Saturday, July 9, 2016

Rule #2: Respect Your Honesty

Last week, I wrote that the first rule in my classroom is to "Respect your grade." I devoted this post to a number of ideas to get the students to strive for an A in every class, and how I would give copious rewards to the "heroes" who earn the top grades in my classes.

But by placing such an emphasis on high grades, I'm tempting the students to cheat. If Rule #1 is all about earning good grades, then Rule #2 needs to address cheating somehow. Just as in my last post, I will draw from the ideas of the traditionalists -- at least when I agree with what they're saying.

1. Have I Ever Cheated Before?
2. The Traditionalists on Cheating
3. How to Avoid Thinking
4. Other Comments From Traditionalists
5. Avoid Trying to Get an A Without Learning Anything
6. A Warm-Up Equation
7. What It Means to Show Work
8. Foldable Notes
9. Grading Group Work
10. Rule #2: Respect your honesty.

Have I Ever Cheated Before?

I'd love to be able to say that I've never cheated before in my life. But I am not perfect. My eyes have wandered to other students' work before, especially around the time I got that C+ in my Graduate Analysis class. But I always felt guilty whenever I did that -- and so I always wrote only some of what I saw on the other paper, never all of it. (I even remember once when I wrote down everything I saw on a separate sheet of paper, then copied only some of it onto the paper I turned in!) Because of this, I truly earned all of the A (or A-) grades that I ever received in any math class. The only grades that could have resulted by my eye-wandering were B- grades (and probably that C+ as well), since I only copied a little bit of what I saw.

But that's the thing -- since I succumbed to the temptation to cheat in Graduate Analysis, how much more will my students be tempted to cheat in my middle school classes? And this particularly applies to the students who find middle school math as challenging as I found graduate school math.

The Traditionalists on Cheating

Two years ago, there was an article about an app that solves math problems. (I don't link to it here, since students might see this and try to use it to cheat.) Now Bill (the traditionalist whose writing I plan on quoting the most) replied:

I’d agree, but many students go through school unable to actually do math, and they fail when they get to college and find out that the calculator can’t do it all. Perhaps teachers should give the grade to the calculator or app?

This is the biggest concern of the traditionalists -- too many students are able to get good grades in math class without actually knowing any math.

Now here's a link to a more recent article about this concern. As usual, I will link first to the article, and then to the Joanne Jacobs website where most of the traditionalists are responding:

The article is actually about the controversial 0=50 grading system. I will not repeat here what I've written before about 0=50, as that has nothing to do with my current grading system.

I'm surprised that so far, Bill hasn't commented in this thread (but he might soon). Instead, let's look at what some of the other commenters have to say about 0=50:

lee says:
That’s not the worst of it. I teach at the local CC, and a number of the schools here have a “50% minimum” policy. As might be anticipated, when these snowflakes eventually enroll in community college, they expect the same mollycoddling they’ve been accustomed to. They don’t do homework unless points are attached to it, they bomb exams and then pester you for opportunities for extra credit. Of course, you are the big, bad meanie who stands in the way of their future when you refuse to cater to the nonsense.
Enough. I’m looking to make a lateral move to some other profession ~

Here "lee" doesn't write what subject he teaches at the community college. But I must assume that some students do see math teachers as "big, bad meanies" who block their futures -- especially if they need to earn a degree to work at a job where they don't need to know math beyond arithmetic. The students enter the class thinking "How can I get a good grade without knowing any math?" instead of "How can I know lots and lots of math?"

The next poster has a homework policy that could be similar to my plans for my upcoming classes:

I have no problem with not basing grades on HW; tests, quizzes, and projects that indicate mastery of the lack thereof is all that is needed. But these administrators should be careful what they wish for. Many marginal students who diligently do their HW keep themselves afloat that way (i.e., failing but near-passing test scores and excellent HW records).
I still like giving HW, though, as it gives me an idea what the students understood…and what I need to re-teach.

