Last night, I tutored my geometry student. His school is one of those that observes a five-day Thanksgiving vacation, from Wednesday to Sunday. So naturally, he was not in a mood to study geometry on his first night of vacation. But still, he hadn't done as well on his last quiz, which covered the basics of proof, and so I need to help show him what he did wrong.
So I showed him some worksheets from the blog. I began with Section 3-3, "Justifying Conclusions," but as it turned out, what helped him was Section 4-4, "The First Theorem in Euclid's Elements." In fact, the proof he needed was very similar to Example 2, which I included on my worksheet. The only difference was that he needed to prove AC = BD rather than AB = CD.
Given: B is the midpoint of
C is the midpoint of
Prove: AC = BD
Now naturally, my student would've preferred that I just give him a fish and tell him the missing steps in his proof. But instead, I taught him to be a fisherman and worked on finding the missing steps of the proof together. His teacher was expecting a paragraph proof -- not the two-column proof that the U of Chicago text and I provided. Here is a complete answer, by his teacher's standards:
Proof:
We are given that B is the midpoint of
As it turns out, my student had forgotten the "by the definition of congruent" and the "substituting" steps of the proof. And so I taught my student to become a fisherman, and fish out the steps that often come up in geometry proofs. This is the context in which I want to discuss the Common Core Standards. Are the math standards for grades 4-7 the equivalent of teaching men to fish or just giving them fish?
Before we begin, you may be wondering, why did I choose grades 4-7 as this grade band? Well, back when I was in the third grade, our school had a policy for which students were allowed to play on the jungle gym, or "big toy," during afternoon recess for grades 1-4. Each grade was assigned a different day of the week on which they could play on the big toy -- first graders on Monday, second graders on Tuesday, third graders on Thursday, and fourth graders on Friday. (Wednesdays were common planning days -- students went home early and there was no afternoon recess.) But a strange thing happened -- there were many kids using the big toy on Mondays, Tuesdays, and Thursdays, but on Fridays it was almost empty. This was because fourth graders were "big kids" -- too mature to want to play on something as juvenile as a big toy. And so the teacher who was assigned to watch afternoon recess on Fridays allowed us third graders to have a second day on the big toy!
The academic work became tougher in the fourth grade as well. Up until then, our homework mainly consisted of spelling words, but in the fourth grade -- here in California, the standards require students to learn about our state's history that year -- we had our first extended project. We had to visit a mission set up by Spanish explorers and write a report about it. The report card shows that my grades dropped slightly that year -- I'd earned straight A's in second and third grades, but received a few B's during the fourth grade. (Of course, my math grade was still an A.)
Last week, I subbed at a middle school -- but I very nearly subbed at an elementary school. As it turned out, the only teachers at the middle school who needed a sub were coaches who were only out the last two periods of the day, so they wanted to have me go to an elementary school -- most likely the one that's right next door to the middle school. I observed the school's bell schedule, and noticed that in this district, fourth and fifth graders have a longer school day -- nearly half an hour more than students in grades K-3. And furthermore, there are even a few schools in the district that are only for fourth and fifth graders! So we see that in this district, there is a sharp distinction between the "little kids" of grades K-3 and the "big kids" of grades 4-5. (I ended up turning down the offer, since I'm not accustomed to working in elementary. I'm not used to having to deal with students having to line up for lunch, or supervising them for recess, or dealing with lessons for multiple subjects. Instead, I ended up subbing in a math class for sixth graders -- some of whom were only a few months older than the fifth graders I might have taught instead.)
Now on to the math. What math should students learn in grades 4-7? Well, there's one word that many preteens dread hearing during math class, long before hearing the word "algebra" -- and that word, of course, is "fractions." Dr. Hung-Hsi Wu not only wrote an extensive essay regarding Common Core Geometry. He also wrote about how fractions should be covered under the Core:
http://math.berkeley.edu/~wu/CCSS-Fractions_1.pdf
Why do we make students learn fractions? Well, all the math in grades 4-7 is supposed to culminate in the study of one of the most important sets in mathematics -- the set of rational numbers. This set is so important that mathematics use the bold letter Q to refer to the set of rational numbers. The letter Q stands for the German word Quotient, which means essentially the same thing that it means in English. The rational numbers are the quotients of integers -- that is, they are obtained by dividing the integers. The reason that Q is so important is that it is the simplest ordered field. A field is basically a set in which one can add, subtract, multiply, and divide any two members of the set we want (with one exception -- we can't divide by zero).
There are four fields mentioned in the Common Core Standards. They are:
Q, the field of rational numbers (Grades 6-7)
R, the field of real numbers (Grades 8-Algebra I)
C, the field of complex numbers (Algebra II)
R(X), the field of rational functions (Algebra II)
This last field appears in the following standard:
CCSS.MATH.CONTENT.HSA.APR.D.7
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
(Notice that the phrase "system analogous to the rational numbers" is just another way of saying "field" without using that word.)
