Thursday, May 21, 2015

Math problems of the week: Common Core-inspired "algebra" test problem

From a Algebra II  Performance Based Assessment Practice Test from PARCC (a consortium of 23 states that are devising Common Core-aligned tests).


Extra Credit:

Discuss the relative challenges of the mathematical labels (i.e., for types of methods) vs. the mathematical concepts vs. plain old common sense.

Tuesday, May 19, 2015

Two approaches to math assessment: quantity vs. "quality"

Auntie Ann makes a great point on my last post:

Giving many problems and demanding wordy answers on a test are mutually exclusive. In the time it takes to explain in words one problem, a student could demonstrate their proficiency on several problems with different mathematical concepts. Writing wordy explanations is much slower than giving a student a variety of different questions.
Assuming that the point of making students explain their answers is to distinguish those who really don't understand the math from those who've simply made stupid mistakes, then there are two possible approaches.

1. Assign a smaller number of problems so that students spend time explaining their answers.

2. Assign a larger number of problems.

Back in the day, we got perhaps ten times as many problems per session as students do today.

A student who is prone to stupid mistakes won't get nearly every answer wrong; a student who doesn't understand the math will. The type of answer generated by stupid mistakes often looks different from the type of answer generated by conceptual misunderstandings. Assign enough math problems, and a competent teacher can easily distinguish between the two types of student. Include harder problems that involve more mathematical steps than today's problems do, such that more students will naturally write down their mathematical steps, and it's even easier to distinguish those who understand from those who don't.

Doing lots of math problems (and getting timely feedback on them) is probably also a better way for students to overcome conceptual misunderstandings than explaining a much smaller number of problems is.

And its a great way for everyone to get better (especially more fluent) at math.

Sunday, May 17, 2015

Knowing, Doing, and Explaining Your Answer

Barry Garelick and I have a piece up on Education News.

Some excerpts:

At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked “What do I do?” The proctor read the instructions for the problem and told the student: “You need to explain how you got your answer.” 
The girl threw her arms up in frustration and said “Why can’t I just do the problem, enter the answer and be done with it?”
[For some problems] the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious.
... 
Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus— doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?
Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?

Friday, May 15, 2015

Math problems of the week: Common Core-inspired math vs. Singapore Math

I. The final problem in the Common Core-inspired Engage NY 5th grade Fractions module:

1. Lila collected the honey from 3 of her beehives. From the first hive she collected 2/3 gallon of honey. The last two hives yielded 1/4 gallon each.

a. How many gallons of honey did Lila collect in all? Draw a diagram to support your answer.

b. After using some of the honey she collected for baking, Lila found that she only had 3/4 gallon of honey left. How much honey did she use for baking? Support your answer using a diagram, numbers, and words.

c. With the remaining 3/4 gallon of honey, Lila decided to bake some loaves of bread and several batches of cookies for her school bake sale. The bread needed 1/6 gallon of honey and the cookies needed 1/4 gallon. How much honey was left over? Support your answer using a diagram, numbers, and words.

d. Lila decided to make more baked goods for the bake sale. She used 1/8 lb less flour to make bread than to make cookies. She used 1/4 lb more flour to make cookies than to make brownies. If she used 1/2 lb of flour to make the bread, how much flour did she use to make the brownies? Explain your answer using a diagram, numbers, and words.


II. The last two fractions problems in the 5th grade Singapore Math Primary Mathematics 5A Workbook (in Unit 4, Multiply and Divide Fractions, pp. 98-99):

3. After giving 1/3 of his money to his wife and 1/4 of it to his mother, Mr. Li still had $600 left. How much money did he give to his mother?

4. Lucy spent 3/5 of her money on a purse. She spent the remainder on 3 T-shirts which cost $4 each. How much did the purse cost?


III. Extra Credit

One of the biggest challenges found in Singapore Math problems (and not just in the one that recently went viral) is in figuring out what the first step is.

Compare the obviousness of the first steps in the EngageNY problems to those of the Singapore Math problems above.

Wednesday, May 13, 2015

Five 21st century ways to eliminate the achievement gap (vs. one 20th century way)

1. Tell teachers to arrange students into heterogeneous-ability groups, assign most work to the group as a whole, and give everyone in the group the same grade on this work.

Justify this by saying that this is how things work in the collaborative, 21st century work place.

Justify this to teachers in particular by pointing out how much it reduces the amount of grading they must do.

2. Tell teachers (and educational testing companies) to minimize the cognitive and traditional academic challenge in the various assignments/assessments so that most kids ceiling out on these measures and earn more or less the same number of points on them.

Justify this by saying that in today's world, where you can look everything up on the Internet, and where more and more of the computational and analytical work is done by calculators and computers and technicians in Asia, knowledge and computational/analytical skills matter less and less.

3. Tell teachers to maximize (in the various assignments/assessments) factors based on inherent personality traits like extraversion and sociability ("makes appropriate eye contact;" "engages the audience") and other subjective factors like creativity and outside-the-box thinking ("takes risks;" "shows innovation;" "includes colorful, pleasing illustrations")--where the skills/traits involved are fairly evenly distributed across the academic spectrum. Give these factors an aura of objectivity by making them the headers of columns in quantitative-looking assessment grids called "rubrics."

Justify this first by saying that, since computational/analytical skills matter less and less in the 21st century workplace, creativity and interpersonal skills matter more and more.

Justify this also by touting the inherent virtues of allowing the "type of student" who wouldn't have thrived under traditional measures to shine as never before.

4. Subtly incentivize teachers to use appropriate discretion in assessing the more subjective factors so as to boost the scores specifically of those whom traditional measures might deem the "weaker" of the students.

5. When these erstwhile "weaker" students later fail to thrive in the real, 21st century world, blame it on poverty, prejudice, and the chronic under-funding of public schools, and say that such outcomes are therefore beyond the schools' control.

Looking beyond the "stakeholders" of the 21st century educational-industrial complex, one finds more promising, outside-the-box, 20th century ideas about eliminating the achievement gap. Take linguist John McWhorter, an increasingly prominent spokesperson for disadvantaged children. In an article he wrote for the New Republic over 6 years ago, he reminds all of us about Project Follow-Through:

A solution for the reading gap was discovered four decades ago. Starting in the late 1960s, Siegfried Engelmann led a government-sponsored investigation, Project Follow Through, that compared nine teaching methods and tracked their results in more than 75,000 children from kindergarten through third grade. It found that the Direct Instruction (DI) method of teaching reading was vastly more effective than any of the others for (drum roll, please) poor kids, including black ones. DI isn't exactly complicated: Students are taught to sound out words rather than told to get the hang of recognizing words whole, and they are taught according to scripted drills that emphasize repetition and frequent student participation.
In a half-day preschool in Champaign-Urbana they founded, Engelmann and associates found that DI teaches four-year-olds to understand sounds, syllables, and rhyming. Its students went on to kindergarten reading at a second-grade level, with their mean IQ having jumped 25 points. In the 70s and 80s, similar results came from nine other sites nationwide, and since then, the evidence of DI's effectiveness has been overwhelming, raising students' reading scores in schools in Baltimore, Houston, Milwaukee, and other districts. A search for an occasion where DI was instituted and failed to improve students' reading performance would be distinctly frustrating.
...
...schools of education have long been caught up in an idea that teaching poor kids to read requires something more than, well, teaching them how to sound out words. The poor child, the good-thinking wisdom tells us, needs tutti-frutti approaches bringing in music, rhythm, narrative, Ebonics, and so on. Distracted by the hardships in their home lives, surely they cannot be reached by just laying out the facts. That can only work for coddled children of doctors and lawyers.
But the simple fact of how well DI has worked shows that "creativity" is not what poor kids need. At the Champaign-Urbana preschool, the kids--poor kids, recall, and not many who were white--had a jolly old time with DI, especially when they found that it was (hey!) teaching them to read.
McWhorter was talking, specifically, about the reading gap. But Direct Instruction's efficacy is seen in all subjects, and performance in all subjects, of course, is partly a function of reading skills.

Monday, May 11, 2015

How deeply do UCARE: “Going deep” in 21st Century, Common Core-inspired math

Four feet deep, to be precise.

In the 21st century, a deep understanding of mathematics, and the ability to apply that understanding, is more important than it has ever been. In Montgomery County Public Schools (MCPS), and across the country, mathematics instruction is changing to make sure we provide our students with the skills and knowledge they need for success in college and the workplace. From the MCPS’s math website.
Sound familiar? Yes, it’s all about our friends the CCCSS and the PARCC, along with the collaborative, 21st century workplace all our kids are going to end up in:
The improvements to the math curriculum are in response to several factors and will results in MCPS students having a stronger, more comprehensive understanding of mathematical concepts.  
Reasons include:
• The adoption of the internationally-driven Common Core State Standards (CCSS) and new, more difficult assessments being developed by the Partnership for Assessment of Readiness for College and Careers (PARCC), of which Maryland is a member… 
• The changing demands of the work force, including 21st century skills, such as, collaboration, persistence, critical thinking, and creative thinking…
The CCSS, the website notes, “demand a higher level of thinking in math for all students”:
Computation and procedures were sufficient to reach success in previous curriculum [sic] and assessments. The CCSS requires students to show greater depth by demonstrating their Understanding, Computing, Applying, Reasoning and Engagement (UCARE) in mathematics. As a result, the math content at each grade level is more difficult than previous curriculum [sic; boldface, here and elsewhere, mine].
So difficult that much more is required in order to advance through it:
Following the CCSS, the elementary program is designed to go deeper in the topics of number (counting, addition, subtraction, multiplication, division, fractions, and decimals) to ensure that students have a strong foundation before moving on to more advanced content.  
…Students at all levels are expected to express a deep understanding of the math content they are studying before moving to more advanced content. This means students will need to demonstrate their understanding in multiple ways, beyond just memorizing a formula or single procedure for solving a problem.  
Despite these hurdles, and despite all the ways in which the new curriculum is “more difficult” for all students, gifted students will get even more challenge:
C2.0 [Curriculum 2.0--the new curriculum] includes enrichment and acceleration options added by MCPS to ensure that students who demonstrate understanding of a topic will be able to deepen and extend their learning.
Indeed, MCPS’s C2.0’s enrichment opportunities “exceed the requirements” of the CCSS. In particular, students who “demonstrate readiness” in grade 3 will have the option to enroll in the 4/5 Compacted Math class. However:
due to the increase in the rigor of the grade level curriculum, far fewer students than in previous years will need to skip a grade level in elementary mathematics to be challenged.
Readers who’ve read this far must be burning with curiosity about just how deep a deep new, CCSS-exceeding curriculum goes for those select few who’ve already demonstrated exceptionally deep understanding. So here’s an example from the compacted 4/5 curriculum—a problem so deep that the class spent two weeks on it:
What is the opposite of 4 feet ABOVE sea level? What is the opposite of the opposite of 4 feet ABOVE sea level?"
The parent who shared this problem (on a listserv for MCPS parents) adds that:
Kids who learned negative numbers years before do not get any acceleration through this, and group work is a huge part of every math concept. If your kid learned long division years ago and understands the concept completely, they will still have to memorize all the different, laborious "strategies" and spit them out verbatim…
She concludes:
MCPS is not a place for rapid learners anymore… Our daughter has been bored to tears in math and science, as she has been every year. Most of the kids at the top 1% or so come home from school and then do a private math program in order to keep them engaged in math…
For more examples of what the MCPS website calls “Changing Expectation in Curriculum 2.0 Mathematics,” see this past week's Problems of the week.

According to a 2014 article in the Boston Globe:
many, or even most [gifted kids]... aren’t identified early, and they don’t necessarily get special attention from their schools. [Researchers at Vanderbilt] have .. found that those who weren’t challenged in school were less likely to live up to the potential indicated by their test scores. Other research has shown that under-stimulated gifted students quickly become bored and frustrated—especially if they come from low-income families that are not equipped to provide them with enrichment outside of school.
One of the researchers, Vanderbilt psychologist David Lubinski, worries about the broader impact of shortchanging our most academically capable students.
“We are in a talent war, and we’re living in a global economy now,” Lubinski says. “These are the people who are going to figure out all the riddles. Schizophrenia, cancer—they’re going to fight terrorism, they’re going to create patents and the scientific innovations that drive our economy. But they are not given a lot of opportunities in schools that are designed for typically developing kids.”

Saturday, May 9, 2015

When showing your work means stepping out of your shoes

In the next few days, J will be taking AP exams in BC Calculus and Computer Science. He earned a 4 on last year’s AB Calc exam, and whether he earns 5s, in general, depends on how well he slogs through the verbiage on the open-ended problems, and whether he shows sufficient work on them.

I used to think that the challenge, for J, was exclusively a verbal one. For understanding what to do with verbose problems, it certainly is. But showing your work is different. As far as the AP is concerned (at least for now), showing your work means what it did a generation ago: showing the key mathematical steps that lead to your solution. This requirement isn’t problematic for language-impaired kids in the way that the Reform Math and CCSS-inspired “explain your answer” is.

But for certain ASD kids like J--kids who do math in their heads and have difficulty taking other people’s perspectives--this requirement, I’m realizing, is problematic nonetheless. A neurotypical kid who can do a multi-step problem in their head has some sense of what steps she should write out for others—in particular, for whoever is evaluating her. But this kind of perspective taking (stepping out of the shoes of someone who has just solved the problem and into the shoes of someone who is evaluating your answer) does not come naturally to people with autism.

The open-ended problems on the BC Calc exam can often be broken down into dozens of steps; you’re not expected to write them all down (and doing so takes time). The trick is to figure out which combination of steps to spell out in order to satisfy a particular problem’s requirements, and this is where even the most autism-friendly math programs can shortchange those on the spectrum.