minute inscribed on the outer bevel. It is suggested that you begin with the enhanced clock face and then gradually decrease the level of detail presented on the clock face until your child is able to achieve success with just the regular clock consistently.

Playing the game

Each game consists of 10 rounds in each round a time is displayed on one of the clocks and the player finds (sets) the time on the designated clock (a glowing blue arrow or face reminds your child which clock is being manipulated at the beginning of each game) and presses the ‘check’ button.  

After the check button is pressed, the game timer is stopped until your child decides to move on by pressing ‘next’ – this allows them to take as much time as they need to review their answer and reflect without feeling the pressure to move quickly to the next question.  In this review mode the correct time is displayed on both clocks, but your child may tap and hold the screen to flip from the correct time on both clocks to a second screen that shows the time that they entered on both clocks – with the message ‘you entered’ displayed at the top.  

Helping children to learn

In order for your child to enjoy math – it should make sense!  Children don’t learn about mathematics from the quick ‘just do this’ solution – rather they learn by exploring and understanding.   Start by focusing on the simplest clock values first, emphasize sense-making and accuracy (can they get perfect scores), then encourage your child to improve their speed.  If your child doesn’t seem to be ready or doesn’t respond well to the program – set it aside until they are ready – and try another game or activity.

This game is not designed to replace other activities that will support learning about time – it is just a supplementary tool to support the process.  Regularly discuss with your child what they understand or what they are struggling with as they are playing the game (the feedback they give you can be very helpful), challenge them to read clocks and watches around the house and car and be sure to discuss other ways we use to describe time (e.g., about 3, quarter after 8, ten to 4, am and pm, etc.).

To help your child learn it is best to have them focus on getting good scores rather than completing the game quickly.  Once your child is able to earn close to perfect scores at a given level (say 45/50 or better) then you might challenge them to work on speed while maintaining accuracy. 

You are encouraged to play the game yourself in order to better understand the thinking that your child is doing as they play and to develop a sense of what might be a reasonable goal for your child with regards to fluency.

Classroom activities

As a classroom activity (if you are fortunate enough to have a collection of ipods available for this purpose, or in a small group setting with just one) – you might start by exploring a physical clock first and then moving onto using this app. The practice clock might be used to explore and discuss the relative movement of the minute and hour hands and the corresponding times represented on the digital clock. Carefully examine how the minute hand moves when the hour hand is manipulated (note that we cannot do this with physical geared clocks, but in software we can) it is a useful tool to support understanding.

Next, you might have the students work through a couple of games in pairs (have them set up the game to match their ability/experience first) while sharing their ideas with a partner.  Then bring the class together and present some examples on the overhead (hold power down and tap on the home button to take a snapshot of a screen), and ask them to share some of the challenges that they experienced, strategies they used, and questions that they have found effective.  This sharing of ideas and language in the group setting will reinforce helpful ideas and language usage, allow students to self-correct erroneous ideas and provide students with new words and ideas to use as they continue in the activity.  After the discussion, let the students play a few more games and then challenge them to explain some aspects of the analogue and digital clocks as a journal entry. 

Look for ways to help your students to transfer their learning by making connections between the time that they are reading on clocks and the time that they experience.  Discuss and explore the passage of time during activities and emphasize the use of the clock by setting specific start and stop times.  Bring in activities for examining elapsed time and help them to see that both a timer and a clock can be used to support the measurement of these times.  After students have become familiar with the app, a sponge game might be developed with students challenging their classmates to set a particular time in practice mode - using common, uncommon, accurate or inaccurate terms for time (almost 10, quarter past 5, twenty to 8, bedtime, etc.).

Once your students are familiar with Clock-master, you might challenge each of them to improve their level of mastery and fluency in each of the modes as a ‘center’ activity.  It is important to emphasize to the children that we all learn differently, that accuracy is more important than speed and that the goal of the activity is not to have a race between students, but rather it is a challenge to improve as individuals.   If a child is struggling to master clocks with this application – find another resource that makes more sense to them.