But there's more to Q than simply being a field. The set Q is an ordered field -- by "order" we mean greater than and less than. In order to qualify as an ordered field, the field must have < and > inequalities that satisfy the four "Properties of Inequality and Operations" mentioned in Section 1-7 of the U of Chicago text. Clearly both Q and R are ordered fields. As it turns out, one can prove that R(X) is also an ordered field, but C isn't.
Of course, there is one exception to this -- division by zero. If we're going to work hard to make 1 / 2 possible, why not 1 / 0? Well, making 1 / 0 leads to contradictions. Indeed, there is a classic "proof" that 2 = 1:
Given: a = b
Prove: 2 = 1
Proof:
Statements Reasons
1. a = b 1. Given
2. a^2 = ab 2. Multiplication Property of Equality
3. a^2 - b^2 = ab - b^2 3. Subtraction Property of Equality
4. (a+b)(a-b) = b(a-b) 4. Factoring (officially, Distributive Property)
5. a + b = b 5. Division Property of Equality
6. b + b = b 6. Substitution Property of Equality
7. 2b = b 7. Combining like terms (officially, Distributive Property)
8. 2 = 1 8. Division Property of Equality
But the proof is fallacious. In step 5, we divided by zero (disguised as a - b). Because of this contradiction, we instead conclude that division by zero is impossible. Therefore, in a field we may add, subtract, multiply, or divide any two elements, provided that we don't divide by zero.
Now Q is a field, but it's by no means the simplest field. Q is an infinite set, but the simplest field has only two members -- 0 and 1! In order to be a field, we must be able to add, subtract, multiply, and divide any two members (except division by zero), and here's how we do it in the field GF(2):
Addition:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0
Subtraction:
0 - 0 = 0
0 - 1 = 1
1 - 0 = 1
1 - 1 = 0
Multiplication:
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1
Division:
0 / 1 = 0
1 / 1 = 1
Now, I bet that those preteens who hate fractions would love it if math classes would teach the field GF(2) instead of Q. Compared to Q, GF(2) is so easy. Everyone would get straight A's in math if this were taught instead of fractions! But that would be giving the students fish. We want them to learn how to be fishermen instead. The field GF(2) doesn't describe the real world. In real life, one fish plus one fish is more than one fish. We want 1 + 1 > 1, not 1 + 1 = 0. We want an ordered field, where addition works the way we expect it to. In short, we need Q, not GF(2).
When people complain about the Common Core Standards in grades 4-7, the two most common complaints are of the form:
- Students are not receiving credit for correct answers.
- Students are receiving credit for incorrect answers.
The issue here is on questions that require a student to explain how he or she obtained an answer. I notice what ends up happening is that a student gets an answer perfectly correct, but can't give an adequate explanation to the answer. If the student is in grades 4-7, what ends up happening is that the parents remember when the students earned high marks in earlier grades, then observe a significant drop -- sometimes as much as two or three letter grades -- after the transition to Common Core. They often say that their children loved math before Common Core, and now they hate it. On the other hand, these parents fear that other students can give incorrect answers to the same questions, yet come up with explanations so ingenious that they can receive credit.
What should happen when a student gets a question wrong? To many traditionalists, that student should be told "You're wrong!" as quickly and emphatically as possible. To say anything else is to give the student a fish. Students who aren't told that they are wrong -- the traditionalists fear -- will have misconceptions for years and will never become fishermen.
But there is an underlying assumption made by the traditionalists here. They assume that the students will meekly accept that they are wrong and learn the correct answer. But human nature tells us otherwise. In his work How to Win Friends & Influence People, nearly 80 years ago, Dale Carnegie writes that most people are resistant to being told that they are wrong. Rather than correcting themselves, they will become defensive. This is why one of Carnegie's principles is, "Show respect for the other person's opinions. Never say, 'You're wrong.'"
But now I can hear you saying that this is mathematics, not opinion. Any student who tries to claim that 2 + 2 = 5 doesn't have an opinion that any teacher is bound to respect, because that statement flies in the face of the fact that 2 + 2 = 4. I admit that I was as doubtful as you when I first read Carnegie's book. Yet still, simply telling the student that he or she is wrong and that 2 + 2 is really 4 won't result in the student magically accepting it. A student is more likely to say, "No, you're wrong, 2 + 2 = 5," than "You're right. 2 + 2 is really 4."
One Carnegie principle that can be especially helpful in a math class is, "Get the other person saying 'yes, yes' immediately." Here's an example of "yes, yes" in a math class:
Student: 2 + 2 = 5.
Teacher: Well, if you have 2 apples and add one more, you have 3 apples, right?
