Every new mathematical concept is connected to existing knowledge and experiences. There are several types of important connections for learning mathematics, including connecting ideas within mathematics, connecting math to the real world, and connecting math to other subject areas. Effective teachers help students make these connections by drawing on their prior experiences and highlighting how concepts are related across lessons, grades, and other disciplines.

1. Mathematical Process of the Month: Connections CN
Learning in all subjects is predicated on making connections between new
Understanding Mathematical concepts and existing schema; that is, mental images and ideas, or what the learner
Process Standards
Successful learning in mathematics is
already knows. Powerful learning draws on the learners’ experiences and life context.
achieved through “relational Students with extensive life experience have more pre-existing schema to create
understanding”. A conceptual connections with new concepts. This is called “crystalized intelligence”, as opposed
understanding of math builds on
to “fluid intelligence”, which is the natural ability we are born with. Interestingly,
humans’ natural desire to make sense
of things. It helps learners acquire crystalized intelligence is a greater predictor of student success. This is good news for
new knowledge and apply it in learners, because while fluid intelligence is innate, crystalized intelligence can be
unfamiliar situations. Conceptual controlled. Educators capitalize on this by arranging experiences for students through
understanding improves students’
attitudes toward math because they field trips, interactive learning experiences, projects, and inquiry. Everything a
can see math as an accessible subject student has learned in the past becomes the hook on which to hang new
that can be understood. Effective information. As math educators we are very familiar with this analogy, as we know
mathematics instruction incorporates
seven process to ensure that students
that math concepts build on each other through the years.
develop a conceptual understanding There are several types of connections that are important to learning
of math: Communication, mathematical concepts:
Connections, Mental Math and Connecting ideas within mathematics. Our curriculum helps us do this by
Estimation, Problem Solving,
Reasoning, Technology and organizing mathematical themes which are evident in the strands of the curriculum.
Visualization. Beside the Outcomes These themes tie mathematical topics together so the student can realize general
in the curriculum document are principles at work and how they are related. Students will encounter these big ideas
suggestions for which of the
processes lend themselves well to the repeatedly and in many different contexts as they develop depth of understanding
teaching of that outcome; however, through the grade levels. As educators, we need to ensure that we highlight these
all processes are interrelated and are related concepts to help students build on and expand their prior learning;
part of the everyday business in the
classroom. They are the media
otherwise, math is perceived as fragmented and compartmentalized. Learning is
through which we deliver content. 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
Connections: Because the through the years so that we can understand ideas that are nested within each other,
learner is constantly searching and concepts that are threaded and integrated. Ideas must flow naturally from
for connections on many levels,
lesson to lesson and grade to grade.
educators need to orchestrate
the experiences from which
Connections between math and the real world. All learning is achieved
learners extract understanding through anchoring new concepts to existing ideas and experiences that are existing
…. Brain research establishes understandings. The things a learner already knows become the “pegs” on which to
and confirms that multiple pin new information. A teacher’s task is to illuminate the relations between the
complex and concrete known and the new. Teachers must always seek opportunities to draw on students
experiences are essential for past experiences and understandings to introduce new topics. When students are
meaningful learning and
encouraged to contribute their own understandings into the learning, they are more
teaching. (Caine & Caine, 1991,
p. 5) (Excerpt from Sask engaged and have a sense of ownership of the learning. Math must be understood
Foundations PreCalc 10 as intrinsic and enmeshed in the fabric of life, physics, and society, not an elite,
Curriculum) unobtainable, isolated topic. Teachers and resources must draw out the

2. (Continued from P. 1)
connections between mathematics in the classroom and mathematics in the real
world.
A simple model for talking about By engaging many senses we create memorable experiences to which concepts
understanding is that to are linked. This is the basis of inquiry-based, hands on instruction. Exploring
understand something is to mathematical topics through experiences, manipulatives, collaborative discussions,
connect it with previous presentations, debates, and multimedia create much more memorable learning
learning or other experiences…
events than pencil and paper seat work, though the actual content and topics may be
A mathematical concept can be
thought of as a network of the same.
connections between symbols, Connections to other areas of learning. Helping students connect to math in
language, concrete experiences, their lives involves highlighting connections to other subject areas. Our curriculum
and pictures. (Haylock & documents give suggestions for helping students transfer mathematical knowledge
Cockburn, 2003, p. 18, as cited in to other disciplines. Some examples are shapes and tessellations in art, data
Saskatchewan Math 8 interpretation and probability in health and social, data and graphing in science
Curriculum)
education, fractions and music education, timing and statistics in phys ed, and graphs
Existing knowledge and as models of behaviours in physics, logarithms as necessary to chemistry and physics,
experiences are the anchors to and calculating and measuring in trades classes. In the same way that math
which we tie new concepts 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.
Communication works together NCTM Web Site, http://www.nctm.org/
with reflection to produce new 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
relationships and connections. Manitoba Mathematics Curriculum Framework, Grade 8 Curriculum Support Document,
Students who reflect on what http://www.edu.gov.mb.ca/k12/cur/math/support_gr8/full_doc.pdf
they do and communicate with
others about it are in the best
position to build useful
connections in mathematics.
(Hiebert et al., 1997, p. 6)
Every new idea is connected to pre-existing knowledge and
experiences
SUM conference: May 3-4, Sciematics: The Changing Face
GSSD Divison-wide PD day,
Saskatoon. Featuring Dan Meyer of Education. Saskatoon, May
Feb 1 2013. Math topics:
and Marian Small. 9-11, 2012, College of
Benchmarking with Susan
http://www.smts.ca/sum- Agriculture and Biosciences, U
Muir, Exploring Math
conference of S.
Instruction with Cindy Smith
http://www.sciematics.com/

