- Cubes Problem Solving Worksheet
- Cubes Strategy Problem Solving Equations
- Problem Solving Strategy Worksheet
- Problem Solving Skills
The CUBES Math Strategy is a great tool for students to break down and solve those tricky word problems. Includes a free CUBES math strategy template! Angie Olson at Lucky Little Learners Math for Second Grade. Problem solving the new mathematics national curriculum has three aims relating to problem solving reasoning and fluency and as discussed above using well chosen tasks which involve cubes can certainly help to achieve all three it also contains more challenging content compared with the old version so having mental images to draw upon.
- Through our choice of task
- Through structuring the stages of the problem-solving process
- Through explicitly and repeatedly providing children with opportunities to develop key problem-solving skills.
NRICH offers a wide range of rich tasks to engage learners in the problem-solving process. We want all our tasks to be used in such a way that they enable learners to explore and work from their own level of understanding, and then build on this towards new understandings. We believe that this approach offers the opportunity for rich, embedded learning.
Some of our NRICH tasks are simple games that can be played time and time again to develop numerical fluency as well as problem solving and reasoning, such as Spiralling Decimals and Totality (see also our article Developing Number Fluency - What, Why and How), whilst others may be non-routine problems that have a specific solution,such as Shape Times Shape. Some may take a short time, like Shut the Box, whilst others may intrigue and challenge over more than one lesson, like Dice in a Corner.
The key is to be clear how you want to use a particular problem to support the development of the children’s skills to problem solve. It is tempting to choose a problem that relates to the mathematical content that you have been working on in class. However, whilst children need to be fluent with the mathematical content demands of any problem they tackle, it may be more productive to choose aproblem that builds on a specific element of problem solving that you are working on as a class, and uses content that they are very familiar, and more confident, with.
The Primary National Strategy (May 2004) suggested that there are five different types of problem:- Finding All Possibilities (for example Half Time)
- Logic (for example Teddy Town)
- Visual (for example Baravelle)
- Rules and Patterns (for example Ip Dip)
- Word Problems
The stages of the problem-solving process
The problem-solving process can usually be thought of as having four stages:
- Stage 1: Getting started
- Stage 2: Working on the problem
- Stage 3: Digging deeper
- Stage 4: Reflecting
We can helpfully spend time with children concentrating on one of these stages explicitly, in turn, as they learn to become confident problem solvers.
Stage 1: Getting started will mean offering them strategies to help them engage with the problem. These could be prompts such as:
- Tell me/a partner what you think the problem is about.
- What would help you understand the problem?
- You might like to draw a diagram, act it out or represent it with a model.
- Could you try something out and see what happens?
- What other problems have you seen that are ‘a bit like’ this one?
- What mathematical skills have you got that could be helpful here?
- Try making a simpler case to get an idea of how the problem works.
- Trial and improvement
- Working systematically (and remember there will be more that one way of doing this: not just the one that is obvious to you!)
- Pattern spotting
- Working backwards
- Reasoning logically
- Visualising
- Conjecturing.
The children will benefit from becoming proficient in each of these skills and working on one of them as a key focus in a lesson or series of lessons could be a useful strategy. Our article Using NRICH Tasks to Develop Key Problem-solving Skills offers further guidance.
Stage 3: Digging deeper usually happens when the problem has been explored and then it is possible to look for generalisations and proof. Here’s an example from our NRICH problem Make 37:
Stage 4: Reflecting is the part of the problem-solving process where we support the children to:
- interpret their findings so far in the context of the problem
- explain their solution both verbally and in writing
- evaluate their method and compare different strategies.
Written recording could be in the form of a photograph, diagram or written explanation. Here are some children’s solutions to our NRICH problem Domino Square that illustrate how recording may develop:
Children will need support to develop their proficiency with written recording. Here’s an example of a child explaining their thinking for Eggs in Baskets. She has used a trial and improvement approach.
You can read more about types of recording in this article.
As children reflect on the problem-solving adventure, they will need to be supported to compare different strategies that were used to solve the problem in order to consider the efficiency of the method and the elegance of the solution. This will enable them to see how they might refine their own methods or adopt a different one next time they encounter a similar problem.
The skills needed for a problem-solving task
By this we mean the problem-solving skills listed above in Stage 2: working on the problem. It will help the children become fluent in these if you take every opportunity to explicitly talk about them and use the appropriate language when they occur in games or larger problem-solving activities. You may like to focus on developing one or two at a time. The article UsingNRICH Tasks to Develop Key Problem-solving Skills unpicks what we mean by these skills and draws attention to activities which will help learners develop them.
