Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
And so, even though I only sub at middle and high schools, I still come in contact with elementary math standards. The above standard is one that traditionalist Common Core opponents dread. As I've already stated on the blog, this is one of those standards that mentions "strategies" and "algorithms" other than the standard algorithm. Once again, this is a third grade standard, and the corresponding standard that requires the traditionalist algorithm appears a year later:
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
My proposed standards placed the traditionalist algorithm one year earlier, in third grade, and dropped the other standard about "strategies" altogether. But the fact that I found this in a special ed classroom brings up another issue: My proposal endorses traditionalism up to the third grade, and then a mixture of traditionalist and progressive methods for fourth grade and above -- the thought that preteens and teens don't respond well to direct instruction from a "sage on the stage."
Now this is a third grade standard, so it's intended for students young enough for traditionalism, but it's being taught to teenagers in a special ed class, students too old for pure traditionalism. So which philosophy rules, the intended age of the audience or the actual age of the audience?
My rule of thumb is, when students stop learning a subject just because a teacher -- "the sage on the stage" -- tells them to, and starts questioning what they are learning, as in "Why are we learning this?" and "When will we ever need this?" -- that is when pure traditionalism should end. I suspect that this aligns more with chronological age than with the level of material. This means that a young student who is well above grade level can learn more advanced material via direct instruction than my proposals recommend, only because they can reach such advanced material before reaching the age of questioning authority. Conversely, students who are well below grade level are still learning basic material well past the age of questioning authority. They may benefit from extra "strategies" that are suggested in the Common Core Standards.
This appears to be consistent with my observation. Clearly this group of students that I saw this week would not learn just because I told them to. I had to cover a science lesson where I played for the students a video about the 4.5-billion year history of the earth. It was a simple assignment where all the students had to do was take notes, but it was difficult to get the students to show any interest in the material at all, answering the questions I asked them with silence. This is consistent with their chronological age and shows the problems with purely direct instruction at that age.
On the other hand, I read anecdotes from traditionalist homeschooling parents about how their second grader learned fifth grade math effectively via direct instruction. I don't consider this to be a valid argument that the fifth grade math standards should be written from a purely traditionalist view -- because fifth graders, more than second graders, will start questioning why they have to learn how to compute with fractions.
But this does mean that the traditionalists' favored standard algorithms and memorization of basic math facts are to be taught as soon as possible, and not delayed a year as in Common Core. One common complaint among traditionalists is that students are never made to memorize basic multiplication facts. Questions such as six times nine or seven times eight should be considered very easy questions that take no more than a second to answer. But not only do many people consider such problems to be difficult, but it has become fashionable to consider those who have difficulties with such problems to be normal and those who find such problems easy to be outliers -- nerds.
It's often pointed out that people would feel deeply ashamed to admit that that can't read at a third grade level, yet are proud to admit that they can't do third grade math. Since I've stated that third grade math is something that students should have learned traditionally -- that is, have memorized -- I should do something about it in my classes.
The thought is that, rather than have those who find single-digit multiplication to be easy be outliers who get the label nerd, it's those who can't multiply by the time they reach middle and high school who should be considered outliers -- just as someone who can't read at a third grade level is taken to be an outlier. But of course, it's improper for me, a teacher, to start calling my students derogatory names such as idiot, no matter how low their understanding of math is.
So I need a word that criticizes the student, yet is proper for me to use in a classroom. Well, since I want my word to have the opposite effect of the word nerd, I briefly mentioned at the end of one of my posts a few months back that I made up my own word, by spelling the word nerd backwards, to obtain "dren."
My plan is to use my new word "dren" in such a way to make it sound as if a "dren" is not what a student wants to be. For example, when we reach the unit on area, students will need to multiply the length and width to find the area of a rectangle. So I might say something like, "A dren will have trouble multiplying six inches by nine inches. Luckily you guys are too smart to be drens, so you already know that the area is ...," and so on. Similarly, if a student, say, starts to reach for a calculator to perform the single-digit multiplication. I can say, "You're not a dren. You know how to multiply six times nine ...," and so on.
Notice that in these examples, I don't call anyone a dren directly. But every time I say the word "dren," I want to be annoying enough so that the students will want to do what it takes to avoid my having to say that word.
I coined the word "dren" to be the word nerd spelled backwards. But ironically -- according to my new Simpsons book -- the word nerd is already spelled backwards! Originally, the word was "knurd," which is drunk spelled backwards. The net result is that my word "dren" is basically just an abbreviation of drunk. Of course, the word drunk isn't a word that I should use in the classroom!
Here are the answers to today's test.
1. Using the distance formula, two of the sides have the same length, namely sqrt(170). This is how we write the square root of 170 in ASCII. To the nearest hundredth, it is 13.04.
2. The slopes of the four sides are opposite reciprocals, 2 and -1/2. Yes, I included this question as it is specifically mentioned in the Common Core Standards!
3. Using the distance formula, all four sides have length sqrt(a^2 + b^2).
4. Using the distance formula, two of the medians have length sqrt(9a^2 + b^2).
6. From the Midpoint Connector Theorem,
7. From the Midpoint Connector Theorem,
9. (0.6, -0.6). Notice that four of the coordinates add up to zero, so only (3, -3) matters.
10. At its midpoint.
11. 49.5 cm. The new meter stick goes from 2 to 97 cm and we want the midpoint.
12. Using the distance formula, it is sqrt(4.5), or 2.12 km to the nearest hundredth.
13. sqrt(10), or 3.16 to the nearest hundredth.
14. 1 + sqrt(113) + sqrt (130), or 23.03 to the nearest hundredth.
15. sqrt(3925), or 62.65 to the nearest hundredth. (I said length, not slope!)
16. -1/2. (I said slope, not length!)
17. (2a, 2b), (-2a, 2b), (-2a, -2b), (2a, -2b). Hint: look at Question 5 from U of Chicago!
18. (0, 5).