Teaching old dogs new tricks

The first draft of Chapter Eight addresses the toughest of questions in adult numeracy: How do we teach old dogs new tricks?

The quote directly below the title: "So how you gonna teach a old dog new tricks?" comes from an adult learner discussing his frustration with the mathematical demands of his programme.

The experiences of 'old dogs' (adult learners) who have developed negative beliefs about mathematics are devastating. The cuts run deep. When a 46 year old man can recount unpleasant conversations he had with a maths teacher when aged 10, you know those early experiences matter. Being positioned by your peers and significant others as 'not good at maths', and then coming to accepting this as fact, does not just change. It sticks.

Learning mathematics is entirely to do with how you engage with it. Do the beliefs, developed during early experiences, influence how 'old dogs' engage? Undoubtedly.    

As a nation we invest large sums of money predicated on the notion that an adult who struggled to learn mathematics during school, and developed negative beliefs, will re-engage and learn in a vocational setting.

How did we arrive at our conclusion?
And what do we do if we were wrong?

The answers are far more complex than I ever guessed at the beginning of this journey.  The stakes are also higher for the learners than I ever imagined.

The Magic Square

The magic square can be used to generate great thinking.  The trick again, is not to try to get it done as quickly as possible, but to use it to generate thought.

Here is how

You have engaged in analysing how many ways certain numbers can be produced using two dice. This activity builds on this.

The magic square has 9 squares.  The numbers 1-9 need to be entered so that they equal 15, horizontally, vertically and diagonally.

Before launching in and beginning to guess, have a go at this instead.

Task one

First think about how many equations each individual square will be involved in. For example, the top right square will be involved in 3 equations.

How about the middle right square? It'll be involved in only 2 equations.

Now work out and put in order which will be involved in how many equations.

Task two

Now work out and order the quantity of 3 number equations that equal 15, that each number is included in. For example:

The number 1 is only able to be in two equations that equal 15. 1+8+6=15 and 1+5+9=15.

Therefore, the number 1 can only go into a square that is involved in two equations.  You also know the numbers (8, 6, 5, 9) that must be in the equations with the 1. Read this again if you're not getting it.  

This is how mathematicians solve problems. They deduce their answers - like Batman. They do not go random. Try not to succumb to the temptation to guess.  Use those deductive reasoning skills.

You can do it!

Next challenge

Once you have nailed the three by three square use your new powers of deduction to solve the four-by-four square.  This one is a beast, and well worth taking a deductive approach.

Enter the numbers 1-16 so that horizontally, vertically and diagonally each row and column, and diagonal equals 34.



Good luck.

The joy is in the struggle.

Thursdays' whiteboard challenge

What's the most likely score when you roll two dice? Do you know, and do you know why? What's the second most likely score? The third?

If you were gambling and someone said pick any number except seven, and if that comes up I'll double your money, what number would you pick?

When it comes to rolling two dice there are more chances of a seven being rolled than a two. Have you ever explored why this is so? The ramifications are possibly greater than you realized.




Have a go at this challenge with your learners.  Although its not that exciting, it will set you and them up for the next challenge which is well worth it. This and the next have the potential to really shift your learners mathematical thinking - in terms of thinking like a mathematician.

The challenge

Play a little gambling game with lollies or match sticks.  Take two dice, pick a number each, and play at least 50 rounds. Track which numbers come up the most using tally points.  I.e over 50 throws the 6 came up 17 times etc.

Here's a good game to play with the above. Everybody bets a lolly or matchstick each round.  If a seven, a five or an eight is rolled they lose all their lollies (modify this to just a 7 if you want). Each round they stay in they win one extra lolly. They can retire at any time and cash out their lollies. The longer they stay the greater the risk. There are lots of different ways to play.

 Click here for another good game that's fun, quick, and easy to play.  

Next activity
Work out if you are rolling two dice, how many ways each score can be rolled.  For example, how many ways can a three be rolled? It can be rolled only two ways: 1+2 =3, 2+1= 3.  How many ways can a five be rolled? 1+4 =5, 2+3=5, 3+2=5, 4+1=5, so four ways - twice as many as a 3 (so which would you bet on?)

Rank the possible scores in order of highest chance to the lowest chance.

To get you started:
A twelve can be rolled in only 1 way
An eleven can be rolled in ______ ways
A ten ...
A nine ...
An eight ...
A seven can be rolled in 6 different ways
A six ...
A five ...
A four ...
A three ...
A two can be rolled in only 1 way.

If you want to you can extend this by asking your learners to identify the chance of each being rolled. Check this video out and see if you can apply it to the two dice scenario.

Your answers might look like this:
The chances of a 7 being rolled are 6 in ?
The chances of an 8 being rolled are ? in ?

On Thursday night I will post the follow up whiteboard problem which is really interesting and useful. Get this one sorted first.

Whiteboard Problem: Algebra

Below is a typical algebra problem. Keep in mind - word problems are tools to generate thought, not tasks to be completed.

Problem:

Water is being drained out of a tank through 2 pipes at the rate of 330L/min. We know that one pipe releases 50L/min more than the other. How much do the 2 pipes drain each?

Solution below if you want it - but remember, the joy is in the struggle.

Solution:

Reasoning. If one pipe releases 50L/minute more than the other then the difference between them is 50L/min. If I subtract the 50L/minute from the higher capacity pipe, both pipes will be releasing the same quantity. This will also remove 50L/m from the total. So together both release 280L/m.  That means I halve 280 to get 140 for each pipe. Now we add 50L/m to one pipe bringing it to 190. So 140 plus 190 equals 330L/m.

This is harder to follow than actually do! Hope that makes sense.  A good activity would be to get the learner (and you) write your own approach in your own words. Quite difficult!  

Algebra:

330= x+x+50

330= 2x + 50

Cancel 50 from both sides

280= 2x

x=140

140 + 50 = 190

You will discover that it is the written explanations that make mathematics daunting. The concepts are actually pretty easy to grasp, it's translating the concepts from other people that's the real challenge.

Ten bowling pins problem

For the whiteboard this week.

The ten bowling pins are pointed toward the bottom of the page. Move any three of them to make the arrangement point up toward the top of the page.

A good problem for learners to chew over.

This problem is totally solvable but takes some real thinking.  Write (or print) it out and let the brainstorming begin.

The radiation problem
You are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die. There is a kind of ray that can be used to kill the tumor. If the rays reach the tumor in sufficient intensity the tumor will be destroyed. At lower intensities the rays are harmless to healthy tissue, but they will not harm the tumor either. What type of procedure might be used to destroy the tumor with the rays without destroying healthy tissue?

Keep in mind that you are trying to destroy something inside the body without destroying healthy tissue on the way in.

Damon's class friendly version
A second question of the same type that I developed (while thinking about avoiding the whole tumor thing) makes the whole thing easier I think.  Here is the same problem but couched in friendlier terms.



First, check these out.











These are small glass blocks with beautiful designs embedded within them.  If you haven't seen them then drop on by your local tourist shop - they are everywhere.  They are also really amazing, particularly when you begin to question HOW they were made.

I handed out several to my class, and then posed the question - how were these made?

Once they solved it, I handed out the top problem, and they totally solved it in five minutes.

So, what's the answer?

Mathematical problems are not tasks to be completed - they are tools to generate thinking.

A couple of weeks ago I suggested that parents and adult tutors begin the daily practice of giving mathematical problems to learners in order to cultivate a culture of mathematizing. The clip included some strategies and problem types to begin the process.

The clip below demonstrates how to use these problems to do so.

Here is the theme: Mathematical problems are not tasks to be completed - they are tools to generate thinking. This one gets better about half way in - hope it's helpful.

YouTube link here.