Learning in all subjects is predicated on making connections between new concepts and existing schema; that is, mental images and ideas, or what the learner already knows. Powerful learning draws on the learners’ experiences and life context. Students with extensive life experience have more pre-existing schema to create connections with new concepts. This is called “crystalized intelligence”, as opposed to “fluid intelligence”, which is the natural ability we are born with. Interestingly, crystalized intelligence is a greater predictor of student success. This is good news for learners, because while fluid intelligence is innate, crystalized intelligence can be controlled. Educators capitalize on this by arranging experiences for students through field trips, interactive learning experiences, projects, and inquiry. Everything a student has learned in the past becomes the hook on which to hang new information. As math educators we are very familiar with this analogy, as we know that math concepts build on each other through the years.
There are several types of connections that are important to learning mathematical concepts:
Connecting ideas in mathematics. Our curriculum helps us do this by organizing mathematical themes which are evident in the strands of the curriculum. These themes tie mathematical topics together so the student can realize general principles at work and how they are related. Students will encounter these big ideas repeatedly and in many different contexts as they develop depth of understanding through the grade levels. As educators, we need to ensure that we highlight these related concepts to help students build on and expand their prior learning; otherwise, math is perceived as fragmented and compartmentalized. Learning is through memorization which is low-level and not lasting. Our first job as educators is to become very familiar with the curriculum, especially at our own level but also through the years so that we can understand ideas that are nested within each other, and concepts that are threaded and integrated. Ideas must flow naturally from lesson to lesson and grade to grade.
Connections between math and the real world. All learning is achieved through anchoring new concepts to existing “schema”, or ideas and experiences that are existing understandings. The things a learner already knows become the “pegs” on which to pin new information. A teacher’s task is to illuminate the relations between the known and the new. Teachers must always seek opportunities to draw on students past experiences and understandings to introduce new topics. When students are encouraged to contribute their own understandings into the learning, they are more engaged and have a sense of ownership of the learning. Math must be understood as intrinsic and enmeshed in the fabric of life, physics, and society, not an elite, unobtainable, isolated topic. Teachers and resources must draw out the connections between mathematics in the classroom and mathematics in the real world.
By engaging many senses we create memorable experiences to which concepts are linked. This is the basis of inquiry based, hands on instruction. Exploring mathematical topics through experiences, manipulatives, collaborative discussions, presentations, debates, and multimedia create much more memorable learning events than pencil and paper seat work, though the actual content and topics may be the same.
Connections to other areas of learning. Helping students connect to math in their lives involves highlighting connections to other subject areas. Our curriculum documents give suggestions for helping students transfer mathematical knowledge to other disciplines. Some examples are shapes and tessellations in art, data interpretation and probability in health and social, data and graphing in science education, fractions and music education, timing and statistics in phys ed, and graphs as models of behaviours in physics, logarithms as necessary to chemistry and physics, and calculating and measuring in trades classes. In the same way that math classrooms draw on literacy and social skills, so should other disciplines require students to apply mathematical reasoning and value mathematical literacy.
Connections between symbols and procedures in math. Part of our work in establishing mathematically literate students is helping them gain an understanding of the representations of mathematical ideas. Students must be actively engaged in the work of mathematics to be immersed in the language of math. Word walls and front-loading vocabulary are strategies to assist with connecting to the language of mathematics, as are frayer models, carrol diagrams, and other concept attainment activities and graphical representations. Teachers promote mathematical literacy by introducing many representations, modelling different approaches, and arranging opportunities for students to compare, explore, reason with and talk about mathematical approaches and representations.
Saskatchewan Renewed Mathematics Curriculum
Glanfeild, F. (2007). Reflections on research in school mathematics. Toronto, Pearson.
NCTM Web Site, http://www.nctm.org/
Ontario Association for Mathemamtics Education, http://www.oame.on.ca/main/index1.php?lang=en&code=home
New Jersey Mathematics Curriculum Framework (1996) Standard 3-Mathematical Connections
Manitoba Mathematics Curriculum Framework, Grade 8 Curriculum Support Document, http://www.edu.gov.mb.ca/k12/cur/math/support_gr8/full_doc.pdf
Mathematical Connections for Benchmarking
Math to self — recalling an experience that relates to the problem
Math to real life—recalling or imagining a situation where you may require math similar to the problem. Students may recall working on similar problems
For primary grades these are often the only type of connections we see, as these young children haven’t experienced enough math to make connections within mathematics.
Math to other subjects—for instance, statistics in social studies, or graphing in science
Connections within the math:
Noticing patterns or relationships
Noticing how certain math concepts are related (like multiplication to repeated addition, or money to percents, percents to fractions or ratios, probability to fractions)
Connecting symbolic notation to pictures or diagrams, like a fraction symbol to a fraction diagram, an area model, a graph to an equation (this is also a representation and a communication)
Providing a key to diagrams to help the reader understand representation (this is also a representation and a communication)
Finding alternative strategies to solve (this may also be reasoning and proof)
Connecting appropriate vocabulary to diagrams and algorithms (also communication)
Providing and identifying a proof (also reasoning and proof)
Creating multiple representations (connecting a table to an equation or graph, a fraction of a whole model to fraction of a set, drawing manipulatives and providing written descriptions)
Extending a solution
A true extension of a solution is when a student recognizes a mathematical relationship, pattern, or rule that could be applied to efficiently solve the problem and solve for any unknown. For example, two years ago there was a problem about roofing. There was a linear relationship between the number of shingles needed and the number of houses to be done. Many students drew a linear graph, or a table of values where they identified a problem. Expert problem solvers wrote the rule for the linear function, and could then use that rule (equation) or graph to determine the number of shingles needed for any number of houses.
Problem posing: Many students “extend” their solution by creating a new problem that deals with similar math. This is one type of connection but maybe not the strongest mathematical connection, and can be contrived, though the way the rubric is worded we did feel it was valid and encouraged our students to do it. We recognize this as a way of making a mathematical connection.
How many, what type, and what level of connections are appropriate depends on the grade level and problem. Certainly problems that require sophisticated mathematical vocabulary or diagrams and algorithms allow students to showcase the mathematical connections they are making.
Please refer to the exemplars website and examine the criteria and rational in the rubrics attached to the summative assessment tasks. http://www.exemplars.com/education-materials/free-samples