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I'm currently a research mathematician, getting involved in more and more outreach activities. One of these involves delivering lessons for elementary school math teachers (K-6). The purpose is two-fold: to motivate them about the coolness of math, and to give them more fodder for their own classroom lessons.

I have taught a lot of different types of audiences before, but not this one. I am wondering what would be safe to assume as far as the mathematical background of someone who teaches math in a US public school, grades kindergarten through 6th grade. What would be the minimum highest level math course a person in that profession would have to have taken? What would they be likely to remember, mathematically, from when they were in college?

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    $\begingroup$ Case study: In a survey of 50 K-6 teachers in New Jersey, none (zero) knew how to find the area of a rectangle (Kenschaft, 2005).: madmath.com/2010/04/in-new-jersey.html $\endgroup$ – Daniel R. Collins Dec 2 '17 at 16:58
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    $\begingroup$ "The purpose is two-fold: to motivate them about the coolness of math, and to give them more fodder for their own classroom lessons." Think about things from their angle, not from yours. IOW, don't push your research, don't push some set theory rigor stuff that you love. Think about things that will (a) delight the student (e.g. little games) or (b) conveying the importance of math for future in high school, jobs, etc. $\endgroup$ – guest Dec 2 '17 at 19:19
  • $\begingroup$ That's ridiculously sad and... common. My guess is that from their days as elementary students through the textbooks they use as teachers, they've only been exposed to pseudo math definitions of "area" such as a place, a location, a size, a shape, etc. It's possible they've never been exposed to the definition of area as the number of square units required to cover a plane figure. So, they might not even think of area as a number. This would make it hard to connect it to multiplication. $\endgroup$ – WeCanLearnAnything Dec 2 '17 at 19:20
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    $\begingroup$ What does "delivering lessons" mean? How much time are you going to spend with them? // If I could do one thing to influence elementary teachers, regarding math pedagogy, it would be to steer them away from rote arithmetic drills delivered by computer, and to steer them towards interactive games for drilling arithmetic facts. $\endgroup$ – aparente001 Dec 4 '17 at 2:28
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    $\begingroup$ @j0equ1nn - Do you only gt them for a one-off session, or will you be meeting with them regularly? I suppose the best influence you could have on them would be to help them resolve their math phobia, get set with some resources they'll be comfortable using on their own in the future, and start to see themselves as competent math learners. They each need to have an aha moment where they discover that they can think successfully about math without hitting a brick wall of frustration and confusion. But to be competent math teachers they have to be willing to spend time doing math on their own. $\endgroup$ – aparente001 Dec 5 '17 at 15:56
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Supporting the answer by Brian Borchers, and also the comment by aparante001: K-6 elementary teachers in the U.S. will know effectively zero math, or even a negative amount of math in many cases. It's well established that for over a century in the U.S., the very weakest college students, and in particular the ones with the highest level of math anxiety, self-select for K-6 education (often presuming that no math is required for that):

http://www.madmath.com/2016/02/hembree-on-math-anxiety.html

Taking CUNY as an example, K-6 majors are not required to take intermediate or college-level algebra (as of next year, they will not even need to demonstrate proficiency in elementary, i.e., 8th-grade, algebra). Consider a case study at CUNY Brooklyn college -- requirements for major and the two math courses in question are linked below.

The two required courses are described as follows:

MATH *1401 Elementary Mathematics from an Advanced Standpoint

Mathematics content needed for teaching major strands in the early childhood and elementary school mathematics curriculum: Problem Solving; Sets; Number Systems; Geometry; Probability and Statistics.

MATH *1406 Mathematics in Education

Concepts and principles of mathematics underlying the elementary school curriculum. Taught in coordination with Education 3206 [44]. a. Early childhood education section: emphasis on topics relevant to teaching children from prekindergarten to grade 3. b. Elementary, bilingual, and special education section: emphasis on topics relevant to teaching children from kindergarten to grade 6.

What is somewhat obscured is that the MATH 1401 course may or may not be taken under CUNY's "Pathways" policy. That is: If a student starts at one of the community colleges, they may take an entirely different liberal-arts math appreciation course which satisfies that same requirement.

