Friday, November 2, 2012

Math problems of the week: 5th grade Investigations vs. Singapore Math

I. The first number puzzles in the 5th grade TERC/Investigations Student Activity Book, from the very beginning of the book [click to enlarge]:

II. The first number puzzles in the 5th grade Singapore Math Primary Mathematics 5B Workbook, from the very beginning of the book [click to enlarge]:

III. Extra Credit

Compare the relative challenges of figuring out the directions vs. doing the math in each of the problem sets.

Anonymous said...

Even as a teacher, I really don't understand the point of the first two pages, much less even understand how to do those exercises! The Singapore book is directly understandable and upon completion of the problems gives a sense of accomplishment.

R. Craigen said...

I agree with Anony above. The Inestigations sheets are uninteresting "busy work", not "puzzles". Whoever made them up are seriously unimaginative and don't know what makes math interesting for children. The language in the instructions is so verbose because they are trying to avoid any mathematical jargon. But this begs the question: If you don't learn mathematical jargon in math class ... where DO you learn it? Isn't that the point (partly at least) of math class?

I have one complaint about the Singapore sheet, and it's a minor one. The first two problems are worded so as to make the answer unique, through the number range specified. However, the other two do not have unique answer (eg 28 is also an answer for the last one). This will induce some uncertainty in students as to what is inspected, or whether they have missed something. I'm not saying that's bad, only that it may be an unintended negative consequence, particularly for the very first such sheet in grade 5. As I understand Singapore it works to instil confidence in students; throwing them off balance right off the bat might not help toward this goal. The question "What number am I?" gives no hint that the answer may be non-unique.

R. Craigen said...

I suppose one might also contend that the first Singapore problem has a bit of ambiguity. What does "between" mean? Is it inclusive or exclusive? Perhaps this is made clear elsewhere in the text. But if it is inclusive, then one would have to accept that there are 3 correct answers. Would a student answering "20" receive full marks?

R. Craigen said...

To be fair, I should be more explicit about my problems with the Investigations sheet.

"Draw rectangles?" Uh, unless the student is reading the textbook writers' minds, there is no way they will understand that a particular rectangle is intended here (or one of two), based on the number chosen. If the student happens to pick 94 and 86 they will have a very hard time fitting the answer into the space given -- and of what value will the time taken to do so be to the student?

"Label the dimensions?" Category errors and confusing instructions. What they mean is "label the sides with the dimensions" -- but even then this is a poor use of the word "dimension". Why not say "provide the length and width". But if the students are supposed to do this exercise presumably they have already shown that they know length & width by drawing a grid .. or are they not expected to? See, it's not clear at all what is expected here. I assumed that they were supposed to do so. If they draw "generic" rectangles and label the sides, what if it looks like a square and they label it as 1x17? Is that wrong? (Is the student expected to be an artist?) Also, the instruction to draw the rectangle comes BEFORE the instruction to label it, which comes BEFORE the purpose of the labels. If a student is following these tasks in sequence they are missing critical information for the performance of each step. "Here, I drew a rectangle ... oh, I'm to label the sides ... um, the labels are supposed to show something ... wait a minute -- are these rectangles supposed to be connected to the numbers I chose?" Should the instruction not be to draw a PARTICULAR rectangle based on the number chosen, making clear which one is intended? Or ones, since in some cases there are more than one. I do like that they are asking the child to demonstrate understanding beyond the instances selected, however, the redundancy between #3 and #5 suggest to me that something is wrong in their expectations. If a child has come up with a few answers to #3 they should already be thinking "Prime numbers". If they are supposed to have produced several examples without jumping to the category, this is holding back students from the "aha!" moment that is supposed to be at the heart of such exercises, and killing the main attraction of the problem. The presence of question 5 suggests to me that they may not even want students to make this leap at all. Then #5 will be a tiresome, pointless exercise. If a student writes down "Primes" for the "List other numbers" step in #3 -- as I think they should in spite of the instruction -- then why doesn't #5 just say "Find all the numbers up to 50 that fit the description given in #3"?

R. Craigen (WISE Math co-founder) said...

Further there is a serious problem with the instructions for the sheet. They say "For EACH number puzzle", which means, evidently "For ALL..." However it is clear that the instruction applies ONLY to questions 1-4, not to #5 and #6.

Also, what does "1 Clue" mean here? I stared at that for quite a while, and still do not know. It bugs me. It will bug students who have a hard time reading. They will struggle to understand it, before they even go on to read the rest of the sheet. What is it even doing there? Is this the title? I have 0 clue.

Plurality/singularity mismatch. In the instructions "two numbers" are asked for, and in each case it is made clear that other numbers "fit the clue". So why does the clue say "THIS number..." as if there were only one? Any student with a modicum of logic in their head will stare at that and say "...uh, wait a minute ...?" Yes I get that this is "riddle style" but even then it's wrong. When a riddle says, at the end "... who am I?" the understanding is that a unique answer has been specified. The game is not, "I've made up some catagory. Find SOMETHING that fits." At least, not the way I ever played it when I was a kid.

There's more, but that should be enough to start.

R. Craigen (WISE Math co-founder) said...

Er, no, there's more.

The blue rectangles, which are unexplained, were a mystery to me. Finally it occurred that these contain the "descriptions" in question. But then should this not be stated in the first instruction -- "Find two numbers that fit the description found in the blue box".

Also "EACH clue" in instruction a. is poor language. This is, after all an instruction to be done "For EACH number puzzle". But then, in EACH number puzzle there is only ONE "clue". Why do they say "EACH"? That is confusing! I realise that it is an attempt to be super-clear, but it does the opposite, and it gives the impression of talking to babies. Babies who will end up grammar-challenged at this rate.

Also it is not, strictly, a "clue" that is being "fit". It is a "description".

Anonymous said...

These are practice pages. It is possible they did something similar in class. Maybe with real tiles. And drawing them afterwards. Workbook are supposed to be follow up to lessons in class. It can be impossible to have enough instructions if these were totally out of the blue unrelated to anything students did in class. The Investigations page is looking at factors and it seems like it is based on an activity that was done in class. The answers given for the teacher for the SM page do accommodate more than one possibility for the workbook page (e.g. 28 or a multiple of 28) with the suggestion that teacher asks students to find lowest common multiple (which is covered in the lesson for this page) but it would be better if there were a unique answer.

The concepts being covered in the Investigations was taught in grade 4 Singapore math and such a lesson would probably include a hands-on lesson where they also arrange tiles or objects into rectangles. The textbook for Primary Mathematics 4A does show pictures of items arranged in rectangles when it is introducing factors. The textbook for 5A shows one such picture, but that is just a reminder.
I think the main difference is the level where the concept is introduced, and the fact that in Primary Mathematics students do move on to the abstract rather than staying stuck and regurgitating a hands-on lesson. Also, the Investigations page never mentions the word factor, keeping it all in the realm of juvenile.

Auntie Ann said...

I Singapore the use of "between" is consistently presented as exclusive of the numbers stated. By the time a student has seen this page, they would have already learned to use "between" as intended.