Hello my name is Ahsan and this is my presentation on "Why is this useful? A
search for meaning in mathematics education." So students will ask questions to this effect,
when is this ever going to be useful, why is this important, things of that nature all the time and
the way that I'm looking at this is or the focus of my research is how teachers respond to this
question and in particular like I'm treating important useful meaningful and kind of the same
way and so the way that I'm kind of seeing this play out then without I'm suspecting this will
play out is that teachers' goals beliefs and values will interact with their context and
background to influence well how they respond to this question instructionally if they choose to
but if I'm treating these words of like useful relevant
important meaningful all as synonymous then I'm particularly interested on an instructor's beliefs
on what's meaningful or important and how that influences what they how they end up teaching
and so if I'm looking at ways to construe meaning or meaningful or important
I'll set out this framework for kind of what I'm thinking or kind of like the glasses i'll use to
kind of frame the different ways that meaningful or important can be construed don't worry about
understanding it now i'll kind of explain it as i go. Brownell talks about two different senses
of meaning he talks about the meaning of something as being like the understanding of thing in itself
it talks about the meaning for something as understanding the relevance of something
so for instance he talks about not knowing the meaning of the atomic bomb because he doesn't know
the engineering and the physics beyond it but he does know the meaning for the atomic bomb
in terms of its implications for war and peace the destruction of society things of
that nature and so then extending that idea then to mathematics we can talk about the meaning of
mathematics as like understanding math itself um like connecting concepts things of that nature
or understanding like the relevance of math with the meaningful mathematics and there
that can be like procedural computations because physicists might need it or something like that
um or you know making critically informed citizens and promoting like critical thinking skills
but it doesn't always have to be as grandiose as that sometimes it can just be skills you need
for the next test or the next course um what's important in all of that though is that all of
those previous ideas kind of focus or all those previous ways of construing meaning talk about
meaning from the perspective of understanding and understanding is something that lives in our hands
um but we all go to grad school we've been to school we know the purpose of our education isn't
just for you know passing on information in a way our grad school is a way to kind of prepare us for
the careers that we'll end up being in so we can kind of think about it in terms of a bit of an
apprenticeship in a way and so a lot of uh i guess a lot of people who take a social perspective on
learning kind of take um kind of point to this aspect of education needing to be attended to
in terms of thinking about education not only as information but also just practices and from their
perspective um meaning or at least from some some social some people who take a social perspective
for them meaning is that experience of everyday life uh within particular communities
and so i can talk about like taking that conceptual understanding and taking the social
analog of that they'll be talking about like the social meaning of mathematics and that will be
like practices that mathematics uh mathematicians engage in so this could be like proving um
defining things of that nature doing mathematics and i can take that aspect of relevance and take
the social analog of that and that will be at least according to how i synthesize things
um talking about practices that non-mathematics communities engage in
um so this could be procedural computations or critical thinking skills like historically some of
the goals of our education system has been to push critical thinking skills on elites while pushing
uh procedural computations on lower class citizens as a way to kind of maintain social uh social
stratification and kind of maintain social order or maintained the status quo to say so to speak
um and so i'm using these uh kind of taking those initial like cognitive views of what's
meaningful and important and i can extend those socially um and i'm using this then to frame
how instructors answer this question of when is this ever going to be useful and why should i care
um and here are just some examples of kind of how each of these ways of construing meaning can
be demonstrated or can manifest so you know that conceptual understanding can be pushed
um by having teachers really focus on like connecting ideas or concepts um if you're
talking about relevance it could be just like you need to know how to do you need to know
how to do math to do your taxes or something like that um in terms of that social analog of
like conceptual understanding there's that functioning as mathematicians so i mentioned like
doing mathematics proofs things like that um and if we're taking that idea of relevance
and taking the social analog there um that would be things like critical thinking and procedural
skills for societal maintenance or even like being able to engage in routine calculations
as a to be a physicist and so then i'm focused on in this in response to this question of when
is this ever going to be useful i'm focused on like what do instructors understand to be
meaningful or important um and how do they focus on those in their classrooms and why um and i'm
thinking things like their background effective factors things like that might influence that
um so uh in terms of looking at actual instructors then um i talked about instructors in multiple
