Problem solving is a crucial piece to learning, and is integral to every strand of math education. Developing problem solving skills should start at a very young age. Though Tiffany is not yet in school, she is learning how to solve problems every day. Doing things like opening up a box, expressing what she wants or finding a lost toy develop problem solving skills. It is important to let Tiffany work through these problems on her own, so she builds the confidence she needs to solve more complex problems as she encounters them. It is good practice for Tiffany to receive two-step directions, like “brush your teeth and put the toothpaste in the drawer”. At this age, Tiffany may need some reminding, but she should be able to follow these directions, and they will encourage positive development of problem solving. Another aspect of problem solving at this age is an understanding of rewards and consequences of actions. Consistency in these will promote awareness and understanding of cause and effect.
As Tiffany starts school and develops her concrete math skills, she should also be developing her problem solving abilities. Learning math vocabulary is essential for figuring out problems. The development of reading and problem solving are intrinsically related at this stage. Word problems, as they become more common with reading and vocabulary fluency, promote problem solving because they do not provide algorithms or formulas but push the students to figure out what tools to use. Playing games like checkers and tic-tac-toe will help Tiffany think deductively and critically, not just knowing her next move but also thinking of her opponent’s moves. With her brain developing so rapidly, it is crucial that Tiffany develop strong problem solving skills, not only for her mathematical success, but for her accomplishment and confidence in all Areas of her life. The National Council of Teachers of Mathematics points out in the “Curriculum and Evaluation Standards” that “students need to view themselves as capable of using their growing mathematical knowledge to make sense of new problem situations in the world around them” (1989, p. ix)
At the end of 2nd grade, Tiffany should be able to solve this problem:
I am thinking of a number. My number has 3 digits. It is between 110 and 130. The sum of its digits is 9. My number is even. What is my number?
As Tiffany strengthens her reading and comprehension skills, mathematical word problems can include more steps and become more complex to encourage multi-step solutions. Reading and math problem solving are intertwined, and it will likely become obvious which of Tiffany’s math classmates are struggling with reading. These students might be able to easily solve a visual problem like a game, but when the same problem is presented as a story or word problem, they may flounder. It is helpful for these students to be able to represent the problem visually before trying to solve it. Word problems should focus on problem solving skills like working backwards, portraying the problem as a picture or graph, and approaching the simple steps first. Games like Mancala or Connect 4 are a good way to exercise Tiffany’s ability to develop game strategies. Her ability to think more deeply and view the “whole” to apply parts is key to solving problems.
The Common Core State Standards for Mathematics states that teachers (and parents) should seek to develop basic mathematical practices in each student. These practices rest on important “process and proficiencies” in mathematical education. These practices are as follows:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
At the end of 5th grade, Tiffany should be able to solve this problem:
When she arrived at the airport, Kerry checked her watch. She noted that she had one hour to wait until her scheduled flight time. As soon as she looked at her watch, she heard an announcement that her flight would be delayed by two hours. Later, an additional one and one half hour delay was announced. Kerry’s plane finally took off at 9:00 pm. At what time had Kerry arrived at the airport?
Richard Rusczyk had an eye-opening experience at a math competition in 10th grade. Armed with his formula sheets and confident he was “good at math” from memorizing formulas, Rusczyk did not solve a single one of the 60 math problems presented to him at his competition. It was a humbling and perplexing experience for him, but he soon learned the reason: “true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems.” Rusczyk has since started his own math competition, writes math books and resources, and provides math instruction. He encourages educators to focus more on problem solving and less on what he calls the “memorize-use-forget” approach.
So that Tiffany does not have to endure that same trying experience, it is crucial that her math education focus on allowing her the room to find the solutions to complex math problems on her own. This may be the most important time in Tiffany’s math education, since the math found in many careers is the math learned in an 8th grade curriculum. With a strong foundation in necessary algorithms and complex problem solving, Tiffany will have the skills she needs to advance and succeed, not only in math, but in every aspect of her life. As Rusczyk says,
Problem solving is crucial in mathematics education because it transcends mathematics. By developing problem solving skills, we learn not only how to tackle math problems, but also how to logically work our way through any problems we may face. The memorizer can only solve problems he has encountered already, but the problem solver can solve problems she’s never seen before. (https://www.artofproblemsolving.com/Resources/articles.php?page=problemsolving)
At the end of 8th grade, Tiffany should be able to solve this problem:
Ashrita Furman, who holds 19 Guinness World Records, walked 80.96 miles in about 24 hours carrying a bottle of milk on his head. He also bounced about 23 miles on a pogo stick in 12.5 hours. What is the difference in rate per hour in these two feats?