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I'm looking for studies on how and to what degree a teacher's conception of what mathematics is influences their way of teaching and, in the case of students, how this conception influences their learning.

here are some free thoughts:

A mathematics teacher's conception of what mathematics is can profoundly influence their teaching approach, strategies, and interactions with students. Here's how:

Perception of Mathematics as a Set of Rules vs. A Conceptual Framework:

If a teacher sees mathematics primarily as a set of rules and procedures to be memorized, they might focus more on rote memorization and practice of algorithms. This could lead to a more procedural-based teaching approach. In contrast, if they see mathematics as a conceptual framework, they might emphasize understanding underlying concepts, problem-solving, and making connections between different areas of mathematics. Fixed vs. Growth Mindset:

A teacher who believes that mathematical ability is innate might not invest as much time in students who they perceive as "not math people." They might also not encourage a culture of making mistakes and learning from them. Conversely, a teacher with a growth mindset might believe that every student can develop mathematical skills with the right support. Such teachers would emphasize the importance of effort, resilience, and learning from mistakes. Mathematics as Static vs. Dynamic:

Viewing mathematics as a fixed, unchanging subject might lead to teaching from the same textbook year after year without adapting to new educational research or integrating real-world applications. Seeing mathematics as a dynamic and evolving field would encourage a teacher to integrate current events, technology, and interdisciplinary connections, making the subject more relevant and engaging for students. Relevance of Mathematics:

If a teacher believes that mathematics is only important for passing exams, they might not draw connections to real-world applications or other disciplines. On the other hand, if they believe that mathematics is a tool for understanding and interacting with the world, they might consistently integrate real-world problems, interdisciplinary projects, and discussions about the broader importance of mathematical ideas. Student Engagement:

A teacher who sees mathematics as a series of abstract, disconnected topics might struggle to engage students or might rely heavily on extrinsic motivators like grades. Conversely, a teacher who views mathematics as an interconnected web of ideas, relevant to everyday life, might utilize project-based learning, discussions, and exploratory tasks to foster intrinsic motivation and curiosity. Role of Technology:

A teacher's view on the place of technology in mathematics can influence whether they integrate tools like graphing calculators, computer algebra systems, or dynamic geometry software into lessons. Attitudes Toward Collaboration:

If a teacher believes that mathematics is a solitary endeavor, they might prioritize individual work and discourage group projects or discussions. A more collaborative conception of mathematics would lead to more group work, peer teaching, and cooperative problem-solving.

but instead of wishful thinking I am looking for more serious studies and references AND SOLID STATEMENTS

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    $\begingroup$ "Seeing mathematics as a dynamic and evolving field would encourage a teacher to integrate current events, technology, and interdisciplinary connections, making the subject more relevant and engaging for students." - that's more or less the synopsis for The Mechanical Universe and Beyond. A 52 part, 80s telecourse from Caltech that I've been watching since I was a child, and why I can consider myself a polymath while having never seriously attended upper education. If you can teach like Goodstein, I'm in. $\endgroup$
    – Mazura
    Aug 5 at 22:10
  • $\begingroup$ "Each episode opens and closes with bookend segments in which Caltech professor David Goodstein, speaking in a lecture hall, delivers explanations "that can't quite be put into the mouth of our affable, faceless narrator". After more than a quarter century, the series is still often used as a supplemental teaching aid, for its clear explanation of fundamental concepts such as special relativity." $\endgroup$
    – Mazura
    Aug 5 at 22:20
  • $\begingroup$ "the earlier Feynman course had sought to make physics exciting by relating each subject, wherever possible, to contemporary scientific problems. The new course took the opposite tack, of trying to recreate the historical excitement of the original discovery." - Math for the sake of math is pretty droll. They plan on taking physics, right? Otherwise prob don't try so hard. $\endgroup$
    – Mazura
    Aug 5 at 22:21
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    $\begingroup$ matheducators.stackexchange.com/questions/9608/… $\endgroup$
    – Alexander Woo
    Aug 7 at 9:50
  • $\begingroup$ This makes me think of Jo Boaler's book, "Mathematical Mindsets", which came out last year. Especially the last 3 chapters, which are all about growth mindset. $\endgroup$
    – Adam Rubinson
    Aug 11 at 10:02

3 Answers 3

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Teachers' conceptions and beliefs about mathematics is a broad topic which has been an active sub-field of mathematics education research for several decades. There are thousands of studies that you can find on Google Scholar! I suggest that you search there if you are looking for research references and not an explanation or an experience-based answer.

If you want broad overviews of this topic, here are a few review articles that can also point you to more references: Dossey (1992), Thompson (1992), Wilson and Cooney (2002), Philipp (2007), Francis, Rapacki, and Eker (2014).

As an extra, here's a paper (not a review article) that you may find thought-provoking. Beswick (2012) presents case studies of two teachers and discusses how teachers can simultaneously hold different beliefs about "school mathematics" and "mathematicians' mathematics."

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    $\begingroup$ "school mathematics" and "mathematicians' mathematics." although I didn't read the paper you mentioned but from personal experience I can tell that way mathematician talk, behave and conduct their business is way different than how Math Teachers do it. And I felt that difference in these two sites "Mathematics Educators" and "Mathematics". $\endgroup$
    – Ashish Shukla
    Aug 6 at 5:12
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In The Teaching Gap, James Stigler references studies of teaching and learning math in different cultures. Sadly, better teacher beliefs aren't enough to change the situation because classroom culture is so hard to change.

According to him, much of it is deeply imprinted in us, as what school is. “The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom.” (p.87)

Many teachers who’ve tried teaching with more of a problem-solving focus can attest to how much resistance students put up: “That’s not how math class is supposed to work! Just tell us how to do it!”

But I do think you'll find his work enlightening.

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    $\begingroup$ belief is a quite important keyword! thanks! $\endgroup$
    – Humberto José Bortolossi
    Aug 7 at 11:46
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    $\begingroup$ I had a look at some of the sections of this book today. This is indeed fascinating! $\endgroup$
    – Jochen Glueck
    Aug 7 at 23:08
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    $\begingroup$ @Sue: "Many teachers who’ve tried teaching with more of a problem-solving focus can attest to how much resistance students put up." This book addresses that issue: Tolman, A. O., Kremling, J., & Tagg, J. (2017). Why Students Resist Learning: A Practical Model for Understanding and Helping Students. Sterling, Virginia: Stylus Publishing LLC. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Aug 13 at 16:25
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I introduce Mathematics as a Language. Also I tell them that there are two different Universes so to speak, one is this one where we live and the other one is Mathematical Universe. Mathematics is a language of the Mathematical Universe. And then I give them a word problem, which is the problem of "this world" and then how does that gets transformed into an equation which is Mathetical Language of Mathematical Universe. Also I give them an equation (problem of Mathematical Universe in Mathematical Language) and then ask them to translate that into word problem, which is a problem of this Universe. Students should be well versed with translation both ways.

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    $\begingroup$ What you do to teach has nothing to do with the question $\endgroup$
    – trula
    Aug 6 at 20:19
  • $\begingroup$ @trula maybe you are right but no one mentioned Language and just wanted to point that Math can also be looked as Language. Sometimes the answers are not direct but inferred. :) $\endgroup$
    – Ashish Shukla
    Aug 7 at 5:35

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