Review: ‘Column Subtraction’

If your goal is to help your child to master the traditional subtraction algorithm then Column Subtraction may be the ideal App. I first became aware of this app a couple of weeks ago after reading an article and searching out a news release on prMac.com. In the release the developer, Esa Helttula, explains that he could not find any apps which would support his daughter in practicing column addition, and as a result set out to generate apps to provide users with an opportunity to practice of traditional algorithms for column addition, column subtraction and long multiplication. He explains that his focus was on minimizing button presses and not penalizing wrong answers. This review focuses on only the Column Subtraction app – but the other apps appear to have similar construction, behaviors, benefits and limitations.

The interface is clean – perhaps even sterile – but that is not such a bad thing if you are working one-one with a child and trying to discuss the important details (and a good thing if your child is easily distracted). A reference copy of the question is presented at the top of the working space in horizontal format and then on the right side a working version is presented in a vertical format, finally the current sub-operation is presented at the bottom of the working screen. Below the working screen there is a ‘borrow’ button and buttons for 0-9. The feedback is minimal but adequate – when your child selects the borrow button and borrowing is appropriate, then an animated borrowing occurs in the vertical/working version of the expression and when a numerical value is selected then it is shown at the bottom of the white space and only floats up into the answer space if it is correct. Thus your child is generally provided with feedback that their response was registered and they are given immediate confirmation if it is correct.


The learning
If your child is familiar with the traditional subtraction algorithm they will likely be able to improve their mastery of the procedure using this application. With the ‘show current operation’ option turned on, they are constantly reminded of the next step, which can lead to efficient practice of the procedure. Then by turning off the current operation your child is able to practice remembering the steps on his/her own. Because no wrong entries are allowed the app provides a passive form of positive reinforcement and your child will likely improve in both confidence and competence with respect to the procedure. However, to support learning and understanding in mathematics you need to take an active role in assessing understanding and challenging your child to explain their thinking.

You can support your child in using this application, by first confirming that they have an understanding of subtraction (see caveats), then working through a couple of examples with the current operation showing. By working through a series of questions with you at their side - challenging them to explain what the question might mean (understanding with a context), estimating the outcome, and then showing and discussing each of the steps and checking the answer - your child can have a productive and positive learning experience. Once your child seems to be comfortable using the application and is able to correctly identify when borrowing is necessary, then it would be best to turn off the ‘show current operation’ option. Sitting with your child and asking them explain their thinking will allow the parent to assess whether any further support is needed for them to master both the concept and the process. Once the parent is convinced that the child has mastered both the concept and process, then continued independent use of the app will allow your child to improve their fluency with the operation.

Besides allowing you to turn the current operation on or off, the settings page allows you to choose to have 2 or 3 numbers in each expression presented and 2, 3, 4, or 5 digits in each number. In the BC curriculum guide children are expected to be proficient in subtracting whole numbers up to 10000 in grade 4. Thus the app supports a reasonable progression of starting with 2 digit numbers and moving up through 3, 4, and 5 digit numbers as the child demonstrates mastery (i.e., is able to understand, estimate, complete and check the answer in a moderately efficient manner with sub-operations feature turned off and your quiet observation). I don’t see any particular value in using the 3 values option (it's adding too much complexity for young children and a distraction), and would steer you (parents) away from it unless your child is particularly comfortable with mathematics.

A recommendation, some caveats and suggestions
As a parent I like the potential that this Column Subtraction app has to help my child to master the traditional subtraction algorithm, but I have some caveats and suggestions to add to my recommendation. I urge you (parents/tutors/teachers) to support your children as they use this app and confirm that your child is understanding the process, that they are able to efficiently estimate and check the results and that they are able to visualize an appropriate context for the mathematical expression. I also encourage parents and teachers to investigate and discuss alternative approaches and algorithms with their child.

It is important that children understand what they are doing before learning rote procedures – otherwise the learning achieved will be unconnected to other knowledge, fragile and irrelevant to their lives. Children should be able to represent and solve subtraction related problems in the physical and visual (i.e., pictures) realms before they are pushed to begin manipulating symbols. To check this, ask your child if they can come up with a reasonable story or two to interpret the symbolic expressions presented in the app, and see if they can show you the process with some physical representations (beans, base-ten blocks, etc.). If they can’t easily generate an example, provide them with some and then spend some time helping them to make sense of where these expressions might come from (e.g., given 73-58 the child might say “I have earned $73 doing chores over the past month and spent $58 on a LEGO set – how much do I have left over?”).

