I think the point of the exercise in this image is for the student to decide whether or not the question is correct and to explain why they think so. He or she has to analyse each step and justify his or her conclusions. This is a lot more work than a question which asks you to find the answer in the simplest way possible.
Now don’t get me wrong: I also think it’s important for students to know basic skills as well like how to subtract numbers the traditional way. It’s a very effective strategy because it’s quick and simple to use. But we don’t take enough time to teach the processes behind those steps, such as reasoning and problem solving. Give that student the question, 427 – 316, and if he memorized the traditional way of subtracting he’ll probably get the answer right.
But change that question into a problem solving question such as “find the error on a number line” and most students who have only learned the traditional strategies might find it more difficult because they only see one way of solving subtractions.
Often, the traditional way of teaching meant teaching the most effective way to get to the answer. It would be often modelled as follows:
- The teacher would define the problem and the strategy,
- The teacher would prove how the strategy works.
- The teacher would give an example on how to use the strategy on a given problem step by step.
- The students were then expected to work on class problems and the homework.
- The next day, the teacher would review a few homework questions for clarification.
- The teacher would also give quizzes and tests to grade whether the students understood the lesson, and have done the homework.
Do we see a pattern here? I’m sure this method was pretty familiar for who took math a generation ago and beyond. The teacher is doing most of the work by directly telling the students what to learn, as though they were reading a living, breathing instruction manual, and it’s expected that the students follow these instructions and complete the work. No one questions if there are any other methods because it’s again expected that the teacher is ultimately giving his or her students the correct and only steps on how to find the solution. It’s also expected that the students mimic and memorize the model the teacher has established.
Let’s look at the type of questions Math teachers often use to model their strategies in the traditional way of teaching. These are the “What” type of questions – which mostly answers direct questions: “What is 427 – 311?” This direct method of teaching is great for those who find it easy to follow instructions verbally and written down step by step. And has that method worked well for everyone in the past?
It’s comparable to an instruction manual from Ikea. The steps are drawn as pictures and they can be followed gradually. What’s the common complaint from Ikea manuals? There are no words or labels for each piece, so there can be difficulty interpreting each step, or which piece to use. People get frustrated and pieces get broken or missing – which ends with them swearing off buying DIY furniture ever again. My question for you: Have you ever felt frustrated when you couldn’t understand a particular step? Do you always have to depend on the instructions or can you find an alternative method to solve your problem? Does it matter to solve it exactly like how the instructions describe? Sometimes a futon or a bed doesn’t need all the pieces or look exactly like the store model to be an effective piece of furniture.
We often expect Math to have a universal truth: For every math question, there must be a given answer. It’s a very rigid way of thinking because it gives a quick answer to one type of problem without having to reflect too much on the question because we have faith in the process. We don’t also take enough time to ask the “Why” questions: “Why is 427 – 311?” “Is there a different way to see this answer?” “Why is subtracting the traditional way often the most effective strategy?”
This type of questioning goes further than just knowing how to subtract. We are now growing up in a society that is changing socially, technologically, financially at an exponential rate. In the last 30 years, we grew up with several generations of technological devices that have already outgrown society: For music devices, we went through vinyl records, 8-tracks, cassettes, compact discs, to digital media (mp3s). When I was a child of the 80s, I was able to reprogram my VCR clock, work a remote control and understand how to program software on pre-Windows computers. Children today are able to work on tablets, SMART phones, and video games; they can browse the Internet, and create apps, all at a very young age.
The types of jobs are also rapidly changing – we are training students for jobs that haven’t been invented yet until after they have graduated. We are also teaching for jobs that will become obsolete. Yes, students need to learn basic numeracy and literacy skills so that they can be used as tools to help them solve problems. But they should also be encouraged to learn other processes such as problem solving skills and reasoning so they can learn to adapt in an every changing world.
Parents like the one in this image lived in a world where life decisions were black and white. You followed a straight forward path – you had a problem: here’s the solution. Most people had jobs with simple tasks – such as inserting parts in a car factory. With the advent of the 21st Century, society has been fortunate to have many options offered to us – but not without many more difficulties as well.
As well, our society is also now competing financially with other countries that are growing and flourishing. Workers in car factories get laid off because robots can do the work 100 times better and faster. It’s also much cheaper to hire labour in countries with lower GDP if we want more t-shirts, more TVs and more tablets available in our homes. Students need to be trained to think critically as opposed to just basic skills. We also want to teach students to decide which strategy would be the most effective for them to use. If we’re only teaching students to follow instructions and not to think critically, there could be serious consequences in the long run. If we’re only teaching students to memorize the formula as opposed to understanding it, are we training them responsibly?
Mathworkz 