The Importance of simple Problem Solving

One struggle for many students in solving problems is that the problems are just too cognitively demanding. 

If a student has just begin to grasp a new process or idea, is it fair to then ask them to dive into a open ended task that requires skills beyond mathematical understanding, such as perseverance and communication?

This is why one of my most importnant strategies for problem solving is to aim for the maths involved to be at least one academic year below the current topic. Multiplying 3 digit by 2 digit? Explore 3 digit by one digit problems. Learning about prime numbers? Do an investigation into odd numbers first.

By simplifying the problem, students can achieve more success and build confidence to attempt further problems, or simply learn to enjoy the subject and its beauty.

Take for example the simple prompt:

The answer is 10. What is the question?

This is accessible for Year 2, but equally just as good for Year 5 and 6. We can build on this through rich conversations. Let the students set the parameters. Are negative numbers allowed? Can we use fractions? Decimals? It has to be whole numbers? Is addition the only operation we can use?

We can keep extending it. What if the question was a diagram, a picture? Are there infinite possibilities? How do you know? 

How about restricting answers to a question so there are manageable finite solutions:

Here one student said to me "There are so many answers, it will take forever!" 

I replied by asking, if we forget about prime numbers for a second, what numbers would work?

I also gave the prompt, what is 10 x 30? 300? Ah that would be too big. This question and response was enough for some of my students to go off and complete the problem. I wasn't totally happy that I gave them that much help, but at an early stage of learning to problem solve there is still a lot more for them to explain here and so guiding them will still lead to great discoveries.

Basically, problem solving should be treated like fluency - grab some early wins for the students, have them access and feel comfortable with the topic and then stretch when they are ready and engaged.

Key tips:

• Set up problems that involve maths from the previous academic year or two
• Model how to share explanations to a problem, including giving sentence starters
• Scaffold as much as necessary to allow students to work on the problem. Don't leave them struggling. Reduce the possible answers if possible
• Share answers - show them it wasn't impossible (some students won't want you to reveal answers, but realistically they very rarely carry on trying to solve it, better to just show the answer.
• By encouraging these tips, after two or three school years, students will face problem solving with a positive approach, a bank of successes to think back on, and a wealth of experience in similar tasks.

Multiplication Game

 

I love simple mathematics games that can be reused over and over. Also ones that combine strategy and fluency, such as this multiplication game. 

Fairly self explanatory - two players, each take turns to choose a number from the top and the bottom and multiply them together. The answer will be in the grid, and the aim is to get four in a row on the grid.

Link to PDF Here

One observation is watching students reverse their thinking, "Oh I need 3204, that is an even number and it ends in 4, which numbers will help me achieve that result"

Completeness - Fraction Multiplication

 

I recently came across a fun question (I think from White Rose Maths) but I thought it had missed a trick, so I amended it to allow for more explanation:

This first question proved much more difficult than the second one, given you have less numbers to use. I think I would have preferred the other way around first.

It was fascinating to see the different approaches to solving it. Most dived in with brute force, trying as many numbers as they could. One student was adamant it was impossible, so I asked her to prove it was impossible.

"How do I prove that it is impossible"

"Well, start with why you think it is impossible?"

"Because it is hard"

"What exactly is hard?"

"There are no fractions that work, I've tried them all"

"Show me you have tried them all then"

At this point she realised she hadn't considered improper fractions and so was focused in on a more rigid approach.

One student made an excellent discovery (we have only taught up to multiplication of fractions). 

"We can just flip the numbers and it still works"

"Which numbers did you flip? Do you have to flip all of them?"

"Just the fractions, but they both have to be flipped"

"Does that help when finding non-solutions"

"Maybe. Maybe all the answers are in pairs"

Turning a question into one of completeness and asking them to PROVE and EXPLAIN really gets the students thinking about how to strategise and solve complex problems. It also sowed the seeds of reciprocals and division, but I did not need to explain this here, it was of no significance to their process at this time and would cognitively overstretch them and take them away from their own strategies.

 This year I've been working with a small group of students who are aiming for any level of pass in their upcoming IGCSE.

In order to simplify the study, we've concentrated our efforts on about 45% of the paper. Pretty much all of the algebra (apart from completing the square and graphing and lines of equations) and FDP, as well as data.

Some of the students have difficulty with processing complex information or struggle with cognitive load, so most of the topics we cover are the straightforward questions that can be solved with a a simple algorithmic approach.

Anyway each week se spend 45 minutes on only algebra, and 45 minutes on everything else, leaving the extra double lesson to look at past papers or teach/refresh a new topic.

The constant spaced repetition seems to be working - so sharing the sheets I created, questions, taken from a variety of places, including the amazing Dr Austin and Boss Maths to name a couple. When I get to the end, I simply start again from the start! I'm adding more to the Everything Else sheets as we teach it.

Algebra Qs

Answers

Everything Else Qs

Answers

Division Games

Here are some activities for division, particularly useful in a primary setting.

The 'Race To One' sheet is particularly liked by my recent year 3 class.

Slideshow of them (So I can dynamically add to the slides over time) or you can download as PDF version below.

Download Slides as PDF

Circles and Pythagoras

I've not used Geogebra for a while - I forgot how fun it is!

This is similar to a problem from many sources, such as this excellent University of Georgia article

Find Me... Grids

A simple but rich activity.

Find Me Grids

There is a template version available to create your own - although it isn't hard to create the model yourself.




Depending on age and ability, they could create their own - or explain understanding.

The document is visible on google slides, you can print any page you need - there are a couple that are slightly different just because it is hard to create fractions in google slides.


I will keep adding to the slide link below so keep this page bookmarked.

Template Link