3. Formative Assessment Feature
Frayer Model: Like many strategies that may be introduced as
“formative assessments”, The Frayer model (one example is shown at
left) is not only useful for formative assessment but also as a routine
instructional practice. It has been described as a concept map, graphic
organizer, and vocabulary development tool. It can be used to
introduce topics or concepts, front load vocabulary, or check for
understanding. It can also be used for clarifying mathematical symbols.
The Frayer model helps students make sense of words or concepts,
and connect them to pre-existing understandings. It requires critical
thinking to establish deeper understanding, and it creates a visual
reference for concepts and vocabulary. Students can work on a Frayer
model individually, in pairs, or collaboratively in groups. You could
have students include pictures to help make connections with the
concept. After making Frayer models have students examine each
other’s work, compare and discuss. Frayer models have more to them
than meets the eye!
Customizable, Downloadable Frayer Model
Templates are available here:
http://www.worksheetworks.com/miscellanea/
graphic-organizers/frayer.html
Shown at left is a table of instructional
practices (Marzano) that are shown to have
significant impacts on student learning. The
Frayer model incorporates several of these!
“Contextualizing and making connections to the experiences
of learners are powerful processes in developing
mathematical understanding. When mathematical ideas are
connected to each other or to real-world phenomenon,
students can begin to view mathematics as useful, relevant,
and integrated.” –WNCP 2006
K-W-L Chart: This is often used as a pre-assessment or
entrance slip. A chart has three columns, where students
record what they already know (K), what they want to know
(W) and what they have learned (L). This can be done on a
small piece of paper, for entrance slips, or on three separate
poster papers for a more collaborative activity. This activity
requires students to activate prior knowledge, and apply
higher-order thinking strategies in order to construct
meaning. Having students do this at the start of a lesson or
unit can be a focussing activity, and having them repeat it at
the end gives a visualization of learning and progress, which
encourages motivation by creating an awareness of
A downloadable K-W-L Chart is available at achievement.
http://www.educationworld.com/tools_template
s/kwl_nov2002.doc

4. Relational Understanding: Instrumental Understanding:
 Conceptually based  Rule based
 Knowing both “how” and “why”  Knowing “how” but not “why”
 Acquired by sense-making  Acquired by rote
 Interconnected knowledge  Isolated knowledge
 Easier to remember  Harder to remember
 Involves fewer principles of more general  Involves a multiplicity of rules
application  Inflexible, not readily adaptable to new tasks
 Flexible, more adaptable to new tasks
Cool Stuff to try: Have you
heard of three-ring? It allows
Prototype Departmentals for WAM
30, Foundations 30 and PreCalc 30 you to quickly create digital
are on line at folders for all your students,
http://www.education.gov.sk.ca/pro and upload documents, screen
totypes
snips, photos and videos to
each file. It’s a way of creating a
digital portfolio. What a cool
Did you know? You have access to way to give teachers a snapshot
“Destination Math”, a web-based learning of their child’s work in your
tool for k-9 student practice. You can tailor
assignments to fit curriculum concepts and to class, or to track formative
differentiate. Destination Math is a assessment data.
responsive program, so it tracks student
answers, gives like problems to reinforce
concepts it finds a student is having trouble
with, or introduces a tutorial to get the
student back on track. It can also track time Learn how to set up destination math for
on task and correct answers.
your class from the Destination Math
Webinar and other Webinars by Michelle
Morley, which can be viewed at
http://central.gssd.ca/math/?page_id=1
520
The link to Destination Math Destination Math is popular
is http://success.gssd.ca/lms as a Pod activity.
Math Webinars. SMART Math Tools – Gary, Jan. 23 ~ Screen Casting – Michelle, Math Coach
March 6 ~ Photo Story – John, April 17 ~ Building a Personal Learning Community - Michelle. These Please visit my blog at
webinars are free. See Michelle Morley’s blog for log in info www.blogs.gssd.ca/csmith/
This site has useful resources,
but it is a work in progress.
Please email me if you have
Web Resources: ideas or requests for this
PreCalc, Foundations resources and screen captures newsletter.
http://arthurmathwarman.blogspot.ca/p/pre-calculus-30.html
Quick Draw: Class starter activity that generates good discussion and math vocabulary:
http://pages.cpsc.ucalgary.ca/QuickDraw/
Effective use of math Word Walls Rubric http://blogs.gssd.ca/smuir/?tag=word-walls
Ipad math Apps for middle years: http://www.teachthought.com/apps-2/12-of-the-best-
math-ipad-apps-of-2012/