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This question is about references in current best practices in teaching math to future elementary teachers at a university level. I am asking it because I do not see any question so far on this site regarding this specific type of course (other than a textbook request, here), and I think it is worth discussion.

These courses are generally focused on helping future elementary teachers understand CCSS (Common Core) language (edit: or other equivalent standards-basedf language) and rephrase it into child-friendly language, helping them understand arithmetic through arithmetic in other bases and understand and be able to explain visual models of arithmetic, including understanding and identifying mistakes that people can make when using these models. The general focus tends to be on mastery of addition, subtraction, multiplication, division, divisibility rules, fractions, negative numbers, geometry, proportional reasoning, and equations, among other things in the same vein.

Arithmetic in alternate bases is a common part of these courses. While I strongly believe that this helps future elementary teachers conceptualize the learning of mathematics, I have noted (while tutoring students taking similar courses) that this message does not get through very clearly. So, my question is truly about "best practices" in this kind of course, but for clarity about what I mean by "best practices" I pose the following more specific questions, all of which are within my scope of asking about "best practices":

  • What are some references regarding good pedagogical approaches in this type of course? For example, I would lean to, but do not currently have evidence for:

    • De-emphasis of lecturing in favor of activity-based, constructivist approach: Students already "know" the material (though I recognize that they often do not quite "know" it, but feel like they do), and are asked to make sense of it in a deeper way, and so should rarely need to be 'told' material (there are arguments against ever lecturing in any math class, but that is a different question entirely)
    • Use of different bases to help students understand the underlying structure of their number systems instead of rote

  • What are additional types of exercises which help elementary teachers process this information and be prepared to use it when they teach mathematics to children?

  • What do elementary teachers really need to know, vs. simply be able to do? What evidence can I provide them that they need to know it rather than simply be able to do it?

Clarification: This is NOT about teaching a "methods" course: That is for experienced elementary educators to teach. This is instead about teaching a math course TO elementary teachers, where the elementary teachers are expected to have a firm grasp on the basics before they take their "methods" course on how to teach the basics.

Opal E
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    I've taught this type of course once (but it was science, not math). The fundamental problem was that almost all the students didn't want to be there and weren't willing to do any work outside of class. My school even went so far as to hire a tutor specifically to help these students. The tutor complained that the students were refusing to read the book and showing up without having made any effort on the work. There is no pedagogy or set of best practices that can overcome that barrier. –  Apr 16 '19 at 02:34
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    Is there really evidence that, "Students already 'know' the material"? Note that elementary education majors (in the U.S.) perennially have the lowest math skills, highest math anxiety, etc. (link) I know several of my community college students who say they've chosen elementary education precisely because it's a career path they believe requires the least math. – Daniel R. Collins Apr 16 '19 at 03:31
  • I say "know" in quotes precisely for that reason. In any case, most of the ones I've interacted with feel like they know arithmetic and so lecturing feels like a waste of time to them. – Opal E Apr 16 '19 at 03:35
  • This is quite broad. 2. Given the reference to Common core, which to my understand in quite USA-specific, are you interested in USA in particular or also other countries? You might want to specify this in the question, either way.
    • – Tommi Apr 16 '19 at 07:14
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    @DanielR.Collins Reading articles like this https://www.charlotteobserver.com/news/local/education/article223260005.html where about a half of North Carolina elementary teachers could not pass high school test, I am losing my faith in teachers and in ed schools. What is the point of graduating high school and then graduating a college, if you cannot pass a high school test? – Rusty Core Apr 16 '19 at 17:27
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    It baffles me that OP looks for solutions instead of using a standard program. Why the need to re-invent the wheel? "Teaching math to elementary students" should be a standardized course in any ed school across the country. – Rusty Core Apr 16 '19 at 17:30
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    I ask because this forum is a resource for math instructors, and it would be good to have references collected in this forum. I am aware standard programs exist, but standard programs also exist for teaching calculus, and this forum is happy to discuss them because there are reasonable disagreements on best practices. – Opal E Apr 16 '19 at 17:36
  • Ideally, a bunch of professors, researchers and teachers should have joined together and created a program that works. Then everyone uses it. If you screw up in a self-designed calculus course, then a couple of hundred students won't know calculus. If you screw up in a math pedagogy course, than countless number of kids will not be taught as well as they might have if you used a standard working program. The cost of error is much higher, IMO. – Rusty Core Apr 16 '19 at 18:13
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    So, is this forum not a bunch of professors, researchers and teachers, and is this forum not capable of collating the information created by previous professors, researchers, and teachers? I'm under the impression that that is precisely what this forum is for. I'm asking for research and references, not anecdotes, anyway.

    Additionally, this is NOT about "methods" courses (i.e. "teaching math to elementary students" courses) but about "Math for elementary teachers" courses, (i.e., "teaching math to elementary teachers courses").

     – Opal E Apr 16 '19 at 18:30
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    @RustyCore: Fear of deviating from an already-baked program is in my experience often a sign that a teacher lacks appropriate knowledge of the subject. The OP is teaching at the college level, where it's normal that the instructor puts significant effort into an original presentation of the material. Teaching preservice elementary teachers is not the same thing as being a preservice elementary teacher, and should typically not be done using the same methods that would be used with children. E.g., kids in lower grades may not be able to read yet, so you don't assign textbook reading. –  Apr 16 '19 at 18:48
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    A question about how to teach math "to future high school math teachers": Is there an elementary way to explain that a map of the earth cannot preserve distances? – Jasper Apr 16 '19 at 21:38
  • @OpalE "Math for elementary teachers" - no need to explain what has already been spelled out in the question. But maybe it was my comment that was not clear. The "countless number of kids" mentioned by me are those that will be taught by the would be teachers you teach. Hence, the cost of error for you is much higher than teaching non-educators. – Rusty Core Apr 17 '19 at 00:41
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    Ok, I get you -- I can follow a canned program, and I have a curriculum I am following which covers the content I am to teach. I don't intend to change the content. But I can choose how I emphasize the time in class and how to ask my students to engage with the material, e.g. the types of exercises they should do, the type of support I provide, the types of reflection questions I ask of them. I'd like these choices to follow best practices in the discipline, and "just pick a course and follow it" still leaves lots of room for interpretation. – Opal E Apr 17 '19 at 03:34
  • Not an answer but relevant, require a reading of some of the best material, such as A Mathematicans Lamnent, which can be found free online, and then discuss it in class! – Aaron M May 05 '19 at 19:37
  • @RustyCore, "standard programs" are the textbooks that come from publishers. They throw in whatever they think anyone buying copies of the textbook will want. This is an important question for any math ed professor to be thinking about. – Sue VanHattum Jul 08 '19 at 00:51
  • @SueVanHattum I did not mean kitchen-sink mish-mash of unrelated content, I meant a coherent program that can be taken and used directly top to bottom. I would be surprised if such a program did not exist, after all teachers, and math teachers in particular, have been trained for hundreds of years in normal schools. My understanding was that creation of such programs is a primary task of colleges of education. Well, maybe ed schools should be using them, and their parent universities should be creating them. In European countries this would be curated by the Ministry of Education. – Rusty Core Jul 08 '19 at 16:14