The Consequences of Letting “Where’s the Math” Set ...



The Consequences of Letting “Where’s the Math” Set Washington State Mathematics Policy 

Five perspectives on how “Where’s the Math” is hurting kids and compromising Washington State’s long-term economic viability

Steve Leinwand, Washington, DC

February, 2008

 

Provided as a public service on behalf of rational policy, informed s thdecision-making, and the long-term best interests of the students of Washington

 

Introduction.  There certainly are “math wars” raging across the United States.  The NCTM Standards (both old and new) are vilified by some and cast as the holy grail by others.  Instructional materials like Everyday Mathematics, Investigations and Connected Math, funded by NSF to support reform along the lines of the NCTM vision, have spawned as much enthusiasm as venom, but many forget just how ineffective the very textbooks they were designed to replace have been and continue to be. 

 

No one argues that there is a crisis in American mathematics education.  A confluence of data shows clearly that:

•        the U.S. scores relatively poorly in all international comparisons and this is cause for serious concern and thoughtful action;

•        there are only rare instances where one can show that U.S. mathematics programs are working for more than about 40% of our students; and

•        gaps between white and Asian on the one hand, and black and Hispanic on the other, are stark and portend disaster, if not addressed.

 

However, despite the misguided advocacy of Where’s the Math and their Mathematically Correct California cousins, an ounce of common sense tells us that the answers to these problems are not more traditional “show and tell” instruction, nor more mindless computational drill and practice or the rote regurgitation of mathematical procedures without meaning or understanding. And given global competition, the answers are not an increasingly obsolete curriculum that keeps us mired in international mediocrity.  Rather, the answers are a coherent, internationally-benchmarked, forward-thinking curriculum, implemented with effective instruction, and supported by powerful assessments – all carefully aligned and reflective of the kind of elementary level mathematics found in Singapore and Finland, two of the world’s highest performing countries. 

 

Five perspectives shed light on how buying into the perspectives of Where’s the Math, without first taking a hard look at the data and the research, leads Washington down a primrose path that serves neither the state’s children nor its future.

 

1. The NAEP Perspective.  So much has been made of the California Math Standards by Where’s the Math.  So much of the formative work of Where’s the Math was based on Mathematically Correct and the advice of Stamford’s Jim Milgram and Cal State Northridge’s David Klein.  However, the question isn’t where’s the math, it’s where are the results?

 

The tables below show that:

• Contrary to all the doom and gloom descriptions, Washington State students consistently perform above the national average;

• While the national average increased between 1996 and 2007 by 17 and 9 scale points for grades 4 and 8 respectively, Washington students increased by 18 and 9 scale points respectively.

• Despite all the claims for the strength of the California Math Standards, after ten years the California scores, ranking, and improvement are dismal and provide little for Washington to envy, emulate or replicate!

• Instead of looking south to California, if Washington seeks to emulate one state, it ought to be Massachusetts.

 

NAEP Grade 4

|  |1996 |2007 |

|  |Score |Rank |Score |Rank |

|Washington |225 |17 of 43 |243 |15 of 50 |

|California |209 |41 of 43 |230 |46 of 50 |

|Massachusetts |229 |6 of 43 |252 |1 of 50 |

|U.S. |222 |- |239 |- |

 

NAEP Grade 8

|  |1996 |2007 |

|  |Score |Rank |Score |Rank |

|Washington |276 |14 of 40 |285 |18 of 50 |

|California |263 |31 of 40 |270 |45 of 50 |

|Massachusetts |278 |10 of 40 |298 |1 of 50 |

|U.S. |271 |- |280 |- |

 

It is easy to see that the California Math Standards have been a failure.  If one really cares about effective school mathematics for Washington State, look to Massachusetts.  While the Exemplary Mathematics Standards: 2008 cut and pasted together by a committee of Where’s the Math pays lip service to Massachusetts, it’s the Dana Center’s Washington K-12 Mathematics Standards (January, 2008 Review Draft) that indeed captures the spirit of what has worked in Massachusetts.  Massachusetts, interestingly enough, is also where the penetration of Everyday Mathematics (40% of the districts in Massechusetts use EM), Investigations and Connected Math is among the highest in the nation, but that’s a correlation Where’s the Math doesn’t see fit to publicize.

