Authentic Inquiry Maths - Revisited

This was my very first post, written over a year ago.

Time to revisit and see if it still holds true...

Authentic Inquiry Maths

Did you ever read a book called “The Number Devil”  by Hans Magnus Enzensberger? If you ever see a copy of it, grab it and have a look. In chapter one, Robert, the hero of the story, meets the Number Devil. The Number Devil explains to Robert that knowing a little bit of arithmetic, such as addition and subtraction, is quite useful for when the batteries of your calculator run out but really it has little to do with mathematics.

How often do teachers fail to grasp the distinction? How often do we overstate the importance of the “skills” that we fail to recognize the importance of their application? It’s like a football team that focuses so much energy on their training sessions that they forget to turn up to play their game.

I had some parents come to me with a few questions last week about the importance of skills and learning multiplications facts. I agreed - yes, that's important - but it's also important to know why we do something.

The place of skills training

Make no mistake – a good level of competence and fluency with operational skills is useful in producing fast and accurate calculations.

Make no mistake – the skills are not an end in themselves.

TRAIN for the GAME

Just like a sports coach, I believe that students need to train to develop skills.

By “TRAIN” I mean:

Tedious Repetitive Activities Involving Number

Just as in a sporting context, what we do in training is useful. A football coach will drill players over and over until they get mastery of the skill.

But this is not the end of the process. There are no trophies for the team that trains the best or has the best drilled skills session. The skills need to be employed and articulated in the game context.

Yes, I still agree that all players need to learn skills. It'd probably also important to know that great players keep practicing when they get home. 

By “GAME” I now refer to:

Genuine Application of Mathematical Experience

Children have a great capacity for learning and replicating very sophisticated processes and operations. Does this necessarily imply understanding? I’d be inclined to say no – I can’t really observe understanding until I see how students apply their knowledge or skills to a real-world problem.

Just as the coach wants to see his players apply their ball skills in the game, I want to see my students apply their maths skills in the real world.

We're heading into grand final season for all our winter sports here in Oz. Now the players get to really demonstrate their abilities. It's too late to start learning how to kick the ball - all of that should have happened at training.

So what is Authentic Inquiry Maths?

Mathematics is a language that helps people understand the world in which they live.

That previous statement could almost have been lifted straight from "Making the PYP Happen" - "

"In the PYP, mathematics is viewed primarily as a vehicle to support inquiry, providing a global language

through which we make sense of the world around us." (pg 81)

When we get students who are keen to apply their knowledge and skills to make sense of the world around them, then we see authentic inquiry learning happening.

To get to this point, there are a few things that I feel are central to authentic inquiry in general and maths in specific:

1. Inquiry is in response to a provocation or problem drawn from real-life experience.

2. The provocation may be chosen or decided by the teacher but the direction and focus of the inquiry is determined by the student.

3. Inquiry is an open-ended process that follows a cycle – good inquiry leads to further provocations and problems to investigate.

4. Collaboration is central to effective inquiry to ensure multiple perspectives and relevance to a diverse audience.

5. An inquiry moves the student from the known into the unknown. It needs to challenge the student to question his or her own understanding.

6. An inquiry is transdisciplinary. In maths, this means a shift in thinking away from “pure maths” to a focus on “real maths”.

On this framework I intend to develop my own exploration into the world of maths and maths education. As a teacher and learner, I aim to follow my own advice – to TRAIN for the GAME.

Yep - I'm still thinking about it, trying to make sense of the world around me, still learning...

Working with Decimals

This week we have started working with decimals. Specifically, this has been linked to our National Curriculum ACMNA130 - Multiply and divide by powers of 10.

There have been some interesting conversations about the process of multiplying and dividing by 10, 100 and 1000.

We started by asking the kids what they thought a decimal was. Here are a few of their responses:

A decimal is....

• less than a whole and represented by the decimal point in it

• like a fraction

• a fraction of an integer (number) which can be divided and multiplied to make a number

• another way of representing fractions and percentages e.g 25% = 0.25 of a whole

• part of a whole number

• a dot that indicates the number after it is less than one whole

• a decimal is a fraction of a whole number usually used in complex maths problems or as an alternative to percentages

• a decimal is a way to to show numbers smaller than a whole and a way to show exact numbers

So next we started to play with some numbers.