different courses just because there was more of a chance of seeing different ways of looking
or understanding meaning show up um and so yeah um in terms of my like i guess how i was kind
of getting a view of what instructors were doing um i did i took interviews with each instructor i
did classroom video recordings which i brought into interviews and asked instructors like you
know what were your goals in the moment how did those connect to your larger goals and
like now what are you trying to do here um and that was pretty much done through uh my analysis
process and i'll leave the details out um so in the discussion that i'll follow i'll talk about um
the background of each instructor and then how um each one of the ways that i'm framing or
construing meaning showed up in the instructor's goals um so the first instructor was a
uh was a math for liberal arts instructor um they had an identity kind of rooted in
mathematics like they were good in math they grew up good in it they made friends that way by
tutoring others in math and so it was kind of natural for them to go on and major in mathematics
um and they developed this view where mathematics they saw math as kind of like this pure thing like
you know you can get anything you can do anything with math you don't really
the other all the other subjects are subservient to that but as they grew older um they saw a lot
of inequalities in our society either through news their friends were realizing their own privileges
um and so they ended up kind of shifting away from that idealistic view of mathematics to
seeing mathematics as like a practical means of upliftment in a society that really values stem um
the next instructor was a stats instructor they had a lot of difficulty with math um because they
moved around a lot so they had no main intention of majoring in math when they gone to college
uh they went in as a business major and when they were doing that they went into a business
scout class where they were able to really see how math could be applied not really appeal to them
and so they decided to switch their major into mathematics despite a lot of gender stereotypes
and barriers that were up against them um then they ended up going into grad school
doing a master's in statistics they went into the business life a little while and then they decided
to come back to teaching where they really threw themselves into looking at education
research and how to teach better the next instructor was a calculus instructor
um he as he was growing up he was curious about why math rules were what they were um
but their family really wasn't well wasn't really able to answer those kinds of questions for him
but they were well off enough to be able to send him to a private school where he was able to get
ahead and so he really did well in math and he liked it um but once he was done with um k-12 he
kind of left school had no intention of going into math wandered a bit um after kind of traveling and
seeing he kind of seeing things he kind of came back to school um where he got a job as a tutor
and there um he ended up turning people in math and he was really he really enjoyed being able
to talk about mathematical details with his students and so he ended up deciding to or that
he wanted to end up teaching math um and because of kind of his own background he mentioned how he
mentioned having really strong beliefs about how math has its own kind of inherent beauty
and it doesn't need to be relevant the same way we don't talk about poetry or
playing the piano as being relevant those just kind of have their own natural beauty
the last instructor was a teacher educator um she had a very love-hate relationship with math like
she liked doing that and especially teaching it um but she didn't like being identified good at
math um and despite that though she had a lot of supportive individuals who kind of pushed her um
so that she ended up majoring in mathematics in college and when she was there she came across
a class where the teacher showed her how all the mathematical ideas that she was doing were just
particular cases of larger mathematical ideas um and that and being able to see those connections
really appealed to her and so that love hate relationship changed her pure love relationship
um and she decided to continue forward and go into teaching uh where she ended up uh
teaching a lot of underserved students and um kind of uh in really hard-hit areas and
she was really able to see how math could be a gatekeeper issue for a lot of those students
so in terms of the different ways that mathematics can be meaningful
if for looking at meaningful and important from that perspective of functioning as a
mathematician that actually barely showed up um and so like for instance with the calc instructor
um he knew some of his students would be math majors and so he felt like he needed to talk about
certain ways of doing mathematics like proving or like ways of approaching problems just because he
felt like some of them would need that in their future in their future careers um the national
liberal arts instructor um they they were trying to use math for actually applying it to things
but first they needed to kind of understand how mathematicians were communicating certain ideas
and so they really had to talk about notation a lot before they actually ended up using math
um in terms of that importance or meaningful in terms of like professional functioning or
societal functioning um or like you know practices that apply to nothing non-mathematics communities
um there's a lot of diversity here but all of them actually have like
an underlying trend of being influenced by their student demographics in various ways
um so for instance um like the count constructor they have stem measuring students so they were
