Thus the only way I'd sub today is in my old district. Calls from this district are rare -- and I'd just subbed there last Friday, so it was unlikely that I'd be called to sub today. And I wasn't.
Today is the review for tomorrow's Chapter 9 Test. In many ways this is a light chapter. While the modern Third Edition includes surface area in Chapter 9 (Lessons 9-9 and 9-10), my old Second Edition stops after Lesson 9-8. Then again, students who have trouble visualizing three dimensions will struggle on tomorrow's test.
Well, at least our students won't have any Euclid on their test. Let's return to his next proposition:
Yesterday we construct the perpendicular from a point not on the plane, and now we construct the perpendicular from a point on the plane. This construction uses yesterday's as a subroutine.
This construction is even sillier than last Friday's to perform inside a classroom. This time, we have a point A on the floor and we wish to find a point directly above it. First, we label point B on the ceiling and use yesterday's construction to find a point C directly below it. Then we construct the line through A that is parallel to
And so there's no way that our students can physically perform this construction. There will be nothing like this on tomorrow's test, even though ironically, it would be easier to answer test questions about this construction than physically perform it.
Notice that the U of Chicago text doesn't actually provide the construction for drawing a parallel to a line through a point not on the line (which is a simple plane construction). The only way implied in the text to perform this construction is to make two perpendicular constructions (which we did in yesterday's post).
Many texts that teach the construction of parallel lines use copying an angle (as in corresponding or alternate interior angles). Lesson 7-10 of the Third Edition is on constructions, and duplication of an angle is given, but still no parallel lines. (I also see some DGS "constructions" mentioned there -- I wonder whether this is similar to Euclid the Game, as alluded to in last Friday's post.)
If you must, here is a modernization of the proof of Proposition 12:
Given: the segments and angles in the above construction.
Prove:
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2.
The proof is trivial since both
On Friday, I mentioned that Barry Garelick had posted. And even though my next traditionalists' post isn't scheduled until tomorrow, that might change depending on the comments at Garelick's blog. My prediction was that some major traditionalists (SteveH and at least one other) would comment during the weekend on Friday's blog entry.
Well, my prediction turned out to be only half-right. There were some comments on Friday's entry, but none of them were from SteveH. And in fact, the main commenter there, Ray, is actually trying to challenge Garelick's original post. But in addition, Garelick blogged again yesterday -- and that's the post where SteveH and the other traditionalists are leaving comments!
And so I will, in fact, write about traditionalists both today and tomorrow. Today I'll discuss the debate between Garelick and Ray, and tomorrow we'll look at Garelick's second post where he's just preaching to the choir.
Let's start by linking to Garelick's Friday post:
https://traditionalmath.wordpress.com/2019/01/25/worked-examples-and-scaffolding-dept/
In teaching procedures for solving both word problems and numeric-only problems, an effective practice is one in which students imitate the techniques illustrated in a worked example. (Sweller, 2006). Subsequent problems given in class or in homework assignments progress to variants of the original problem that require them to stretch beyond the temporary support provided by the initial worked example; i.e., by “scaffolding”. Scaffolding is a process in which students are given problems that become increasingly more challenging, and for which temporary supports are removed. In so doing, students gain proficiency at one level of problem-solving which serves to both build confidence and prepare them for a subsequent leap in difficulty. For example, an initial worked example may be “John has 13 marbles and gives away 8. How many does he have left?” The process is simple subtraction. A variant of the original problem may be: “John has 13 marbles. He lost 3 but a friend gave him 4 new ones. How many marbles does he now have?” Subsequent variants may include problems like “John has 14 marbles and Tom has 5. After John gives 3 of his marbles to Tom, how many do each of them now have?”