What happens when you multiply a decimal by 10?

Here's a few responses from the kids...

0.37 x 10 = 0.370

Here is a very common mistake. We know that when we multiply by 10, you just put a zero on the end. This indicates that the student doesn't really understand what is going on with the process, that to increase by a factor of 10 we move the decimal point one position (since we have a base-10 number system). With whole numbers, this might involve adding a zero to the end of the number. With a decimal, you need to move the decimal point.

0.37 x 10 = 30.7

This indicates that the idea of place value is still not fully developed. Why does the zero suddenly appear in the middle of the number? The student has remembered that something happens with a zero so slots one in where it doesn't belong.

0.37 x 10 = 0.47

This is a bit more like addition than multiplication. An interesting way to solve the calculation but failing to grasp the idea that multiplying by 10 will increase a value tenfold.

The Importance of Context

Many of the issues that arose from these examples of errors and misunderstandings came from a lack of context. I had presented the "question" 0.37 x 10 in a purely symbolic form, without a background, context or even a diagram. I did this deliberately to see what would happen. 

I found out....

Yes - we need to teach the skills.

But - the skills need to be presented in a meaningful way - a context that explains what is being asked.

Probability Acitivities

Yesterday was the Canberra Mathematical Association Conference. We had about 120 people attend, which was pretty good for Canberra I think. Sadly, I might have been the only primary teacher there.

Note to primary teachers - find your local Maths Association and join it. You will get a lot out of it. I came away with a bottle of wine for being a presenter, a set of 3D shapes that I won as a lucky door prize, a cool tie with numbers and maths symbols that I bought for $10, a free "CMA 50 Years" lapel pin and a wireless printer/scanner/copier/fax that one of the reps was selling off for $40. 

But I also was able to attend some really interesting presentations, mostly from secondary teachers - but interesting none the less.

However, one session that was aimed at Primary teachers was called "50-50" presented by Theresa Shellshear from Australian Catholic University. 50-50 was the theme for the conference, 2013 being the 50th anniversary of the CMA and also International Year of Statistics.

Theresa presented a range of probability games and activities for using in classrooms. Here they are, supplemented with a few of my own.

1. Greedy Pig

To Play: A game consists of 10 rounds by default (see options below). All players are 'standing' as each round begins. Players roll the die (take turns or have a specified 'roller', as desired). If a two, three, four, five, or six is rolled, all standing players add that number of points to their scores for the current round. A player can 'sit down' at any time. When a player sits, he or she safeguards all the points he or she has earned in the round, but is not able to earn more points until the next round. When a one is rolled, all 'standing' players lose the points they have accumulated in the current round. The player with the most points at the end of the game wins.

There are lots of variations on this game - change the number of rounds, change the "greedy pig" number from 1 to something else, include a "Double Whammy Special" round where all scores are doubled etc
These rules taken from http://www.math.usu.edu/~schneit/CTIS/GreedyPig/ 

2. Black or Red

Use a deck of cards with the picture cards removed. Everyone starts on 50 points. Take turns turning over a card from the deck. If it is red, add it to your 50 points, if it is black then subtract it. Winner is the person with the highest score after 10 rounds.

3. Four in a Row

A game for 2 players. You need 2 six-sided dice, 2 sets of coloured counters and a chart similar to below. Take turns to roll 2 six-sided dice. Multiply the numbers and cover the product on the chart with your coloured counter. If the other player already has their counter on that number, replace it with your counter. Winner is the first to get four in a row.

Get the kids to chose other types of dice to use and redesign the board. How does this change the game? What other variations can you include?

4. Number Line

Roll 2 dice. Use these numerals to create 2 two-digit numbers. eg 1 and 5 = 15 and 51.

Plot the numbers on a number line. What is the difference between them? Who has the biggest difference?