Our youngest learners can start thinking about ‘working systematically’ in contexts such as choosing two toppings out of sprinkles, sugar stars or flakes to go on top of iced biscuits they are making. The key question is – ‘how do you know that you have got them all?’. This comes after, ‘I can find some solutions’ and ‘I can recognise ones that are the same’. For example, is havingsprinkles and sugar stars the same as having sugar stars and sprinkles on top of my iced biscuit?
Children are often quite good at having a random guess as to how to solve a problem. To become fluent at trial and improvement they need to be able to think about how to adapt their first guess so that it is more likely to become a solution rather than throwing the first one out and starting again. When starting to explore Dice in a Corner children may well put the dicetogether at random and be surprised when they get the magic total of 18. Those children who are becoming fluent at trial and improvement will then want to adjust the dice to see if they can make 18 in another way, rather than trying another random arrangement.
There are lots of NRICH problems that will help you develop these skills with children in our collections.
Being a competent and confident problem solver is central to the mathematical development of all our learners. It is also the major aim of the mathematics national curriculum in England. This article has detailed the individual elements that teachers can focus on to support children to gain this level of proficiency. We trust you will find it useful and we are always interested in your feedbackand experiences as you explore problem solving together with the children in your class.
References
Primary National Strategy (2004) Problem solving A CPD pack to support the learning and teaching of mathematical problem solving. DfES Publications
Here is a pdf version of this article: DevelopingExcellenceInProblemSolving.pdf
The right problem-solving strategies can make the difference between putting a challenge behind you and wallowing in an endemic issue.
Branford and Stein presented the IDEAL problem-solving method in 1984.
This easy-to-remember heuristic device represents the 5 steps of this evergreen problem-solving method.
IDEAL problem-solvers I-dentify the source of the problem, D-efine its context, E-xplore solutions strategies, A-ct on the best solution, and L-ook back and evaluate the process:
1) IDEAL First Step: Identify
You need to get to the bottom of the issue, not play the blame game.
Have your team members (perhaps after a team-building game like #7) write all the causes they can think of on a large white board. Go through them one-by-one, asking one question: what caused this cause?
By backtracking to the ultimate root cause, you create a solid foundation for further discussion.
2) IDEAL Second Step: Define
Get to know the problem by asking questions and getting all pertinent information out in the open.
Ask your team leading questions that will get them to open up and share:
- When was the last time this system worked correctly?
- When did this issue arise?
- What do we know/not know about this problem?
Wrap up this step by creating a one-sentence definition of the problem.
Revise/edit this sentence (as a group) until it clearly and succinctly represents your challenge/goal.
3) IDEAL Third Step: Explore
Just like the previous steps, you must brainstorm possible solutions without judgment.
Solicit inputs from all meeting attendees, not just those closest to the problem – or those who like to talk the most. Have a contest to see who can write the silliest solution on the white board.
When you have a good crop of solutions, pick the best and examine what it would take to implement them.
Ask your team to brainstorm a new round of solutions that address the difficulties of each approach.
Once you’re sure you have a clear picture of the logistics involved in each possible solution, make your choice.
4) IDEAL Fourth Step: Action
During the action phase, carry out the solution you chose in accord with your plan.
As a team leader/project manager, make sure everyone knows and implements their part of the solution. Identify miscommunications and unexpected complications early – before they create drama and further setbacks.
If everyone knows their role—and how they fit into the bigger picture—you can quickly put this problem behind you.
5) IDEAL Fifth Step: Look Back
Of all problem-solving strategies, this makes the most long-term impact.
Another issue will arise, soon enough – are you ready?
Have you learned how to work with your team to handle difficult situations?
Did you use this challenge to learn about your employees’ strengths and weaknesses?
Have you developed your team’s communication skills and increased your trust levels?
You can retain millennial employees by making them feel connected to the mission (and decision-making process) of your business.
With the problem-solving activities below, you can develop your team, help people communicate, give everyone a voice, and spur creativity.
Try these quick problem-solving games when your group needs a collective brain boost to tackle a difficult challenge:
6) Group Communication/Decision-Making Game: Lego Master
Split your group into two or more teams. Before the event, create a few quick Lego structures.
Choose one “Lego Master” from each team – only this person can look at the team’s structure. The Lego Masters must describe their original structures to their teams without touching the ones under construction.
Choose Lego Masters who need to develop clear and precise communication techniques; pick construction team members who need to improve their active listening skills.
The Lego Master game opens up new lines of communication and leads well into brainstorming sessions.
In idea/problem-solving meetings, you want the input of all attendees – not just those who like to hear themselves talk!
7) Group Communication/Collaboration Game: Escape Room
As teams develop and grow, members work out their conflicts, identify each other’s communication styles, and discover how to function as a unit.
