How to Solve …
Routine
Word Problems
Word problems in mathematics often pose a challenge because they require that students read and comprehend the text of the problem, identify the question that needs to be answered, and finally create and solve a numerical equation. [1]
Non Word Problems
Non word problems in mathematics often pose a challenge because they require that students comprehend the mathematical concept behind the problem.
You understand a piece of mathematics if you can do all of the following: [3]
- Explain mathematical concepts and facts in terms of simpler concepts and facts.
- Easily make logical connections between different facts and concepts.
- Recognize the connection when you encounter something new (inside or outside of mathematics) that’s close to the mathematics you understand.
- Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)
By contrast, understanding mathematics does not mean to memorize recipes, formulas, definitions, or theorems.
Non-Routine Problems
Non-routine problem solving serves a different purpose than routine problem solving. While routine problem solving concerns solving problems that are useful for daily living (in the present or in the future), non-routine problem solving concerns that only indirectly. Non-routine problem solving is mostly concerned with developing students’ mathematical reasoning power and fostering the understanding that mathematics is a creative endeavour. From the point of view of students, non-routine problem solving can be challenging and interesting. From the point of view of planning classroom instruction, teachers can use non-routine problem solving to introduce ideas (SET SCENCE stage of teaching); to deepen and extend understandings of algorithms, skills, and concepts (MAINTAIN stage of teaching); and to motivate and challenge students. [4]
Learning At All Levels
No matter what grade level you are, or what job you are in, the information on this web page will help you to solve problems of almost any kind. That is one of the things that mathematics teaches – logical thinking.
“He who learns but does not think, is lost!
Confucius
He who thinks but does not learn is in great danger.”
I highlight the CUBES solving strategy because the steps are what I have used to solve math, and other, problems in my career. (Having never heard of CUBES until the 2022-2023 school year.)
CUBES
CUBES is a Math word problem solving strategy. Using the CUBES strategy help students tackle difficult and challenging math problems, by using each letter in CUBES, to help them understand what they need to do to solve a word problem.
Circle any NUMBERS. If they are written in word form, write the standard form above he words.
Underline the QUESTION. What are you being asked to solve.
Box any math ACTION words. These key words will tell you what to do to solve the problem.
Evaluate what STEPS you take to solve the problem. Eliminate any information you do not need. Draw a picture!
SOLVE and CHECK. Does my answer make sense? How can I double check my work?
The following video lesson explains and demonstrates the CUBES Method, which is great for visual learners. It allows them to interact with a word problem by dissecting it and critically analyzing it.
Why use the CUBES math strategy?
- It gives students a starting point when they are faced with word problems. We know many kids fear word problems and many times they have no idea where to start.
- It makes kids aware of what the problem is asking – this might seem simple but forcing kids to underline the question is a good way to make them read what is the problem asking me to do.
- It brings out all the numbers in the problem – sometimes word problems present numbers in numerical form, word form, or any other tricky form. Circling the numbers forces kids to look through the problem to find numbers in any form. [2]
References
[1] “Reading and Understanding Written Math Problems”. 2023. Reading Rockets. https://www.readingrockets.org/topics/content-area-literacy/articles/reading-and-understanding-written-math-problems.
[2] Denis, Addys. 2021. “CUBES Math Strategy for Word Problems”. Loving Math. https://lovingmathresources.com/2021/01/cubes-math-strategy-for-word-problems.html.
[3] Alfeld, Peter. “Understanding Mathematics”. 2023. math.utah.edu. http://www.math.utah.edu/~alfeld/math.html.
[4] “Routine and non-routine problem solving”. 2023. ion.uwinnipeg.ca. https://ion.uwinnipeg.ca/~jameis/EY%20course/Readings/R%20general/rnrPS.pdf
Additional Reading
“12 Math Tricks To Help You Solve Problems Without A Calculator”. 2021. Medium. https://towardsdatascience.com/12-math-tricks-to-help-you-solve-problems-without-a-calculator-704fdd663286.
Blankman, Richard. 2023. “What Is Conceptual Understanding in Math?”. hmhco.com. https://www.hmhco.com/blog/what-is-conceptual-understanding-in-math.
