Tag Archives: math

Multiply like a Roman

The Romans used weird, weird math. To multiply two terms, they repeatedly doubled one of them in one column while halving the other term in another column, throwing away any remainders that came up, crossed off half the numbers in the first column, added up what remained, and — voila! — got the right answer.

How on earth did they come up with it? And why on earth does it work?

Here’s what it looks like. If you’re not a mathy person, don’t worry, I’m not asking you to do any calculations–just to enjoy the strangeness.

1. Start with two numbers

536 * 42

= (500 + 30 + 5 + 1) * (40 + 2)

= DXXXVI * XXXXII

2. Multiply the first column and halve the second

Multiply these Divide these
DXXXVI XXXXII
MLXXII XXI
MMCXXXXIIII X
MMMMCCLXXXVIII V
(8M)DLXXVI II
(17M)CLII 1

3. If a number in the second column is even, cross it off in the first column.

XXXXII
MLXXII XXI
X
MMMMCCLXXXVIII V
II
(17M)CLII I

4. Add up all the remaining numbers in the first column.

(22M)CCCLLLXXXXXVIIIIIII

= (22M)DXII
= 22,000 + 500 + 10 + 2

5. And here’s the answer!

= 22,512

That’s all I have. If you want to know why it works, one of these links should satisfy.

A Different Kind of Multiplication

Roman Arithmetic

Or, if this is too much math for you, just be glad I didn’t tell you about multiplying infinity.

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I do want a standardized test, BUT

I opted my kids out of the state Smarter Balanced Assessment Consortium (SBAC) test last spring, joining the thousands of students and parents who felt it was fatally flawed — but the truth is that I do want my kids to take a standardized test that measures mastery of Common Core standards in math. Ironically, though, despite the millions upon millions of dollars that have been spent on Common Core assessments, the kind of test I think my kids should have does not seem to be available.

Let me back up a minute and talk about what the Common Core standards are. They’re a set of national standards, pushed on states across the country essentially by the private sector, that are clearly delineated. In the opinion of many education activists, they’re a complete mess. I won’t comment on that just now. I will say that having clearly delineated national standards in math makes sense to me. If everybody knows what’s supposed to be taught and when, that is a win. You can build off that knowledge to differentiate instruction, and you can keep track of what each child does and doesn’t know.

But oddly, that’s not what’s happening. The general concept was for the Common Core standards to be adopted by the states and for the private sector to start making textbooks and assessments and online curricula and so forth. In other words, free-market chaos.

So on the national and state level, our government is pouring enormous amounts of money into tests whose only practical purpose is to compare the performance of teachers, schools, school districts, and states according to whether or not the students have mastered Common Core standards. These are high-stakes tests. Because of the high stakes, their content is not open for public inspection, and we can’t evaluate their quality. Also, they are summative tests, given at the end of the school year to determine what was taught that year. In the case of the SBAC, the results were not available until well into the next school year.

What schools actually need are formative or interim assessments that can be used in the classroom, like we had in the olden days when schools had textbooks and the publishing company provided tests and quizzes.

But Common Core adoption has broken that this year, at least for Seattle Public Schools elementary kids. It shouldn’t have. Washington State has always had standards, and textbook companies have always adapted their curriculum to those standards, more or less. When Common Core came out, the same thing happened — apparently. The district spent several years on a textbook adoption process for elementary school and ultimately chose Math in Focus, which was supposed to be aligned with the Common Core.

This year, though, the school district has dictated that teachers abandon the “scope and sequence” (the content, plus the order in which subjects are taught) of Math in Focus and instead use a district-made “scope and sequence” document.

That breaks all the classroom tests from the Math in Focus textbooks–but doesn’t replace them with anything.

Why did they mandate the new scope and sequence? I would hazard a guess that they were told, or decided, that Math in Focus wasn’t the right textbook to prepare schools for the SBAC. Were they right? Were they wrong? Who knows! I bet that, as often happens, the private sector got way too much input in our schools.

This is the third year in a row that elementary school curriculum has been changed at our school. The district spent two years planning for a curriculum adoption. Two years ago, our school piloted a textbook called My Math. Then the district adopted Math in Focus, and we used that. Now we’re still kinda using Math in Focus but only as far as it matches the “scope and sequence.” How the teachers can adapt to all these changes, I have no idea.

