How to make math fun ! Part 2

We hear all the time now about how math is not fun. I firmly believe that math is fun, and the way to show it to students of today is to expose them to the problem solving side of mathematics.This series of blog posts lays out several ways to illustrate the fun side of math, with some concrete illustrated exercises.

There is a second very important objective behind these posts. We are at a new turning point in our usage of computers – they are now coming into our homes, cars, everywhere and automating much of the repetitive portions of our lives. This means that the jobs of tomorrow are going to demand more and more serious problem solving skills.

In this post, we talk about helping kids discover beautiful patterns in math.

Help kids discover beautiful patterns in math

One of the things that makes math really fun and engaging is the discovery of amazing patterns, some of which are not always immediately obvious. It’s what creates beauty in mathematics. And it is possible to create exercises that allows or helps kids to discover these patterns. The learning created through self-discovery will stay in their mind for a long long time. Below are some examples of such exercises. We’ll start with a medium difficulty example, and then illustrate some simpler activites that are possible for younger kids.

  1. Question to ask students : What is the sum of the first 2 odd numbers, first 3 odd numbers, first 4 odd numbers, first 5 odd numbers and so on.

1+3 = 4

1+3+5 = 9

1+3 +5 +7 = 16

1+ 3+ 5 + 7 +9 = 25

1+3 + 5+ 7+ 9 + 11 = 36

Do we see a pattern here ? What pattern do we see ?

Answer : The sums are squares of 2,3,4,5,6 .. and so on. Isn’t that amazing !

Follow-up question : Can you illustrate why that happens ? Below is a diagram that shows it beautifully, in a proof without words

sum of first few odd numbers

2. Here’s another example. This is a really fun way to practice multiplication, and observe patterns, and learn about prime numbers.

Step 1 : Let’s start by creating a hundred board – this is the numbers 1-100, is rows of 10.

Step 2 : Then let’s color in all multiples of 2, let’s say with the color yellow. Do you see a fun pattern on the board? Answer : All the multiples form alternate straight lines down the board.

Step 3: Now let’s color all the multiples of 3 on the board – with the color blue. The color blue forms a fun pattern on the board.

Question : What about all the cells that are the color green ? Can you tell what they are multiples of ?

Answer : They are multiples of 6

coloring multiples of 2 and 3 on the hundred board

Question : Why do you think that is the case ? You didn’t deliberately set out to color the multiple of 6 green. It just happened.

Answer : Because 2X3 makes 6. And the color yellow and blue is on the squares with numbers that multiples of 2, and multiples of 3, and yellow and blue combine to make green. They may of course observe other patterns e.g. every alternate multiple of 3 is a multiple of 6, when you look down the columns, every 3rd number is a multiple of 3, just like in the rows. Discovering some of these patterns by yourself is very much why this is a fun and engaging exercise.

It is possible to continue to color the multiples of different numbers upto 10 this way, and make many different observations.

Step 4 : E.g let’s color the multiples of 4 orange ?

Question : Observe, do you color any white squares ? No ! Why ?

Answer : Because all the multiples of 4 are also multiples of 2, and hence they are already colored when we did the multiples of 2.

It is a similar observation with multiples of 8.

Similarly, you can observe patterns with 2 and 5 both colored.

Step 5 : Let’ s finish coloring all the multiples of 1-10.

Question : Are all of the squares from 1-100 colored ? No ! What can you tell about the numbers that are still white ?

coloring all multiples upto 10 on hundred board

Answer : They are prime numbers. They are divisible only by themselves and 1.

We hope you and children you teach enjoy these exercises. If you get a chance to try any of these exercises out, or like the approach or would like to see more ideas, do drop me a line via the contact page.

About the Author : The author is the founder of Minds On Play LLC, which creates educational logic puzzles games for kids and adults to develop critical thinking and problem solving skills.

https://itunes.apple.com/us/app/logiccity-jr/id936822232?mt=8

How to make math learning fun ! Part 1

Math was my favorite subject in school. And thankfully it never even struck me that that is odd. And now it is my daughter’s favorite subject. Yet I hear all the time now about how math is not fun. So I decided to put some thought into what makes math fun, and what to do to so that learning math is fun !