By the way, the Los Angeles charter school mentioned the article is not -- I repeat, not -- the charter school where I'm going to teach in the fall. (The article mentions a high school, but I'll teach at a middle school in the fall.) The student interviewed for the article said that she actually opposes the school's former retake policy, since colleges won't let students retake tests.

Recall that in my classes, the only assessments students can retake are the Dren Quizzes -- and this is only because 90% correct is required to pass them.

How to Avoid Thinking

I recall some traditionalists who said that the real problem is that students don't want to think long and hard about math problems. The avoidance of thinking leads to several problems cited by traditionalists, including:

-- Students enter a number incorrectly into the calculator, obtain a result more than an order of magnitude away from the correct answer, and barely flinch.
-- Students groan when asked to solve a multi-step problem or one that takes more than a minute.
-- (Here's one I've seen myself, rather than mentioned by any traditionalists.) Students cheer when an equation has no solution.

Notice that this last one runs counter to the history of mathematics. Mathematicians hate -- I repeat, hate -- it when an equation has no solution. Indeed, they hate it so much that they invented new numbers for the sole purpose of preventing an equation from having no solution -- beginning with fractions, irrational numbers, negative numbers, and finally imaginary numbers.

But this is part of the problem -- students don't want to think. An equation without a solution means nothing to think about. Equations with solutions means something to think about -- and students especially find those invented numbers mentioned above difficult to think about.

I point out that if someone views math as a barrier, as many students do, then it's reasonable to want as many math problems as possible to have either no solution or a quickly found solution. That way the students can pass the barrier as soon as possible and they can engage in non-math activities that may be more relevant for their futures.

But if math is actually a door, as many traditionalists do, then it's not reasonable to hope that problems have no solution. If someone really wants to save time, then look for faster and better solutions rather than hope that a problem has no solution.

Other Comments from Traditionalists

Before I get to my own Rule #2 to stop cheating in the classroom, let me comment on a few other posts I've seen around the web. Unlike most Jacobs posts, the following links to a video rather than an article:

This video discusses the Next Generation Science Standards (i.e. "Common Core Science"). As I will be teaching in an integrated STEM program, the science standards will be relevant.

The poster Dennis Ashendorf comments on the Next Gen Science Standards:

Dennis Ashendorf says:

Frankly, the curriculum people face unfortunate choices in high school:
1. NGSS is a four-year program. California has a two-year requirement. UC wants three lab years
2. Making NGSS compatible with AP science takes consideration. NGSS intentionally ignored AP in its design.

This reminds us of the traditionalists' complaints regarding Common Core Math and AP Calculus. So just as Common Core Math doesn't fit with AP Calculus, NGSS doesn't fit with AP science.

Let's skip down to the end of the Ashendorf's comment:

1. Earth Science, a course that provides no benefit to getting into college, becomes a senior elective. This decision places pressures on Earth Science teachers to get more official credentials, but they and we have known this for three years.
2. The easiest standard path that is most compatible with AP would be either:
(a) 9th-Chemistry, 10th-Biology, 11th-Physics or (b) 9th-Physics, 10th-Biology, 11th Chemistry
The “best” sequence (see Leon Lederman) that sadly conflicts with the two-year CA requirement would be: 9th-Physics, 10th-Chemistry, 11th-Biology
3. Simplest solution would be Integrated Physics/Bio/Chem for 9, 10, 11, where Biology needs completion by the end of the second year.
Students could opt out after two years and take any AP Science course or stop taking science, and three years completes the UC requirements, which is useful for CSU schools!
Notice that what Ashendorf calls the "best" sequence is often known as Physics First. I've mentioned Physics First a few times earlier on the blog.