As a variation, you could roll the dice twice and record the numbers as fractions, like 1 and 6 becomes 1/6. Find the difference between the fractions. More challenging for those who want to go a bit further.

5. 9 Piles

Remove the picture cards and 10s from a deck, leaving 36 cards behind. Deal these cards into 9 piles of four cards, face up, set up in a 3 x 3 grid.

Choose a target number (such as 12). Players take turns picking up cards that add to make that number. When a pile is exhausted, get a card from another pile. Winner is the player with the most cards at the end.

Variations - choose different target numbers, allow multiplication as well as addition, include all operations

6. Beat the Bank

This is a game I came up with. You start by giving everyone in the class $100 - it doesn't have to be real money, they can just write down $100 on a piece of paper.

Each individual decides how much of the money they are going to risk (ie bet) on the outcome of the roll of a pair of dice. If the class rolls a higher number than the teacher, then each individual gets the money that they risked. If the number is less than or equal to the roll of the teacher, they lose that amount. 

The teacher (ie the bank) gets all the money that is lost.

A good way to show that the odds are stacked in favour of the "house", as the teacher will come out well ahead of most of the students even though a few individuals might get lucky.

Here's an example of how we set it out. The students keep a running tally of their individual cash total and how much they are prepared to risk on each throw of the dice.

7. Other resources

Here's a few other resources for chance and probability activities:

• Paul Swan has a range of books (RIC Publications) that have lots of games and things to do

• The Monty Hall Problem or the 3 Door Problem http://www.youtube.com/watch?v=mhlc7peGlGg

• Probability Pond  http://www.pearsonschool.com/live/images/custom/envisionmath_ca/games/pond.html

Good Luck!

The Human Graph

As our culminating activity for a Week of Maths and to celebrate National Maths Day, the whole school got together to create a human graph and to answer the age-old question...

What Fruit Did You Have For Breakfast?

Very sneakily, this combined maths (data collection and representation) with a healthy diet.

Here are a few pics of what it all looked like:

Getting ourselves organised - yes, it took a bit of time. 560 kids to organise.

That's the "apple" line disappearing into the distance - very popular with the kids. "Bananas" is the next longest and the one you can see further across coming in third place is "strawberries" - they're very cheap at the moment.

Yep, that's the "strawberries" group with "apples" and "bananas" on the far left of the picture.

I got some Year 6 kids to design the data categories. They chose:

1. Apple
2. Avocado
3. Banana
4. Fruit Salad 
5. Grapes
6. Kiwi fruit
7. Mandarin
8. Orange
9. Peach
10. Pear
11. Rockmelon
12. Strawberry
13. Tomato
14. Watermelon 
15. Other
16. None

Results

 1. Apple was the most popular fruit for breakfast, followed by banana and then strawberry.

2. Only 25 children had no fruit for breakfast.

3. Only 1 person had tomato for breakfast and 1 had avocado. These were the two smallest categories.

4. There was a protest that bananas are a herb. 

5. There was an interesting selection of fruits in the "Other" category  - star fruit, cherries, pineapple, etc

6. Everyone had some fun.

So what?

Over the week, have we achieved anything?

I think so.

• We have focused the kids' attention on data collection. 
• We have got them to think about some of the issues behind working with data.
• We have constructed graphs.
• We have made statements based on information represented in graphs.
• We have made some sound generalisations about how to handle data.
• We have participated in an activity across the whole school P-6.
• We have had some fun.

Can't wait till next year...

More Data Handling

Day 3 of our Week of Maths and we got to look a bit more closely at some data that compared the average intake of various food groups in four countries - Australia, Italy, China and Ecuador.

The data was presented on the AAMT website in graphs produced on an Excel spreadsheet. It looked a bit like this:

You could enter a value from 1-4 and the data for each country would appear.







The students were asked to get the data from one category and compare it across the four countries.

Comparison of grain consumption as a percentage of total

Comparison of drink consumption as a percentage of total.

Do they really not drink anything in Ecuador?

Comparison of average meat consumption in $US per week.