The first and most important thing to do is to approach it conceptually : Math is based on concepts that can be applied to many many situations and form the foundation. A lot of parts of math are then just logical derivations of the basic rules. Eg multiplication is a basic concept – when you add the same number several times, you are multiplying the number. Practice and quick recall of math facts is useful, but not as important as being able to understand the concept, and derive any multiplication answer from that concept. The same is true for all basic math operations. Once you know the concept, then anytime you are stuck, you can derive the answer from the concepts. And the concepts apply to different situations e.g multiplication as repeated addition for integers can then be applied to multiplying fractions by integers. It’s the same concept.

The question then arises of how to do this. We may all agree that this is important, but to teach, we need the tools and lessons that can help us do this. Below are a few examples and links to tools below that hopefully provide a more concrete approach. The key to teaching several math concepts is to visualize it. And once that visualization is used a few times, the mathematical concept can settle down in the mind of the person/child learning it.

Here’s a wonderful article by Stanford Professor Jo Boaler on the value of teaching math visually

https://www.youcubed.org/think-it-up/visual-math-improves-math-performance/

Specific tools that help in teaching maths through concepts :

  1. A set of tools that teaches math concepts really well through visual and physical manipulatives is the Montessori materials – which are fabulous at teaching math with manipulatives that do a good job of illustrating the concept through physical materials. I once went a Montessori teacher training and there were trainee teachers there who were in tears because suddenly math operations made simple sense, instead of being this long list of numbers and operations (math facts) requiring rote memorization.

Below are links to a couple of videos that illustrate a few of the materials and approaches used in Montessori math training.

For teaching multiplication via skip counting :

https://www.youtube.com/watch?v=eKuVCeg7zCc

Here’s a wonderful example of visualizing algebra – the binomial cube which is a visual illustration of  (a+b)= a3 + b3 + 3a2b + 3ab2


https://www.youtube.com/watch?v=crjItXrt43Q

I once did the Montessori teacher training for mathematics, and it was amazing to see some teachers in tears because math finally made sense to them as they saw it conceptually through the Montessori math materials.

  1. Another great source of visual math lessons is Stanford professor Jo Boaler’s site youcubed.org.
  2. Last but not least, it’s not hard to make up your own. A wonderful exercise I once saw on introducing fractions was to ask students to cut a square piece of paper into 4 equal pieces in as many ways as they can think of. Below are some creative examples. This is a lot of fun, and the kids can compete on who has the most creative shapes that are equal. It also helps understand the concept of fractions as each of these methods leads to pieces that are a 4th of the whole.

 

fractions picture 1

And of course you can continue. Eg what happens when you cut another one of these into 2 pieces – it then becomes 1/8th. A student went on to explore and did the series shown in the picture below.

fractions picture 2

Doing math conceptually is a lot of fun, and the approaches often allow for further discovery with the tools as we saw above in the fractions example. We hope you agree. If you have additional examples or tool sets, please do email them to us.

We will continue this blog post with several other approaches and examples of how learning and teaching math can be a lot of fun.

The inspiration behind the LogicCity series of apps

When I meet people at conference, local events and mention our apps, I often get asked why I created them. Here’s my story.

I am not an app developer, or an educator by profession. I am trained to be an engineer, and then an MBA, and I worked in strategy consulting and banking in highly analytical jobs. And I loved them for many reasons, one of which is how much problem solving and innovation was needed on a daily basis and how much of an impact I could have because of it. Having always loved puzzles and board games, they were a natural fit.

Fast forward to when I had kids, and got increasingly involved in their education. I loved the Montessori schools that my children went. They follow the children’s interests in a structured and safe environment, and helped them build a broad array of skills, independence and concentration. While doing that, they taught my older one, a girl, the concepts of math upto division by the time she was 6.