And how is it going to be assessed? Probably with the SBAC. We’ve been promised formative assessments, but the tests the district got weren’t available to all schools, had serious privacy issues, and didn’t satisfy most teachers.

It’s a muddle for everybody, advanced learners included. Our school offers ALOs (advanced learning opportunities) for kids who qualify or just need extra challenge. By the end of last year, I figure that my fourth grade student had mastered all of the fourth grade curriculum, most of the fifth, and some sixth and seventh. In other words: all over the map. Next year, under normal circumstances, she’d enter sixth grade doing seventh grade math. But will she be ready?

Without appropriate assessments, I don’t see how her teachers could possibly know the answer to that question, or even what to teach her. Getting a handle on what parts of fifth grade curriculum she has and has not mastered is hard enough.

It didn’t have to be this hard. The SBAC and Amplify tests were designed top-down, to be sold to upper (mis)management. But for a whole lot less money, tests that gave the information we need could have been designed from the bottom up by teachers who are actually in the classroom with kids.

Here are the qualities I wish a Common Core assessment could have:

  1. Open and transparent (no high-stakes);
  2. Easily administered and quickly scored;
  3. Quickly given;
  4. Able to measure the curriculum in discrete chunks;
  5. Able to measure content above and below grade level;
  6. Easily modified for disability accommodations.

If such a test existed, then chances are, I wouldn’t opt my kids out.

Should I hold my breath? I don’t know. I doubt the ability of the Smarter Balanced Assessment Consortium to bring it to me, but efforts are being made by other organizations. For my own child, Khan Academy is a reasonable choice. But it’s not for everyone, especially kids with certain types of disabilities or kids without access to computers. There are also organizations like OERCommons and others that collect educational materials that are licensed for free use (though with no guarantee of quality). And the New York Department of Education appears to have done something clever in designing and delivering its own educational content.

But on the other hand, even in the best of times, good tests are hard to design and time-consuming to take and to grade.

All I know is, my family needs appropriate assessments for sixth grade math. And we need them now.

– Kristin

(Post updated 12/4/2015)

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My daughter, the guinea pig

We’re a very mathy family. We all learned math the way human beings learn things — that is, by making it a fun part of our everyday activities. But this year, my daughter is a guinea pig for the new national experiment that is called the Common Core standards. Her class is using a new curriculum designed specifically for the Common Core, “My Math,” and she hates it. I’m hearing her say, “I don’t like math,” and I’m concerned. She still likes math, really, but only when we mess around with it as a family. Her homework is procrastination and resentment and sometimes tears.

The Common Core standards came to us by a fairly strange route. Sixty people wrote them without public input, and only one of them was a teacher. And there’s a lot of controversy surrounding the Common Core, particularly the assessments designed for them.

I’m trying to keep an open mind. As one of my kids’ teachers told me, they’re simply one more set of standards. I do like one thing about these new standards: they tell us what our kids are supposed to be learning.

But what exactly are they supposed to be learning? Mathematical reasoning, the boring way. What makes it boring is the frequent testing to make sure kids have learned each individual concept, which appears to mean that each individual concept takes priority over a holistic understanding. Even worse: there is national pressure to base teachers’ job evaluations on kids’ mastery of these standards.

Also, it looks a lot like math fact fluency is being pushed aside. I’ve looked at the standards many times, and they do say what kind of fluency kids are supposed to have, but as one standard among many. And I’m not seeing it in the curriculum. My kids, and other kids whose parents I’ve talked to, are not getting the five minutes of math fact practice per day that they need to learn the facts.

Here are the standards, if you want to know more.

And here are some examples of math lessons that teach to these standards. The National Education Association (NEA) teamed up with some company or other to bring together lots of teachers to share the curriculum that they use in classrooms. So this curriculum is free online for browsing, which makes it a godsend for parents who want to understand what their kids are learning in class. I keep looking and looking for math fact fluency exercises, but I’m not seeing them.

