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 supplement their K-12 standards for content with eight standards for
- 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 accompanying solution pages from Mathematical Reasoning™ Middle School Supplement. 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.