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From The Development of Algebra in the Elementary Mathematics Curriculum of V.V. Davydov (by
Jean Schmittau, State University of New York at Binghamton , and Anne Morris , University of Delaware ) one briefly understands about the mathematics curriculum Davydov prepared for Russian school:

The curriculum itself (which Schmittau used as written in the implementation with US students) consists of nothing but a carefully developed sequence of problems, which children are expected to solve. The problems are not broken down into steps for the children, they are not given hints, and there is no didactic presentation of the material. There is nothing to read but one problem after another. The third grade curriculum, for example, consists of more than 900 problems. Teachers, in turn, present the children with these problems, and they do not affirm the correctness of solutions; rather the children must come to these conclusions from the mathematics itself. The children learn to argue their points of view without, however, becoming argumentative.

How peculiar ! I would like to have a look at the text books.

However, seems nowhere could I find them, namely:

  • Davydov, V. V., Gorbov, S. F., Mikulina, G. G., Saveleva, O. V. (1999a). Mathematics Class 1. J. Schmittau (Ed.). Binghamton, NY: State University of New York.

  • Davydov, V. V., Gorbov, S. F., Mikulina, G. G., Saveleva, O. V. (2000a). Mathematics Class 2. J. Schmittau (Ed.). Binghamton, NY: State University of New York.

  • Davydov, V. V., Gorbov, S. F., Mikulina, G. G., Saveleva, O. V., & Tabachnikova, N. L. (2001). Mathematics 3rd Grade. J. Schmittau (Ed.). Binghamton, NY: State University of New York.

May I ask, where could I find them?

athos
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1 Answers1

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This is not an answer, but more like a long comment.

I could not easily find a free PDF of these textbooks - the measures to prevent book piracy work better now in Russia than a decade or two ago. Anyway, I found several screenshots, which I post below. The method has New Math-y feel to me, that is, instead of comparing and counting things it starts with comparing abstract quantities, and one has to make a leap to realize that letters mean quantities.

While I like some of the ideas in these textbooks, there are enough things that irk me, similar to the stuff that Common Core-aligned programs push, like the exercise 149 below for 2nd grade: two-dimensional arrangement of things, letters, arrows, and I really cannot make sense of the K -> _ -> M thing. Maybe if it were explained by a teacher, I would get it, but why use non-standard notation when one can use a standard one, or just write in plain words?

Same with exercises 110 and 113 for the 3rd grade, here the idea of what the arrows depict is clearer, but then again, why not use standard notation? Taking exercise 113 as an example, the arrows on the left mean multiplication: $ 8 \times 5$ and $8 \times 4$, but the arrows on the right mean addition: 32 + 40.

So, before anyone starts hunting for these textbooks (and there are a whole bunch of them: teacher's edition, student textbook, student exercise book, student exercise book with solutions, etc) one should take a second look at this method. I am not very impressed.

At the same time I want to point out, that usage of segments ("rods"), boxes, counters is traditional for Russian/Soviet elementary math education. Also, traditionally pupils did not use worksheets, and there were very few "fill the box for answer" exercises, mostly in the 1st grade. The screenshots below show the placeholders for answers, something that is more traditional for American textbooks.

Word problems are the staple of Russian/Soviet math curriculum, so whatever method you take, you'll see problems about trains, ships, pipes that fill a pool, workers that work in parallel on the same big assignment, etc. In this regard Russian programs have many common elements. I personally prefer a more traditional approach that starts from abstracting counters from real things (cows, cars), then comparing counters, then counting counters, then writing these tick marks as digits, etc. This approach does not exclude algebraic-like exercises.

For example, here is a 5th grade word problem: a child care center purchased two kinds of candies of the same price: 9 kg of the first kind and 13 kg of the second kind. The cost of the second kind of candies is $16 more than of the first kind. What is the total cost of the candies?

Another very important thing to keep in mind: Russian school curricula is very structured with tight relationships between subjects. There is more freedom nowadays in choosing this or that program to teach this or that subject, but still there are only three to five different programs that are popular, and they must cover specific topics in specific sequence. Algebra and geometry start from 7th grade and continue until the final grade, in parallel. Physics starts from 7th grade and goes for five years. Chemistry goes for at least two years. All these word problems with ships and trains use uniform motion first, but when physics starts, uniformly accelerated motion is used too, plus basic trigonometry - in 7th grade! So, I think it is not very productive to take just a single subject like mathematics/algebra from Russian program and use it elsewhere.

I don't like how American school program is structured, it basically does not exist. There are some vague conventions that algebra and geometry start in high school, that geometry, which is only one year course, must follow Algebra 1, that physics is usually not taken in 9th grade, etc. Different districts, schools and even individual teachers choose whatever courses they feel like, and it is impossible to plan, say, learning about right triangle and Pythagorean theorem a couple of weeks before solving a problem in a physics class about a boat, crossing a river. I see that some courses like Core-Plus try to incorporate basics of physics into math course only to fail miserably.