I'll now skip to the part of Garelick's post to which Ray is about to respond:
Once the foundational skills of addition and subtraction are in place, alternative strategies such as those suggested in Common Core in the earlier grades may now be introduced. One such strategy is known as “making tens” which involves breaking up a sum such as 8 + 6 into smaller sums to “make tens” within it. For example 8 + 6 may be expressed as 8 +2 + 4. To do this, students need to know what numbers may be added to others to make ten. In the above example, they must know that 8 and 2 make ten. The two in this case is obtained by taking (i.e., subtracting) two from the six. Thus 8 + 2 + 4 becomes 10 + 4, creating a short-cut that may be useful to some students. It also reinforces conceptual understandings of how subtraction and addition work.
Now let me introduce our dissenter, who goes only by the name of "Ray":
Ray:
I suspect that you are one of those people for whom rote memorization comes easily. It is difficult for you to believe that a child could have trouble with just memorizing their math facts. Trust me, a child can spend hours with flash cards trying to rote memorize their facts and still be unable to tell if 7 + 5 equals 12 or 13. The only reason for teaching about making tens is to help children with the more difficult math facts. Both Singapore and Shanghai maths use the making tens method beginning in first grade to help children learn their hard math facts. The idea that you would wait until a child had already learned their math facts by rote to introduce making tens makes no sense. If a student can recall all of the math facts by rote, and some children are very good at this, then there is no point in making tens. I work with students who have struggled for years trying to rote memorize their math facts. When I show them how to make tens, their first reaction is always, “Why didn’t my teacher tell me about this to begin with?” There is a reason why Singapore and Shanghai teach math this way.
And I agree with Ray wholeheartedly. I think back to our discussion of different number bases and how to learn the tables in these bases. A dozenalist, for example, would try to make dozens of course rather than tens. In both bases, our goal is ultimately to learn all the facts, but some facts are easier to learn than others. When we introduce making tens (dozens), we're assuming that the student is at a stage where some of the facts -- the tens (dozens) -- have already been learned. Then that student can use those facts to learn the ones he hasn't learned yet. So in Ray's example, 7 + 5 must be two more than 5 + 5 = 10, hence 7 + 5 = 12. Here's a similar example in dozenal -- 9 + 6 must be three more than 6 + 6 = 10, hence 9 + 6 = 13.
I like how Ray mentions Singapore Math in his comment. We already know that the traditionalists love Singapore Math and find it superior to American texts, and now Ray is using the traditionalists' beloved text against him. I'm glad that someone is finally standing up to the traditionalists as opposed to the usual pattern -- Garelick posts, and all his commenters fall in line. (Disclaimer: I am not Ray.)
Let's see how Garelick responds to Ray:
Singapore does both; making tens is offered as a method, but doesn’t require students to do it for weeks on end. Yes, there are strategies for memorizing the math facts, but the idea that straight memorization without the strategy is eclipsing “understanding” is how people are framing the practice. As I say in the piece, “Teachers should therefore differentiate instruction with care so that those students who are able to use these strategies can do so, but not burden those who have not yet achieved proficiency with the fundamental procedures.” Insisting that all students use these procedures can hinder some–and Singapore does not insist upon it.
And no, memorization did not come easy for me. But we had drills/quizzes just about every other day when I was in 2nd grade (which was when arithmetic started to be taught when I was in school), and by the end of the first semester, I had all the facts down.
Actually, let me address Garelick's second paragraph first. His remark that he didn't even begin arithmetic until the second grade reminds me of another country that neither he nor Ray has mentioned yet -- Finland. It's often brought up that in Finland, formal education doesn't start until around age 7. We see that likewise, Garelick didn't start formal arithmetic until age 7.The trend in recent years has been to start formal education earlier and earlier. For starters, first grade becomes a year of formal education, and to prepare for first grade, kindergarten is a year where school lasts for a half-day and is mostly play. Then K becomes a year of formal education, and to prepare for K, Head Start is a year where school lasts for a half-day and is mostly play. And now I've even heard of some schools trying to extend Head Start to a full day!
(Notice that the California concept of "Transitional Kindergarten" is part of this trend. Back when K was more play-based, starting K at 4 years 9 months was OK. But now with a more rigorous K, it was decided that students should start it at 5 years, and so TK was introduced.)