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    $\begingroup$ +1 for "or even a negative amount of math in many cases" I know it's kind of cruel, but it made me laugh out loud. It reminds me of some of the times, when grading a student's solution, I felt that what the student wrote deserved negative points because it suggested less understanding than I would have ever guessed had they just left it blank. $\endgroup$ – Dave L Renfro Dec 3 '17 at 10:15
  • $\begingroup$ Would you happen to know if the CUNY situation is typical nation-wide? I did mean to ask about this on a national level. (You noticed in another comment that I'd worked at CUNY, but currently I'm actually at UNAM in Mexico, and applying to my next job, which will likely involve broader nation-wide outreach activity.) $\endgroup$ – j0equ1nn Dec 5 '17 at 3:32
  • $\begingroup$ @j0equ1nn: Good question; in truth I'm not sure. But Benjamin Dickman's answer regarding BU -- showing a similar two courses with similar topics -- suggests that this may be fairly normal. $\endgroup$ – Daniel R. Collins Dec 5 '17 at 13:59
  • $\begingroup$ lol for negative amount of math $\endgroup$ – user2139 Dec 5 '17 at 22:29
  • $\begingroup$ I perhaps should say that "negative knowledge" is a term I use to mean "scores lower on a multiple-choice test than one would expect from random guessing". $\endgroup$ – Daniel R. Collins Dec 5 '17 at 23:33
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Based on my experience in the US, most school districts teachers teach all subjects including mathematics in the elementary schools. There typically aren't "mathematics teachers" at this level although there are some school districts that have designated mathematics teachers.

You should probably not assume anything beyond basic arithmetic- they might have taken a traditional college algebra course, but many have them have forgotten whatever they might have learned in such a course. Many of them will have taken specialized courses in mathematics for teachers and in math education, but the actual mathematical content of these courses varies tremendously. Furthermore, many of these teachers have been traumatized by their own experience in mathematics courses and are afraid to engage with the subject- getting them to participate in any kind of problem-solving activity is extremely challenging.

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    $\begingroup$ Many of these teachers have been led to believe that they're not "good" at mathematics (e.g. because of poor grades in courses that they took) and as a result have a negative emotional reaction to the subject. When that negative emotional reaction to the subject interferes with the teacher's efforts to teach the subject, students suffer. $\endgroup$ – Brian Borchers Dec 1 '17 at 20:05
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    $\begingroup$ True. I'd go farther. The choice to go to teachers college and specialize in elementary ed is based on several factors of course, but a biggie is math phobia. I think the correlation between math phobia and being an elementary school teacher in the U.S. must be huge. // I recently mentioned to my 14yo's Algebra 2 teacher that learning to sketch graphs by hand (rather than letting the graphing calculator do all the work) is helpful preparation for analytical geometry, and he asked me what that is. I said, "Precalculus." He still wasn't with me. I wonder what he did in college. $\endgroup$ – aparente001 Dec 2 '17 at 4:03
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    $\begingroup$ At CUNY (where OP does work), K-6 education majors are not even required to take intermediate or college algebra. I have one in my current college algebra class (challenging themselves in higher course), and this person was in tears last week at the difficulty level. $\endgroup$ – Daniel R. Collins Dec 2 '17 at 17:03
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    $\begingroup$ @aparente001 I would argue that your interaction with the teacher is a neutral one, if not actually good. When presented with a topic he/she did not understand, the teacher told you so and was open to further discussion. This is exactly the skill that your 14yo needs to learn, which is much more important than analytic geometry. $\endgroup$ – Chris Cunningham Dec 3 '17 at 23:27
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    $\begingroup$ Additionally we as educators are very vulnerable to "one time a math teacher said X which was dumb" stories, and we should not proliferate them on the site. $\endgroup$ – Chris Cunningham Dec 3 '17 at 23:28
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Despite what the negative answers, there are elementary teachers who are good at math. I count myself among them. The problem is that there are a broad range of abilities and attitudes among those who will you be talking to.

My suggestion is that you look for cool things about math that don't require a knowledge of higher math.

One example might be my and other's answers to Imbuing a six year old with a sense of mathematical wonder

Another example is the many cool patterns in Pascal's Triangle or some properties of Fibonacci numbers.

I also direct you to books by Marilyn Burns on mathematical education as well as many NCTM publications. These will give you a starting point for approaching teachers who don't have a math background.

Good luck

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    $\begingroup$ +1 for some needed constructively positive comments in this thread. (And yes, this was written AFTER my comment to Daniel R. Collins.) $\endgroup$ – Dave L Renfro Dec 3 '17 at 10:20
  • $\begingroup$ There's also Cynthia Lanius's web pages, which I remember stumbling across back around 1996 or 1997. $\endgroup$ – Dave L Renfro Dec 3 '17 at 10:26
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    $\begingroup$ Teaching arithmetic well is facilitated by a "knowledge of higher math". $\endgroup$ – Dan Fox Dec 3 '17 at 12:01
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    $\begingroup$ @DanFox I agree but the reality is that many elementary school teachers became elementary school teachers without a good background in math or confidence in their ability to learn math $\endgroup$ – Amy B Dec 3 '17 at 15:48
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Here are the requirements for Teacher Certification in Massachusetts:

An image pasted from the latter link:

enter image description here

The examination in Massachusetts is called the MTEL. To prepare for it sufficiently (especially in teaching "elementary" as opposed to "early elementary") I can speak to the two course sequence at a particular teacher-education program, that of the Boston University School of Education (where I previously worked as a postdoc and have taught both of these courses):

enter image description here

enter image description here

Even in Massachusetts, there are a range of alternative paths towards teaching (depending, for example, on the needs of a particular school or school district). Looking to the United States as a whole, the number of possible paths to teaching elementary school is finite, but barely so.