S: Yes.
T: And if you add one more to that, you have 4 apples, right?
S: Yes.
T: So now that you added 2 apples to 2 apples, how many apples do you have?
S: 4.
According to Carnegie, by getting the student to say "yes, yes," the teacher has led the student to the inescapable conclusion that 2 + 2 = 4, and this is more effective than getting into a one-on-one argument, student vs. teacher, with the teacher saying 2 + 2 = 4 and the student saying 2 + 2 = 5.
Even though nearly everyone is subject to the human nature of rejecting "You're wrong," I suspect that preteens -- and teens, of course -- are especially resistant to "You're wrong." It is because of this that while I support traditionalism in the lower grades, I cannot be a full traditionalist for fourth grade and above -- instead I support a mixture of traditionalist and progressive methods.
Sarah Hagan, a high school teacher from Oklahoma, discusses how her Algebra I students have trouble with overgeneralizing -- which is one of the most common errors made at this level. The students tell her that since two negatives make a positive, the sum of -3 and -5 must be +8. And -- despite what the traditionalists believe -- Hagan's telling the students that they are wrong and that the sum is -8 didn't lead to the students correcting themselves:
Because as soon as I start reteaching something that they have heard before, their minds shut down and start ignoring me. I guess they are thinking, "I don't have to listen. I already know this!" But, the problem is that they don't know this. They think that a negative exponent means that you need to change the fraction to its reciprocal to make the exponents positive. In some cases, this works. But, they are overgeneralizing. They've been told that two negatives make a positive. So, -3 + (-5) must be +8. Again, they've taken a rule for multiplication and division and overgeneralized it. And, don't even get me started on the order of operations. No matter how many times I say that multiplication and division must be performed from left to right, I have a student who will argue with me that multiplication comes before division in PEMDAS so we must always do it first.
I repeat this for emphasis -- upon being told that they were wrong, Hagan's students either ignored or argued with their teacher. As Carnegie warned, the students certainly didn't correct themselves. I would point out that one easy way to get the students to realize that they are wrong without raising their defenses would be to have them add -3 to -5 on a calculator. As soon as they saw that -8 smiling back at them on their calculator screen, they'd know that they were wrong. But many traditionalists are also opposed to calculator use at any time prior to pre-calculus.
Now I want to write some math standards for grades 4-7, but what should they accomplish? Notice that the traditionalists often fear that their children's grades will drop significantly with the new Common Core Standards, but notice that they aren't actually opposed to huge grade drops. They just want the grades to drop for the right reasons. Students who can give all the correct answers, yet can't give the correct explanation that the teachers are looking for, don't deserve grade drops. But students who've been skating along by giving clever explanations and excuses without knowing the correct answers can and should have their grades drop fast.
But I say that it's neither those who know all the right answers, nor those who can give the best explanations, but those who can apply math to real-world situation, who deserve the rewards. So a student who can multiply 58 times 46 perfectly using the standard algorithm, but can't tell me the area of their backyard that measures 58 feet by 46 feet, deserves a grade drop. I'd much prefer the student who can tell me the area, but uses a nonstandard algorithm (such as the much-maligned "lattice method") to perform the multiplication. Explanations matter only to the extent that someone who explains why they multiplied 58 and 46 to find the area is more likely to know how to find area for years to come than someone who can't explain why. There's zero value in getting right answers if they answer the wrong question.
So what this all boils down to is, are the assessments -- that is, PARCC, SBAC, and worksheets created in the name of Common Core -- authentic enough that only those who can apply the math that they learned get the highest scores? I don't know -- but I suspect not, only because truly authentic assessments are extremely difficult to write. I want the standards that I propose here to lend themselves to such authentic assessment, but I will almost certainly fail.
So instead, I look online and read about the texts that homeschoolers -- especially parents who say that they took their students out of public schools specifically to avoid Common Core -- use to teach their preteens math. One text that's often mentioned is published by John Saxon. And I was able to purchase a Saxon text -- specifically, Saxon Math 65 (Second Edition) -- for $1, from the same library where I found the U of Chicago text.