Another poster in this thread is Mike:

Mike says:

Ocean waves ARE very similar to radio waves. And the sine waves generated by spinning the radius in a unit circle demonstrates how we can model them mathematically. Teaching wave phases teaches how waves can add and cancel, something that can be seen and demonstrated in a tank or a resonance tube.
I’m a big fan of hands on science and hands on math. A lot of phenomena ARE related within and across disciplines. Similar principles, different expressions of energy. Hands on science also teaches students they can reason from doing. It teaches them they can teach themselves.
The Illinois State text from which I'll be teaching takes this "hands on math" approach. And so I lean more towards Mike and away from the traditionalists in this regard.

Last week I mentioned most of my usual traditionalists. But in that post, I didn't mention Katharine Beals, who was inactive that week. I don't want Beals to feel left out of my traditionalist party, so here's a link to one of her posts from this week:

Many of us education bloggers have complained about people who advocate policies for other people's children that they would never inflict on their own. My own litany includes these types:

I've addressed this issue in the past when discussing "Presidential Consistency," but here Beals is referring to various education experts. Let's just skip down to the types that are the most relevant to what I will be teaching in the fall:

5. Parents who say charters and vouchers are destroying public education, but opt their own kids out of public education.

Well, technically I will be teaching at a charter school in the fall so this is relevant, but the list item I wish to discuss is:

7. People (typically education experts or education software developers) who would like to see the latest education fads--heterogenous group work, child-centered discovery, Everyday/Investigations Math, online, project-based learning--applied to children in general, but send their kids to more traditional schools that evade these fads.
[emphasis mine -- dw]

And here's the problem -- the Illinois State text from which I'm teaching is project-based learning. Of course we already know that Beals, like most traditionalists, dislikes project-based learning. But here she's mentioning PBL as an example of inconsistency -- those who advocate PBL know that it's "inferior" to traditionalism, so much that they prefer traditionalism to PBL for their own children.

Here's my response -- what about those kids who are actually sitting in those traditional schools, especially the middle and high schools, particularly in the math classes? Do those students actually enjoy their traditional Algebra classes? Or do they hope that the equations all have no solution so they can be done with the homework quickly -- deep down, do they wish they were in the PBL and other nontraditional classes for which their parents advocate?

The latter are the students I'm preparing myself to teach in my classroom. My goal is to get them to learn something rather than try to get a good grade without learning anything. So I need to have a rule that goes something like:

Avoid trying to get an A without learning anything.

And so here I will discuss my plans for how I will encourage students to learn actual math, rather than attempt to get a good grade without the math.

As is common in many math classes today, I'll begin each day with a warm-up. This will consist of a single question whose answer is the date. If this sounds familiar, it should -- the idea comes directly from Theoni Pappas and her Mathematical Calendar 2016.

At this point, you may be asking, doesn't this make it easier to cheat, not harder? After all, if the answer is just the date, any student who knows the date can just write down the answer without doing any math. And any student who doesn't know the date can just ask another, "What's today's date?" So far, this appears to accomplish the opposite of what we want.

But here's the thing -- since the students already know that the answer is the date, they'll have to show more work in order to to receive credit. So the typical student complaint, "How come I can't get credit just for writing the answer?" has a ready-made response: "You don't get credit just for knowing what today's date is!"

Today is July 9th, Let's look at the question Pappas wrote for today on her calendar:

To write 39065.21 * 10^3 in scientific notation, the decimal point must be moved in front of the digit ____.

This very well could be a warm-up question that I'll ask my eighth graders. After all, scientific notation appears in Learning Cycle 5 of the Illinois State eighth grade text: "H20 + ?: Measuring Using Parts Per Million (ppm)." This is the first cycle in Unit 1 of the text. (Recall that Unit 0 is Tools for Learning -- the first four modules are identical for all three grades.)

