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The problem you describe is well-known in mathematics education research. I cite the paper of De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334. and give some ...
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I don't view these common mistakes as 'universal linearity' assumptions. The mistake that $(a+b)^2=a^2+b^2$ is just a visually appealing statement. It is mistaken to be correct because it looks nice. Our brains tend to like things that look nice. Similarly, $\sqrt{a+b}=\sqrt a+\sqrt b$ is visually appealing and it resembles the correct formula $\sqrt {ab}=\...
23
Perhaps the key-word needed here is not just struggle but productive struggle.
Hiebert and Grouws (2007) discuss two key features of mathematical teaching/instruction for "promoting conceptual understanding" (p. 383). Their paper can be found here:
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ ...
20
For those with a background in mathematics, perhaps the following source would be of use:
Curtis C. McKnight (Ed.). (2000). Mathematics education research: A guide for the research mathematician. American Mathematical Soc.. Link.
A review of the book can be found on the MAA website here.
From this latter link:
In a nutshell, this book is a guide for ...
20
This became to big to be a comment.
Layman's opinion.
Where does it come from?
It comes from the fact that universal linearity is useful to move forward in calculations even if it's wrong. Psychologically this is very attractive. The other option is being stuck. Moving forward has the added incentive that it can be right, that maybe the student can get ...
17
As a student myself, I'd say that, while I'm not representative of all students, some of it is the intimidating or dismissive way some lecturers or teachers might treat questions relevant to simple things like these, or not teach off the exam specification. This leads to a subconscious or even conscious bias against asking these sort of questions which we're ...
13
I guess you have already got plenty to read for "Where does it come from?" part of you question. Thus I just shortly introduce my favorite strategy that I use for "What can we do about it?" part, when there is a case to work on:
Encourage your students to find conditions that linearity assumption indeed works!
For examples, for which values of $a$ and $...
13
From Clark, Richard, Paul A. Kirschner, and John Sweller. "Putting students on the path to learning: The case for fully guided instruction." (2012):
Even more disturbing is evidence that when learners are asked to
select between a more-guided or less-guided version of the same
course, less-skilled learners who choose the less-guided approach tend
to ...
13
Abstract from Alfieri, Brooks, Aldrich, "Does Discovery-Based Instruction Enhance Learning?", Journal of Educational Psychology, 2011:
Discovery learning approaches to education have recently come under
scrutiny (Tobias & Duffy, 2009), with many studies indicating
limitations to discovery learning practices. Therefore, 2
meta-analyses were ...
12
I very much welcome connections of research and teaching, although results from research still need or be interpreted in your specific situation. Let me give an example.
It might happen that your students can state the correct definition of a subspace of a vectorspace, but if you ask them if a specific set is a subspace, they "try guessing". There is no ...
12
Researchers have philosophized about and demonstrated that there is a specific kind of knowledge that teachers need that is likely different from just knowing the content itself.
So, there is data that demonstrates that knowledge of content alone is not what good teachers need. Rather, they need knowledge of things like how students make sense of the ...
12
I believe the classic reference from the mathematics education literature is:
Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. The Journal of Mathematical Behavior, 15(4), 375-402. Link (no pay-wall).
The authors are all out of EDC (Cuoco and Goldenberg on the linked page; Mark on the ...
12
Yes, there is one called MathEduc.
Quote:
MathEduc (formerly MATHDI) is the only international reference database offering a world-wide overview of literature on research, theory and practice in mathematics education. MathEduc also covers education in computer science on the elementary level. The scope includes literature for all school levels up to ...
11
Why does it matter what the research shows?
At best, a study might have found that students who were blindly fed recipes did better in the short term on examinations for calculus.
But then, what is the ultimate goal of education? For students to pass tests? I really hope not. The goal of any teacher, especially a teacher in mathematics, should first and ...
11
Course-taking beyond the level of advanced algebra (what is normally called "Algebra 2" in the United States) has been strongly linked to future academic and economic attainment; for example, a U.S. Department of Education study (Adelmann, 1999) found that students who complete mathematics coursework beyond the level of a course in advanced algebra are more ...