There are 140 lessons in the Saxon text. Presumably, one lesson is to be given per day. There are 180 days in the school year, and we can assume that 40 days are given up to first/last day of school bureaucratic issues, assemblies, or testing (both teacher directed and state/Common Core). Here are a few randomly chosen lessons mentioned in the Saxon text:
Lesson 49 - Reading an Inch Ruler to the Nearest Fourth of an Inch
Lesson 56 - Making Equal Groups to Find an Average
Lesson 57 - Multiplying by Two-Digit Numbers
Lesson 65 - Dividing and Writing Quotients with Fractions
Lesson 75 - Writing Hundredths in Decimal Form
Lesson 82 - Adding and Subtracting Decimal Numbers, Part 1
Lesson 90 - Identifying Prime Numbers
Lesson 121 - Multiplying Decimal Numbers by 10, 100, and 1000
Lesson 126 - Probability and Chance
Lesson 131 - Dividing Decimal Numbers: Keep Dividing
We see a mixture of traditionalist and progressive lessons in this random cross-section. Lesson 57 is traditionalist as it focuses on the standard algorithm for multiplication, and Lessons 65 and 131 show long division with fractions and decimals. But other lessons are more progressive and focus on the practical uses of math. The title of Lesson 49 makes its real-life use explicit, and Lesson 126 focuses on probability, which has many real life uses (weather reports, games of chance, etc.). Lesson 56 sounds very progressive -- a traditionalist would just add up all the values and divide by the number of values to find the mean -- until we see that the making equal groups is just there to motivate the definition of mean.
Therefore, I recommend the Saxon text for teaching math to preteens. Notice that the title Math 65 indicates two grade levels -- the text is appropriate for below average sixth graders and above average fifth graders. At the school-age level, the Saxon books are numbered with a single grade, but at the preteen level they are numbered with two grades. At the teen-age level, Saxon originally based his texts on an integrated sequence (despite naming his texts Algebra 1, Algebra 2, Advanced Math), but in the Fourth Edition, the texts follow a traditional sequence, with Geometry thrown in.
But recommending a text isn't the same as giving standards. It's difficult to convert the Saxon lessons into standards because the text jumps around topics so often. (Indeed, for all the praise the Saxon texts receive, this is a frequent criticism of them.) Notice that Lessons 56 and 57 above aren't directly related at all. This is more striking when we see Lessons 55-58 in sequence:
Lesson 55 - Solving Two-Step Word Problems
Lesson 56 - Making Equal Groups to Find an Average
Lesson 57 - Multiplying by Two-Digit Numbers
Lesson 58 - Identifying Place Value Through Hundred-Millions
although 54 ("Finding Information to Solve Problems") is clearly related to 55, just as 59 ("Naming Numbers Through Hundred Millions") obviously follows after 58.
Then again, it's instructive to see how the Saxon lessons correlate with Common Core. Lesson 49 is actually a third grade standard in Common Core:
CCSS.MATH.CONTENT.3.MD.B.4
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
Lesson 57 is on multiplying two-digit numbers. The standard algorithm for multiplying is given as a fifth grade standard:
CCSS.MATH.CONTENT.5.NBT.B.5
Fluently multiply multi-digit whole numbers using the standard algorithm.
Fluently multiply multi-digit whole numbers using the standard algorithm.
Lesson 121 is on multiplying decimals by 10, 100, or 1000. Many teachers give horror stories about students who are asked to multiply decimals by a power of ten and say "I don't know!" instead of "That's so easy -- just move the decimal point!" Indeed, I'd follow that up with, "I wish all decimal multiplication problems were by easy numbers like 10, 100, and 1000 instead of hard numbers like 7, 58, or 3.14.") Many traditionalists lament the Core's emphasis on ten and its multiples and powers, such as asking first graders to "make ten" when adding 8 + 6. What do I mean by "make ten"? It's best just to look at the relevant standard directly:
CCSS.MATH.CONTENT.1.OA.C.6
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
I tend to agree with the traditionalists here (as I usually do regarding the first grade). But multiplying decimals is definitely a proper use of ten and its powers. It's a fifth grade standard:
CCSS.MATH.CONTENT.5.NBT.A.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
As we expect, most of the lessons in Math 65 correspond to fifth and sixth grade standards. A few are introduced slightly earlier in the Core. But probability, Lesson 126, is a seventh grade standard:
CCSS.MATH.CONTENT.7.SP.C.5
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Finally, notice that assigning two grade levels to this content can lead directly into tracking -- Math 65 is intended for sixth graders on a lower track and fifth graders on a higher track. I'd discussed tracking in a previous post and would like to continue that discussion, but not in this post since it's already bloated longer than I want it to be. The issue with tracking is, can one truly be fair to the students on the lower track?
The answer is, go back to the proverb at the beginning of this post. If one is denying fish to those on the lower track in order to teach them to become fishermen, then I support it, as long as one is genuinely teaching them to become fishermen. But if one is merely saying that he is teaching those on the lower track to be fishermen only as an excuse to deny them fish and let them starve, then I am opposed to it. And I believe that both of the above are occurring in various tracked schools.
Happy Thanksgiving, everyone! My next post should be on my new computer -- hopefully it will be on December 1st, but it depends on when the Internet is connect to the new PC. Once again, I'll begin Day 70 on December 1st with Section 7-1 of the U of Chicago text, but if I must start late I'll cover Section 7-2 no matter what.
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