--

1st grade:

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2nd grade:

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3rd grade:

enter image description here

Rusty Core
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    physics is usually not taken in 9th grade --- To show how different things in the U.S. can be, physics at my high school was a 12th (last) grade course, and as far as any of the teachers could remember, no one had ever taken it before 12th grade until I took it during the 1976−1977 school year when in the 11th grade (and no one did so for several years afterwards either, including this person who took it in the 12th grade 5 years after I took the class), and also the class enrollment was around 5 to 8 students (for a high school of around 500 students). – Dave L Renfro Jun 28 '21 at 11:48
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    Oops, this should have been "... took it during the 1975−1976 school year". – Dave L Renfro Jun 28 '21 at 14:48
  • @DaveLRenfro I am in the U.S., my son wanted to take physics in his freshman year, but the counselors told him off, saying that this is unheard of. I don't know what can be so complicated in an algebra-based mechanics course. They suggested him to take bio, so he took AP Bio, which has some chemistry and statistics. How one can explain chemical bonds before studying chemistry first, or chi-squared distribution before studying probability theory and statistics, I don't know. Very unsystematic. – Rusty Core Jun 28 '21 at 18:58
  • In the case of my school, no one took algebra prior to high school, so it was alg. 1, geo., alg. 2, advanced math (= precalculus) for years 1, 2, 3, 4; and for science it was physical science, biology, chemistry, physics and/or advanced biology (with adv. biology being much more popular than physics) for years 1, 2, 3, 4. No AP courses at my school back then. The only acceleration was that every year 1-2 students doubled up on geo. and alg. 2 in 10th grade, took adv. math 11th grade, then calculus 1 & 2 at a (then small) university 30 miles away in 12th grade (continued) – Dave L Renfro Jun 28 '21 at 20:31
  • (and usually these 1-2 students were among the top handful of student(s) enrolled in calculus 1 & 2 at that university for that year). It wasn't really a case of not allowing students advance faster, but rather no one really trying to double-up and everyone having to take alg. 1 first year of HS. Most anyone who was in the top 10% or so took every college prep course offered, so there wasn't really much opportunity to double up (and no summer classes and such to get ahead). This had the unusual situation (which I didn't realize as unusual until years later) that at my HS no one (continued) – Dave L Renfro Jun 28 '21 at 20:32
  • who was reasonably good in math (say top 5% for entire school population) had any trouble what-so-ever with math classes, unlike the case at many high schools I later learned, where even good students might struggle with AP calculus a bit. Curiously, for about an 8 year or so period, my HS (smallest and most rural located in the region) (continued) – Dave L Renfro Jun 28 '21 at 20:35
  • had several students pass through who in math outperformed by a very, very wide margin everyone else in the region (besides the person I mentioned in an earlier comment and myself, there is this person, and 3 others who were also fairly strong), which had (from what I later heard) teachers from other schools wondering what was in our drinking water ("drinking water" is the actual phrase I heard someone use). – Dave L Renfro Jun 28 '21 at 20:35
  • By the way, I might delete some of these comments in a few days given the personal nature of them and the fact that in at least two cases I've identified two people who might not wish to be mentioned here (but I don't know either way). – Dave L Renfro Jun 28 '21 at 20:42
  • I always chuckle reading something like this: "Since the 1990s, a major thrust of the effort to improve mathematics achievement in America has been moving an authentic Algebra I course from the high school and into eighth grade. This would be similar to what high-achieving countries have been doing for a long time." Well, Russian kids start algebra and geometry in 7th grade. They used to start in 6th grade when school took 10 years (11 years now, no K class). This did not help the economy though :) So maybe it is ok to keep most people undereducated? But why spending 13 years on that? – Rusty Core Jun 28 '21 at 20:55
  • As for "Russian Math" in the U.S., here is a quotation: "Rifkin says the school's curriculum is based on Russian teaching traditions that emphasize reasoning and deeper understanding early on, not just memorization and practice drills. "The child should be brought to abstract level as soon as possible," she says, "meaning early introduction of algebra and geometry, not only arithmetic," and helping children figure out principles for themselves rather than spoon-feeding them." So you can see, the ideas are the same. – Rusty Core Jun 28 '21 at 21:00
  • From the article linked above: a puzzle presented on the chalkboard:

    _First line: A blank box shaped like a triangle + a blank box shaped like a circle = 12. Second line: A triangle box + a triangle box = 16.

