1. Sciematics: The Changing
Face of Education.
Saskatoon, May 9-11, 2012,
College of Agriculture and
Biosciences, U of S.
http://www.sciematics.com/
SUM conference: May 3-4,
Saskatoon. Featuring Dan
Meyer and Marian Small.
http://www.smts.ca/sum-
conference/
“Posing conjectures and
trying to justify them is an
expected part of students’
mathematical activity.
“(NCTM, 2000, p. 191)
“Teaching mathematics as an
exercise in reasoning should also
be commonplace in the
classroom. Students should have
frequent opportunities to engage
in mathematical discussions in
which reasoning is valued.
Students should be encouraged
to explain their reasoning
process for reaching a given
conclusion or to justify why their
particular approach to a problem
is appropriate. The goal of
emphasizing reasoning in the
teaching of mathematics is to
empower students to reach
conclusions and justify
statements on their own rather
than to rely solely on the
authority of a teacher or
textbook.”
http://www.nctm.org/standards
/content.aspx?id=26596
Mathematical Process of the Month: Reasoning R
Reasoning describes our ability to make sense of things. Our philosophy of
teaching mathematics for deep understanding is founded on the premise that
students must make sense of the mathematics, not just memorize procedures.
Providing students with opportunities to explore, manipulate, demonstrate,
make conjectures and explain approaches and understandings are ways that we
help students to reason for themselves.
Mathematical reasoning underlies logical thinking and sense making and is
the mental process of making connections and drawing conclusions based on
what is already known. Mathematical reasoning involves knowing which
procedures apply to which purpose, and being able to identify useful strategies.
Reasoning is a natural component of mathematical proof, as an explanation of
one’s reasoning and understanding is the premise of mathematical proof. Logic,
reasoning and proof are of pivotal importance in developing mathematical
literacy. Students who learn math through understanding will be more
confident in applied mathematics in further education and in life.
The rigorous demand for proof is the essence of mathematics and the
cornerstone of humanity’s advances in accumulating knowledge and
understanding of our world. If we are helping students gain an understanding of
Math as a Human Endeavour (one of the goals of math education), then we must
also reveal mathematics as understanding relationships, not just procedural
knowledge. Procedural fluency is important but cannot be sought at the expense
of conceptual understanding. Because students today need to be able to think
critically, analyze and interpret information, math instruction needs to
encourage prediction, reflection and justification of results. Students that can
communicate their reasoning will develop confidence in their thinking.
Math instruction must involve tasks that engage all students in thinking
about, discussing, and making sense of mathematics, not just practicing
procedures. By organizing the curriculum around central ideas, students see a
coherence that contributes to their reasoning and helps them make sense of the
mathematics.
Focus in High School Mathematics, Reasoning and Sense Making, National
Council of Teachers of Mathematics.
Saskatchewan Renewed Mathematics Curriculum
“Classroom instruction in
mathematics should always
foster critical thinking –
that is, an organized,
analytical, well -reasoned
approach to learning
mathematical concepts and
processes and to solving
problems.”
Ontario Grade 9 and 10
Curriculum
http://www.edu.gov.on.ca/eng/curr
iculum/secondary/math910curr.pdf

2. Possible Curriculum connections
that support some discussion of
Pi
Math 7 SS 7.1 Circles, radii, diameter
Math 8 SS 8.2 Surface area of shapes
including cylinders
Math 9 SS 9.2 Surface areas of shapes
including cylinders
Great Pi Day Ideas
I teach 7th grade Honors Pre-Algebra. I
had each student measure the
circumference and diameter of a circular
object to the nearest tenth of a cm the
day before pi day. They then recorded
the class data on a chart. I had them
calculate C/d. They also found the class
average which they discovered was very
close to pi. Then they created a scatter
plot to see the correlation between
diameter and circumference.On pi day
we ate pie and discussed our findings. I
also read them the book “Sir Cumference
and the Dragon of Pi”. However the big
hit was the photo story I made for them
with picture of our class set to the music
“Lose Yourself (In The Digits)” This song is
great (clean) and a free download.
By S. Browning
http://www.piday.org/2008/2008-pi-day-
activities-for-teachers/
www.exploratorium.edu/learning
_studio/pi/
Send a pi day greeting card
http://www.123greetings.com/ev
ents/pi_day/
Some examples of Pi day activities:
Younger grades: Create a Pi
caterpillar, with 3 on the head,
each paper circle segment created
by students has a separate digit of
pi. This caterpillar can be added
to year after year. Having
students write their name on each
digit provides a legacy of past
students.
For older students, try graphing
the circumference of circles vs
diameter. Guess what the slope
will be!
Sir Cumference, his wife, Lady Di of Ameter, and their son
Radius use geometry and problem-solving techniques to help
King Arthur. A math adventure by Cindy Neuschwander
You tube http://www.youtube.com/watch?v=OU_O8PdDJpI