Getting her to learn math after the Montessori environment was not as easy, and of course being an engineer mom, I wanted her to be good and like it. When she was in 2-3rd grade, we tried doing math operation practice with some creative rewards (e.g. the answers could be put together to create a picture etc.) And we tried some well known math books and online programs. They were dull and boring and she was least interested in doing it. And I really wanted her to like what she did, not just do it because I told her to.

Along the way, I bought her some logic puzzles books. And I would find her solving them for hours on end, and loving it. And to my surprise, she was doing some that I would find hard. And that was my moment of epiphany. It showed me that she could do pretty hard problem solving even at that age, and that this was the way to help her build those skills while she was still learning the math operations that are needed as a foundation for mathematical problem solving.

As a lucky chance, I also discovered some puzzle board games by Michel Lyons, and his math books for elementary schools that have whole sections on logic. He has the same philosophy of approaching math through concepts, and problem solving and his books do a stellar job of that. He’s been doing it for over 30 years in Canada, which interestingly scores very high on the PISA test.

We partnered to create LogicCity Jr for elementary school kids to develop deep problem solving skills even before they master all of the mathematical operations. It’s been a labor of love as I re-taught myself programming after many many years. It took a year to get the first one published.

And then we developed the advanced version, LogicCity, in which we managed to create some very challenging puzzles that even adults find both fun and challenging. Although middle school kids can sometimes do better at them than adults 🙂 It took another 6 months to get the second one out the door. And I continue to learn about gamification in education, user experience to make it seamless, and  how to do better graphics etc.

Both apps have won some awards, and that has been very encouraging. We are still waiting to see if enough parents and teachers will choose to get them for their children, and if we should continue to build more. So dear reader, that is your decision to make.

Some of our favorite apps

It’s a dull winter morning, and as I get ready to create some more puzzles for Logic City – our puzzles app – I started thinking back to how excited we were when the first iPad came out, and what apps have stood out for us as a family.

There are a couple that really stand out in language skills. Our now 10 year old loves “A Year of Riddles”. We’ve spent many a flights trying to solve them together, and at this point, she is better than either of her parents at finding the answers. And both our kids have loved Starfall. The older one picked up reading from it on the computer (it wasn’t yet on the phone), and the younger one knew all his letters by 2 years of age – all because of the star fall app.

These have been some apps that have inspired me to create more for kids around the world, because kids are so amazing at what they can learn if they are given the opportunity to do it in a way that is fun, challenging, and yet allows them to explore at their own pace.

LogicCity Jr Launched on the Apple App Store

“Education is not the learning of facts, but the training of the mind to think” Albert Einstein

That’s why we made LogicCity Jr – an app with logic puzzles, for children ages 7+ to practice and build problem solving and logical thinking skills while having fun.

LogicCity Jr is a pure logic puzzle game for ages 7+.  It’s simple in concept and easy to learn, and then gradually gets really challenging (even for adults) making it all the more fun !

IMG_3221            levels screen

The goal is simple, to fill the grid with shapes, such that there is no shape repeated in a row or column (yes, like sudoku !). Depending on the level, the colors also follow specific rules (a two level sudoku!). Introducing two properties to manage simultaneously creates all kinds of cool possibilities in the puzzles.

The puzzles have been developed by Michel Lyons and Arunima. Michel is an educator with over 35 years experience. He’s written math books with a focus on problem solving, and created several highly rated and award winning board games (Metaform, Equilibrio, Architecto …).

For our very young users, the puzzles develop very useful problem solving skills to an advanced level without needing to get into any complicated formulas – just pure logic, and deduction. In our tests with children age 7+, they loved solving them. It helps build their sense of achievement and self-esteem. As parents, it’s nice when you don’t have to push your kids to do them, they naturally enjoy playing with them.

For adult puzzle lovers, it’s just fun to solve them. It’s a nice break when you want to put your feet up but keep your mind engaged or do something different from everyday life. You can jump to the later levels faster if you like, no unlocking necessary.

Questions, comments, suggestions – you can reach us at arunima@mindsonplay.com

Even more challenging puzzles coming soon … If you like it, we’d love to hear from you. It’s been a journey to create them, and we sure would love some encouragement !