Since today's the ninth, we already know that the correct answer to the question is 9. So when I check the students' warm-up papers, I obviously need to see more than just the number 9 written down. The most obvious thing to have the students write is the actual scientific notation of the number, which in this case is 3.906521 * 10^7. Here knowing that the date is the 9th serves to provide the students with a hint that the decimal place should be moved. When I check the papers -- and I'll need to check 30 papers within a few minutes -- I won't look at the decimal points (since they already know that it belongs in front of the 9) and instead look for the exponent, which is 7 (not 9). Thus giving students the date hint helps the students out and allows me to check more papers in less time.

The problem of converting 39065.21 * 10^3 to scientific notation is tricky (bordering on deceptive), since the given form is not the standard form either (that would be 39,065,210). In the actual classroom, I'm likely to ask the students just to convert 39065.21 to scientific notation -- then the right answer is 3.906521 * 10^4. I'm not bound to use the exact same question as Pappas -- in fact, I'll almost never do so as it's rare that the question from the Pappas calendar is on the exact same topic that I'm teaching in any of the three classes. (For example, tomorrow's question involves finding the major axis of an ellipse. This is definitely not a middle school question -- it belongs in Algebra II, if not Precalculus.)

Here's another good scientific notation question that I could ask on the 9th of a month:

-- (2 * 10^2)(5 * 10^6) = 1 * 10^____

Here the students must write something to show why the correct exponent is nine. In this case, I'll be scanning the papers for an intermediate step such as 10 * 10^8.

A Warm-Up Equation

Of all the questions that appeared on the actual Pappas calendar this week, the only one that might be appropriate in a middle school classroom was the one from the 6th:

3/5 - 1/5 (14 + 9x) = -13

Let's look at all the steps written out:

3/5 - 14/5 - 9x/5 = -13
3 - 14 - 9x = -65
-9x - 11 = -65
9x + 11 = 65
9x = 54
x = 6

For this problem, I'll glance at the papers for key numbers, such as -65 all by itself on the right hand side of the equation. If I don't see -65, I'll know that the student didn't really do the work.

Then again, I'm not sure whether I'd ever assign this exact equation to eighth graders. If the lesson is all about clearing fractions in equations, then it's not quite obvious how to clear the fractions in:

3/5 - 1/5 (14 + 9x) = -13

We need to multiply every term in the equation by five, but it's not evident why -1/5 (14 + 9x) counts as only one term:

3/5 - 1/5 (14 + 9x) = -13
5(3/5) - 5(1/5)(14 + 9x) = -13
3 - (14 + 9x) = -65

Here I think it's better to distribute the -1/5 before clearing fractions. But then again, we tell the students to clear fractions in order to avoid calculating with them, yet we must calculate with them in order to perform the distribution! This is why I don't necessarily like to teach equations with both fractions and distribution required.

Moreover, the question contains many negative signs -- in fact, every term turned out to be negative, so I multiplied every term by -1 in order to make the terms positive:

-9x - 11 = -65
9x + 11 = 65

So this question really assesses three different topics: clearing fractions, distribution, and negatives. I don't believe that warm-up questions should assess three difficult topics -- especially when this increases the danger of "two wrongs make a right," such as:

3 - (14 + 9x) = -65
3 - 14 + 9x = -65
-11 + 9x = -65
9x = 54
x = 6

which they'll know to be correct if the question is asked on the sixth of the month. I can even imagine a situation where students make several mistakes along the way:

3/5 - 1/5 (14 + 9x) = -13
5(3/5) - 5(1/5)5(14) + 9x = -13 (multiplies wrong values by 5)
3 - 70 + 9x = -13 (fails to distribute anything to 9x)
-67 + 9x = -13
9x = 54
x = 6 (Hey look, today's the 6th!)

So I will avoid this giving sort of problem on the warm-up. My warm-up questions will focus on a single concept, which students must understand in order to get the right answer. So to assess clearing of fractions, I might give a question such as:

9x/5 + 11/5 = 13

where issues with distribution and sign have been eliminated.