11
In addition to other good answers and comments... I think it should be noted that "linear mathematics" at higher levels is the part of mathematics that we (collectively) understand relatively well, while "non-linear mathematics" is often intractable... except to the extent we can usefully approximate it linearly.
It is both symbol patterns and the ...
11
This question seems like a big opportunity for casual social science conjecturing which may or may not be productive. I hope I can clarify a couple of things in my response.
According to your supporting statement, your question is "what are the possible explanations for why so many mathematicians you look up on Wikipedia are Jewish?" For the purposes of ...
11
Yes, there is a growing literature at the nexus of mathematics education and creativity. The main name to know is Bharath Sriraman (google scholar) though the classic pieces to read for mathematical creativity are the book by Hadamard (see the answer of Lucas Virgili) as well as the chapter Mathematical Creation by Poincare that served to inspire the ...
11
Given 100% control, I would have one-to-one instruction. One instructor meeting individually with each student. That instructor can change the approach, the speed, the order of topics, the method of instruction, based on that individual student.
But of course hiring (and training) enough instructors for that is probably way beyond any reasonable budget. (...
10
I've read all the existing answers long ago but still feel that none have gotten to the heart of the issue. We obtain mathematical results through a process of reasoning. That reasoning must be logical and enough to convince anyone that our results are correct given our initial assumptions. That is the actual purpose of a proof. It does not matter what form ...
10
These books will be of interest to you:
Theorems in School edited by P. Boero
Teaching and Learning Proof Across the Grades, ed. by Despina Stylianou et. al.
My subjective sense is that the majority of researchers in math education agree with your point of view: learning proofs requires hands-on practice with writing proofs, and not only improves the ...
10
As an earlier comment (How can a research mathematician transition into a mathematics education researcher?) indicates, I speak from the perspective of someone who has done a smidgen of math-education research, but not in any sense as an expert.
With this caveat, I think that it is important to realise that math-education research is much closer to science ...
9
Here is an NPR article that discusses how teachers' efforts to engage learners in productive struggle (or not) may be culturally situated. (Of note, Benjamin cites Jim Hiebert above, who has written The Teaching Gap with Jim Stigler, interviewed in this article.) Jim and Jim have conducted research on how instructional approaches differ culturally between ...
9
The answer you are giving to your students overlaps with the so-called "Mental Discipline" theory (also "Theory of Formal Discipline") for justifying mathematics education. As you search for research related to your favored justification, this theory may help you focus in on more relevant work.
One researcher who has written about this is George Stanic. You ...
9
From the article:
Grade level was a categorical factor consisting of three levels: preschool–kindergarten students (or its equivalent if outside of the United States), 1st‐ to 6th‐grade students, and 7th grade and above. This variable evaluated whether the effectiveness of mathematics game‐based learning varied across different grade levels as to ...
9
Without directly answering your question, you don't seem to have the background you need to be "improving" the undergraduate experience yet, and have some work to do. I think you're right in sensing that your question is too general
Don't talk about the "difficulties in forming such a course", and spend your initial time finding out what aspects of the ...
9
I have some personal experience.
I taught in a school that has Israeli students whose families had moved to the US. These children had Hebrew as their native language. Hebrew is written right to left. I never saw these students struggle with math because they learned to write from right to left.
Furthermore all students in my school studied both Hebrew and ...
8
I think the precise wording of the question accidentally prejudices the potential answers. Namely, at all levels of fanciness-of-math, and at all levels of development of kids'/peoples' thinking, the thread of "how can we do these things" and the thread of "why did that work to accomplish our goals?" and the thread of "should I be curious about both ...
8
I am a mathematics educator who conducts research on teaching and learning, and I also teach mathematics courses that are mathematics content courses for future teachers. I endorse the references given here! Great ideas.
Another option: Mathematics and Mathematics Education: Searching for a Common Ground.
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