"Young people today can't make change" just took on a whole new meaning

About 20 years ago, I was standing in the checkout line at a cafeteria-style restaurant.  Everyone in line was over the age of 30 because that style restaurant mainly appealed to the older generation.  The checkout clerk, who was probably in her early twenties, rang up the man in front of me, and he handed her a 20 dollar bill.  She punched in the 20.00 and hit return.  Just at that instant, the electricity flickered.  The cash register momentarily went dead and then came back on, but the transaction was lost.  The clerk didn't know what to do.  At first the people in line thought she wasn't sure if the transaction had been recorded or if she needed to repeat it, but that wasn't the problem.  She didn't know how much change to give him.  She remembered the price of his meal.  She knew he had handed her $20.  She couldn't tell what the change should be unless the cash register told her.  Everyone in line knew exactly what the change should be, but she couldn't risk taking our word for it.  A succession of managers was called to the scene until they found someone old enough to know how to make change for a twenty without electronic aid.  Until yesterday that was my "young people don't know how to make change" story.

Yesterday, I was at the cash register paying for an item that cost $7.42.  I reached into my purse, grabbed a ten-dollar bill and then poked around in my change pocket.  I pulled out a quarter, a nickel, a dime and two pennies.  I dropped the change into the palm of the checkout girl and then handed over the bill.  She set the bill aside, spread the coins across her palm and stared at them for 20 seconds or so.  She stared for so long, I began to worry:  Had I grabbed a third penny instead of the dime I was aiming for?  Is she trying to find a tactful way to tell me I've given her the wrong amount?  Finally she looked up, held out the coins and asked, "Is this 42 cents?"

My first thought was:  what in the world would have happened if she had needed to give ME 42 cents in change?  If she can't count out 42 cents, what would she have done?  Grabbed random coins, and if I complained add more random coins?  Call a manager? I wondered how she had been handling this problem up to now, but then I realized:  I may have been the very first customer she has ever had that paid for something in cash.

We are rapidly becoming a cash-less society and as we do, the ability to count change is becoming obsolete. Money-counting lessons will eventually be dropped from the school curriculum, and my question is:  Will this have unintended repercussions?

I frequently have this exchange with my students:

Student:  (staring at 75 ÷ 25)
Me:  How many quarters are in 75 cents?
Student:  Oh. Right.  Duh. (writes 3)

Coinage is an excellent medium for practicing all sorts of math skills that have applications in other places, but how many of those applications will continue to be relevant?  Do we need to be able to do things like divide 75 by 25 in our heads?  Much of the math in the current high school curriculum is unnecessary for the vast majority of adults.  We teach it anyway because we want to leave the door open for our students to choose those careers that need math.  However, there are other things going on when we study math.  What we learn shapes our brains.  For a long time we justified teaching geometry proofs by telling students that proof by deduction teaches us "how to think."  Geometry proofs have largely been dropped from the Common Core curriculum.  (You can read my elegy here.)
Will we see changes for the worse in students' overall cognitive functioning?  Will becoming a cash-less society have an adverse effect on our brains?

It's hard to solve the triangle if you don't know what a "guy wire" is.

Happily all of my students currently studying trigonometry know that one would find a shadow on the ground, and all of them know how a kite works!  However, I did have to tell three students in a row what a "guy wire" is.

I'm not sure how we should be trying to address this problem.  On the one hand, we don't want a student who knows how to solve a triangle to miss the question because he doesn't know a non-math vocabulary word.  (And what about the ESL kids?) One solution might be to provide a labeled diagram with each question.  However, many would argue that being able to model the problem involves the student drawing his own diagram. Is there even a description of a guy wire that doesn't essentially tell the student how to draw the diagram?  And if we restrict ourselves to vocabulary that was used for examples in class, then how do we ever present a student with a novel problem?

The fact is that a word problem has to be about something.  And if the student has no experience with that "something" then it's a lot harder to work the problem.

(For more context, see this earlier post.)

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.

The new SAT: Math

This is the second of a series of posts about the redesigned SAT.  Yesterday's post addressed comments made about vocabulary.

About the adjustments to the math section:

When Daniel Coleman announced a year ago that the SAT would be redesigned, he listed 3 reasons:

1.     To better serve the needs of college admissions officers.

2.     To improve equity.

3.     To better align with the Common Core curriculum.

At the time, the prediction among many was three-fold:

1.     The SAT would become more like its rival the ACT.

2.     The SAT would become less of a college entrance exam and more of a high school exit exam.

3.       The new SAT would be “easier” and thus do little to distinguish between top students and above-average students.

For the math section in particular, speculation was that, like the ACT, the SAT would test topics currently included in pre-calculus.  The current SAT doesn’t test topics past Algebra II.

Since the recent announcement giving more details about the SAT that will be rolled out in March 2016, some pundits are still predicting that the new SAT will be very similar to the ACT, but I disagree.

It’s no secret that the redesign was inspired in part by the ACT’s increasing market share.  Among other things, several states have contracted to give the ACT in schools as a measure of high school achievement.  That has to be a tempting market to tap. 

One detail in the announcement seemed designed to lure back students who would have chosen the rival test.  The current SAT "penalizes guessing" by subtracting points for incorrect answers.  The ACT does not.  Students have actually chosen not to take the SAT for that reason alone, and eliminating this practice is a much-needed marketing move.  It is also a move I heartily approve.  Less time spent on this useful-only-for-the-SAT test-taking skill means more time we can spend on content and problem-solving.

HOWEVER:  In a very bold move, the College Board also announced that part of the new test will be "calculator inactive."  This will be VERY unpopular, and since calculators are allowed on the entire ACT math section many students will choose to take the ACT on this basis alone.  The College Board may have shot themselves in the foot on that one.  I, on the other hand, am pleased as punch.

In Wednesday’s announcement, they said that the math section would include fewer topics, not more.  The topics are vaguely divided into “Problem Solving and Data Analysis”, the “Heart of Algebra”, and “Passport to Advanced Math.” One presumes that each topic would then be tested in more depth.  This would be consistent with the Common Core goal of studying fewer topics in more depth and of promoting critical thinking skills.  The list of topics was very vague, but nothing in the list specifically pointed to the inclusion of pre-calculus topics.  As for the difficulty level of the questions, it’s much too early to tell.

I am ambivalent about a reduction in the number of topics.  On the one hand, does a college student really need to be able to apply the Hinge Theorem – a topic on one of the Official Guide practice tests?  On the other hand, a student who didn’t remember the theorem could reason through the problem as long as he understood the basic principles of geometry, and isn’t the ability to do that exactly what the test purports to measure?  Won’t reducing the number of topics just put a limit on the questions that can be asked?  Or would this question continue to be asked because the student can reason through it using……wait.  Geometry isn’t in the list of math topics.  Unless it is included in “passport to advanced math” – and what IS that, anyway? – then GEOMETRY IS NOT ON THE REDESIGNED SAT. 

Well, that would solve their market share problem, right there.  On the other hand, if I’m a college math, physics or engineering department, I definitely want to know if the kid remembers geometry.  So, if they leave it off the SAT, will colleges start requiring the ACT instead?  And if this test is designed to influence high school curriculum, then will geometry no longer be taught in high school?  I guess we won’t really know what the topics are until we actually see some sample questions, so I’ll try to avoid getting my panties in a wad between now and then.  Still.  It’s something to keep an eye on.

I’ll be posting about local high school students’ experience with an implementation of the Common Core math standards in a week or so once I get the SAT stuff said, but at this point it’s relevant to remind the reader that the “architect” of Common Core and the leader of the College Board are the same person.  When the announcement first came out that the SAT would be redesigned, aligning with the Common Core standards was specifically mentioned.  But in the announcement last Wednesday, College Board spokespeople and the media were strangely quiet on that point.  Perhaps because Common Core’s approval rating is currently in the toilet?

But, I digress.  One other announcement was indirectly math-related.  Oddly, the one change that seems to have resonated most with the general public is that the SAT will go back to scoring on a 1600-point scale.  I thought this was a silly thing to focus on, but then someone pointed out that this causes your math score to represent half of your overall composite instead of one-third.  That’s true.  The individual who pointed it out was bemoaning this fact (she was an English teacher) but as a math major I am pleased.  I also think it is appropriate given the current emphasis on STEM.

So, changes to the math section:  Good or Bad?  It’s too early to tell, but I confess I’m worried.