Of course, notice that it's still possible for students to copy all the work from each other. But this takes more effort than just copying a single-number answer. And besides, a student who copies all the work could actually learn more than a student who just copies a number. The emphasis now is on the process, not just the correct answer.

It's consistent with this idea that I want students to focus on the process and not just the answer for me to assign odd-numbered exercises from a traditional text for homework. This refers to the idea that many texts contain the answers in the back of the book. So the students know the all the answers and so they realize that they need to write down the process in order to get credit. But this idea doesn't apply to the Illinois State text, which doesn't provide any answers at all.

What It Means to Show Work

Let's go back to the traditionalist Beals now. Lately, her "Math problems of the week" series has been focusing on scored free-response questions from various Common Core tests. The idea behind these recent posts is, students can know lots of math, yet receive low scores for these questions because the Common Core graders don't like their explanations.

When I say that I want the students to show work, I mean that I want them to show me just enough to indicate that the understand the math rather than copied a cheap answer. I assume that traditionalists like Beals don't like it when students just copy answers rather than learn the math (though sometimes they forget that with their emphasis on individual problem sets for homework, many students will just copy instead of do the work).

Indeed, when I'm checking warm-ups quickly, I don't want to see Common Core-type explanations. I want to see something that shows that the student really knows the math. In my examples above, I'm looking for a specific number or value that the student can't reach without understanding the material, such as -65 in the equation example or 10 in the scientific notation example (first 10 in 10 * 10^8).

Foldable Notes

When I was a student teacher, my school used the Glencoe text for Algebra I. This text encourages teachers to use Foldable notes in class. Students would take several sheets of construction paper and fold them together to make a miniature "notebook," or "foldable," on which to take notes.

Several members of the MTBoS use foldables, and I've linked to some of their blogs before. Back in March, the blogger "Math Easy As Pi" used a foldable to teach surface area. In January, the blogger "Math Milla" (Mrs. Miller) used a foldable to teach systems of equations. And last year, the blogger Sarah Hagan used a foldable to teach HOY-VUX, a mnemonic to help students remember the properties of horizontal and vertical lines.

(Oops! Of course by "Sarah Hagan" I mean Sarah Carter, the newlywed and one of the best-known members of the MTBoS. Recently she wrote a post to make a big deal about how much trouble she went through with her marriage and name change -- especially as her new husband is Australian. So the least we can do is call her by the correct last name.)

Meanwhile, I can't see the traditionalists liking foldables one bit. In March, there was a post on the Jacobs website about a Day 100 project at an elementary school (as usual for Jacobs links, I give both the article and the comment thread):

I won't quote the comments here, but looking at them, we can see what the complaints are -- the project (in which students had to create a poster with 100 of something) was just artsy busywork with no educational value. And so if traditionalists don't even like this sort of assignment for young kindergartners and first graders, how much less then will they want to see middle schools creating foldables in math class? They'd deride foldables as pointless artsy busywork as well.

But here's the problem -- any traditionalist will say that many students don't take notes properly. I have the students create foldables -- sure, the time the students spend making the foldable is better spent writing notes in a college-ruled notebook. Yet -- this is par for the course for traditionalists -- they assume that the students will just diligently take notes in the notebook. I know firsthand that some students are willing to take notes on the foldable but won't take them in a notebook -- and in some cases, won't bother even to purchase a notebook to class. As usual, I'd rather a student spend a few minutes making a foldable and taking notes in it than spend the entire period staring blankly at a traditional notebook.

I mentioned in my Father's Day post that I might consider giving my students open-note tests. When I was a student teacher, my master teacher suggested that we let the students use the foldables on some (but not all) of the tests. And so I may do the same in my upcoming class this fall.

Grading Group Work

Of course, no discussion of trying to get a good grade without knowing the math is complete without addressing grading during group projects. With group projects, the traditionalists' biggest fear is that some students don't even attempt to contribute to the group.