 

2. The International Perspective. All the discussion about how poorly U.S. students perform relative to their international peers often misses the key point about exactly what is being expected and assessed in other countries and on international measures.  Too often people believe that it is the computational prowess of students in Holland or Finland or Korea or Singapore or Japan that must be emulated.  However, a closer look reveals that the missing element in the U.S. is reasoning and problem solving, not more multi-digit multiplication, division of two fractions without context or factoring 30 trinomials for homework. 

 

Consider the following item from the Singapore Grade 6 Primary School Leaving Exam:

 

Lee and Chan drove from Town P to Town Q. They started their journeys at different times. Lee drove at an average speed of 45 km/h and took 40 min. Chan drove at an average speed of 72 km/h and reached Town Q at the same time as Lee.

a)  How far was Town P from Town Q?

b)  How many minutes later than Lee did Chan start his journey?

This problem, and most of what is important for workplace readiness, does not require high levels of computational prowess.  Rather, it requires a thorough understanding of the concept of rate. Students must organize five steps of thinking and solve for an intermediate unknown (how long Chan drove) as they translate Lee’s rate from minutes to hours (or vice versa) and use this information to calculate distance. 

 

Or consider the following percent problem from the same Singapore test:

 

At Mrs. Ong’s shop, there were two vases for sale at $630 each. She sold one of them at this price and earned 40 percent of what she paid for it. She sold the other vase later at a 20 percent discount. If the two vases had the same costs, how much did Mrs. Ong earn altogether?

This non-routine problem requires students to represent earnings as the sum of two prices minus two costs and requires a full understanding of percent increases and decreases.

 

Or consider the following real-world released PISA (Program for International Student Assessment) item:

 

Internet Relay Chat

Mark (from Sydney, Australia) and Hans (from Berlin, Germany) often communicate with each other using ‘chat” on the Internet.  They have to log on to the Internet at the same time to be able to chat.  To find a suitable time to chat, Mark looked up a chart of world times and found the following: 

Greenwich  12 midnight

Berlin 1:00 a.m.

Sydney 10:00 a.m.

 

Question 1: At 7:00 p.m. in Sydney, what time is it in Berlin?

Question 2:  Mark and Hans are not able to chat between 9:00 a.m. and 4:30 p.m. their local time, as they have to go to school.  Also, from 11:00 p.m. till 7:00 a.m. their local time, they won’t be able to chat because they will be sleeping.  When would be a good time for Mark and Hans to chat?  Write the local times in the chart.

 

                                            Place                                       Time

                                          Sydney                                    ________

                                          Berlin                                       ________

Every parent and mathematics educator should turn to the PISA Released Math and Problem Solving Items () and ask themselves a few key questions:

• Is this an item that I would hope my child/students could get correct at age 15?

• Do my school district’s and state’s curriculum standards adequately prepare students to be successful on items like this?

• Are there items like this on my state’s high stakes assessments and my district’s end of grade and end-of-course assessments?

Unfortunately, in most cases, the answers to these questions are Yes, No and No respectively.

 

Anyone who truly cares about the degree to which our students and our country are ready to compete needs to ask one simple question:  What are our schools doing to ensure that our students can successfully solve problems like this?

 

Where’s the Math’s answer is very evident.  Only 17 of 149 Grades 9-12 performance standards in Exemplary Mathematics Standards: 2008 are noted P, which we are told represents standards that “demonstrate an added opportunity for students to reason, communicate and connect concepts in problem solving.”  It doesn’t seem like the remaining 132 performance standards that are proposed will adequately address the real problems we face.  Alternatively, at every grade, the Dana Center’s Washington K-12 Mathematics Standards (January, 2008 Review Draft) presents a set of key core processes that explicitly address reasoning, problem solving and communication. 

 

3. The Instructional Materials Perspective.  It’s truly amazing how much the anti-reform movement focuses on Everyday Math, Investigations and Connected Math. It is just as amazing how these critics consistently and conveniently ignore the facts that:

• All three programs have been extensively studied and all have shown to have a positive impact on student achievement – particularly in the domains of conceptual understanding and problem solving; and

• None of the four major brother-sister traditional programs (Houghton-Mifflin/McDougal-Littell, Harcourt/Holt, Scott Foresman/Prentice Hall, and McGraw Hill/Glencoe) have been subject to any kind of similar evaluation.