I do realize that there are lots of things I don’t understand about the standards, the way they’re being implemented in curriculum, and the way my kids’ teachers are handling them in class.

But I sure am sorry my daughter has to be the guinea pig. 

Are our schools teaching arithmetic?

My dad the mathematician taught me something important about math. Mathematics has two aspects that go hand in hand: arithmetic and concepts. Arithmetic is basic addition, subtraction, multiplication, and division. It’s hard to learn the concepts if you don’t have a good grasp of arithmetic, and it’s boring to learn arithmetic without the concepts. 

Math can be fun. I’m serious here! How come so many people hate it? How come so many women feel like they’re no good at math? 

There are a lot of reasons, but let’s start with ground zero. Arithmetic. How many people have their math facts memorized? I assumed everybody who gets to adulthood has their math facts memorized, and I assumed it was being taught in schools. Now I’m not so sure. 

Here in Seattle there are a lot of the parents I know are paying for math tutoring, particularly for arithmetic practice. Check out this map of Kumon tutoring centers. There are ten centers within nine miles of downtown Seattle. 

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Kumon tutoring centers near downtown Seattle

People are paying for something that really should be the job of schools. However, it’s hard for teachers to “squeeze it in.” How could that possibly be? Is it because high-stakes standardized testing has squeezed it out? 

Now take another look at that map, bearing in mind that northeast Seattle, Bellevue, and Redmond are the most affluent and whitest parts of this area. Where are these tutoring centers concentrated, and where are they completely absent? There are none in southeast Seattle. Guess it’s not so profitable there. 

This is the “opportunity gap” in action. 

In addition to paying for tutoring, many Seattle parents are practicing math facts at home. This would be fine if kids were also getting it at school. But if they’re not getting sufficient math fact practice at school, then parents who are practicing math facts at home and paying for tutoring are masking a significant deficit in our children’s education. 

This is the opportunity gap in action. 

So I’ve been asking around. Math fact practice is not necessarily a daily event in the classroom. If kids haven’t mastered arithmetic and subtraction by the end of grade 2, they’re not necessarily going to get any more practice, but they will be expected to start learning multiplication, division, and fractions. If they are significantly below standard, they will be eligible for some kind of pullout service, where they get math help but miss the regular curriculum other kids are getting. Plus, the stigma of “not being good at math.” So they fall farther behind. 

This is the opportunity gap in action. 

Now, there’s a lot of talk of “accountability.” High-stakes standardized testing is supposed to hold teachers and schools accountable, isn’t it? Well, it doesn’t work. It punishes teachers if they spend too much time teaching what’s not on the test. And I don’t think that basic math facts are on the test. 

Through elementary school, there are two tests Seattle kids get. One is the Measures of Academic Progress (MAP), and it’s district mandated. The other is the Measures of Student Progress (MSP), and it’s state-mandated. The MAP test is untimed, which means that if kids are counting out their math facts on their fingers, that’s fine. As for the MSP, I’ve looked at a practice test  and it hasn’t got much in the way of math facts. It focuses on math concepts and generally uses easy arithmetic. And yet enormous amounts of time and money are spent on these tests. Schools have to give up their libraries for weeks in order for these tests to be given. Schools are rated based on these tests, and the trend is for teacher evaluations to be based at least partly on the results of these tests. 

If parents want math facts to be taught in schools on a daily basis, then we kind of have to stand in line, behind the demands of these tests. 

What’s the answer? To me it’s blindingly obvious. Do these three things.

  1. Take five or ten minutes out of every school day to drill on math facts. But some kids are farther along than others, right? Won’t the kids who already know addition and subtraction be bored? Actually, I think it’s okay for kids to be bored for five or ten minutes a day.  
  2. Take some of the pressure off the teachers. Cut down on the amount of concept material that is required to be taught and eliminate the high-stakes component of standardized tests.
  3. If you’ve got to have insanely expensive standardized tests, at least take part of that test time and use it to measure math fact mastery.

Of course, everybody and their dog looks at schools and thinks they see a massive problem that is blindingly obvious to them. Most people are wrong. Am I wrong? Am I missing something? 

All I know is I’m out of time for today. I’ve got to go drill my son on math facts.