I'm thinking back to my own days as a student. I didn't have group assignments that often in my math classes, but I remember one in particular that I had back in my Geometry class. As it turns out, the blogger Sarah Carter (yes, I just mentioned her earlier in this post) recently wrote about a group assignment that she plans on giving the first day of school -- and this assignment is very similar to the one I had in Geometry twenty years ago:

Students are placed in groups of 3-6.  Each student is given an envelope that contains 2-3 puzzle pieces.  The objective of the activity is for students to put their pieces together in such a way that each student has a complete circle.  

There are some very specific rules that must be followed, though.  

1.  No talking.  I think this will be the hardest rule for my students to follow. 

2.  No point or hand signals may be used at any time.  This will also be very tricky for my students.  As soon as they realize they can't talk, this will be the next thing they want to do. 

3.  Each player must put together his or her own circle.  No one may show another player how to put together his or her circle or do it for him or her.  

4.  Students may not take pieces from another student.  However, they may give one or more of their pieces to another student.  They may not put the piece in another person's puzzle.  Instead, they must hand it to the other person or lay it down on their desk.  

And I remember not faring particularly well on this assignment. I was placed in a group where all of the other students were older girls (recall that I was the only eighth grader in the class). I was looking around to see whether I could give my pieces to anyone, but the others just assumed that I wasn't going to participate fully. So they just took my pieces away -- and they claimed that they weren't breaking Rule #4 because one girl was taking my pieces to give to one of the other players. I wanted to tell them that I really was looking to give them pieces, but I couldn't say anything, because of Rule #1 of course. (Note: Unlike Carter, my teacher didn't give this on the first day of school.)

Throughout my years, I've had a mixed record with group assignments. I recall my fourth grade teacher saying that I had trouble with "cooperative learning groups," and there were some group assignments in high school history classes that I completely bombed. On the other hand, I usually did well on group lab assignments for science.

At this point, the traditionalists will point out that just as I didn't always do well on my group assignments, many of my students won't succeed either. So they conclude that I shouldn't torture my students with group projects.

In fact, I now think about a group assignment I once gave while student teaching. The rules for this assignment came from the district, and it was given near the first week of school (like Carter's). The students were divided into groups of four. The first student was supposed to build an equation by starting with a simple equation and performing the four operations on each side of the equation:

x = 1
2x = 2
2x + 4 = 6
(2x + 4)/3 = 2
(2x + 4)/3 - 1 = 1

Then the first person would give the second person in each group only that final equation, which now must be solved:

(2x + 4)/3 - 1 = 1
(2x + 4)/3 = 2
2x + 4 = 6
2x = 2
x = 1

This way, the students would see that to solve an equation, they must perform the inverse of each of the operations that they see in the equation, in reverse order. But here was the problem -- the first person in each group didn't build the equation correctly! This ended up frustrating the second group members, since their success was contingent on the first students' work. And I believe that many of these students disliked my teaching method the entire time I was there. When they had trouble with a later lesson, some didn't want to ask me for help because they thought I wouldn't really help them -- not after I'd refused to help them on this equation builder activity.

So these are the things I must consider when assigning group assignments from the STEM book. I will state the second rule of my classroom as:

Rule #2: Respect your honesty.

I must always be on the lookout for cheaters. The warm-ups and foldable notes should be set up to reward learners and punish cheaters.

And how should I handle group projects? I should monitor them to make sure that the students are actually learning something from the assignment -- and they aren't, find out why? If the activity is too complex (as that equation builder activity was), then I should simply it -- even if my simpler activity deviates from the instructions given by the district or the textbook. (So for the activity above, I should have had the students solve first one-, then two-step equations, even though the district insisted that the students build and solve four-step equations.)

And if the reason for the lack of success is a particular group member, then I should address it. I can break up groups to divide workers from slackers, to allay the traditionalists' fear that slackers can get a good project grade without learning any math.

My next post will be in about a week, when we'll return to spherical geometry.

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