 

So if we seek to honestly explain the dismal NAEP and TIMSS performance, it is far more likely the result of the fact that 80% of the classrooms in the U.S. use traditional materials than the minimal intrusion of reform programs.  School officials have learned the hard way that, when it comes to mathematics and the “math wars,” it is lot easier to just go along with mediocrity than to rock the boat with something different, even when it’s better.

 

4. An Instructional Perspective or a Tale of Two Classrooms.  When all is said and done, every educator knows that the one paramount variable is the quality of instruction when classroom doors are closed. But once again, the difference between what Where’s the Math is seeking and what will best serve students is as stark as day and night. 

 

The clear message that emerges from the Washington Exemplary Math Standards: 2008 and the compelling video of the adorable and articulate meteorologist is “fifth grade priorities are to master long division with whole numbers” and that means “teaching” the standard “guzinta” and “bring down” algorithm for four-digit numbers divided by two-digit numbers. Despite the fact that almost no one uses pencil and paper to divide by two- and three-digit numbers anywhere in the real world, despite the fact that many 5th graders, even in the best of school systems, have not yet mastered multiplication and division facts, and despite the fact that most people are sensible enough to recognize that one size rarely fits all, the mathematically misguided turn mathematics into a series of mindless rules, devoid of any sense-making or conceptual understanding, that everyone is magically expected to master. It sure is easy to conjure up some imaginary vision of the good old days when everyone sat quietly and compliantly in their rows practicing and learning long division and to think that this is the answer to today’s problems. Unfortunately, such good old days never existed and worse, such instruction will only exacerbate the problems we face today.

 

Anyone can show a class of students the process for how to divide a four-digit number by a two-digit number using a traditional textbook and a slew of worksheets.  The almost certain result will be that about a third of the class will memorize and master the procedure the first time, while the remaining 2/3 become convinced they’re dumb and that math is nothing but a set of meaningless rules. It takes a trained teacher, with the support of powerful instructional materials, to teach all students when and why a situation requires division, how to find a reasonable estimate for the problem, how to interpret the decimal that appears on their calculators, and finally, how to use the solution to make a meaningful judgment.  Consider then the class that begins with the following data:

 

|Big Macs     $1.79 each |

|You have $10.00 |

 

In a class that makes mathematics come alive, fosters and reasoning and communication, values thinking, draws from students’ own experiences, and makes judicious use of technology the teacher will engage students with questions like:

 

•        Where would you find this sign?  How do you know?

•        Can you buy ten Big Macs?  Convince the class. How else did you know that?

•        About how many Big Macs could you buy?  Who has a different estimate?  Explain how you arrived at your estimate.

•        Exactly how many Big Macs can you buy?  How do you know?  What order do you enter the numbers in your calculator?  Why?  Why do you need to use the “division key”?

•        How do you interpret a digital readout of 5.586592?

•        How much change do you get?  How did you figure that out?

•        What about tax?  How is tax “added on”?  Does it change your answer?

•        Oops, sorry, the Big Macs are now on sale for $1.59 each and you now have $20.  What are some important questions you can ask and what are the answers to these questions?

 

This is the type of rigorous mathematics instruction for which we all ought to be fighting. 

 

5. The Mindset Perspective.  However, as noted above, the biggest problem when dealing with finding ways to make mathematics work for all students is the chasm that separates those whose conception of mathematics overwhelming involves a focus on procedures and abstractions, preparation for a traditional, and increasingly obsolete calculus course, an unwarranted wariness of technology and a belief that since teaching by telling and explaining worked for them, surely it must work for everyone else, so long as we tell it precisely and explain it correctly.  Unfortunately, this is exactly the conception of mathematics that will continue to result in mediocre achievement and success by a minority of students.  However, there is an important alternative conception that many who have always found mathematics easy to learn refuse to acknowledge or consistently disparage.  It is a conception of mathematics as the development of skills and concepts that empower students to solve practical and real-world problems.  It is a mathematics that makes judicious use of technology, like every other productive segment of our society, to enhance learning and preclude the waste of precious time devoted to pencil and paper skills that are no longer used in the workplace. It is a mathematics that builds coherently over time when students are confronted with interesting and complex problems and expected to struggle as part of the learning process.  It is a mathematics that effectively prepares students to solve the problems found in Singapore and on PISA, and not coincidentally, it is the mathematics embedded in programs like Everyday Mathematics, Investigations and Connected Math, when they are taught effectively by trained teachers. 

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