You may have better luck homing in on a particular school, and investigating how the teachers presently at that institution were prepared (as well as the sort of ongoing professional development required/attended). Your final two questions, if answered literally, will do little to provide insight into the planning/implementation of outreach activities.

What would be the minimum highest level math course a person in that profession would have to have taken?

Across all possible paths: None.

What would they be likely to remember, mathematically, from when they were in college?

This will vary tremendously, as some elementary school teachers were math majors, some were math education majors, some were education majors with a focus on math, some were education majors who did not focus on math (and may not have completed any relevant coursework), and some may have majored in something unrelated to math or education.

I suggest that you interact directly with a potential site for outreach and the teachers who are working there, and go in with an open mind and a sincere belief that the teachers want what is best for students - even if, in practice, this leads to actions that may appear as irrational from the perspective of someone outside of the institution of k12 education and its associated structures/requirements.

For more information about how some elementary school teachers are honing their mathematical craft, search Twitter for the MTBOS (Math Twitter Blog-O-Sphere) or iteachmath or tmwyk (Talking Math With Your Kids) hashtags, or check out Tracy Zager's book, "Becoming the Math Teacher You Wish You'd Had: Ideas and Strategies from Vibrant Classrooms" (Amazon, Google Books).

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Academic, applied, and research math is the world where you come from.

[Edit: Much of] Elementary "math" is a whole different world.

If you want to know why quotation marks are literally necessary in the prior sentence, check out the writings of Hung Hsi Wu. Then, prepare for something radically worse than that when you ask students to explain their thinking aloud.

Then prepare for something radically worse than that when you ask teachers to explain their thinking aloud.

Despite the variance of teachers' math preparation, the mean, median, and mode are terrifying. For the record, I am not blaming teachers; my mother was a teacher in a high-needs elementary school for much of her career. I consider them, and many others, victims of an incredibly stupid "math" education system.

Teacher certification is similar in Canada and the US so this Toronto Star article should give you a taste of what you may be in for. Short version is that many teachers-in-training :

  • Cry when asked about their own experience with elementary school math.
  • Believe that a quotient of 1 with a remainder of 3 means the same thing as a 1.3.

So, with this in mind, what do you plan to do with those elementary school teachers?

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    $\begingroup$ +1 Likewise (from the same Kenschaft article I cited before), a case of a teacher thinking that 1/3 was "Near three, isn't it?" (i.e., 3.1?). madmath.com/2016/12/observed-belief-that-12-12.html $\endgroup$ – Daniel R. Collins Dec 2 '17 at 19:50
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    $\begingroup$ -1; I do not think this is a good answer and in fact I think it is actively harmful. Instead it attacks elementary school teachers repeatedly from the same angle by implying that they are not real math educators. If I am incorrect about your intent with things like the quotation marks around "math" and references to crying, I would be happy to be corrected. However, this answer in its current form is nonconstructive and insulting. There is a way to say "your audience is not expert in mathematics" without belittling the subjects of your statements. $\endgroup$ – Chris Cunningham Dec 5 '17 at 1:01
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    $\begingroup$ Insulting and belittling my own mother's heroic profession was not the intent at all! The intent was to respond candidly to the original question of what he should expect regarding his audience's knowledge of math. As for the examples of crying and remainders vs decimals, those are ~identical to what I've heard from many elementary teachers themselves, from both admissions officers of education faculties at both universities in my city, at workshops, and in academic literature (linked to elsewhere in this thread). I think it's pretty reasonable to list them as examples. $\endgroup$ – WeCanLearnAnything Dec 5 '17 at 8:59
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    $\begingroup$ Re: Quotation marks around "math". Check out the Hung Hsi Wu link near the quotation marks for the real justification. For now, here is a common example of "math". (1) A ~VERBATIM quote from curriculum docs: "A fraction is a part of a whole." (2) "part of" implies "smaller than", so every fraction is less than a whole. (3) The next 3 years of textbooks confirm this; in all fractions, numerator < denominator. (4) Kids then learn that 9/4 = 2 + 1/4, but are baffled by 9/4 and mixing pizza slices and numbers, so they revert to rote memory instead of actual mathematical reasoning. $\endgroup$ – WeCanLearnAnything Dec 5 '17 at 9:41
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    $\begingroup$ psychic plus 1. (Can't vote.) P.s. Chris, you are nice guy but be wary of minus one for things you disagree with, but are still well thought out expressions of a point of view (that you disagree with). $\endgroup$ – guest Dec 5 '17 at 23:28

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