A and B are the intersections of the two circles. AM and AN are tangents to the circles. P is where BM extended meets one circle, and Q is where BN extended meets the other circle.
Show that PM is the same length as QN.
If you want a hint, have a look at this Geogebra worksheet.
I'd love to know how you spent the first 5 minutes trying to solve these problems:
Here is a possible hint.
I have been sitting in some lessons in the hope of observing what some students do when they don’t know what to do next. Today, the teacher asked me to help a student who has been having some difficulties with the work they have been doing recently.
There were occasions when I felt the teacher intervened before it was necessary. This is a judgement, but an honest one. I felt the solutions would have been within her grasp, given more time. I made the decision to intervene as infrequently and as unobtrusively as possible.
On completion of a problem, I avoided evaluating her solutions, asking only: ‘How can you check whether your answer is correct?’ I sometimes asked some variant of: ‘Does this make sense?’, regardless of whether the answer was correct or incorrect. This is another form of checking, but with it comes the opportunity to make connections.
Checking is a primary means of coming to know for oneself, with which comes confidence, and the development of what Gattegno called 'inner criteria'.
'Teacher lusts' are ever-present; they come from a variety of sources and manifest in a variety of ways.
In my short time working with this student, I felt the desire for the student to be 'successful', but instantly became aware that this was based in pride, bringing with it the desire to intervene more than is necessary, and the possibility of disappointment.
Some degree of non-attachment is required. With this comes a movement away from wanting to having faith, thus allowing the student time and space to come to know for themselves. Taking the wanting out of the waiting.
With this comes the possibility of the equivalent in teaching of a ‘moment of grace’: the student solves a problem which previously seemed beyond their grasp and, upon checking their solution, they come to know that they are correct, without any intervention.
I was once asked if I was 'still struggling' on a maths problem, which was a bit annoying. Yes, I was (still) working on it, it was difficult, and it was taking some time. But I wouldn't say I was struggling.
I also wonder about the word 'stuck' as applied to solving (maths) problems. To me it implies something unfavourable, of wanting to get out as soon as possible, of being constrained and uncomfortable.
Might it be beneficial to remove 'stuck' and 'struggling' from the vocabulary of learning mathematics? No longer stuck and struggling, we may become more accepting of the (possibly lengthy) state of not knowing as necessary for coming to know.
There are various types of not knowing: from not being able to recall facts and procedures, through not being able to integrate the past with the present, to the always unknown future.
Ways of improving recall are well documented and perhaps more easily addressed. Less well understood are ways of improving integration of the past with the present, and ways of coping with the future.
What can be done to help students be comfortable with these various ways of not knowing, to assist them in coming to know?
For the student, there are various sources that may provide assistance in coming to know. A student may come to rely on the teacher, his/her friends, the textbook, the internet.
But whilst other sources may be useful in providing information, turning to another may be a lost opportunity to learn something for - and about - ourselves.
It is essential that students spend some time not knowing, and coming to know. It is time spent in these states that we develop what Gattegno calls criteria for truth.
One may be aware of oneself as time.
This cryptic quotation, taken from Gattegno's Science of Education, has remained with me. I currently take it to mean that we are formed by the way we use ourselves, and the energy available to us, over time.
In this post, I will attempt to describe how time is a crucial element in solving problems, and learning how to solve problems.
Try this problem, taken from the book Five Hundred Mathematical Challenges:
This problem took me around 30 minutes. I tried 7 different approaches and made a completely fresh start 3 times before arriving at a solution.
We do what we can in order to prepare for insight, but we must wait for a moment of grace. Knowing that solving problems takes time is a simple but important awareness.
It might be useful, perhaps essential, for teachers to work with students on remaining with problems for longer and longer periods. If time isn't taken, elusive solutions will not be reached, and perhaps more importantly, awarenesses regarding what to do when we get stuck will not be developed.
We are formed by the way we use ourselves, over time.
Here is a second problem you might like to try:
I worked on this for two days and one sleep. I was stuck for long periods. There were glimmers of hope, a number of what seemed like good ideas, followed by dead ends. There were long periods of doubt, and eventually I gave in and looked at the answers.
Sometimes the moment of grace occurs, and sometimes it doesn't.
Having the answers at hand is sometimes just too tempting. For this reason, I consider it preferable that answers are withheld from students for as long as is suitable. Only through not knowing for long periods of time do we bring everything we have to bear on a problem; only through time do we learn about the dynamics of being stuck.
Solutions are most informative having worked on a problem for a long time and exhausted the energy available.
Even though I did not reach a solution to the Problem 58, it was a rich learning experience, in terms of dynamics and content. Upon reading the solution, I created this set of questions that might help you access the mathematics involved in this problem if you became stuck, like me: