You want the 700, but you don't have number sense: try this!

This post is a follow up to yesterday's post which describes the difference between a student who scores in the 700's on the math section of the SAT and a student who scores in the 500's.

So I've told my 500's student that getting a 700 on the SAT will be "a lot of work."  But what should that work consist of?  I have some activities we could do and some problems we could work, but we need to do the activities and work the problems over a long period of time so that she can actually internalize the lessons.  Unfortunately, working one-on-one with me for that period of time would be prohibitively expensive for a lot of people.  Is there a cheaper alternative?  Can someone do it on her own?

I have been using materials from Art of Problem Solving with a handful of elementary school students, and I am struck with the fact that they place more emphasis on the number sense than they do the algorithms.  They walk students through steps toward comprehension that focus on the meaning and assume that the algorithm will come on its own.  That process works best for the advanced kids, but that is their target market.  Still, I thought they might have something useful for the high school student who has learned the algorithms, but would like to retro-actively work on the number sense.

Introducing Alcumus.  Alcumus is a free online problem bank.  Once you register, it will give you a math problem.  Get that right, and you will get a more difficult problem. Complete enough problems in a row, and you will move on to the next topic. It is somewhat similar to the practice modules on Khan academy with a couple of notable differences:  First, you will be given an explained solution even if your answer was correct!  In fact, to get anything out of this exercise, you need to carefully read every explanation to see if there was a different, more intuitive (as opposed to algorithmic) method of solving the problem.  There are a few videos to watch for more instruction, but Art of Problem Solving believes in a problem-first approach.  There are also references to chapters in Art of Problem Solving math text books. (The books can be a bit pricey.  If you have the means, buy a couple of copies and donate one to your school library.)

Try to see if these methods lead to being able to solve complicated-looking problems in your head.  (It goes without saying that you should NOT be using a calculator.)  You will be led through addition, subtraction, the distributive property - simple stuff, but there are lessons here for how to think about these problems differently.  How to use your head instead of that hand-held machine you have been using as a crutch.

Try it!  I'd love to hear how it works out!

What does a 700 student look like?

Yesterday, a student who last scored in the 500's on the math section of the SAT asked me what it would take to score above 700 by fall.  She asked me during a class change at school, so there wasn't much time to say more than, "a lot of work." However, once she had gone on to class, I asked myself:

What is the difference between this young lady, and that hypothetical person who scores above 700?

Both students have taken all of the math courses listed as prerequisites and then some.  (This is true of all of my tutorees.)  Both have good grades in math (A's and B's.)  What is true of that 700 kid that isn't true of everyone else?

The short answer:  all of the kids can tell me that a certain math fact is true, but the 700 kid behaves as if the math fact is true.  For example, all of the kids can give me a definition of an even number.  They can all recognize one when they see it (if it is written as a number and not an expression.)  The 700 kid can glance at a problem, see that 2 will have to be a factor of the answer, and eliminate the answer choices that are not even.

A 700 student can promptly tell me that 1÷ (1/16) is 16.  A 500 student will either labor through the algorithm for dividing by a fraction or, more often, sit stymied because s/he doesn't recognize that the dividing-by-a-fraction algorithm is relevant. (Note:  this most often happens when the above exercise is expressed as a compound fraction in the first place.)

A 700 student understands additive inverses and therefore doesn't sit gaping in horror if I ask him or her to add all of the integers from -25 to 26 inclusive without a calculator.

A 700 student can tell me that the square of the square root of 2 is 2 without laboring through the algorithm for multiplying square roots.  (She or he also remembers from one day to the next what fractional and negative exponents represent.)

A 500 student may or may not be able to recite the commutative properties of addition and multiplication, although he or she will confirm that numbers can be added or multiplied in any order.  A 700 student may or may not be able to recite the commutative properties of addition and multiplication  depending on whether or not she or he remembers which property goes by the name "commutative."  However, she or he will not hesitate to rearrange addends or factors to find the most efficient way to compute the answer.

A 700 student recognizes that every integer has a unique prime factorization and understands that any factor of that integer must be the product of some combination of those prime factors. If asked, the 500 student can find the prime factorization of a positive integer, but he or she will not recognize those occasions when finding the prime factorization of an integer would be useful.  If the student has found the prime factorization of an integer (with or without prompting) to be 11 x 17, and if you ask the student if the original integer is divisible by 6, the 700 student will say, "no."  The 500 student will whip out a calculator and do the computation.

In short, the 700 student has a characteristic called "number sense".  The 500 student does not.  These two students might be in the same math class, at the same school, earning the same grade, but the 700 student is only working half as hard.  Furthermore, this will have been true for years.  I am currently working with two elementary school students - brothers.  One has number sense, the other doesn't.  I can already predict what their first SAT scores will be (or perhaps would have been if the test weren't changing.)

So now you're a junior and you don't have number sense, but you want that 700.  AND you're willing to do the work.  What should you do?  Check in tomorrow for instructions.

Study Materials Review: Ultimate Guide to the Math SAT

This book does not include anything new in the way of advice.  Its formatting is not as accessible as some of the other options, and a student working on his or her own may find it intimidating.  However, it does include plenty of great problems.  There is an art to writing good SAT-style math questions, and Richard Corn (an SAT tutor in New York) has the technique down.  I still prefer Top 50 Skills or PWN the SAT for students working on their own, but if you've worked through those, and you would like some additional targeted practice, this is an excellent option.

Euclid is rolling over in his grave.

Standardized testing has killed geometry.  All that’s left to do is plan the funeral.  True, geometry had been ailing for some time and was too weak to put up a fight.  Still, theoretical mathematicians should pause for a moment of silence and then figure out what to do next.

The objective of geometry was never understood by most modern folks.  They tended to dismiss it as the study of “shapes” and to wonder why it was included in the curriculum.  But geometry was never about shapes.  Shapes were merely intended as the vehicle for making deductive reasoning more accessible to students.  Students tended to find formal proof to be very challenging, and as the self-esteem movement grew and grade inflation ran amuck, math teachers were under more pressure to gloss over the proofs that made the subject so difficult.  Eventually, many, if not most, high school students went off to college without ever having done a formal mathematical proof.

Still, geometry problems tended to involve informal deductive reasoning:  “I know these two lines are parallel, therefore these angles must be congruent.  If that’s true, then this thing is a parallelogram and these two line segments are congruent.”  In addition, geometry continued to be a class where you had to be careful and precise about how you talked about something.  Definitions were important.  Leave out a phrase, and everything changes.

The problem is that formal deductive reasoning can be difficult to test.  Informal deductive reasoning is easier to test, but requires a great deal of background knowledge about “shapes.”  Thus the layperson thinks that “shapes” was the concept being tested in the first place, and does anyone really need to remember that a midsegment of a triangle is half the length of the side to which it is parallel?

So now the Common Core Standards and the SAT have essentially gutted geometry from the curriculum.  Only the bits about shapes that are essential to trigonometry and to transformations (since there is an increased emphasis on graphing functions by transformations) have been kept. Formal definitions and proof are no longer included.  For true mathematicians this means that real math is no longer taught in kindergarten through 12^th grade at all.  What’s left is just the arithmetic and modeling needed for science and statistics.  Where will our future mathematicians come from?

The High School Common Core Math Sequence Is Broken, part II

This is a follow-up to the previous post, which you may be able to read by scrolling down.  If that doesn't work for you, check the blog archive in the right-hand column.

In the last post, I wrote about problems with the Common Core Standards.  I argued that trying to brush off criticism by saying that parents are responding to other fears – fear that my child isn’t smart enough, fear that my child won’t grow up to be like me, fear that my child’s standard of living won’t be as good as mine – fails to address very real issues in today’s classrooms.  Today let’s look at a specific example.

I am a professional tutor.  I work with some students in math, and I help others to prepare for college entrance exams.  Most of my students attend elite private schools in the Research Triangle area of North Carolina.  I expected the second week of March to be very quiet.  I planned to do some extra housecleaning and some curriculum preparation for an upcoming evening class.  On Monday evening the phone rang. 

Could I help a student enrolled in a course titled “Common Core Math 2?”  Probably.  This is the first year that this course has been taught, and I wasn’t sure what was in it, but I’m familiar with most high school-level math, so I figured I could help.  Word got out, and I am currently working with a number of students all from the same class.

Common Core Math 2 is currently taught to students who, under the “old” system, would have been taking geometry.  If you will recall, I had been hopeful that certain geometry topics would be pushed to a later course, that there would be fewer topics overall, and that the topics included would be covered in greater depth.

Typically, when I get a new math student, my first question is, “Who is your teacher?”  Often that tells me all I need to know.  A handful of individuals have accounted for the bulk of my tutoring clientele.  However, the teacher this time is a veteran and a star.  She knows her stuff, both mathematical and pedagogical, so if there’s an issue, it probably doesn’t lie with her.  The reason for the sudden increase in business was apparent as soon as I looked at the homework packets.

This veteran teacher is incredibly well organized.  You can get online and see what students will be responsible for each day of the semester.  A few clicks, and the entire course was laid out before me.  I’ve never seen such a mess.  First, there are too many topics to be covered.  The list for the students to review for their midterm listed 47 topics.  FORTY-SEVEN.  Forty-seven topics had been covered in forty class periods. Some are topics that I would have voted to leave out altogether.  (Which is the incenter and which is the orthocenter?  I don’t remember from one day to the next and I teach this stuff!  Seriously, is there anyone who actually needs to know?)  Some are topics that have been pulled in from pre-calculus (Common Core 4 will replace this), and given the brain maturity required to understand them, should have been left there.  The topics don’t flow.  They don’t relate well to one another.  While I can see some of the basic principles that lessons are trying to address, it might be better to use different topics to address them.  In short, it is no wonder that students are floundering.

What went wrong?  The overall process has been remarkably opaque.  Stakeholders who should have been pulled in at certain levels of the process clearly weren’t, and it is difficult to figure out exactly what happened or where the whole thing broke down.  However, I have been paying attention to this story from the beginning.  I’ve done some poking around, and I have pieced together a story that seems plausible.  Here it is:

The setting:  For those of you who may not live here, the North Carolina educational system is fairly centralized.  Teachers are state employees and in addition to funding teacher salaries, the state gives local school systems money for busses and other expenses.  Local governments are responsible for building and maintaining school property, but the bulk of the money comes from the state.  In addition, the state jumped on the high stakes testing bandwagon before it became popular and has written and administered it’s own standardized exams since sometime in the 1980’s.  This effectively means that the state has been in charge of local curriculum for decades.

In 2009, when the Race for the Top grant was announced, we were in a recession and the state was broke.  The powers that be were scrambling for revenue sources that wouldn’t involve raising taxes, and the grant money looked awfully juicy.  Sure they had to agree to adopt some standards and then test to see if they were meeting them, but weren’t they pretty much doing that already?  Count us in!

We were awarded the grant in 2010. Now, keep in mind that the state was looking for money for everyday operating expenses.  So having spent the money on teacher’s salaries, there wasn’t much left for implementing the standards.  Best I can tell, the National Common Core Standards lump all of the high school math standards in “high school.”  It is up to the states to design the sequence of the high school curriculum. So the state wrote lists of what would be tested at the end of each year and pushed the work of curriculum design to the districts. 

Since the late 1980’s the state had developed a rich bank of curriculum resources that districts could use.  The scope and sequence of each course were spelled out with suggested pacing.  There were sample worksheets, examples of activities, and banks of test questions.  All of this was now obsolete.  In the summer of 2012 local districts were faced with having to design math courses based on lists from the state of what would be tested as early as January of 2013 (for block schedule high school courses.)  They didn’t have any money for curriculum design, either, so they dumped the work on the teachers who scrambled to write each piece in time to use it in the classroom.

As you can see, at every step there were ample opportunities for the process to break down.  Where can we pin the blame for this particular fiasco?  Again, because the process has been so opaque it’s hard to say.  I do believe that the teachers at the district level have done the best they can, given the mandate from the state and the time constraints.  I actually think the state Department of Instruction did the best it could, given the tight deadline and lack of money.  Where did the deadline come from?  Who said we had to have everything in place and the first tests administered by winter 2013? (And whose bone-headed idea was it to promise that we would adopt the standards without spending enough money on the task??) We don’t know.  Parents see “Common Core” in the title of the course and so it’s the Common Core Standards they rail against in letters to the editor and at school board meetings.

Regardless of who is to blame, the situation is this:  The North Carolina Common Core high school math curriculum is broken.  The scope and sequence of the topics do not reflect what we know of how students learn or of when concepts should be introduced.  Topics and concepts in each course are so numerous, that it is impossible for concepts to be studied in depth, but they will be tested as if they were. 

Arne Duncan would have you believe that the resulting poor scores mean that our little darlings just aren’t as smart as we thought they were.  Ms. Boylan would have you believe that we aren’t really upset over our little darlings’ failure to learn anything in math class, but rather that our little darlings might learn something we didn’t, while some editors in Long Island think that we are really upset over the general state of the economy.  How insulting!  North Carolina parents can recognize a mess when we see one, and this is a mess.  We can’t be faulted for misplacing the blame when the mess was made behind closed doors.  The fact remains that someone needs to clean it up.

The High School Common Core Math Sequence is Broken

There has been a great deal of criticism of the Common Core Standards in the past year, and the discussion is getting heated.  On one side we have accusations of a federal take-over of local educational systems and myriad examples of nonsensical homework assignments.  Some of the responses from the other side have been interesting.   First we have Arne Duncan accusing “white suburban moms” of fearing that their little darlings “aren’t as brilliant as they thought they were.”  Then there’s the piece by Jennifer Finney Boylan (“A Common Core for All of Us”) claiming that Common Core opposition is rooted in the fear that our children might not turn out to be carbon copies of ourselves.  Now most recently we have an editorial from Long Island Newsday that tells us our concerns about Common Core are really “the fear that the fundamental promises of American society are eroding, that the next generation will not be better off.”

At some point I need to poke around and see if anyone is writing about why Common Core proponents are trying to make their case by insulting millions of parents.  (Surely they realize they’re outnumbered.)  But for now, lets just say that parents don’t have their panties in a wad over some vague angst. They are upset over what they see happening to their children in the classroom.

When the Common Core Standards were first announced, I saw two promises that gave me reason to cheer:  First, there would be fewer topics taught in greater depth.  Second, school districts would have the option of eliminating the traditional algebra I, geometry, algebra II sequence in favor of a system where the algebra and  geometry topics are mixed and taught in an order that recognizes some things are better left until student’s brains have matured.  I have spent years watching the math curriculum (geometry in particular) get watered down until it was meaningless because we tried to teach too much too soon, and I was looking forward to adopting a model that had long been used in Europe and Canada.  I knew there would be bumps in the road on the way to implementation, but I thought that if we could all just hang in there, we would eventually end up with something really effective.

Then reports started to flood in from parents and teachers.  Children are miserable.  Teachers are exhausted. The failure rates on assessments are high.  Whom do we blame? It’s hard to say because what we are looking at is likely a mixture of the standards themselves, local implementation of the standards, and a culture of high stakes testing that pre-dates the standards.  And yes, a bit of resistance to change to season the stew.

Part of the problem is that the path from the standards to the day-to-day workings of an actual classroom has been very opaque.  Some of the blame that has been cast at the national Common Core Standards might be more accurately pointed at a more-local agency.  That doesn’t change the fact that something will have to change.  Brushing away the criticisms by libeling parents and teachers is not a valid or viable solution.

Next up:  A look at a particular course in a particular school district as an illustration of what may have gone wrong.