Finally, in many regions the use of the traditional algorithm is not mandatory and other processes are equally valid. If your child is frustrated in their attempt to master this traditional algorithm and cannot master it for 2 digits easily, you may want to consider refocusing your efforts toward further building up ofestimation strategies and using alternative algorithms (e.g., Partial difference subtraction)

Interesting apps to consider for your iPod Touch or your iPhone

Here is a partial list of the apps I have on my Touch right now. These seem to have some potential to support thinking and learning and useful play:

Math Dice (a fairly basic game with some potential to support student exploration in the classroom).

Cute Math looks like it has some potential for preschoolers.

MakeNum - seems OK, but I have only looked at the light version so far.

For basic fact flashcard apps I like the Brain Blaze series - not perfect, but better than most - especially the progress screens.

Another neat app is the Bell Curve. It's not a game but as a tool to understand the relationships associated with the bell curve it's cool.

Touch Physics is a fun physics related game - gravity, levers, etc.

Number Line challenges players to sort out a set of decimals, fractions and percents.

Codewords is an interesting challenge - another app I'd like to take into the classroom for a discovery learning lesson.

Parking lot is a good problem solving game.

Word Party is a good fun 'guess my word' game for a small group.

Catch a mouse is a fun problem solving game and may even have some potential for supporting a discussion on graph theory.

Lemonade is a classic app that can help students to begin to understand the potentials and complexities that develop when you can manipulate more than one variable - but I haven't had a chance to form any judgement on this version.

Finally,Zentomino and Tanzen are neat puzzle games that support spatial reasoning and development of geometry sense.

I don't have a favorite calculator - any suggestions?

Nine Gaps promotes thinking and learning

The Nine Gaps game has been around for a couple of months now, and has managed to climb up near to the top of the heap in math applications. Indeed, as I browse through the 10 math apps above it, I think it deserves to be on top.

In the game, the player is challenged to position the digits 1 through 9 onto a 3x3 equation grid such that all of the equations formed (horizontal and vertical) are true. Although it has a passing similarity to Sudoku – it is much more worthwhile. The user interface is clean and simple and a pleasure to use. Everything seems to work as it should and players are never distracted by the interface as they work on solving the puzzle.

I like Nine Gaps because it is a creative game that challenges the player to problem solve and think through multiple possible expressions as they search for solutions for each of the 6 equations on the board. It provides simple but very effective feedback in the form of sounds (positive and negative) and color (green means the completed equation is true and red means that the completed equation is false). The goal is to minimize the time taken to complete the game, but because players can’t really achieve a good time by grinding or guessing, they are encouraged to think through their choices systematically and even undo correct equations if further explorations identify a dead end. This type of game – where the players are challenged to examine and evaluate the multiple possible expressions is a great tool for helping students to think logically, refine accuracy and build fluency in basic facts.

The only gripe that I have with Nine Gaps app is that the order of operations convention is not followed. Players are forced to ignore the precedence rules and follow the game rule of calculating from right to left or top to bottom. Although this quirk may not a big problem for adults who have mastered the convention or even perhaps only minimally so for children who have not yet been introduced to it, it may be distracting and unhelpful to students who are just learning to apply such (typically ages 11-12). Perhaps the authors of this program could add a setting whereby parentheses could optionally be included on the game board to support the learning of this important convention.

Nine Gaps has 3 levels of play - the easiest mode gives you a bit of a hint by preloading two of the digits onto the grid in order to provide the player with an obvious next step. At the medium level, a single digit is preloaded into the grid providing the player with two expressions missing only two values reducing the number of combinations that must be tried in order to discover the first workable guess. Finally, the hardest level leaves all 9 positions unfilled – giving the player no explicit hints.

