By Donna Wall
Assistant Professor, Mathematics at American Public University
In College Algebra, we include mixture problems, distance problems, interest rate problems, work problems, wind and air speed problems, etc. We provide math problems with a variety of options so that the students learn how to pick the best problem-solving option. One can reasonably argue that we are teaching thinking skills since a student would need to know when to use which equation and how to apply the variables to the equation.
Maybe we can learn something from the early 1900s about critical thinking. My grandfather understood this concept. He was born in 1896 and lived during a time when the teacher presented a problem to the class and the class would try to figure out how to solve it. As a result, he became an expert in critical thinking. He surprised us at age 85 by acing the GED so he could enroll in a college course after he was not able to track down his high school diploma.
Can we learn from the past? What if students were presented with a problem or problems in a forum, were given time to brainstorm, and later were given tools to solve the problem? Would it help with critical thinking skills and challenge them?
One scenario that comes to mind is the problem that was originally coined by the popular TV game show ”Let’s Make a Deal” and is a classic example of critical thinking. In the show, three doors are presented to contestants with a prize behind only one of the doors. If the contestant choses “door #1” but then “door #2” is opened showing no prize, is it better to stay with the original chosen door, or go with what is behind “door #3?”
If switching to “door #3” is the correct way to go, one must assume that the contestant must always have a second chance and a door is always revealed. What makes that statement true? Can you think of how you could apply this concept to a similar problem?
Do you think problems like this would make successful discussion questions? Why or why not?
About the Author
Donna Wall is an assistant professor with American Public University where she currently teaches College Algebra, Trigonometry, Contemporary Math, Statistics, Discrete Math, and Math Modeling. She received her BS degree in mathematics and MSIS degree with a concentration in mathematics from the University of Texas at Tyler. She has done some postgraduate work at Trident University. Originally she worked as a mathematician and programmer for Teledyne Geotech but now enjoys teaching and writing courses. Over the last 10 years she has written more than 15 courses in research, statistics, and mathematics.
Tags: challenging students, critical thinking, math, promoting critical thinking
Good Mathematical Habits for Young Adolescents
Mathematics content is best learned in a way that fosters good habits of mathematical thinking. The Common Core State Standards in Mathematics (www.corestandards.org) supplement their K-12 standards for content with eight standards for mathematical practice:
- 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. These mathematical practices, and how they relate to content, mean very different things depending upon student age.
Middle School as a Critical Transition Period
The middle school years mark a critical transition in a child’s cognitive development – how a child thinks and learns. Generally at age 11 or 12 children enter the fourth and final stage in Piaget’s four stages of cognitive development, called the formal operational stage. During this time children show significant growth in their ability to think abstractly, use advanced reasoning skills, make hypotheses and inferences, and draw logical conclusions. Ideally, the middle school years provide educators with new opportunities to foster good thinking habits and mathematical practices.
The Balance Between Mathematical Content and Practice
Students begin middle school exposed to mathematics as a very broad subject covering a wide array of topics: 2D geometry, probability, percentages, number theory, logic, patterns, statistics, graphing, number operations, proportions, elementary algebra, 3D geometry, and so on. They finish middle school and begin high school usually embarking on year-long studies of content-intensive mathematical subject areas: a year of Algebra 1, then a year of Geometry, then a year of Algebra 2, and so on. Though young adolescents begin middle school ready to think with more power, creativity, and independence, the accompanying increase in content expectations means that a balance between mathematical content and practice can be difficult to achieve. Developing good thinking and learning habits requires investment of time and patience, and well-intended educators can be drawn away from quality mathematical practices when the drive to learn content becomes too formidable.
Committing to Critical Thinking at the Middle School Level
Content can be learned in ways that ask young adolescents to harness and develop their new cognitive abilities. For example, a traditional 2D geometry question might ask:
Calculate the perimeter and area of a rectangle with a 15-inch length and a 9-inch width.