Conceptual understanding is difficult to teach, far more so than teaching straightforward procedures, like the standard algorithm for multiplication. The teacher must possess a deep background in the content, not only to recognize when students lack conceptual understanding, but also to know which tools would best remedy the problem. This is no easy task and asks a lot of teachers, many of whom are already strapped for time or focused on meeting state assessment requirements.
However, the benefits of effectively teaching conceptual understanding are far-reaching. A student who knows a procedure, such as the standard algorithm for multiplication, has the tools needed to multiply two numbers. But when a student knows a concept behind multiplication, then they are ready to apply multiplication to any problem they’re faced in life.
Brack, Tyne. 2023. “Math Problem Solving Strategies”. Maneuvering The Middle. https://www.maneuveringthemiddle.com/problem-solving-strategies/.
How many times have you been teaching a concept that students are feeling confident in, only for them to completely shut down when faced with a word problem? For me, the answer is too many to count. Word problems require problem solving strategies. And more than anything, word problems require decoding, eliminating extra information, and opportunities for students to solve for something that the question is not asking for. There are so many places for students to make errors! Let’s talk about some problem solving strategies that can help guide and encourage students!
“How to Solve Math Problems Faster: 15 Techniques to Show Students”. 2023. prodigygame.com. https://www.prodigygame.com/main-en/blog/15-techniques-to-solve-math-problems-faster/.
“How To Teach Math Word Problems – CUBES Math Strategy – Caffeine Queen Teacher”. 2014. Caffeine Queen Teacher. https://caffeinequeenteacher.com/how-to-teach-math-word-problems/.
The CUBES math strategy teaches concrete steps and strategies to students – especially struggling students. Without concrete steps, I’ve found that strugglers have no idea about solving story problems, so they give up – and that’s exactly what we DON’T want them to do! All of those words are scary and overwhelming. This step-by-step strategy teaches students how to deal with all that information.
Step 1: Read the problem out loud.
Step 2: Circle the numbers AND LABELS!
Step 3: Underline the question
Step 4: Box the keywords
Step 5: Eliminate unnecessary information
Step 6: Draw a picture
Step 7: Determine if the problem is multi-step
Step 8: Solve
“How to Solve Math Problems”. 2023. wikiHow. https://www.wikihow.com/Solve-Math-Problems.
Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall.
“How To Work Through Hard Math Problems”. 2023. artofproblemsolving.com. https://artofproblemsolving.com/blog/articles/dealing-with-hard-problems.
We ask hard questions because so many of the problems worth solving in life are hard. If they were easy, someone else would have solved them before you got to them. You can’t learn how to do that without fighting with problems you can’t solve. If you are consistently getting every problem in a class correct, you shouldn’t be too happy — it means you aren’t learning efficiently enough. Here are a few strategies for dealing with hard problems, and the frustration that comes with them:
Do something.
Simplify the problem.
Reflect on successes.
Focus on what you haven’t used yet.
Work backwards.
Ask for help.
Start early.
Take a break.
Start over.
Give up.
Come back.
“What is Conceptual Understanding in Math and Why is it Important for Children”. 2022. BYJU’s Future School Blog. https://www.byjusfutureschool.com/blog/what-is-conceptual-understanding-in-math-and-why-is-it-important-for-children/.
The term “conceptual understanding” refers to a comprehensive and practical grasp of mathematical concepts. Children who understand concepts are aware of more than isolated facts and methods. They comprehend why a mathematical concept is significant and how it can be applied in various situations. They organize their knowledge into a logical structure that allows them to learn new concepts by connecting them to what they already know. Understanding the concepts also helps with retention because facts and methods are easier to remember and apply when learnt through understanding, and may be reconstructed if lost.
Videos
Imagine where you want to be someday. Now, how did you get there? Retrograde analysis is a style of problem solving where you work backwards from the endgame you want. It can help you win at chess — or solve a problem in real life. At TEDYouth 2012, chess grandmaster Maurice Ashley delves into his favorite strategy.
The featured image on this page is from the Art of problem solving math website.
Mathematical Mysteries 