3. Formative Assessment Feature
Always, Sometimes, or Never True:
This formative assessment strategy involves having the students examine a set of statements, and decide if they are
always, sometimes or never true This formative assessment task promotes reasoning because students must justify
their answer in writing.
Allow students to respond to the statements individually, then have them discuss and compare in small group.
There will be debating, and this is to be encouraged!
Example of an Always, Sometimes, Never activity:
A right triangle is isosceles
□Always □Sometimes□Never
Justify your answer:
An isosceles triangle is an equilateral triangle
□Always □Sometimes□Never
Justify your answer:
A triangle can have only one obtuse angle
□Always □Sometimes□Never
Justify your answer:
A triangle can have more than one right angle
□Always □Sometimes□Never
Justify your answer:
The angles in a triangle sum to 180 degrees
□Always □Sometimes□Never
Justify your answer:
A triangle can have three acute angles
□Always □Sometimes□Never
Justify your answer:
A right triangle is an isosceles triangle
□Always □Sometimes□Never
Justify your answer:
A right triangle is an acute triangle
□Always □Sometimes□Never
Justify your answer:
In this activity, students must
understand that conjectures need
to be justified. It encourages
mathematical thinking because
students imagine examples and
non-examples to support their
answer. Dialogue and
mathematical argument are
encouraged.
Always, Sometimes, Never can
be used as a pre-assessment, to
gain understanding of students’
prior knowledge, or to check for
understanding after instruction. It
may uncover misconceptions and
reveal the extent to which
students understand the concepts.
Create the Problem
This is the reverse of problem solving. The teacher provides students
with a simple equation or math fact, such as or ½ of 10 is 5,
and students create a problem around the mathematics. For example: I
have a three sided rectangular pen with a perimeter of 5. If the front edge
of the pen is three meters, how long are the two congruent sides of the
pen? Or: If I doubled the number of loonies in my pocket then added the
three from my wallet, I would have $15. Students share their stories and
discuss how well the story matches the equation or expression. Having
students write the wording to a problem to match presented mathematics
is a powerful way to help them become better problem solvers because
they must reason through the wording translation themselves. This is an
example of a problem where students can really focus on the process,
because the mathematical answer is already provided. It encourages
mathematical communication because students must write logical
sentences that require mathematical thinking, and then discuss their ideas
with each other. This assessment activity allows insight into student
understanding because it reveals if they know why a certain procedure
may be required, rather than just the process used to perform the
computation.
“Students recognize that
reasoning is a fundamental
aspect of mathematics.
Reasoning can be nurtured at a
very early age by asking
students to explain and justify
their observations with
questions such as ‘Why do you
think that’s true?’ and helping
students distinguish between
real evidence and non-evidence.”
“Students should develop the
habit of providing an argument,
reason, rationale, or
justification for every answer
they provide.”
Florence Glanfield, (2007). Building
Capacity in Teaching and Learning.
Reflections on Research in
Mathematics. Pearson Education
Canada, p. 24,25

4. Web Resources:
https://www.khanacademy.org/exercisedashboard
If you haven’t checked this out yet, take a look. There are thousands of video lessons
on this site.
http://www.livebinders.com/play/play?id=598492 This is a livebinder with resources
on how to teach with manipulatives, virtual manipulatives, and SMARTboard
lessons.
I just downloaded this webmix of high school math sites.
By the way, Symbaloo is an awesome way to keep your bookmarks and links
organized.
National Library of Virtual Manipulatives:
http://nlvm.usu.edu/en/nav/grade_g_2.html
http://www.symbaloo.com/mix/highschoolmath
Fostering Reasoning through Open Ended Questions:
Open ended questions have multiple correct solutions, such as when students are asked
to find different methods or discuss approaches, but openness is lost if the teacher
proceeds through only one method. Categorizing items is an open ended task when the
categories are developed by the students. Asking open ended questions promotes higher
level thinking in mathematics, promotes collaboration and communication, and requires
students to justify their reasoning. Open ended tasks allow all students to participate by
providing multiple entry points, and in this way they are naturally differentiated. Students
can choose their own approach, employ their own learning style, and make personal
choices. They provide opportunity for success and challenge for all students.
Because open ended problems require students to explain their reasoning, the teacher
gains insight into student learning styles, their understanding, and their use of
mathematical language. Allowing students to discuss open ended tasks can help students
develop confidence in their mathematical ability.
http://books.heinemann.com/math/reasons.cf
http://www.hightechhigh.org/unboxed/issue2/opening_up_to_math
Becker & Shimada, (1997). The Open Ended Approach. NCTM, Reston, VA.
Arlene Prestie, Preeceville School
“Reasoning
mathematically is a
habit of mind, and
like all habits, it
must be developed
through consistent
use in many
contexts.”
-NCTM