- It is important to recognize the different learning needs of advanced learners as they are likely to become future scientists. However, advanced learners are often overlooked in classrooms. - Teachers need support to better identify advanced learners through things like professional development on their characteristics. They also need resources to provide appropriate enrichment activities. - Twice-exceptional students, who are both gifted and have learning disabilities, are particularly likely to go unnoticed. Teachers require guidance on strategies like focusing on strengths and accommodating individual differences.

1. Everybody Loves Math
How can we enhance learning
experiences
for advanced learners
and capture the interest of students not
inclined toward math?

2. Why focus on advanced learners?
“Without a science-literate population
[including the mathematical sciences], the
outlook for a better world is not
promising.”(AAAS, 1985)
In order to prepare the scientists of tomorrow,
it is important to recognize the different
learning needs of advanced learners as they
are likely to be the scientists of tomorrow.

3. Why focus on advanced learners?
The central function of the educational system
is providing each and every student,
regardless of his/her social and economical
status with learning opportunities that match
their potential and promote it to the
maximal extent.
“Learning opportunities are the most critical
factor for the realization of human
intellectual potential.” (Leikin, 2010)

4. Also timing is very important
“If suitable learning-stimulating tasks are not
given at the right moment [research suggests
by middle school at the latest], then some
intellectual abilities may not have the
chance to develop.”(Sierpinska, 1994)
Also if their learning needs are not met
students may be frustrated and consequently
they become disruptive or aggressive
elements in the classroom environment.
(Fetzer, 2000)

5. Who are advanced learners?
Students who are intrinsically motivated,
persistently work hard, and achieve
top grades, but may not qualify as
"gifted.”
Gifted students without an adequate
educational plan in place.
Gifted students who are for whatever
reasons not recognized as gifted.

6. Providing enrichment for advanced
learners will require:
Willingness to focus on the learning
needs of advanced learners.
Ability to recognize advanced learners.
Providing teachers with adequate
support.
Including the wider learning community
to support teachers, parents, and
students.

7. Is there a willingness to focus on
advanced learners?
Teachers are teaching classes that are too
big and have challenging compositions.
Through their own schooling teachers have
learned to prioritize supporting students
with low math skills.
Math enrichment resources are plentiful yet
disorganized and appear limited.

8. Reasons that advanced learners are
overlooked
For example Leikin (2011) claims that people’s
views about education of gifted [or advanced
learners] is strongly dependent on their
personal experience and histories related to
the education of the gifted.
While Al-Hroub (2010) points out that there are
students who are gifted along with having
learning difficulties, whose abilities in
mathematics are easily overlooked by
teachers and parents.

9. First reflection point
What is the composition of your present
and/or past classes?
Can you think of examples of students
who you have known who may have
been advanced/gifted?

10. Our hidden censor (Keith Payne, 2013)
Is it really possible that we are constantly
failing to notice things right in front of us?
Yes, hundreds of studies have backed up the
idea that when attention is occupied with
one thing, people often fail to notice other
things.
What we do or do not see depends on the
biases of “unconscious selective attention”
of our mind.

11. How to identify advanced learners?
Advanced learners should be seen as
potentially gifted.
Parents and teachers are important allies
in the identification of students who
need gifted programming.
Research indicates that parents are best
suited to identify gifted behaviours in
young children.

12. How to identify advanced learners?
But, if parents have not initiated setting
of an adequate educational plan then
it is the teacher’s responsibility to
advocate for the child.
Teachers can provide an important
perspective by indicating how a
student performs compared to other
students of the same age.

13. How to identify advanced learners?
Teachers who see students needing a
higher level of challenge in the
classroom have an obligation to
document their findings and provide
appropriate learning opportunities.
It is advisable that teachers look for
support of a trained gifted education
specialist in the school or the district.

14. Obstacles to identifying
advanced /gifted learners
Race/Culture/
Ethnicity
Socio-economics
Instructional Practices
Gender Inequality
Sexual Orientation
Asynchrony
Student Behaviour
Hiding Gifts and
Talents
Affective
Characteristics
Other Exceptionalities

15. Twice-exceptional students
Among all others, gifted students who
also have learning disabilities are the
most frequently overlooked because
their exceptionalities mask each other
and for that reason they are most
frequently “hiding” in our classrooms.

16. Twice-exceptional students
It is even harder for students if only one
of their exceptionalities is recognized.
Usually it is students’ disabilities that
are noticed and frequently education
plans do not address their talents.
This can cause further emotional
distress.

17. Twice-exceptional students
While it is important to address students’
academic needs, recognizing and
supporting the social and emotional
needs of twice-exceptional students
can be even more important.

18. Twice-exceptional students
Because “students with the potential to
be high achievers are showing a rate of
dropout that approximates low
achievers” (Phillips 2008) it is
important that teachers learn the
characteristics of twice-exceptional
students.

19. Possible characteristics of
twice-exceptional students
Discrepancy between
verbal and written
work
Creativity
Excel on tasks requiring
abstract concepts
Difficulty on tasks
requiring memorization
of isolated facts
Anxiety
Depression
Acting-out behavior
Poor organization
Poor motivation
Active problem solvers
Analytic thinkers
Strong task commitment
when topic is
personally meaningful
Withdrawal/shyness
Discrepancy between
out-of-school talents
and classroom
performance

20. Second reflection point
In your experience as a teacher were
there students that were overlooked?
What could be done differently?
What did work?
What kind of support do teachers need in
order to provide better learning
opportunities for advanced learners?