As a math teacher educator, I see Nine Gaps as something much better than a flashcard program for learning basic facts. It is a great tool to offer to teachers and parents seeking to help children to improve their accuracy and fluency with basic facts while simultaneously promoting the development of problem solving and reasoning skills. I’m impressed with Nine Gaps and I intend to keep it on my Touch for a while.

Surveying the App Store for Math Apps

I have been rummaging through the ~65000 apps recently reported to be available on Apple's App Store and have found a few apps that I think deserve further investigation. But before I move onto my next reviews I decided to survey the field with respect to math and education.

Here are some observations I made while searching the whole store (both free and paid apps) for the following terms:
Math ~600 apps
Math and Learn ~400 apps
Geometry ~100 apps
Triangle ~150 apps
Protractor 11 apps
Cartesian 7 apps
Multiplication ~170 apps
Subtraction ~140 apps
Measure ~460 apps
Calculator ~1500 apps
Note that some apps will have more that one of these terms in their description and that not all of these apps will be directly related to mathematics education or found in the Education category.


Scanning through the top 400 apps in the Education category (10% of a total of ~4000 apps in the category) I found the following distribution of apps:
Art 3% (drawing, dance, music etc.)
Early Childhood 17% (matching, animals, words, counting, number words and numerals)
Foreign Languages 30%
Language Arts: 5% (lots of vocabulary)
Math 12%
Science 5% (lots of astronomy)
Social Studies 5% (countries, cities, history, etc.)
Test Prep 5% (GRE, SAT, etc.)
Other 18% (quotes, religious, school access, utilities, IQ, blank flashcards, etc.)

Within the top 400 apps in the Education category (10%) there were 47 Math focused applications, and of these:
40% were primarily focused on drilling basic facts
23% were utilities (e.g., graphing calculators)
16% were reference content (e.g., geometry theorems, math tricks, etc.)
the rest were games, manipulatives, and explorations.

However, only about 1% of the top educational apps that I scanned met the penultimate threshold of approval – i.e., I decided to load them on my iPod Touch to examine them further. If we extrapolate from the apps that I scanned and generously assume that I will find a similar proportion of interesting apps throughout this category, then we might expect that I would find about 40 interesting apps for mathematics education (along with a few other relevant apps in other categories). This is probably a reasonable number especially given that I’m not particularly keen on most of the drill, utility, and reference apps that I have seen thus far, and because I only need one or two calculators.

The apps that are the most interesting to me are those that go beyond looking things up, memorizing facts and procedures, or calculating without understanding. I'm looking for apps that seem to have the potential to help children and adults to explore, make sense of, apply and enjoy math. I’m looking for apps that I (and others) might use in the classroom to support exploration, discovery, discussion and application. I’m a philomath and I’m looking for apps that might spark a similar love of learning in others.

Thinking of how we can engage kids in activities that help them to learn math...

Happy New Year!

As a parent, I'm always looking for applications that would draw my child in - to engage him in an activity that would provide me with an opportunity to challenge him, or tutor him, or play with him as he develops various understandings and skills.

Every time I think of a strategy that we use in the classroom to help children master concepts and procedures, I can work up an idea for a game to support parents and children in mastering that concept or process. Now I just need a bit more time - so I can write a few. In the mean time I'll keep looking for apps that do more than basic drill and kill.

The next apps I intend to evaluate are the Brain Blaze series for basic fact mastery.

Review: Zen of Ten

Overview
The game is aptly named – it is meditative and calming. There is no beeping timer, no flashing lights, just a simple board where you may strive to focus and improve your recognition of pairs of complementary numbers.

Players are presented with a screen containing 48 squares – each of which is marked with a single digit number (1 through 9). The goal is to clear the screen by selecting pairs of squares that add to 10. My first try (with some fiddling around to test the interface) took more than two minutes, My best time so far is just under 33 seconds – and I will be back to improve.

There is one setting adjustment – allowing you to switch the game play from having to simultaneously tapping on two numbers that add up to 10 – to a single touch game where you tap on one number and then tap on the complementary number. I prefer this single-touch mode because it removes some of the frustration and distraction associated with trying to touch two objects on the screen at once. If the touch process was perfect, and never missed, then perhaps the simultaneous mode would be more satisfying, but even then I think I would prefer the sequential selection of numbers because it allows the player to play in a way that is more consistent with their thoughts and reduces the amount of time that the screen is obscured.

