Bangkok, Thailand
April 2011
When Math Makes Sense!
NESA Conference 2011
Rationale: Many students lack the conceptual understanding of different mathematical concepts,
and as a result, teachers suffer all the irrational answers students give on assessment. In this
workshop, we will discuss the importance of emphasizing “checking for reasonableness of
answers” and provide several methods of doing so.
Objective: to stimulate a feeling of urgency to teach reasoning and checking for reasonableness
of mathematical situations.
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Definition of a Reasonable Answer:
The literature has identified two main criteria used for judging the reasonableness of an answer:
(1) Number relationships and the effect of operations and
(2) The practicality of the answer
(Gagne,1983; Hiebert andWearne, 1986; Johnson, 1979; Reys, 1985; Willis, 1992;
Rela4ng Graphically Number Sense Math Making Sense Prac4cality of Answer Rela4ng to Defini4ons Visual Representa4on Working Backwards Es4ma4on McIntosh and Sparrow, 2004).
Bangkok, Thailand
April 2011
1) Number Sense & Conceptual Understanding
“Number relationships and the effect of operations are critical components of
reasonableness (Wyatt, 1985). Unless students conceptually understand numbers as quantities, the
different ways of representing them, how they relate to one another, as well as the effects of operations
on them, math will never make sense to the students. As a result, educators will continue to see
unreasonableness in the answers students provide to different mathematical situations.
2) Importance of Teaching Students to Check for Reasonableness of Answers
“Practicality of the answer is the second criterion for the reasonableness of an answer. It is equally
essential for students to test the results they get and check their practicality. A practical answer is one
that makes sense in real life. For example, 150 Poodles is not a practical answer when given, 30% of
100 dogs are poodles. This criterion is the focus of this workshop. What’s important to realize is that
checking for reasonableness or practicality of an answer cannot be assumed as naturally acquired
knowledge. It has to be taught as a skill.
Some Skills Students Need to Acquire:
 Relating Graphically: statistics, systems of equations, parallel & perpendicular
lines, table of variations of functions, integration...
 Visual Representation: Fractions, perimeters, areas...
 Relating to Definition: Absolute Value, Slope...
 Working Backwards: Writing geometric proofs, Solving Equations...
 Estimation & Prediction: Operations, Percents…
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Since you were 3 years old you used estimation! “I am as tall as Chris” and “I
have more marbles than you”…
A concept that is most of the time either overlooked or its’ purpose is not
identified.
Have you thought about how much time is spent teaching estimation?
Have you thought about how much you estimate in your daily life?
Bangkok, Thailand
April 2011
How to Administer These Skills:
 Open-Ended Questions
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No fixed method
No fixed answer/many possible answers
Solved in different ways and on different levels (accessible to mixed abilities).
Offer pupils room for own decision making and natural mathematical way of
thinking.
Develop reasoning & communication skills
Open to pupils’ creativity and imagination when relates to real-life context of
children experience.
How to Administer These Skills (continued):
 Real-Life Applications & Word Problems: Examples: Taking measures at a
historical site and generating a scaled down drawing. Checking for proportionality
in fire exit maps in relation to the real building, painting a room and approximaring
how much paint is needed…
 Use the Terminology: “Wow Karim, good estimation of the answer”, “Can you give
me an approximation to…”, “Make a prediction of the graph after ….”
 Test Checklists: provide detailed revision checklist at the end of tests/exams.
Example of item: “4) I have expressed my answer in a full sentence and my answer
is reasonable according to the problem.” “ 2) I have estimated the answer before or
after doing the calculations and my answer is close to the estimate.”
 Reminder Posters in class: formulas, rules, common errors…
 Testing for Reasonableness: When teaching skills, it is only right to check if they’re
acquired.
Conclusion:
To make a difference in how students view math in general, we need to look at our
teaching strategies and revisit what is essential. This can be done on two levels: focusing on
conceptual understanding of lessons and checking for the reasonableness of answers to
mathematical situations. Estimation plays a very important role in both. When it comes to
Bangkok, Thailand
April 2011
students connecting with real-life situations, developing a sense of interest and questioning
what they are doing, then math would really make sense to them!
References:
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“REASONABLE AND REASONABLENESS OF ANSWERS: KUWAITI MIDDLE
SCHOOL TEACHERS’ PERSPECTIVES”, A. ALAJMI AND R. REYS, 2007
“Using Questioning to Stimulate Mathematical Thinking”
“Relationship between teacher knowledge and student achievement in middle grades
mathematics” Mourat A. Tchoshanov
http://nrich.maths.org/
http://www.iched.org/cms/scripts/page.php?site_id=iched&item_id=estimation
By Hilda Hanania
ACS Math Teacher
[email protected]