21. S. M. Baum’s guidelines for
teaching twice-exceptional students
Focus attention on developing students'
talents and strengthening their
abilities through enrichment activities.
Provide a nurturing environment in which
students feel valued and their
individual differences are respected.

22. S. M. Baum’s guidelines for teaching
twice-exceptional students
Teach compensation strategies after
efforts to remediate skill deficits have
helped students reach a level of
proficiency.
Encourage students' awareness of their
individual strengths and weaknesses.

23. Math specific guidelines by Nisbet
Talking aloud
Cognitive apprenticeship
Analysis of the processes of argument
Cooperative learning
Socratic questioning

24. Al-Hroub’s and Nevo’s approaches
Al-Hroub distinguishes between
acceleration and enrichment.
Similarly, Nevo distinguishes between
acceleration, broadening and
deepening.

25. Who needs enrichment?
The feedback that Phillips (2008) has
received from educators at all levels,
parents, and indeed students, is that
“we have sacrificed the learning
potential of highly motivated students,
preventing them from breaking away
and advancing in selected subjects -
especially math and sciences - beyond
grade level expectations”.

26. Who needs enrichment?
Many would argue that all students need
enrichment.
Gamoran and Hannigan’s (2000) investigation
supported the benefits of algebra training
for all students, rather than just those who
show aptitude through grades or testing.
Students who were low achievers gained more
from algebra instruction than similar groups
exposed only to general math.

27. Enrichment for all students?
Taking a developmental point of view,
Sheffield (2009) suggests a continuum
of mathematical proficiency through
the development of creative ability in
mathematics: innumeraters, doers,
computers, consumers, problem
solvers, problem posers, and creators.

28. A new risk group
Research shows that students feel social
pressures to "dumb down" in high school to
fit in socially (Colangelo et al., 2004).
This begs the question “Have American [and
Canadian] schools, on the whole, created a
new risk group by applying disproportionate
time and resources to another?”
(Phillips, 2008)

29. Enrichment or Meeting
Requirements
There are highly motivated students who
achieve top grades, but may not
qualify as "gifted".
Focusing only on meeting grade level
requirements neglects the needs of
many highly motivated students
(Colangelo et al., 2004; Davidson &
Davidson, 2004).

30. Courage to advocate for enrichment
Enrichment and advanced information are
academically beneficial for all students.
Raising the bar of excellence will not leave
students behind, it will challenge students to
reach up instead of out.

31. What do teachers need?
Teachers have to be provided with multiple
opportunities to advance their knowledge
about advanced learners’ learning styles and
the importance of mathematical challenges
for meeting advanced learners’ academic
and emotional needs.

32. Also
Teachers should feel safe
(mathematically and pedagogically)
when dealing with this type of
mathematics (Holton et al., 2008).

33. Resources for teachers
Teachers need to be helped with
enrichment tasks by providing them
with appropriate learning material,
making a large number of challenging
tasks available to them, and providing
multiple opportunities to advance their
knowledge, possibly even mentored by
math professionals.

34. Third reflection point
What is your reaction to the suggested
supports?
What other supports would help you to
better accommodate advanced or
twice-exceptional students?

35. Developing students’ math potential
needs additional support
Parental support (not pressure) – intellectual,
emotional and financial
Availability of special settings and frameworks
for highly capable students in schools and
out of schools
The necessity of involving technological tools
that promote mathematical creativity in
students and support teachers' attempts to
scaffold students mathematical inquiry

36. Other components of support
Mathematical challenges as a central
characteristic of a learning environment that
develops creativity and promotes
mathematical talent.
Teachers' proficiency in choosing and managing
mathematical challenges.
Other activities such as math clubs,
competitions, and student conferences found
both in school and out of school.

37. Old stories about math need to
change
Frequently math is seen as hard, boring,
and never needed after high school.
It still causes anxiety in parents and
some teachers which is then imparted
on children / students.

38. In the media
Math is presented as a stumbling block
for all kinds of people.
Only rarely are there programs, like
Numb3rs, that show math in a positive
way.

39. Students’ attitudes can change
Students not inclined toward math
usually also suffer from math anxiety,
while advanced learners find math too
easy and repetitive.
Providing enrichment to support
advanced learners would enhance
math learning experiences for all.

40. Public attitudes can change
Accomplished students could change the
public feeling about math.
Kids might start waking up early not only
for hockey but also for math clubs.

41. Changes in media
When the public perception of math
changes, the media will cover math
related research and discoveries more
often, e.g. one about 'number sense'
that is universal to all animals not only
humans.

42. Ideas for math lessons
Margo Kondratieva (2011) suggests
interconnected problems.
Angela M. Smart (2011) suggests
cryptology and modular arithmetic,
symmetry and the art of Escher, and
Roman numeral arithmetic.
Smart also suggests using ideas and
resources designed for gifted students.

43. Everybody Loves Math?
Not yet but with joint efforts we will get
there soon.
Also, my “Everybody Loves Math” blog
houses enriched Math Magic lessons
and other resources.
http://beyondrequired.blogspot.ca/

44. Bibliography
See the resources page in my blog.
http://beyondrequired.blogspot.ca/p/bib.html