Robert Hansen (RH) had posted as his instance of a 'system' to enable the teaching/learning of arithmetic (see http://mathforum.org/kb/message.jspa?messageID=7903549, dated Oct 10, 2012 1:59 PM): > > Counting to 10, to 20, to 100 > > 1+1 = 2, 1+2 = 3, 1+3 = 4, ... > > 11+11 = 22, 11+12 = 23, ... (multi digit without > carry) > > 15+7 = 22, ... (multi digit with carry) > > Jane has 3 apples, Troy has 6 apples, how many apples > do they have together... > > Subtraction, as the reverse of adding, recall your > addition facts (no negative numbers) ... > > Sinclair has 12 dollars, Mary has 3 dollars, Sinclair > gives 6 dollars to Mary, how many does he have left? > How many does Mary now have? ... > > Multiplication, starting with repeated addition, then > the tables ... > > Problems... > > Division, as the reverse of multiplying, whole > results only, recall your multiplication facts ... > > Problems... > > > During this phase we focus on number, operation and > context (problems). > > Numbers are made of digits that occupy places > (columns). Learn to recognize them, say them, write > them (notationally and with words). > > The four basic operations are addition, subtraction, > multiplication and division. > > Use visualizations only when they are contextual and > developmentally equal to what they represent and what > your are doing, like the number line, rows of > objects, arrays of objects. > > Do not use visualizations that are reconstructions or > proofs, they will only confuse and hide deficiencies > in understanding. > > The same thing goes for problems. Problems at this > stage are language training, not problem solving. > They teach how to map (not model) a literal > situation, involving numbers and operations, to an > arithmetic expression. They start straightforward at > first but then reverse the order as you go on. Jane > is two years older than Mary. Mary is two years > younger than Jane. > > Do not teach things that involve fluency in these > operations before that fluency is obtained, like Time > or Money. Reading a basic clock is ok, but don't > start into the number of minutes in an hour etc. > Whole dollars or whole cents are ok, but decimals, > even just a written example of them, is not. > > Do not teach estimation directly, let it build > through familiarity. How can you teach a student to > estimate addition if you have not yet finished > teaching them addition? > > Do not teach shortcuts in the path before they know > the path first. Talk to them as they arise naturally > but stay on target. > > Do not teach algebra! Math is simply not reasonable > enough yet. Fill in the blank problems are good (3 x > __ = 12), but reasoned solutions are premature at > this stage. As you get further along set aside time > for reasoned discussion but don't conflate it with > the developmental tasks at hand. There will be > students that see further ahead and you should > recognize it and encourage it, but stay on track. > They are not going to see all of the baggage ahead. > > Fractions should be introduced gently at first, in > literal form, "What is half of six?" They shouldn't > be dealt with technically until the students are > fluent with whole number multiplication and division, > probably 4th grade, decimals some time after. And tie > them to arithmetic, not pictures. The pictures are > too algebraic. We are still talking about the number > line here, not algebra. > > Fifth and sixth grade should be used to solidify > fluency in all of these things, technically and > operationally. By sixth grade, a student should look > at a common fraction or decimal pretty much as we do. > They should be able to perform the four basic > operations on any pair of numbers (with finite > digits), with and without a calculator. Mental math > is developed and stressed during this stage, along > with data sense (charts etc). The problems are still > more technical than reasonable although they now have > a hint of algebra. They can involve multiple steps or > simultaneous conditions. > > > Later I will explain my theory as to how it fell > apart and became what we struggle with today. It has > to do with college and I think people are just now > realizing it. > He had posted the above in the vastly mistaken belief that it represented a "system". It is actually a 'curriculum for school arithmetic'. As a "curriculum" it is possibly fairly sound (by the conventional way of thinking) - but it is emphatically NOT a "system"! It could possibly represent an important 'part of a system for school arithmetic'.
In order to create a true 'system', we would need to integrate, with the above-noted 'curriculum', the following considerations:
1a) Is the teacher knowledgeable enough AND capable enough as a teacher should be about the contents of the 'curriculum' to be able to teach it to students who don't know arithmetic?
The above will include matters such as:
i) Is the teacher able to guide the student through the many difficulties that the student will encounter in learning this new topic?
ii) Is the teacher able to help the student out through real difficulties in problems attempted?
iii) Is the teacher capable of making the learning interesting enough for the student to want to learn this arithmetic (as against the manifold other activities that the student may have to occupy his/her time)? [Many other such issues may be involved].
1b) Is there an adequate 'learning+teaching environment' available in the school? As a 'side-bar', we might observe that a recent survey found that a great many schools in India lacked proper toilet facilities (particularly for girls). Now that I would classify as DEFINITELY a "lack of an adequate learning+teaching environment"!! As a 'positive part' of that side-bar, we note that there is, India-wide, a 'free mid-day meal' provided to most school students (at government expense). Though this was not widely understood when the mid-day meal program started, this turned out the part of the 'system' that MOST encouraged and enhanced attendance at school and actual learning! Now, the free mid-day meal program is practically universal here - there are deficiencies reported from time to time, but the program is now understood by all to form part of 'the learning system'.
1c) Is there an adequate 'learning environment' at home that encourages the student to put in enough effort to learn successfully?
1d) Is the student interested enough to put in the sizable effort it will take him/her to learn everything that he/she should? (If not, what can be done to pique his/her interest?)
1e) What can be done to motivate the student when he/she is frustrated? There are a whole lot of other such issues that will develop when we try to create a 'system'.
2) Are the parents adequately enthusiastic about their child learning, do they provide positive encouragement (or, at least, do they refrain from discouraging their child)? [This IS a fairly important issue when the parents themselves are illiterate/unschooled].
3) Should the 'teachers' BE teachers at all for the specific age-levels they are teaching? [It was found, in India, that a great many teachers had no business to be teaching at all!].
4) Is the school administration adequate in all respects to ensure an effective learning system? (It has been found, in India, that a great many schools are just profiteering ventures for some entrepreneurs or the other. Fair enough - but we do need to ENSURE that the 'learning' is the fundamental part of the 'system' - NOT the profits!)
5) Are our governments allocating sufficient resources (money as well as qualified people) to the educational system? (In my view, the answer is currently "NO!" - but that is a matter for us Indians to take up with our representatives of the government of the day. As they learned in the case of the free mid-day meal program, they will surely learn this also).
6) There are a great many other such issues to be considered before that "curriculum" becomes a true 'system', capable of enabling 'learning'
GSC ("Still Shoveling Away!" - with apologies if due to Barry Garelick for any tedium caused; and with the humble suggestion that the obvious way to avoid such tedium is simply to refrain from opening any message purported to originate from GSC)