I assume what Garelick is saying here is that if we use traditional math, then we can delay its start until second grade without the students falling behind. Even though he doesn't mention Algebra I in this post, the implication he makes here is that student can begin arithmetic in second grade yet still be ready for Algebra I six years later, in eighth grade, if we stick to traditional math.
Garelick and Ray also discuss where students are forced to learn the nontraditional methods:
Ray:
Singapore math has children making tens for years. It starts in first grade and continues through the third grade. Both Singapore and Shanghai are known for their insistence on mastery learning. A child cannot pass the end of year tests without the ability to use what they call “number bonds”, a fluency in decomposing numbers that includes making tens but extends much further. This is not a frill. It is essential to the curriculum.
But Garelick counters that Singapore doesn't actually force the students to make tens:
In the Primary Math series, which is what I was going by, the making tens method is indeed introduced in first grade, but the problems that follow do not all require it. I, too, had making tens as my post indicates. It was introduced in 3rd grade in my case. My point is that the foundational mastery can support the alternatives and many times students discover these methods themselves. The point is that Singapore allows both standard algorithms/procedures and the alternate strategies. While they may test proficiency with both, they do not delay the teaching of the standard methods–both are practiced, whereas in the current interpretation of Common Core, there is a belief that students must show “understanding” first before being allowed to do the procedure.
I believe the main idea Garelick makes here is that while Singapore asks the students to make tens or use number bonds, it avoids the Common Core "nightmare." This is where a student answers a problem correctly using a traditional method instead of doing it "the Common Core way" and gets marked wrong -- with the student's preferred method not being taught for another year or so. The idea is that Singapore allows choice, while Common Core uses force.
Actually, Garelick argues that Common Core doesn't really force anyone to use any nontraditional method either:
This goes beyond just the making tens method, but applies to multidigit addition and subtraction. As a result, the standard algorithms for multidigit add/subtract and multiplication are delayed until 4th and 5th grades, even though Common Core does not prohibit the teaching of the standard algorithms earlier, as verified by both Jason Zimba and Bill McCallum, co-writers of the Common Core math standards. See https://edexcellence.net/articles/when-the-standard-algorithm-is-the-only-algorithm-taught
We can see what Garelick would prefer -- lower the standard algorithms by one grade level and then drop all mention of the nontraditional methods. Or better yet, just ask the students to compute multidigit arithmetic in the lower grades (add/subtract in third, multiply in fourth) without specifying an algorithm at all. According to him, it's possible to get from memorizing addition facts in second grade to multidigit addition in third if the standard algorithm is used, but it allows differentiation for those students who are less successful with the traditional methods.
The problem is that there are many levels between Common Core and the classroom teacher. We see that states, districts, schools (via PD meetings, like the one my new district is having today), and texts all interpret the Core differently. And at any level, someone can interpret the Core as requiring the nontraditional methods. And so the teacher is compelled to teach the nontraditional methods, flunk students who use traditional methods, and then blame it all on the Core.
In my last traditionalists' post, I wondered whether the ultimate solution is simply to abolish the Core for the lowest grades. Elementary schools would use district standards, middle schools would use state standards, and high schools would use the Common Core. Then this would eliminate the problem of districts and states misinterpreting the Core. Those districts who believe that their students learn better traditionally can order texts that teach traditionally, and teachers would have the autonomy to differentiate for those students who learn differently.
In the last comment of the thread, Ray agrees with what Garelick is saying. And as I've written before, even I often agree with traditionalists with respect to the lowest grades.
That's it for today's traditionalists' post. But as I warned you, expect more traditionalists tomorrow.
(I actually considered not marking this post "traditionalists" -- after all, today is all about Ray, not the traditionalist Garelick. But I'll keep the label anyway, since it's about the traditionalists' debate.)
Anyway, here is the Review for Chapter 9 Test.
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