    The kids figure out right away that the triangles have to be 8 and so the circle has to be 4. "Why did you start with the triangles first?" - "You should do the bottom one first, because they're the same." - "Right, they're the same number, so you're only guessing one number."_

    (continued)

     – Rusty Core Jun 28 '21 at 21:25
  • But this is wrong! There is no guessing, it is an equation with one unknown, it is a very straightforward algorithmic solution, no guessing. This just shows that non-professional educators cannot be trusted to teach and explain. They may be pretty good in math themselves, but they do not control their vocabulary. OTOH, there are many professional educators who don't know math well enough. Ideally, one hopes to have a teacher who is knowledgeable both in the particular subject and in pedagogy. – Rusty Core Jun 28 '21 at 21:31
  • @DaveLRenfro may I ask, is Algebra 1 taught in Grade 8 (Grade 1 refers to the first year of primary school)? then geometry in Grade 9, alg. 2 in G10, precalculus in G11 -- then what is taught in Grade 12, the last year of high school, before joining University? Or it is Algebra 1 in Grade 9, Geometry in Grade 10, Alg. 2 in G11, precalculus in G12? – athos Jun 29 '21 at 07:41
  • @athos: This varies greatly in the U.S., even when I was a student. However, in the past 20-30 years it has become more common in the U.S. for alg. 1 to be offered in grade 8 (ages 13-14). Of course, this would still be only taken by perhaps 30-40% of the students in grade 8 (lower % in high poverty schools, as well as small rural schools; higher % in low poverty schools, as well as large suburban schools). Assuming such students took math each year (in reality some fail, some stop after required 3 math for graduation -- it was only 2 in many places when I was in school), (continued) – Dave L Renfro Jun 29 '21 at 16:59
  • the student would be taking AP statistics or AP calculus in grade 12 (ages 17-18). I think perhaps 10% of all students take AP calculus (some before grade 12, but the %'s are very low), but I don't know about AP statistics (which is supposed to be much easier for weaker math students). Of course, some schools don't offer either of these AP classes, but at the present time this is much rarer than when I was in school. I think the school I attended began offering alg. 1 in grade 8 during mid 1980s, but I'm not very sure -- might have been late 1980s. (continued) – Dave L Renfro Jun 29 '21 at 17:05
  • Keep in mind that these sequences of math classes are not for ALL students, and in some places not even the majority of students. Standards have risen over the years, and now some states require alg. 2 (rules such as this are usually state-wide, and don't vary from school to school within a state), which would have been unthinkable where and when I was in school (perhaps 30% of the students never reached alg. 1 -- they took 2-3 classes from "general math 1, 2, 3" (below alg. 1 material) along with something called "consumer math", which dealt with % and money computations without algebra). – Dave L Renfro Jun 29 '21 at 17:14
  • @athos In the last quarter century the alg1-geom-alg2 program is being challenged by math1-math2-math3 program, this is what Common Core sees as a preferred approach, although AGA has not been abandoned. Supposedly, M1/M2/M3 program has elements of algebra, geometry and statistics for all three years of high school. This "integrated" course is advertised as being similar to what European schools use. Are you from France? Do you use integrated math in collège/lycée or separate algebra, geometry, statistics, trigonometry, calculus? – Rusty Core Jun 29 '21 at 18:21
  • @DaveLRenfro thanks for the explanation. – athos Jun 30 '21 at 13:10
  • @RustyCore I was from China which is another style again :p When I was in school, ie around 20 years ago, the textbooks are still dedicated to algebra or geometry, but they rotate, e.g. autumn term learning algebra, spring term learning geometry (just for example, i don't recall the exact rotating way now) ps. there's no spring term. Nowadays, it's more like M1/M2/M3, in each book there are both geometry and algebra topics. – athos Jun 30 '21 at 13:22
  • @RustyCore Calculus was in the text book of year 12 in my age, but it wasn't required in GaoKao, actually the last year we study no new things, instead just kept on practicing endlessly :p Now GaoKao's rules have changed, some students could opt for advanced math which includes some basic introduction of Calculus, e.g. only talking about integrals of $x^a$, $e^x$, $\ln x$, $\sin x$, $\cos x$ and $\tan x$. – athos Jun 30 '21 at 13:22
  • @RustyCore I'm now in UK, where M1/M2/M3 approach is taken. However I guess Common Core or AGA, both have the merits. Common Core introduces geometry earlier, which might help to visualize some algebra concepts. AGA seems has a more continuous narrative on Geometry or Algebra. So I gave my kid Discovering Algebra / Geometry / Advanced Algebra series to study. – athos Jun 30 '21 at 13:32
  • @RustyCore My guess is now the idea is to run to calculus asap, so Geometry takes only 1 year, many nice things are reduced, though this is understandable, as time is saved to introduce more useful things as matrix, probability, statistics, vector. Another thing I might complain is probability is explained in Algebra 1, Algebra II, and Precalculus, the narrative is interrupted. Maybe I'll recommend my kid another book. – athos Jun 30 '21 at 13:34
  • @RustyCore sorry i mean there is no summer term in China. – athos Jun 30 '21 at 13:35
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    @athos: My guess is now the idea is to run to calculus asap, so Geometry takes only 1 year --- I don't think more than a year was ever spent on Geometry in the U.S. (outside of rare schools that opted to do so) since the 1940s (earlier?) when solid geometry was commonly taught. Also, back then in the U.S., analytic geometry was taught much more thoroughly than now (but less thoroughly than in England), but this was almost always done early in college and not HS. Beginning in the mid to late 1950s, college analytic geometry courses were phased out and parts were moved into intro calculus. – Dave L Renfro Jun 30 '21 at 14:05