trying to show like they tried showing a lot of stem applications of the calculus they're doing
um the uh the stats instructor um they the courses that their course fed into which were like health
science courses because their classes mainly for like health sciences and business majors
though the depart the departments of the courses that those that that stats course fed into they
really cared about their students kind of being able to interpret uh data and statistical results
but not so much on the mathematics itself um and so they put pressure on the instructor to
really focus on that and so she ended up doing showing a lot of like calculator shortcuts um
as opposed to like going into the mathematics and trying to understand it because of that
influence um in terms of the teacher educate or i know i'll talk about them after liberal
arts instructor first um the national liberal arts instructor he doesn't have stem measuring students
but in terms of projects that he kind of had his students engage in he had them engage in a
data collection project where they were just to figure out how many books in their library are
read and they had a whole debate over like what's considered red um and the point there was really
to focus on how like making his students really critical of things that they might hear news in
the news um and so really fostering that idea of being like critically informed democratic citizens
um the teacher educator they had their teaching teachers um so they spent a lot of time like
having their students like interpret other students thinking looking at student reasoning
and trying to understand that the one interesting case here was like this relevance outside math
um and the the the teacher actually tied this to her own background where she realized that not all
students are going to be engaged by mathematical activity itself and so they need to be teachers
need to be able to talk about relevance in various ways and so she was trying to prepare her uh
pre-service teachers for that in terms of that meaning uh in terms of meaning or importance in
terms of math for its own sake or like conceptual understanding the idea of connections was pretty
consistent across all the instructors or in terms of like different things that they tried to
emphasize um the one interesting case here was the stats instructor who talked about applications um
and she tied this to her own background of having struggled with math she felt like she learned
concepts best by seeing applications first and so she was trying to teach from that perspective of
um what she struggled with learning or how she struggled with learning so in terms of importance
or meaningful in that from that perspective of relevance outside math um there was this idea of
skills across all the instructors that showed up um the one interesting case here for me was the
relevance to non-math that showed up for the stats and the teacher educator um and the reason that
was interesting to me was the two instructors who hadn't mentioned it the math and the
math for liberal arts and the calc instructor both of them had actually expressed discomfort about
going about this kind of this way of understanding meaningful or important um and what stood out to
me there was that both of these instructors were instructors who had backgrounds of like really
appreciating math for its own sake um and liking and like being successful in math and so what that
might suggest for me then is that their background might have actually served as blinders in a way
it might have made answering that question of relevance or importance or meaningfulness
um and diversity very difficult for them just because their backgrounds might have made certain
ways of answering that question more accessible to them than others the one disconfirming case here
was the teacher educator who we'll talk about in the next slide kind of recapping i was i had these
four ways of kind of considering how meaningful or important can be understood and they and i was
using that then to kind of frame instructors goals about what they thought was important um and what
i found was that you know a lot of it a lot of the ways that those meanings or importances kind of
manifested depending on the student demographics but also the instructor's own background
um i mean there was like when i'm thinking background i'm thinking of the calc and the
latin math for liberal arts instructors who like their background in math might have made answering
that question of like relevance and diverse ways it might have made that difficult for them just
because their background might have made certain ways of thinking about meaningful or important
more applicable or relatable to them um the one disconfirming case here however was the teacher
educator who didn't care about math um or didn't care about relevance but she just liked math for
its own sake but because of her experience of like seeing disenfranchised students she realized that
being able to answer this question of importance in various ways was something that was needed the
same thing with the stats instructor she actually had that personal experience of struggling with
math and so kind of looking overall what stood out for me here was that if we want instructors
to be able to answer this question of when is this ever going to be useful in diverse ways
what this might suggest is that we need teachers who've had either that personal experience of
struggling with math like the stats instructor or have that experience of really seeing others
be disenfranchised by that and that can have a lot of really important consequences for equity
and also in terms of engaging our students if we want to really be able to engage our students
perhaps we need more instructors who have these diverse experiences
and are thus able to then answer this question in diverse ways
and these are my references
thank you