Naturally, once we started playing with the data, the questions started flowing and we were able to make some important generalisations:


1. Garbage in, garbage out


You can only work with the data that you have. If you collect poor quality data, you can't improve it by manipulating it. So, if the data provided says Ecuador has no expenditure for drinks, does that mean:
a) they don't drink anything?
b) they don't pay for anything they drink?
c) the data hasn't been collected properly?



2. Your scale needs needs to have all the numbers up to your biggest quantity


The scale need to be cover the range from your smallest value to (at least) your largest value. It also needs the have graduations that are useful, that make the data accessible and that inform the reader.


3. Say the exact amount for each category

When dealing with data, it is important to be as accurate as possible. It was difficult for students to read exact money values off the graph provided. They resorted to estimating which meant that they ended up with different numbers. This made it difficult for them to compare their information.


4. Spreadsheets can give an exact number - roll overs

At this point, one of the bright young things discovered that if you hold the cursor over the individual data column, the exact dollar amount was shown. Now, that's something you can't do with a piece of paper.


5. Zero is data and we need to show all data

And so the conversation came around to the question of "zero". Ecuador had 0 for several categories of data. Students began to realise that if we leave out zero it will change the meaning of the graph. Zero values need to be included in data sets to show that the question was asked, the measure was taken, the observation was made but the result was zero.

Handling Data

So, for home learning last night we had some data to handle.

We had to gather our data from a series of photographs and make some comparisons between the food consumption in three countries - China, Italy and Ecuador - and then compare it with the food consumption in our own families.

The activity is part of the AAMT National Maths Day activities - see the website:

http://www.aamt.edu.au/Activities-and-projects/National-Mathematics-Day-2013/Upper-primary-activities/Your-place-in-the-world-What-do-they-eat-in

So the kids drew up tables of data, comparing different foods groups between the countries and included their own data from their own kitchen cupboards. All part of the Australian Curriculum (ACMSP148: Interpret secondary data presented in digital media and elsewhere).

Job done.

But then came the interesting part - the conversation.

"So," I asked, quite innocently, "Did anyone have any issues with the data they used for this task?"

Indeed, it seemed many of the children did.

1. Being a photograph, it was difficult to see what some of the objects were. How do you classify them? Do you guess? Do you exclude them? Do you make a separate data category for them - Unidentified?

2. Some of the categories that were given in the data table were difficult to understand. Why was there a category for "Meat" and another category for "Fish and Eggs"? Isn't fish a type of meat? Does fish have more in common with eggs than with meat? 

3. Some items could fit into two categories. Is a chicken burger from KFC "Meat" or "Fast Food"? What do you do about that? Do you include it in both? Do you exclude it from both? Do you include a half value for each category? Do you create a new category - Undecided?

4. Some items didn't fit into any category. What do you do about that? Do you force it into a category that it really doesn't belong in? Do you exclude it? Do you make a new category - Unclassified?

5. Why did different people put the same items into different categories? Why didn't we all agree about how items should be grouped?

And this conversation was just as important as the activity of filling out the table. This conversation gave the kids an insight into data, data handling, data analysis and data interpretation. Suddenly they began to realise that there was a whole lot more going on behind the processing of data - it isn't always as black and white as it seems.

Like scientists argued back in the 18th Century, is a platypus a duck or a beaver?

A Week of Maths at Radford

This week is National Science Week in Australia. The AAMT (Australian Association of Maths Teachers) has prepared a range of activities on their website for school to look at on National Mathematics Day (Friday this week). As this is also the International Year of Statistics, the activities are based on data and statistics.

Here at Radford, we have decided to make this a focus for our week. I have drawn on the AAMT resources and developed a set of Home Learning activities for all classes (P-2, 3-4, 5,6) to get some conversation going. 

Each day there is a "take home" activity looking at food resources around the world - it's transdisciplinary and international, two elements of PYP learning. 

Here's a sample of the Year 5-6 activities:

On Friday we are planning a "human graph" on the oval - based on the answer to the question "What fruit did you have for breakfast today?"

And while there is very little in the way of explicit number skills being drilled for "home learning" this week, there is lots and lots of thinking and talking about how we handle data.