This question can be answered by performing a routine calculation using formulas for the perimeter and area of a rectangle. Similar content can be studied with a question that asks for critical thinking:
For what whole number values of length and width will the rectangle have an area of 60 square yards and a perimeter of 38 yards?
This second question (from Mathematical Reasoning™ Middle School Supplement) requires students to develop a strategy to construct a solution. Indeed, a common approach involves making a mental or physical list of pairs of whole numbers that multiply to 60 and then searching for the pair of numbers that add up to 19 (since a rectangle’s perimeter is twice the sum of the length and width). The correct answer is a length of 15 inches and a width of 4 inches (assigning the larger number to length). Note the depth and value of a critical thinking opportunity: the solution strategy connects 2D geometry with the number theory technique of factoring and is a precursor to a more sophisticated factoring procedure used in Algebra 1. The second question requires greater time investment than the first question, but is worth the extra time if one is committed to young adolescents learning content in a way that fully engages their reasoning skills.
The first Common Core mathematical practice standard emphasizes the need to have students make sense of problems and persevere in solving them. The most important ingredient in Polya’s classic four-step problem solving strategy is the act of making decisions, as opposed to simply applying an algorithm that has been instructed. Young adolescent reaction to problem solving and decision making can be decidedly mixed. On the one hand, playing an active role in the solution process – figuring something out and being creative – can be fun, exciting, sometimes even addicting for young minds that are ready to be engaged. However, overcoming obstacles and persevering with a task that requires multiple steps and authentic reasoning can also sometimes be discouraging for early adolescent brains just learning how to tap into their emerging powers. The frustration level can depend on the difficulty level of the problem-solving situation, and a common, safe path is to keep decision making and creative expectations down to a minimum. However, if mathematics education in the United States is to reach a higher standard against a worldwide benchmark, children must be encouraged to persevere with critical thinking and decision making, to embrace both the excitement and occasional frustration of authentic reasoning and creativity.
Enrichment Activities to Stimulate Critical Thinking
The Critical Thinking Co.™ specializes in activities that stimulate use of reasoning skills and creativity when learning content. These enrichment activities challenge students to make decisions and construct solutions – to play an active role when learning content. Variety is favored over repetition, although care is taken to have common themes emphasized and connections reinforced. Presentation is often graphic intensive, resulting in visual appeal to young eyes. Real-world applications are easily identifiable. Problem-solving is supported with clear, comprehensive solutions and explanations. An example is provided with the activity sets Dimension Detective and Linear Patterns and accompanying solution pages from Mathematical Reasoning™ Middle School Supplement. In Dimension Detective students deduce missing dimensions for a variety of geometric shapes by using proportional reasoning, number theory ideas, and connections between 2D and 3D shapes. In Linear Patterns students determine number patterns and geometric patterns, and then deduce algebraic expressions to describe these patterns (a precursor to creating algebraic equations to describe linear graphs). Each activity set is accompanied by needed math facts, strategy tips, and comprehensive solutions that teachers and parents can use to help support student investigations. These sorts of enrichment activities provide middle school students with an opportunity to explore mathematical content, create or reinforce ideas, make connections, and use abstract reasoning. Young adolescents have emerging cognitive powers to accompany their rapid physical growth, and math enrichment can provide middle school students with appealing opportunities to use their maturing reasoning skills.
Free Printable Math Worksheets for Grades 6-9
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Here's what some of our customers have said regarding the use of The Critical Thinking Co.™ math products for middle school enrichment.
"Mathematical Reasoning presents math in a way that makes it not only simple to learn, but also practical to apply to everyday circumstance (obviously making math more meaningful to him)." - Katherine, Mississippi
"My son hated math. I saw your curriculum and gave it a try. Wow, what a great fit for my son! Your Mathematical Reasoning curriculum gave him a way to learn math that made sense to him. Thank you!" - Amy, Ontario, Canada
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