Reflections on learning with Zen of Ten
I chose to review this game first because it embodies many of the ideals that I am looking for in games that might be used in the classroom or at home to help children learn. It is calm but engaging, it is simple but challenging, and it has the potential to help children to a useful concept, fact or skill in mathematics. For Zen of Ten, the learning objective it helps children to master is the automatic recognition of complementary number pairs.

I know it is engaging – because my son (the reluctant mathematician) will play it even though it is educational. With a little bit of encouragement (OK a small bribe) he is willing to engage with the program and strive to improve his time. As he does so, his recall of complementary pairs of numbers is becoming more automatic. As we take turns playing the game, we can watch each other’s strategies and share our thinking

To benefit from this game a child should already have a rational number sense (able to count 1:1 with objects, to recognize that the last number counted represents the number of objects in a set, can tell you one more than a number and one less than a number, can tell which number is greater, etc.). Similarly, the child should already be familiar with numerals, the operation of addition and be able to recognize by sight the number of fingers held up in two hands.

Some strategies for helping a child who is struggling to improve with this game might include
1. If the child has not yet mastered relating groups of objects to numerals or adding groups of objects by counting on they are not ready. They need to focus on counting and adding concretely first (I’ll be looking to review an app to support this soon). Get out some chips, beans, candies, etc. and then move toward playing games involving dot dice, dominos etc.
2. Ask the child to explain what number must be added to 8 to get to 10 and redirect to easier questions if this is too difficult. Work through all of the pairs of numbers.
3. If they struggle ‘seeing’ complementary numbers – draw a ten-frame on a sheet of paper (two rows of 5 boxes) and use some convenient objects as markers. Place 6 objects – one to a box – in the ten-fram and ask the child to explain what they see. If they don’t see 6 and 4 and 10, prompt them with some guiding questions until they do – repeat the process – until they get the idea.
4. Work with the child to create a visual support aid – with 5 different ten frames showing 5+5, 6+4, 7+3, 8+2, and 9+1 (and their commutative analogs – e.g., 4+6). Let them use it for support once or twice as they are playing the game. Help them to recreate it next time if they seem to need or want it later – perhaps by drawing the ten frames for them and prompting them to represent the complementary pairs.

Summary
A nice little program that focuses on only one little skill (automatically recognizing complementary numbers), but that skill is a really useful skill to have. As a testament to my perception of the value of this app I plan to keep Zen of Ten on my iPod for future play and tutoring support.

The price is $0.99 – obviously worth the money…

Usability: 5/5
Educational Potential: 4.5/5
Entertainment Value: 4.5/5

Learning a bit of math with an iPod Touch / iPhone

I've set up this blog as a place to engage with parents and teachers who are interested in math related apps for the iPod Touch or iPhone.

The iPod Touch seems to be an excellent platform for 'apps' that support discovery, creativity, consolidation and practice in mathematics (for both parents and kids). I was hooked as soon as I downloaded and played with my first App. The iPod Touch seems to be an almost perfect 'portable learning device' for supporting instant parent-child shared learning experiences (the 5 minute tutoring session, or the 2 minute exploration, or even the 30 second discovery).

I frequently search through the app store for new applications that have some potential to motivate me or my child or both of us to capture a moment and engage in some mathematically significant activity.

Here I will present my 'finds'. The applications that I have discovered by scrounging through the good, bad and interesting apps in Apple's AppStore. To be clear, I am not looking for apps in only in the education category, I am looking for apps that in my judgement may have some educational merit. These may be found in games, education, utilities, productivity, lifestyle, health&fitness, etc.

For each app that I ultimately download and use I will present a summary description, a screen capture or two, my reflections on the potential uses of the app to support learning and three 'Tap Tap Math' ratings. These ratings will be for usability, educational potential and entertainment value - with one as the minimum number of taps and five as the maximum.

I will leave the comments open to the public and encourage you to participate in this discussion. Please let me know about games or other apps that you think might also support learning in mathematics.