If possible, visit an escape room as a group. If you don’t have the time/resources for such an event, create your own in any lockable meeting room.
Leave clues around the room that lead from one to the other.
For example, if one of your team members loves coffee, you could create a clue saying, “I’m underneath Kathy’s favorite appliance.” The clue under the coffee maker could say, “The combination to the locked box on the table is the sum of everyone’s birthday month.”
Use your imagination or reward one of your creatives with this exciting game-creation task. Don’t forget to tailor this activity to group interaction and collaboration!
8) Group Creativity/Collaboration Game: Marshmallow Challenge
A problem is the difference between what you expected and what actually happened.
Many workplace challenges arise when people indulge in a “but we always do it this way” mentality. The Marshmallow Challenge helps teams confront assumptions.
In this game, teams have a set period of time to create a free-standing structure from 20 sticks of spaghetti, a marshmallow, a yard of tape, and a yard of string.
The trick is, marshmallows seem light but are actually quite heavy (compared to spaghetti).
Observe your team members in this “aha moment”; recount this discovery after the game and ask them to approach your project without assumptions.
9) Non-Verbal Communication Skills: Blind Line-Up
Not all of us use words; many people speak without speaking.
For highly-verbal people, listening to others is quite challenging; simultaneously listening and picking up non-verbal cues can be highly difficult.
Give the talkers in your group a new experience with this fast and fun team-building game.
Give each person in your group a number (1,2,3, etc.). Don’t let people share their numbers verbally or with hand signals. Through body language alone, your employees must communicate effectively and line up according to their numbers.
When the game is over (it’s useful to play a few quick rounds of this game), ask your talkers how it felt to “speak” and “listen” with their eyes – not their ears.
Ask the non-talkers how it felt to use their native communication skills.
This problem-solving strategy can open up unprecedented levels of communication between vastly different personality types.
Bring your team back to the problem at hand and ask them to watch each other closely while listening – and gain an extra layer of communication from non-verbal cues.
10) Advanced Problem-Solving Strategies: Delegate and Separate
Some of your team members work best on their own. Delegate a task (or a subset of a solution) to each of these people and watch them soar.
Give them the opportunity to present their ideas verbally in an upcoming meeting.
Independent thinkers love to be on stage!
Other employees don’t contribute when the most dominant team members get on a roll.
To encourage shy folks to chip in, put them in groups of two or three. Ask them to work on a subset of your team’s challenge and write up a solution.
Without the pressure of having to verbally present their ideas (and get a word in edgewise), these “unsung heroes” can provide massive value.
11) Advanced Problem-Solving Strategies: Divide and Conquer
Instead of wallowing in a complex (and interlocking) problem, separate the problem into individual issues.
For example, say you’re a project manager who inherits an existing team (as a project manager) with a number of unresolved interpersonal dramas.
Please, don’t drag everyone into a meeting and tell them to hash out their differences. Work with individual conflicts, two people at a time.
If you’re tasked with optimizing an inefficient system, identify all the issues, sub-issues, and root causes of your vague overall challenge: a general lack of productivity.
12) Advanced Problem-Solving Strategies: Go to Extremes
When teasing out root causes and the logistical implications of potential solutions, examine both ends of the spectrum.
For example, if you’re considering the effects of humidity on a manufacturing plant, look at what happens under very wet and very dry conditions.
If you’re trying to balance team member workloads, determine the effects of extremely low and high production schedules.
Once you get enough data (and personal perspectives) from all angles of a problem (even extreme ones), you can narrow your focus in on the core issue.
Nothing takes your brainstorming team “out of the box” like going to extremes.
13) Advanced Problem-Solving Strategies: Engage Your Senses
If your team can’t seem to crack the problem-solving process, have them express the problem in new ways.
Have them draw the problem on your white board, role-play as the various components of a system, or teach the concept to someone with no experience in the area.
To see your blind spots and identify optimal solutions, you need a broad and unique perspective. Develop your team’s trust level and encourage shy people to speak up.
Your best solution could be in the mind of that team member who speaks awkwardly (or not at all). As a team leader/project manager, your role is to draw the very best from all your people!
The Bottom Line
Cubes Problem Solving Worksheet
Ultimately, you will find the solution to your current dilemma; these problem-solving strategies will see you through.
Cubes Strategy Problem Solving Equations
Have faith in your team, build their trust in each other, and facilitate open communication.
Problem Solving Strategy Worksheet
And remember, when you get over this hurdle, take a moment to log it (or put a picture of the solution on the wall of your meeting-room wall).
Problem Solving Skills
By tracking and celebrating your progress—and development as a team—you can foster a workplace attitude that says, “Bring it on – we can solve any problem!”