Many students go through their schooling believing that there are only two answers in mathematics education - a right answer and a wrong answer.

Many insecure mathematics teachers also believe there are only two answers as confirmed in the back of a textbook.

Are there more than two answers to a mathematical problem?

As our continental friends would say, “for sure!”

There are many mathematical activities and investigations that require a certain amount of analysis before deciding on which answer best reflects the data. After all, statistics is based on this - trying to make sense of a mass of data. One example of investigations is simulation.

A student once asked me to help him begin a computer studies project. I suggested trying a simple mathematical simulation. I advised him to go to a nearby roundabout, and record the number of cars arriving and leaving, within certain time intervals, the various roundabout feeder roads. I also told him to observe traffic flow at different times of the day, especially around 8.30 am and 4.30 pm, when there was more of a rush hour traffic. He could also count the number of cars waiting to access the roundabout from the different feeder roads in order to find out maximum numbers.

Once he had some observed data, he could first of all construct a chart in a spreadsheet based on that data. After that on a separate worksheet, he could simulate the observed data using the random, and integer functions with the observed maximum numbers. He could also construct a chart based on his simulated data, and compare it with the observed one. Each time he refreshed the worksheet, his simulated data changed (and also the chart), so he could compare different simulations and reject any which were too extreme.

There are quite a lot of activities like this, where you have to make value judgments based on observed data, and mathematical knowledge and experience. There are no books to refer to, no answers in the back to check.

Which method is better to increase mathematical knowledge and understanding? Obviously using set problems with either a correct or incorrect answer is good for employing established methods of mathematical knowledge and logical reasoning. It's the most common method used in exams. However, equally mathematical activities extends knowledge, reasoning and intuition. It teaches the students (and teacher) to rely more on their own understanding rather than an answer in the back of the book.

Footnote:

I gave another student a similar task, to observe cars boarding and leaving a nearby river ferry. He had an uncle who worked in a large firm, and who fancied himself as an ‘expert' in spreadsheets. Some time after I had set the student his task, he brought in a note from his uncle. In the note, the uncle said my spreadsheet model was wrong. I told the student, that the simulation model might be poor, since it did not reflect actual or possible observations, but his uncle couldn't say it was wrong, simply because there is no right or wrong.

It mundanely turned out that the uncle didn't understand random functions (which is why he thought it was wrong), which are fundamental to simulations. I had advised the student that when he achieved a good simulation, to save it, and do the next one in a new worksheet, so he could build up a sequence of possible simulations. Unfortunately for the student, he listened to his ‘expert' uncle who told him to link the worksheets. Consequently, every time the student opened the spreadsheet workbook, all his simulations changed.