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.

The joy of teaching and learning

This meme cracked me up for some reason.  I've been the teacher and the student.

The 'Consolidation of Responsibility' was the tip of an iceberg (Why adult mathematics classes fail)

Early research with adults in mathematics classes revealed a phenomenon called 'the consolidation of responsibility' (Howard & Baird, 2000; Karp & Yoels, 1976). This is when the members of the class cede personal agency to a few members of the class that they believe are more capable. Thus, a small number of the class members account for the majority of interactions between the class and the tutor, while the others remain silent.  If you are a tutor, you will be aware that you will have a learner or two in your class that you end up addressing more than others. It's bad, because we learn by engaging in discussions, not just listening to others. The problem is adult learners letting others take the responsibility for asking questions - however that is the very tip of a nasty iceberg.

My research elaborates on the previous research by allowing us into the private world of learners. This was achieved through the use of microphones that recorded all utterances, at all times.  I am able to recreate every moment of a lesson from various perspectives.  Imagine taking a 3D video of a class which you can freeze, fast-forward or rewind while being able to move around in any direction.  You could take the perspective of any one learner for the whole class, a group, or the tutor. I can do this in an audio sense, and thus fidelity is high. The results are fascinating.

I have been busy putting some theory around this using a symbolic interactionist (SI) framework. A key tenet of SI is that people are constantly interpreting others. They then modify their behaviours in order to achieve some need, usually a smooth interaction or collaboration, often in order to verify their identity in some way. In short, people read others and adapt. We 'read' others based on our beliefs about what their interactions and behaviours mean. Our subsequent adaptions are then 'read' by the others, changing the way they act - and on it goes! If this becomes maladaptive it can create interactions that are negative, which in a classroom will erode learning. This is particularly true for mathematics learning where we know that a particular pattern of behaviour results in learning and another does not.  

It looks a bit like this within a typical vocational embedded numeracy class. Group members assign status to members they believe have a high proficiency in mathematics. They then take somewhat passive or supportive roles and acquiesce to these learners. The learner with higher status is more able to assert their expectations for normative behaviour on the group. These behaviours are not consistent with effective learning behaviours - in fact they are completely negative.  The pattern of behaviour however, reproduces the conditions that produced status, and perpetuates a cycle of high/low proficiency members.

In other words - this pattern perpetuates the failures many learners experienced in school.

In conclusion, the research on the consolidation of responsibility ought to have attracted more attention from the adult mathematics community. It was the top of a very nasty iceberg that potentially negates all the learning potential in an adult class.